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CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning

Xuehang Guo, Pingyue Zhang, Ruiyi Zhang, Zhenhailong Wang, Hanrui Lyu, Heng Ji, Tong Sun, Qingyun Wang, Manling Li

Year: 2026Venue: arXiv preprintArea: cs.CVType: PreprintEmbeddings: 153

Intelligence

Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%

Last extracted: 8/5/2026, 4:34:25 AM

Summary

The paper introduces CURV, a curriculum learning framework for Multimodal Large Language Models (MLLMs) to enhance Chart Question Answering (CQA) by internalizing visual grounded reasoning. It proposes a two-stage training process that couples logical reasoning with dynamic visual grounding, supported by a synthetic dataset CCQA. Experiments show significant performance improvements over baselines on chart benchmarks and out-of-domain tasks.

Entities (12)

CURV · framework · 98%CCQA · dataset · 97%Chart Question Answering · task · 95%Multimodal Large Language Models · model-class · 92%Visual Grounding · concept · 92%Curriculum Learning · method · 90%GPT-4o · model · 90%InternVL3 · model · 90%Qwen2.5-VL · model · 90%ChartMuseum · benchmark · 85%CharXiv · benchmark · 85%MathVista · benchmark · 85%

Relation Signals (8)

CURV → uses → Curriculum Learning

confidence 95% · we propose CURV, a curriculum learning framework

CURV → uses → CCQA

confidence 95% · To assist model learning, we further introduce CCQA... To assist model learning, we further introduce CCQA

CURV → improves → Chart Question Answering

confidence 90% · Experiments demonstrate that CURV achieves up to 20.50% improvements over baselines

CURV → targets → Multimodal Large Language Models

confidence 90% · develops intrinsic visual reasoning capabilities in MLLMs

GPT-4o → evaluatedon → CCQA

confidence 85% · We evaluate GPT-4o... on 60 CQA samples randomly selected from CharXiv... using CCQA

Qwen2.5-VL → evaluatedon → CCQA

confidence 85% · using CCQA (§4) and two models: Qwen2.5-VL-7B

CURV → generalizesto → MathVista

confidence 80% · generalizable to real-world benchmarks... and out-of-domain multimodal reasoning tasks... including MathVista

CURV → generalizesto → ChartMuseum

confidence 80% · generalizable to real-world benchmarks... ChartMuseum

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Abstract

Abstract:Chart question answering (CQA) requires multimodal large language models (MLLMs) to integrate visual comprehension with logical reasoning, yet current models struggle with accurate visual grounding and coherent reasoning chains. While extrinsic chain-of-thought prompting and visual cues significantly improve performance, current MLLMs lack intrinsic visual grounded reasoning capabilities, leading to inaccurate perception and reasoning disconnected from visual evidence. To address these limitations, we propose CURV, a curriculum learning framework that develops intrinsic visual reasoning capabilities by reformulating CQA as multi-step visual grounded reasoning, where each step coordinates logical reasoning with dynamic visual grounding through spatial attention concentration. To assist model learning, we further introduce CCQA, a three-level curriculum dataset with scalable synthetic generation across diverse chart types and reasoning patterns. Our curriculum systematically progresses from basic single-operation reasoning to complex multi-chart compositional tasks. Experiments demonstrate that CURV achieves up to $\uparrow20.50\%$ improvements over baselines and is generalizable to real-world benchmarks (up to $\uparrow12.30\%$) and out-of-domain multimodal reasoning tasks (up to $\uparrow10.20\%$), validating the effectiveness of internalizing visual reasoning with dynamic grounding for enhanced chart understanding capabilities. Code is available at: this https URL.

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CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning CURV: EnhancingChartUnderstanding Through Curricu- lum Visual Grounded Reasoning Xuehang Guo 1 , Pingyue Zhang 2 , Ruiyi Zhang 3 , Zhenhailong Wang 4 , Hanrui Lyu 2 , Heng Ji 4 , Tong Sun 3 , Qingyun Wang 1 , Manling Li 2 1 William & Mary 2 Northwestern University 3 Adobe 4 UIUC Abstract Chart question answering (CQA) requires multimodal large language mod- els (MLLMs) to integrate visual comprehension with logical reasoning, yet current models struggle with accurate visual grounding and coherent rea- soning chains. While extrinsic chain-of-thought prompting and visual cues significantly improve performance, current MLLMs lack intrinsic visual grounded reasoning capabilities, leading to inaccurate perception and rea- soning disconnected from visual evidence. To address these limitations, we propose CURV, a curriculum learning framework that develops intrinsic visual reasoning capabilities by reformulating CQA as multi-step visual grounded reasoning, where each step coordinates logical reasoning with dynamic visual grounding through spatial attention concentration. To as- sist model learning, we further introduce CCQA, a three-level curriculum dataset with scalable synthetic generation across diverse chart types and reasoning patterns. Our curriculum systematically progresses from basic single-operation reasoning to complex multi-chart compositional tasks. Experiments demonstrate that CURV achieves up to↑20.50% improve- ments over baselines and is generalizable to real-world benchmarks (up to ↑12.30%) and out-of-domain multimodal reasoning tasks (up to↑10.20%), validating the effectiveness of internalizing visual reasoning with dynamic grounding for enhanced chart understanding capabilities. Code is available at: https://xhguo7.github.io/CURV/. 1 Introduction How do humans tackle multimodal problems? Inspired by cognitive theories (Baddeley et al., 1974; Johnson-Laird, 1983; Barsalou, 2008; Grant & Spivey, 2003), humans decompose complex tasks into stepwise reasoning chains, interleave each step with dynamic visual grounding, and compose these grounded steps into a coherent solution (Fig. 1). Chain-of- thought (CoT) reasoning has demonstrated its effectiveness in decomposing problems into stepwise inferences (Xu et al., 2024; Zhang et al., 2025a). This ability becomes more critical in multimodal reasoning, where multimodal large language models (MLLMs) are expected to integrate visual and textual information (Fan et al., 2025; Zhang et al., 2025b) while visual perception errors contribute to the majority of multimodal reasoning failures (Wang et al., 2025c). Without external support such as explicit CoT prompting or visual cues (Fig. 1), MLLMs struggle with accurate visual grounded reasoning (Wang et al., 2025d;b;a). This challenge is particularly evident in chart question answering (CQA), where models need to faithfully interpret complex geometric structures, spatial relationships, and quanti- tative patterns to derive correct answers. As a result, CQA requires models to accurately Correspondence to: Manling Li (manling.li@northwestern.edu), Qingyun Wang (qwang16@wm.edu), Xuehang Guo (xguo15@wm.edu) 1 arXiv:2608.02833v1 [cs.CV] 3 Aug 2026 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Figure 1: From Extrinsic Assistance to Intrinsic Visual Grounded Reasoning. Inspired by human ways of thinking, we present CURV to internalize extrinsic CoT prompting and visual guidance to intrinsic capabilities, enabling models to perform visual grounded reasoning through dynamically shifting focuses across targeted image regions. perceive visual details, perform step-by-step reasoning over interconnected components, and dynamically shift focus across different chart regions (Chen et al., 2025). However, existing MLLMs exhibit several fundamental limitations in this setting (§2 & A): (1) De- composition: They struggle to decompose complex problems into coherent chains of reasoning, often producing inconsistent or logically flawed intermediate steps (Fig. 12-13); (2) Interleaved Visual Reasoning: They show limitations in accurately grounding individual reasoning steps in the visual input, such as misreading chart values or attending to correct regions (Fig. 12); and (3) Composition: They exhibit difficulties in integrating logical reasoning with visual grounding across multiple steps into a coherent, interleaved chain, leading to a disconnect between what is perceived, reasoned, and concluded (Fig. 13). Collectively, these limitations lead to inaccurate perception and reasoning that is disconnected from the visual evidence, ultimately causing errors even when the necessary information is present. To address these limitations, we propose CURV, a curriculum learning (CL) framework that develops intrinsic visual grounded reasoning capabilities in MLLMs. Our approach refor- mulates CQA as multi-step reasoning processes where each step couples logical reasoning with dynamic visual grounding. Instead of relying on extrinsic assistance, CURV enables models to internalize the ability through dynamically focusing on relevant chart regions while maintaining coherent reasoning chains across steps. Curriculum learning progresses from single-operation reasoning to complex multi-operation compositions, allowing models to gradually develop both visual perception accuracy and reasoning sophistication. Our main contributions include: •We propose CURV, a curriculum learning framework that develops MLLMs’ intrinsic visual reasoning capabilities by progressively shifting visual attention along reasoning, transitioning from basic single-operation tasks to complex nested reasoning (§3). •We introduce a scalable synthetic CQA data generation method (§4) that systematically increases task complexity via nested reasoning chains, enabling efficient data creation. • We presentCurriculumChartQuestionAnswering (CCQA) (§4) that supports curriculum learning across four different generation modes, three visual grounding strategies, and multi-level metrics for comprehensive performance evaluation (§2 & 5). • Our experiments demonstrate that CoT reasoning with visual grounding provides models with step-by-step alignment between visual perception and logical reasoning, leading to notable improvements across different chart types, complexity, and domains (§5). 2 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning 2 What Prohibits MLLMs From Chart Understanding Success? 2.1 Preliminary Analysis of CQA Failures Mode Error Analysis (count) Acc (%)∆ acc (%) VisionReasoningAnswer A–3443.33– VA 018561.67↑ 18.33 RA176553.33↑ 10.00 RVA05386.67↑ 43.33 Table 1: CQA Preliminary. Preliminary results on GPT-4o using accuracy (%) and relative improve- ments∆ acc (%). What are the bottlenecks that hinder MLLMs from correctly understanding chart images? We evaluate GPT-4o (OpenAI, 2024) on 60 CQA samples randomly selected from CharXiv with four modes: (1) Answer only (A): MLLM directly generates the answer; (2) Vision + Answer (VA): MLLM is pro- vided with human-annotated visual infor- mation to give the answer; (3) Reason + Answer (RA): MLLM is prompted to first generate CoT reasoning, followed by the final answer; (4) Reason + Vision + Answer (RVA): MLLM is provided with the same visual information as in VA, prompted to first generate CoT reasoning and then the final answer. Surprisingly (Tab. 1), GPT-4o achieves significantly higher scores (↑10.00%) when prompted to generate reasoning without visual cues (RA). Moreover, combining perception with reasoning (RVA) yields the best performance (↑43.33%). These highlight MLLMs’ lack of logical decomposition and visual reasoning capabilities. 2.2 Enhancing Chart Reasoning Through Dynamic Visual Grounding Figure 2: Performance Across Reasoning Depths and Modes. We evaluate GPT-4o (up- per) and Qwen2.5-VL-7B (lower) on 1,800 ran- domly selected samples from CCQA (evenly distributed across D i , 1≤ i≤ 3). How to improve MLLMs’ intrinsic visual rea- soning capabilities? Motivated by the effectiveness of extrin- sic CoT prompting and visual guidance (Tab. 1), we aim to internalize these capabil- ities within MLLMs (Fig. 1). To concretize this approach, we expand our prelimi- nary exploration to visual reasoning with grounded focuses (Tab. 2) using CCQA (§4) and two models: Qwen2.5-VL-7B (Qwen, 2025) and GPT-4o (OpenAI, 2024). Simi- larly, we examine model performance un- der four modes using ground-truth rea- soning and visual grounding. Despite in- creased curriculum difficulty (1≤ D ≤3), both models achieve higher performance when equipped with either reasoning or vi- sual assistance, with performance further improved when both are combined. Consistently, extrinsic assistance enhances chart understanding through structured reasoning and visual grounding, motivating our core hypothesis: MLLMs can internalize these capabilities through visual grounded reasoning (§3 & 4). 3CURV: Chart Reasoning with Dynamic Visual Grounding 3.1 Problem Formulation Problem Definition. Given a chart imageI ∈R H×W×C and a questionQ, the goal of CQA is to generate the answer A. However, current MLLMs directly learn the mapping: f θ : (I , Q)→ A(1) Not only does this direct mapping approach lack an intermediate reasoning structure that enables accurate visual perception and robust visual understanding, but it also fails to effectively and dynamically ground reasoning chains in visual space. 3 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Figure 3: CURV Overview. Different visual grounding (§D) and training (§G) approaches to develop intrinsic visual reasoning capabilities via multi-step visual grounded reasoning. Our Approach. We propose to decompose this problem into a multi-step reasoning process with dynamic visual grounding. Specifically, we reformulate the CQA task as: f θ : (I , Q)→(R 1 , V 1 ), (V 2 , V 2 ), . . . , (R T , V T )→ A(2) whereR t represents thet-th reasoning step in natural language,V t denotes visual regions groundingR t in the visual space,Tis the total number of reasoning steps to reachA, and the sequence(R t , V t ) T t=1 forms a structured progressive visual reasoning chain. 3.2 Multi-Step Reasoning With Dynamic Visual Grounding We design a two-stage multi-step curriculum learning framework (CURV) that enables MLLMs to develop intrinsic progressive reasoning capabilities with dynamic visual ground- ing, moving beyond extrinsic assistance toward self-contained visual reasoning (Fig. 1). To further strengthen this process, we implement explicit and implicit visual grounding (§G.5). Stage I: Visual Grounding (RV). In the first stage, the model learns to establish multimodal correspondences between reasoning flows and associated visual focuses. Specifically, given the chart imageIand questionQ, at timet, the model learns to predict the visual focusV t that grounds R t ontoI : V t = f (S1) θ I , Q, R t ′ , V t ′ t−1 t ′ =1 , R t (3) where prior grounding pairs(R t ′ ,V t ′ ) t−1 t ′ =1 provide context for consistent grounding across steps. The training objective supervises the predicted visual focuses: L (S1) = T ∑ t=1 L V (V t , V ∗ t )(4) whereL V (V t ,V ∗ t )denotes the grounding loss between the predicted visual focusV t and the ground-truth V ∗ t at step t. Stage I: Interleaved Visual Reasoning (RVA). Building upon the visual grounding capabil- ity acquired in Stage I, the model learns to proactively leverage visual grounding as feedback during reasoning. Concretely, at each stept, the visual focusV t produced alongside rea- soningR t is applied to the input chart imageIto construct the grounded visual stateI t via one of the visual grounding methods (Fig. 3). This grounded imageI t is then provided as additional visual input when generating the next reasoning step, enabling the model to dynamically shift its visual attention in alignment with the evolving logical reasoning: (R t , V t ) = f (S2) θ I , Q, R t ′ , V t ′ →I t ′ t−1 t ′ =1 (5) 4 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning where V t ′ →I t ′ denotes the new state mapped intoI t ′ . Stage I’s training objective is: L (S2) = T ∑ t=1 λ R L R (R t , R ∗ t ) + λ V T ∑ t=1 L V (V t , V ∗ t ) + λ A L A (A, A ∗ )(6) whereL R (R t ,R ∗ t )is the reasoning supervision loss at stept,L V (V t ,V ∗ t )is the grounding loss at stept,L A (A,A ∗ )is the final answer prediction loss, andλ R ,λ V , andλ A balance the visual grounded reasoning objective against answer supervision. As such, Stage I develops the model’s intrinsic progressive visual reasoning capabilities, enabling it to: (1) dynamically focus on relevant chart regions by updating visual grounding at each step, (2) ground logical reasoning in concrete visual evidence, and (3) maintain cross-step coherence through joint conditioning on both the textual reasoning history and the evolving visual states. Figure 4: From Low-Level Chart Components To High-Level Charts. We decompose all types of charts into low-level components to endow MLLMs with both foundational chart understanding abilities and adaptive generalizabilities to high-level complexities. 3.3 Curriculum Visual Grounded Reasoning Visual Grounding. In addition to explicit and implicit visual reasoning (§G.5), we pro- pose three visual grounding strategies (§D), including applied, boxed, and cropped, to dynamically shift visual focuses during reasoning (Fig. 3). Reasoning Depth. To formalize reasoning complexity in CQA, we introduce two distinct but complementary concepts that characterize the reasoning process: •Number of Reasoning Steps (T): The total number of CoT reasoning stepsR t T t=1 a model goes through to reach the final answerA. Multiple reasoning steps may operate at the same logical complexity level (i.e., reasoning depth tier below) while contributing different pieces of information toward the solution. • Tier of Reasoning Depth (D): The maximum number of nested logical functions required to solve the task, corresponding to the deepest level of functional composition in the reasoning chain. Formally, for a question requiring nested functionsf 1 ( f 2 (...( f D (x)))), the reasoning depth isD. This metric captures the inherent logical complexity of a problem, independent of how many intermediate steps a model uses to reach A. Curriculum Learning. Charts are a unique form of data organized through structured relationships among fundamental meta-elements (Fig. 4). Through two-stage training curriculum, we incorporate chart curriculum to guide the model from understanding low- level visual components to high-level chart structures (Fig. 4) across diverse chart types. With gradually increased task difficulty along both reasoning and visual dimensions, we define data curriculum consisting of three levels (§C.2) across five fine-grained tiers (§C.3). CURV Variants. CURV accommodates multiple design choices along two orthogonal dimensions: First, it supports three visual grounding strategies (§D) that control the rep- resentation and granularity of visual grounding. We further implement explicit visual grounding to develop MLLMs’ intrinsic visual grounded reasoning capabilities (§3.2), while performing ablations of implicit grounding (§G.5) to validate the critical role of visual grounding in CURV as intermediate visual-reasoning alignment guidance rather than an ultimate learning objective (§5.4). Second, CURV is compatible with different training paradigms, including SFT, reinforcement learning (RL), and their combination, enabling flexible instantiations under varying optimization objectives and resource constraints (§5.3 & G.4). 5 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Figure 5: Multi-Level Curriculum Construction. We construct CCQA through reasoning decomposition, interleaving visual reasoning, and reasoning chain composition. Examples of applied visual grounding is shown on the right. More examples can be found in § D. 4CCQA: Curriculum Chart Question Answering 4.1 Dataset Construction Principles Supporting CURV (§3), we introduce Curriculum Chart Question Answering (CCQA), a curriculum learning dataset to progressively develop visual reasoning capabilities. Our dataset construction presents three core principles with rigorous quality control (§C.5): Progressive Complexity. We implement three-level curriculum through systematic variation in reasoning depth (D), chart complexity, and operation sophistication (Fig. 5). Interleaved Visual Reasoning. Each reasoning stepR t is paired with corresponding ground- truth V ∗ t and binary masks M ∗ t , enabling direct alignment of visual grounding. Template-Based Accuracy. We employ synthetic templates (§C.1) to ensure data accuracy and systematic coverage of reasoning patterns, effectively supporting a stable progression of curriculum. As shown below, all chart-specific features are replaced by plotting data: QUESTION: What is the< yaxistitle> of the< objectsingular>? 4.2 Curriculum Chart Question Answering Figure 6: Chart Type Distribution Chart Types. We include 7 chart types to endow CCQA with high visual diversity (Fig. 6 & Tab. 4): bar, histogram, scatter, line, heatmap, pie, and radar. Data Category. We define 30 distinct domain cate- gories (Tab. 4), employing GPT-4o (OpenAI, 2024) to generate plotting data for chart drawing (Fig. 20). QA Types. Our curriculum templates (§C.2) cover various operations (Tab. 5) in atomic or nested forms across different chart components and subplots. Curriculum with Meta Learning. Our dataset construction implements a meta-learning paradigm that maximizes visual reasoning generalization while minimizing visual over- fitting (§C.4). With only 30 unique charts for each chart type (i.e., 7 types×30 categories), diverse query-reason-ground-answer quadruplets are derived from each image via sys- tematic template instantiation (§C.2). Aiming to foster MLLMs’ intrinsic visual grounded reasoning abilities rather than memorizing specific visual appearances, we generate multiple CQA instances(Q k ,D d ,B ∗ d D d=1 ,A k ) K k=1 (K ≫1) for each base imageI j to encourage the model to learn robust visual reasoning skills transferable across diverse chart appearances, data distributions, contexts and domains, as well as task complexities. 6 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning ModelSize Level 1Level 2Level 3 @M@0.0@0.05@0.1@0.2@M@0.0@0.05@0.1@0.2@M@0.0@0.05@0.1@0.2 Close-Source MLLMs GPT-4o-57.6454.0762.0065.3670.5034.0433.7544.9350.5457.9622.1422.2930.2534.0439.14 GPT-4.1-mini-70.8667.4376.2978.7979.9337.6136.5447.1852.9360.8626.1425.4633.1137.7942.32 Open-Source Baselines Gemma-34B38.2128.6432.7937.0041.2918.0712.8618.8223.6428.2911.439.6413.1415.5418.96 Llama-3.2-V11B44.8638.2940.8642.8647.0723.4318.2522.4625.2929.3916.5714.2516.5418.2920.29 InternVL3 1B20.5416.3820.5324.4029.688.517.4412.6115.3720.656.765.878.3110.0910.99 2B33.5332.5237.5242.5247.2913.1113.2219.8927.1435.5510.6911.1814.7118.7224.90 8B46.7944.2952.1457.7163.0025.7525.5735.3641.6450.7917.8417.4824.5728.9334.68 Qwen2.5-VL 3B 45.2543.5251.5456.2561.5422.7522.8631.1437.2145.3616.1816.0021.8925.8231.71 7B54.2150.7960.4364.6469.2928.6828.8239.9345.5452.6119.0119.3826.7131.5036.93 Ours (Stage I) Applied (InternVL3) 1B25.7921.3625.2931.4337.8610.579.2512.8917.1822.217.117.369.2511.0012.93 2B42.6441.7948.7155.3662.1418.6819.2925.7932.9341.5011.3912.5716.6120.7526.32 8B58.8654.6466.2969.1471.7934.4733.9049.6756.4763.9118.8719.8427.9931.9737.41 Applied (Qwen2.5-VL) 3B54.2151.2159.5065.0068.5025.8627.1439.1847.2555.1816.8617.3223.9328.6133.11 7B 65.7959.1471.9375.2978.2936.8234.7950.8256.1862.7521.0421.1130.2134.2539.39 Boxed (Qwen2.5-VL) 3B58.0751.7961.0065.7169.8625.9625.3237.7545.8953.8616.9216.8222.8927.2133.29 7B59.7952.7971.9374.5776.6433.7930.3249.7555.8962.2120.1418.0428.8932.3236.29 Cropped (Qwen2.5-VL) 3B49.7146.7954.8660.0765.2120.6821.1132.6140.3650.8215.0018.8920.9624.7130.25 7B58.9356.5767.7172.9376.7130.0429.7140.4346.7153.7918.6818.7126.7930.3635.11 Ours (Stage I + I) Applied (InternVL3) 1B 28.6426.7931.5736.3643.7914.1812.8616.2522.8627.4310.3910.6413.4615.7917.75 2B47.8646.9354.7160.5767.7923.3623.6128.7936.5743.3614.8915.8220.2924.7931.39 8B67.7162.5770.5775.1477.2939.6837.7952.5458.7165.0424.5722.9630.5434.7940.32 Applied (Qwen2.5-VL) 3B 60.6457.2163.7169.5773.7930.1129.6141.8248.8657.6417.9618.2124.2529.0734.50 7B69.8666.7975.2978.5080.4340.2138.6454.0060.0466.8926.1124.2532.2136.9642.54 Boxed (Qwen2.5-VL) 3B58.5054.7164.2172.5774.7927.6426.8638.7546.8655.3617.4317.5423.5027.9633.46 7B63.2959.5772.1475.8677.0035.3233.5751.5457.5464.3623.5722.1431.9635.0438.25 Cropped (Qwen2.5-VL) 3B 53.0750.7957.7163.5769.5023.2923.5434.2941.4351.5417.1418.0422.4626.1133.04 7B62.1460.7169.5774.7177.8632.2531.8246.3250.5757.3620.1120.3628.4633.8637.11 Table 2: Performance On CCQA. We compare performance across different curriculum levels using five accuracy metrics acc@X where X is MLLM (@ M) or ranges (§ 5.2). Generalize To Real-World Charts & Domains. Bridging visual perception and logical reasoning, we construct CCQA to underline the significance of enhancing MLLMs’ un- derstanding of fundamental visual components and spatial features for accurate visual reasoning (§G.8). CURV finetuned on CCQA are also applicable to real-world chart under- standing and out-of-domain benchmarks (§5.2), validating not only the effectiveness but also the adaptability and generalizability of our approach (§A.1 & C.4). 5 Experiments 5.1 Setup Baseline. We use two close-source MLLMs, GPT-4o (OpenAI, 2024) and GPT-4.1-mini (Ope- nAI, 2025), and seven open-source MLLMs, Llama-3.2-Vision (AI, 2024), Gemma-3 (Google, 2024), InternVL3 (OpenGVLab, 2025), and Qwen2.5-VL (Qwen, 2025) with different sizes, as comparison baselines. Model. We employ five MLLMs as base models, finetuned through CURV for enhanced visual-grounded chart reasoning: Qwen2.5-VL (3B and 7B) and InternVL-3 (1B, 2B, and 8B). Data. We split CCQA into non-overlapped training and test sets. All models are evaluated on the test sets unseen for finetuned models. Implementation details can be found in §G.1. 5.2 Evaluation Evaluation Data. We evaluate on CCQA (§4), CQA, and out-of-domain benchmarks: •CCQA: Highlighting the significance of learning chart basics through increased task difficulty, we evaluate models on three test sets of CCQA covering three increased curriculum levels, respectively. 7 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning • Chart Benchmarks: To assess the adaptability of MLLMs to chart understanding bench- marks, we extend our evaluation to ChartMuseum (visual subset) (Tang et al., 2025), CharXiv (reasoning subset) (Wang et al., 2024), ChartQA (Masry et al., 2022), and ChartQAPro (Masry et al., 2025). •Out-of-Domain Benchmarks: Generalizing to different domains of multimodal reason- ing, we extend our evaluation to multi-discipline multimodal reasoning tasks, including MathVista (Lu et al., 2023) and MMMU-Pro (Yue et al., 2025). Evaluation Metrics. We evaluate different aspects of responses through complementary metrics that take both textual outputs and visual grounding into consideration (§F): (1) Reasoning: For reasoning evaluation, we employ two complementary approaches: ModelSize Chart BenchmarksOut-of-Domain ChartQAChartQA-ProCharXivChartMuseumMathVistaMMMU-Pro Baselines InternVL3 1B41.6811.4815.7010.0135.809.94 2B66.9620.8424.3015.0255.9016.36 8B74.5630.5636.2024.4268.8026.99 Qwen2.5-VL 3B62.3217.0219.5012.2156.1021.45 7B72.4829.7732.5021.6264.6028.21 Ours (Stage I + I) Applied (InternVL3) 1B51.5615.8618.9011.2145.9312.95 2B69.6023.9728.1016.1262.7818.27 8B78.2832.3940.2026.2375.3730.81 Applied (Qwen2.5-VL) 3B72.3624.7431.8019.3266.3022.77 7B75.0432.8036.7025.1272.4132.08 Table 3: Performance on Chart and Out-of-Domain Benchmarks. MLLM-as-judge acc@MLLM (§ 5.2) on baselines and CURV. MicroEvaluation (acc@mic):We evaluate reasoning via a combination of five micro metrics: ROUGE-L (Eq. 17), BLEU (Eq. 18), ME- TEOR(Eq.20), BERTSCORE (Eq. 19), and COSINE SIMILARITY (Eq. 21). Macro Evaluation (acc@mac): We em- ploy GPT-4.1-mini as the judge for macro-level evaluation by assigning quality scores based on three evaluation criteria (§F.2). (2) Visual Grounding: We leverage Intersection-over-Union (IoU) variants CIOU (Eq. 22) and GIOU (Eq. 23), where GIOU (Rezatofighi et al., 2019) is generalized IoUs and CIOU evaluates the cumulative intersection over the cumulative unions (Zheng et al., 2019). (3) Answer: We define CQA accuracy as the mean answer accuracy across all testing samples. MLLM as Judge (acc@MLLM): We employ GPT-4.1-mini (OpenAI, 2025) as the judge to evaluate answer accuracy through True-or-False assessment (pass@1). More details in §F.1. Rule as Judge (acc@range): To mitigate potential biases introduced by MLLM-as- judge (Dorner et al., 2024; Li et al., 2024a), we introduce rule-based evaluation metrics (Algorithm 1) to assess answer accuracy through systematic parsing and rubric judgment. In particular, it incorporates fourrangecriteria that capture different levels of strictness (§F.1), including the absolute accuracy (acc@0.0) and three progressively relaxed thresholds (acc@0.05, acc@0.1, acc@0.2). 5.3 Main Results On Chart & Multimodal Understanding Performance on CCQA. Compared with the baselines (Tab. 2), our finetuned models achieve consistently higher accuracy across all six metrics, with absolute gains of up to 14.85% and 20.92% for single-stage and two-stage training, respectively. The highest perfor- mance is achieved by CURV@Applied (Qwen2.5-VL-7B), with up to 15.65% higher than its base model. Compared with GPT models, it achieves up to 12.22% higher accuracy, demonstrating the effectiveness of CURV in enhancing MLLMs’ visual reasoning abilities. Performance on Complex Chart Understanding. Although trained solely on single-plot charts (1≤ D<3), finetuned models generalize to multi-plot charts (D ≥3). As shown in Tab. 2, CURV@Applied (Qwen2.5-VL-7B) achieves up to↑7.10% improvements, and CURV@Applied (InternVL-8B) also shows↑6.73% across all metrics. Performance on Chart Benchmarks. Aiming for CURV to be not only adaptable across different task complexities but also generalizable to real-world chart comprehension, we 8 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning extend our evaluation to CQA benchmarks ( §5.2). Results in Tab. 3 highlight the strong generalizability of CURV, with improvements≥↑1.20% across four chart benchmarks. Generalizability to Out-of-Domain Multimodal Reasoning. The advantage of CURV remains consistent in out-of-domain multimodal reasoning across diverse multimodal reasoning categories (§5.2), attaining up to↑10.20% accuracy improvements (Tab. 3). Generalizability to Different Training Paradigms. CURV is readily applicable to other training paradigms (§G.4) to enhance MLLMs’ intrinsic visual reasoning capabilities. As shown in Tab. 9, applying CURV to reinforcement learning yields up to 12.58% improvement, while combining SFT and RL further increases gains to 17.04%. Despite even higher perfor- mance with RL, its substantially higher computational overhead (Tab. 8) contrasts with the more favorable performance-efficiency trade-off of SFT. 5.4 Ablations on Visual Grounded Reasoning Figure 7: Explicit vs. Implicit Benefits of Explicit over Implicit Vi- sual Grounded Reasoning. Compar- ing CURV with explicit and implicit visual grounded reasoning (§G.5), Fig. 7 demonstrates that explicit vi- sual grounded reasoning consistently outperforms its implicit counterpart (↑8.78%). This advantage highlights the value of visual grounding as an in- tegral intermediate vision-reasoning bridge, rather than the ultimate learn- ing objective, to effectively develop and enhance MLLMs’ intrinsic visual reasoning capabilities. CURV@Applied Presents More Effective Visual Grounding. Implementing three visual grounding methods (§3 & §D), results in Tab. 2 unveils that boxed grounding stays less beneficial than directly highlighting regions of focus through applied masking, despite its simplicity and straightforwardness. On the other hand, although restricted by the trade- off between zoom-in resolution and computation overhead (§G.2), cropped grounding showcases its strengths (Tab. 2) with up to↑7.93% improvements onacc@ MLLMand ↑ 9.92% gains on acc@0.0, despite a sixteen-fold reduction in resolution. Effectiveness of Curriculum Learning. Comparing across task difficulty levels (§G.6), results (Fig. 30) show that CL consistently improves performance over standard training, which remains inferior to CURV and exhibits diminished gains as task complexity increases. Figure 8: Ablation on Chart Types. We compare ∆ acc among chart types. Consistent Gains across Chart Types. As shown Fig. 8 (top), we com- pare the accuracy improvements (∆ acc ) of CURV@Applied (Qwen2.5-VL-7B) over its base model across chart types. Bar charts benefit the most, followed by line plots and heatmaps. The re- maining chart types also demonstrate positive gains, with radar charts con- tributing the least, reflecting their lower prevalence than other chart types.Furthermore, the compari- son among CURV@Applied, @Boxed, and @Cropped (Fig. 8) shows different distributions across chart types, sug- gesting different effects across visual grounding strategies (§D). 9 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Figure 9: Foundational Learning In Chart Under- standing. We compare Qwen2.5-VL-7B with its ap- plied model trained on curriculum levels 1, 1+2, and 1 + 2 + 3, respectively. FoundationalLearningDrives More Balanced Gains. While all curriculum levels improve overall accuracy, training on levels 1+2 yields the most consistent and sub- stantial gains across different levels (Fig. 9). In contrast, relying solely on level 3 is less effective, highlighting the significance of foundational learning in establishing adaptive and generalizable visual reasoning capabilities to support more robust chart understanding across varying difficulty levels (§G.8). CURV Enhances Multifaceted Visual Reasoning Abilities. CURV notably improves MLLMs’ performance, even on the most challenging level 3 multi-chart QA (Tab. 2). Despite the inherent complexity gap between single-chart (levels 1-2: accuracy up to 80.43%) and multi-chart scenarios (level 3: accuracy≤42.54%), CURV narrows this gap by enhancing both localization accuracy and relational understanding across different chart subplots (§C.2). Figure 10: Performance on Multi-Chart Understand- ing. Top-4 accuracy scores on curriculum level 3. The operational breakdown in Fig. 10 shows that CURV leads to clear improvements in localization (↑4.68%), enabling models to more accurately identify relevant chart subplots within complex layouts. In addition, CURV also effectively improves relation understanding (↑2.42%), facilitating more reliable cross-chart reasoning over dis- persed visual evidence. Collectively, these gains suggest that CURV strengthens MLLMs’ multifaceted visual reasoning abilities to tightly couple visual evidence with logical reasoning. This extends effectively beyond simple CQA settings to more complex multi-chart scenarios and out-of-domain reasoning, indicating that the benefits of CURV arise from improved reasoning grounded in visual structure rather than task-specific adaptation. 6 Conclusion In this work, we present CURV (§3), a curriculum learning framework that develops intrin- sic visual reasoning capabilities via progressive multi-step visual grounded reasoning. To support model learning, we construct CCQA (§4) with three progressive curriculum levels. Through systematic experiments (§5 & G), results demonstrate that tightly interleaving reasoning with visual grounding consistently improves performance across curriculum levels and generalizes effectively to chart understanding and out-of-domain multimodal reasoning. Our work establishes a foundation for developing self-contained visual reason- ing capabilities in MLLMs, moving beyond extrinsic assistance toward intrinsic grounded visual reasoning. 10 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Acknowledgments The authors acknowledge William & Mary Research Computing for providing computa- tional resources and/or technical support that have contributed to the results reported within this paper. This work used DeltaAI at NCSA through allocation CIS260012 and CIS230280 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by U.S. National Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296. This material is based upon work supported by the Google Cloud Research Credits program with the award GCP19980904. References Meta AI. Llama 3.2 11b vision instruct, 2024. URLhttps://huggingface.co/meta-llama/ Llama-3.2-11B-Vision-Instruct. A. Baddeley, G. Hitch, and D. Baddeley. Working memory. In In G. H. Bower (Ed.), The psychology of learning andmotivation: Advances in research and theory, volume 8, p. 47–89, 1974. URL https://api.semanticscholar.org/CorpusId:201393857. L. Barsalou. Grounded cognition. Annual review of psychology, 59:617–45, 2008. URL https://w.ncbi.nlm.nih.gov/pubmed/17705682. 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Deepeyes: Incentivizing ”thinking with images” via reinforcement learning, 2025. 15 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Contents 1 Introduction1 2 What Prohibits MLLMs From Chart Understanding Success?3 2.1Preliminary Analysis of CQA Failures . . . . . . . . . . . . . . . . . . . . . .3 2.2Enhancing Chart Reasoning Through Dynamic Visual Grounding . . . . . .3 3CURV: Chart Reasoning with Dynamic Visual Grounding3 3.1Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 3.2Multi-Step Reasoning With Dynamic Visual Grounding . . . . . . . . . . . .4 3.3Curriculum Visual Grounded Reasoning . . . . . . . . . . . . . . . . . . . . .5 4CCQA: Curriculum Chart Question Answering6 4.1Dataset Construction Principles . . . . . . . . . . . . . . . . . . . . . . . . . .6 4.2Curriculum Chart Question Answering . . . . . . . . . . . . . . . . . . . . .6 5 Experiments7 5.1Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 5.2Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 5.3Main Results On Chart & Multimodal Understanding . . . . . . . . . . . . .8 5.4Ablations on Visual Grounded Reasoning . . . . . . . . . . . . . . . . . . . .9 6 Conclusion10 A Preliminary Exploration & Validation18 A.1 Preliminary Exploration On Motivations . . . . . . . . . . . . . . . . . . . . .18 A.2 Preliminary Exploration on CQA Challenges . . . . . . . . . . . . . . . . . .18 A.3 Preliminary Validation on CCQA . . . . . . . . . . . . . . . . . . . . . . . . .19 B Related Work27 C Dataset Construction27 C.1 Data Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29 C.2 Multi-Level Curriculum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30 C.3 Fine-Grained Curriculum Tiers . . . . . . . . . . . . . . . . . . . . . . . . . .31 C.4 Meta-Learning Supported Curriculum Learning . . . . . . . . . . . . . . . .32 C.5 Quality Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 D Visual Grounding Strategies33 D.1 Applied: Grounding Through Dynamic Visual Focus Highlighting . . . . .33 D.2 Boxed: Grounding Through Dynamic Visual Box Guides . . . . . . . . . . .34 16 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning D.3 Cropped: Grounding Through Dynamic Visual Focus Zooming . . . . . . .34 E Generation Mode37 E.1 Mode A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 E.2 Mode VA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 E.3 Mode RA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 E.4 Mode RVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 F Evaluation Metrics40 F.1Evaluation of Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40 F.2Evaluation of Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 F.3Evaluation of Visual Grounding . . . . . . . . . . . . . . . . . . . . . . . . . .43 F.4Evaluation Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 G In-Depth Analysis44 G.1 Implementation Details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44 G.2 Grounding Method & Computation Cost . . . . . . . . . . . . . . . . . . . .44 G.3 The Role of Visual Grounding: From Extrinsic Assistance To Intrinsic Abilities 44 G.4 Visual Grounded Reasoning via Reinforcement Learning . . . . . . . . . . .45 G.5 Explicit vs. Implicit Visual Grounded Reasoning . . . . . . . . . . . . . . . .46 G.5.1 Explicit Visual Grounding . . . . . . . . . . . . . . . . . . . . . . . . .46 G.5.2 Implicit Visual Grounding . . . . . . . . . . . . . . . . . . . . . . . . .47 G.6 Curriculum Learning: Effectiveness of CL in Chart Understanding . . . . .48 G.7 Two-Stage Learning: Strength of Training Curriculum . . . . . . . . . . . . .48 G.8 Foundational Learning: Trade-off between Robustness & Adaptability . . .49 G.9 Broader Baselines: Comparison Against Chart Specialist & Tool-Use Agents49 H Multi-Step Reasoning With Dynamic Visual Grounding50 H.1 Challenges In Multi-Step Visual Grounding . . . . . . . . . . . . . . . . . . .50 H.2 Inference Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 H.3 Inference Success . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .51 ILimitations & Future Work52 17 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning A Preliminary Exploration & Validation A.1 Preliminary Exploration On Motivations Figure 11: Error Analysis (§ 2) Building on the cognitive perspectives that humans solve multimodal problems through decomposi- tion, interleaved visual reasoning, and composition (Fig. 1), we conduct preliminary studies to examine where current MLLMs fall short (§2). Concretely, we evaluate GPT-4o on 60 CharXiv ((Wang et al., 2024)) samples, categorizing the root causes of CQA failures into three classes: reasoning errors (decomposition), perception errors (interleaved visual reasoning), and integration er- rors (composition). As shown in Fig. 11, perception errors (28.33%) emerge as the dominant reason of failures, reflecting the difficulty of accurately ground- ing reasoning steps in fine-grained chart details. Be- yond perception, models also exhibit weaknesses in decomposition (10.00%), struggling to break down complex problems into coherent chains of reasoning. Additionally, they also show deficiencies in compo- sition (8.33%), failing to integrate grounded visual evidence into a coherent, interleaved chain of thought. These findings reveal systematic shortcomings in human-inspired reasoning stages, motivating our design of CURV that enhances MLLMs’ intrinsic visual grounded reasoning capabilities by simulating human cognitive process of decomposing, interleaving, and composing toward a solution. Inspired by recent findings that models can effectively learn from low-level features (Wang et al., 2025d), we concretize the notion of decomposition in two complementary forms: (1) visual decomposition, where each chart is decomposed into low-level components (Fig. 4) to to guide MLLMs’ attention toward fine-grained and informative details; and (2) reasoning decomposition, where each chart understanding problem is decomposed into a structured chain of reasoning steps to help MLLMs enhance their logical reasoning capacities. Building on these decompositions, we incorporate interleaving insights into our design, enabling reasoning to be interleaved with dynamically shifting visual focuses. Accordingly, the composition process integrates all intermediate learning in a coherent chain: from reasoning composition that consolidates step-wise reasoning into a coherent logical chain, to visual composition that progressively aggregates low-level visual interpretations into holistic chart comprehension. Together, these elements form the foundation of our curriculum learning design (§3), which standardizes two dimensions of progression: (1) curriculum CQA reasoning difficulty, controlled by increasing the number of nested functions; and (2) curriculum chart visual complexity, controlled by increasing the number of low-level components, chart types, and chart subplots. We further support our design with meta-learning (§C.4) to endow MLLMs with adaptability and generalizability in the face of varying chart types, context domains, and task complexity. A.2 Preliminary Exploration on CQA Challenges To investigate the underlying causes of failures in chart understanding, we employ five MLLMs, including three close-source models (GPT-4.1-mini (OpenAI, 2025), GPT-4o (OpenAI, 2024), Gemini-2.5-Flask (DeepMind, 2025)) and two open-source models (Qwen2.5-VL-3B and Qwen2.5-VL-7B (Qwen, 2025)) to identify the root causes of their failures. Specifically, we analyze their CQA outputs case-by-case on different CQA benchmarks (§5.2), noticing several key patterns in MLLMs’ CQA failures: 18 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning •Reasoning Accuracy & Consistency: While prompting MLLMs to do CoT reasoning can guide them toward correct answers in some cases, we still observe notable visual reasoning failures. For example, in Fig. 12, Qwen2.5-VL-3B misaligns line colors with their corresponding labels at the beginning, which propagates this misperception through subsequent reasoning and results in an incorrect answer. On the other hand, in Fig. 13, GPT-4o fails to excluded “Loki” despite having correctly identified it in earlier steps, un- veiling the inconsistency in its evolution of reasoning. Another form of reasoning incon- sistency emerges in recursive self-correction, where it may occur repeatedly throughout the model’s reasoning process, ultimately producing inconsistent or divergent answers (e.g., Gemini-2.5-Flash in Fig. 13). •Visual Grounding Accuracy: MLLMs exhibit significant challenges in precisely capture visual details from the chart images. For example, in Fig. 12, GPT-4o inaccurately esti- mates theW H value of the red “fi” point as approximately 0.105, while the true value is significantly less than 0.1, residing just above 0.0. • Visual Reasoning Effectiveness: In complex reasoning tasks requiring the integration of multiple visual regions and reasoning steps, MLLMs often struggle to effectively link visual attention with logical reasoning. For instance (Fig. 13), although GPT-4o and GPT-4o-mini both perceive accurately in their initial perception, they exhibit distinct failures in subsequent reasoning: GPT-4o incorrectly includes “Roar” while GPT-4o-mini fails to incorporate “Loki”. A.3 Preliminary Validation on CCQA In validating our proposed curriculum learning benchmark, CCQA (§4), we employ the same five MLLMs as our preliminary exploration (§A.2), examining case studies on five fine-grained difficulty tiers (§C.3) of the three curriculum levels (§C.2): • Tier 1: Single-Plot Reasoning (D = 1). The example in Fig. 14 is a Statistics-Mean query (Tab. 5) that MLLMs often fail to correctly answer. Among all three faiure cases, Qwen2.5- VL-3B and GPT-4o-mini fail at accurately perceive the numbers from the chart image, while Qwen2.5-VL-7B encounters calculation errors despite correct visual understanding. •Tier 2: Single-Plot Reasoning (D =2). The example in Fig. 15 defines a specific Subset (Tab. 5) that poses significant obstacles for most MLLMs in arriving at the correct answer. Except for Gemini-2.5-Flask, all other four models struggle to accurately identify the required chart components. •Tier 3: Single-Plot Reasoning (D ≥3). The example in Fig. 16 involves multiple operators (Tab. 5) that unfolds notable challenges for most MLLMs in both visual reasoning and mathematical calculation. Except for GPT-4o, all other four models struggle to accurately calculate the mean, leading to wrong final answers. •Tier 4: Multi-Plot Reasoning (Localization). The example in Fig. 17 involves both Localization and Statistics-Sum (Tab. 5). All five models fail to correctly perceive the visual information, resulting in wrong intermediate reasoning and incorrect final answer. • Tier 5: Multi-Plot Reasoning (Relation). The example in Fig. 18 additionally pay atten- tion to the Relation (Tab. 5) of different subplots. Likewise, the incorrect visual reasoning eventually lead to wrong final answers for all five models. The diverging trends of failure cases (increase) and success cases (decrease) (Fig. 14-18) demon- strate the effectiveness of our curriculum design for progressively increased complexity tiers, meanwhile highlighting the challenges that persist in chart understanding across all reasoning depths and task complexity levels. 19 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning What is the name of the line that is the furthest away from its fi value from the W_H axis? GPT-4o First, locate the “fi” symbols on the graph. These are marked as “x” with different colors corresponding to the lines. Here’s the observation: - The green “fi” (x marker) point is around W_H ≈ 0.15, W_L ≈ 0.03 and it is on the green dotted line (λ_L = 0.13) - The red “fi” point is at W_H ≈ 0.105, W_L ≈ 0.05 and it is on the red dotted line (λ_L = 0.20) - The blue “fi” point is at W_H ≈ 0.27, W_L ≈ 0.08 and it is on the blue dotted line (λ_L = 0.30). Now, to find the line which is the furthest away from its fi point: 1. Measure the distance from each "fi" point vertically to the W_H axis for each line. 2. Check which one is further away from the W_H axis. Considering the given data: - Green line (λ_L = 0.13) fi point: W_H ≈ 0.15, W_L ≈ 0.03. - Red line (λ_L = 0.20) fi point: W_H ≈ 0.105, W_L ≈ 0.05. - Blue line (λ_L = 0.30) fi point: W_H ≈ 0.27, W_L ≈ 0.08. From visual inspection: - The blue fi point at W_H ≈ 0.27 is farther away from the W_H axis than either of the other lines’ fi points. - The red and green fi points are closer to the origin compared to the blue fi point. Hence, the graph shows that the blue line (λ_L = 0.30) is the furthest away from its fi point from the W_H axis. GPT-4o-mini To identify the line that is furthest away from the fi value along the W_H axis, we first need to locate the fi value on the plot. The fi value is represented by a star marker (★) and corresponds to the coordinates approximately at (0.1, 0.02). Next, we analyze the lines representing different lambda (λ) values: 1. **λL = 0.13** (dashed line): has points that are closer to the fi point. 2. **λL = 0.20** (dotted line): this line has some points that are slightly further from the fi. 3. **λL = 0.30** (dashed-dotted line): shows points that are significantly higher in W_L compared to the fi value. Visually assessing the distances from the fi point for each line, it seems that the line for **λL = 0.30** is the furthest away from the fi value due to the combination of both its W_H and W_L coordinates. Therefore, the final answer is: The line corresponding to λL = 0.30 is the furthest away from fi. Qwen2.5-VL-3B 1. **Identify the fi values:** - The fi value for the green dashed line is approximately 0.30. - The fi value for the red dotted line is approximately 0.20. - The fi value for the blue solid line is approximately 0.13. 2. **Determine the position of each line relative to its fi value:** - For the green dashed line (fi = 0.30), it is positioned at W_H \≈ 0.4. - For the red dotted line (fi = 0.20), it is positioned at W_H \≈ 0.25. - For the blue solid line (fi = 0.13), it is positioned at W_H \≈ 0.1. 3. **Compare the distances from the fi value to the W_H axis:** - The distance for the green dashed line is \$|0.4 - 0.30| = 0.10\$. - The distance for the red dotted line is \$|0.25 - 0.20| = 0.05\$. - The distance for the blue solid line is \$|0.1 - 0.13| = 0.03\$. 4. **Conclusion:** The green dashed line is the furthest away Qwen2.5-VL-7B To determine which line is the furthest away from its "fi" value on the W_H axis, we need to follow these steps： 1. Identify the "fi" values on the plot. 2. Determine the distance of each line from its corresponding "fi" value on the W_H axis. From the legend, we can see that: - The "fi" value corresponds to the red "X" marker. Now, let's analyze the lines: - The green dashed line (λ_L = 0.13) starts at approximately -0.2 on the W_H axis. - The blue dotted line (λ_L = 0.20) starts at approximately -0.1 on the W_H axis. - The blue dashed line (λ_L = 0.30) starts at approximately 0.0 on the W_H axis. The "fi" value for the "fi" marker is at approximately -0.05 on the W_H axis. Comparing the distances: - Green dashed line: Approximately -0.2 - (-0.05) = -0.15 - Blue dotted line: Approximately -0.1 - (-0.05) = -0.05 - Blue dashed line: Approximately 0.0 - (-0.05) = 0.05 The green dashed line has the largest negative distance from its "fi" value, indicating it is the furthest away. Final answer: The green dashed line (λ_L = 0.13) is the furthest away from its "fi" value on the W_H axis. Gemini-2.5-Flash CQA Benchmark - CharXiv Understanding the Plot The plot shows three different lines, each corresponding to a different value of λL (0.13, 0.20, and 0.30). Each line is composed of several points, and four of these points are labeled with different shapes representing specific values (ap, sm, fi, and ni). We're interested in the fi value, which is marked by an 'x' . The x-axis represents the WH values, and the y-axis represents the WL values. The prompt asks which line is "furthest away from its fi value from the WH axis." This phrasing is a bit ambiguous. It could mean either the furthest vertical distance from the WH axis to the 'fi' point for each line, or the furthest horizontal distance from a 'fi' point to the WH axis. However, the WH axis is the horizontal axis, and the distance from a point to a line is the perpendicular distance. So, the distance from a point to the WH axis is its WL value. Analyzing the Data For the λL = 0.13 line (green dashed line): The 'fi' point is at approximately WH ≈−0.2 and WL ≈0.025. The distance from the WH axis is about 0.025. For the λL = 0.20 line (red dotted line): The 'fi' point is at approximately WH ≈0.02 and WL ≈0.05. The distance from the WH axis is about 0.05. For the λL = 0.30 line (blue dash-dotted line): The 'fi' point is at approximately WH ≈0.12 and WL ≈0.07. The distance from the WH axis is about 0.07. Comparing the WL values for the 'fi' points, we see that 0.07 > 0.05 > 0.025. Therefore, the 'fi' point for the λL = 0.30 line is the one furthest from the WH axis. The line associated with this point is the one for λL = 0.30. Figure 12: Examples of Preliminary Exploration on CharXiv. We conduct case study on CharXiv samples by prompting the model to generate CoT reasoning and final answer. CoT reasoning prompt is omitted in the input for clarify. 20 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Figure 13: Examples of Preliminary Exploration on ChartMuseum. We conduct case study on ChartMuseum samples by prompting the model to generate CoT reasoning and final answer. CoT reasoning prompt is omitted in the input for clarify. 21 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Figure 14: Examples of Validation On Curriculum Level 1 (D =1). We validate our CCQA on Tier 1 (§ C.3), i.e., Curriculum Level 1 (D =1), by prompting the model to generate CoT reasoning and final answer. CoT reasoning prompt is omitted in the input for clarify. 22 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Figure 15: Examples of Validation On Curriculum Level 2 (D ≥2). We validate our CCQA on Tier 2 (§ C.3), i.e., Curriculum Level 2 (D ≥2), by prompting the model to generate CoT reasoning and final answer. CoT reasoning prompt is omitted in the input for clarify. 23 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Figure 16: Examples of Validation On Curriculum Level 2 (D ≥3). We validate our CCQA on Tier 3 (§ C.3), i.e., Curriculum Level 2 (D ≥3), by prompting the model to generate CoT reasoning and final answer. CoT reasoning prompt is omitted in the input for clarify. 24 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Figure 17: Examples of Validation On Curriculum Level 3 (D ≥3). We validate our CCQA on Tier 4 (§ C.3), i.e., Curriculum Level 3 (D ≥3), by prompting the model to generate CoT reasoning and final answer. CoT reasoning prompt is omitted in the input for clarify. 25 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Figure 18: Examples of Validation On Curriculum Level 3 (D ≥3). We validate our CCQA on Tier 5 (§ C.3), i.e., Curriculum Level 3 (D ≥3), by prompting the model to generate CoT reasoning and final answer. CoT reasoning prompt is omitted in the input for clarify. 26 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning B Related Work Chain-of-Thought Reasoning.CoT reasoning has emerged as foundations for enhancing the interpretability and performance of large language models (LLMs). Even prompting LLMs to perform CoT reasoning before answering can improve performance (Wei et al., 2022; Wang et al., 2023). CoT reasoning is particularly beneficial for MLLMs in complex visual reasoning tasks where attentions are interrelated to both visual and textual features (Zhang et al., 2024b; Shao et al., 2024; Zhao et al., 2025; Qi et al., 2024). Recent work enhances CoT via rationale-augmented training (Zhang et al., 2024b) and feedback-based refinement (Chen et al., 2024), as well as incorporating simple visual operations into reasoning (Qi et al., 2024). However, most approaches remain largely text-centric or loosely grounded, failing to tightly integrate visual evidence into each reasoning step. Visual Grounding. Visual grounding aligns language with visual regions, enabling spatially-aware localization of entities, attributes, and relationships. Through explicit visual localizations (Gou et al., 2024) or implicit coordinate-free grounding (Kang et al., 2025), recent advances in segmentation (Carion et al., 2025) and GUI grounding (Cheng et al., 2024; Wu et al., 2025) significantly improve text-visual alignment. Nevertheless, grounding is typically regarded as a standalone objective or terminal prediction, rather than an integral component of the reasoning process. In contrast, our work leverages visual grounding as an intermediate and iterative medium within the reasoning chain, where visual changes dy- namically inform subsequent reasoning steps and vice versa, shifting visual grounding from a passive alignment objective to an active component in interleaved multimodal reasoning. Multimodal Chart Understanding.CQA represents a specialized task that requires accu- rate understanding of structured visual representations and complex reasoning over visual and textual elements. Recent benchmarks have focused on real-world chart complexity and diversity, such as ChartQA (Masry et al., 2022), ChartQA-Pro (Masry et al., 2025), Chart- Museum (Tang et al., 2025), CharXiv (Wang et al., 2024), etc. Prior work addresses CQA via structured pipelines (Kim et al., 2024), compositional reasoning (Zhang et al., 2024a), improved chart representation (Zheng et al., 2024), and synthetic data generation (Li et al., 2024b). However, these methods often decouple perception from reasoning or rely on static representations, limiting adaptive visual dynamics throughout reasoning. C Dataset Construction Chart Metadata. Extending our introduction to CCQA (§4), we elaborate on the seven types of charts, the domain categories of the source plotting data, and fundamental operators that support multi-layer nested functions, as summarized in Tables 4-5. Data Augmentation. To effectively support curriculum learning with meta-learning in- sights (§C.4), we design a comprehensive set of chart-specific data augmentation strategies implemented through chart rendering functions. These augmentations introduce controlled variability in both the structural layout and visual presentation of charts, thereby enhancing model generalization across diverse chart types. Specifically, we consider the following transformations: (1) chart rotation at different angles (e.g., 0°, 30°, 45°, 60°, 75°, 90°); (2) ori- entation adjustments between vertical and horizontal layouts; (3) axis placement variations, such as shifting thex- and/or y-axis among left, right, top, and bottom; (4) color setting across chart elements; (5) legend positioning (top, bottom, center, left, or right) and visibility; (6) label positions and visibili(ty such as axis labels, tick labels, and numeric annotations); and other augmentation strategies tailored for specific types of charts (e.g., ‘explode’ settings for pie charts, marker styles for scatter plots, etc.) Collectively, these augmentation strategies form a systematic approach for generating richly diverse chart appearances using a small set of metadata, ensuring robustness and adaptability of models trained under meta-learning supported curriculum learning. 27 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning CategoryDescription Chart Types BarHistogramScatterLine HeatmapPieRadar Domain Categories 1. Media & Entertainment 2. Geography & Demography 3. Education & Academia 4. Business & Industry 5. Major & Course 6. Animal & Zoology 7. Plant & Botany 8. Biology & Chemistry 9. Food & Nutrition 10. Space & Astronomy 11. Sale & Merchandise 12. Market & Economy 13. Sports & Athletics 14. Computing & Technology 15. Health & Medicine 16. Energy & Environment 17. Travel & Expedition 18. Arts & Culture 19. Communication & Collaboration 20. Language & Linguistics 21. History & Archaeology 22. Weather & Climate 23. Transportation & Infrastructure 24. Psychology & Personality 25. Materials & Engineering 26. Philanthropy & Charity 27. Fashion & Apparel 28. Parenting & Child Development 29. Architecture & Urban Planning 30. Gaming & Recreation Table 4: Metadata of Chart Plotting. We employ seven commonly used chart types across 30 different domain categories to construct the meta images for CCQA. 28 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning OperatorDescription ReadRead or estimate the value of chart component(s) that meet given requirement(s) Statistics - Sum/Mean/Median Calculate the sum/mean/median of a group of chart components that meet given requirement(s) Statistics - CountCount the number of chart components that meet given requirement(s) Extrema - Value - Min/Max Calculate the minimum/maximum value (which may be combined with nested functions, e.g., the minimum mean value of two groups of chart components) of chart components that meet given requirement(s) Extrema - PositionLocalize chart component(s) that meet given requirement(s), e.g., the leftmost bar in the bar chart Sort - Ascending/De- scending Sort a group of chart components that meet given requirement(s) Compare - Value/Diff/Position Compare the value/difference/position of two groups of chart components based on the given requirement(s) FilterFilter chart component(s) based on the given requirement(s) ThresholdIdentify chart component(s) based on the given threshold condition(s) SubsetIdentify the subset of chart component(s) that satisfy the specified requirement(s) LocalizationLocalize specific chart components and/or subplots RelationUnderstand relations between or among different chart components and/or subplots Table 5: Foundational Operators. Our CCQA incorporates 12 basic operators to query different aspects of chart components, facilitating comprehensive understanding of each chart elements. Figure 19: From Template To CQA Data. A template-based data generation example that illustrates how question and reasoning templates are converted to CQA data based on the chart data. C.1 Data Structure Our CCQA (§4) encompass seven basic chart types, including bar chart, histogram, scatter plot, line chart, heatmap, pie chart, and radar chart (Tab. 4). Only the chart plotting data (i.e., the value and label of each chart component, along with the axis and image titles) are generated by GPT-4o (see an example in Fig. 20). We construct the CQA data of each chart type and curriculum level across 30 domain categories (Tab. 4). The number of samples for each chart type is influenced by chart features (e.g., scatter plots depend on both X- and Y-axis features, whereas heatmaps depend on cell values and labels), CQA types (Tab. 4, 5), and properties 29 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning of the source plotting data (e.g., the number of bars, scatter points, or cells; variations in label angles; etc.). To ensure high data quality, we implement template-based CQA generation that guarantees not only the diversity of CQA tasks but also the accuracy and reliability of intermediate reasoning, visual grounding, and final answers. Specifically, all question-answer pairs in CCQA, together with their corresponding reason- ing steps and dynamic visual grounding coordinates, are generated using human-defined templates and functions (§4.1). An example is shown in Fig. 19 to illustrate the template- based CQA data generation process. GPT-4o Prompt For Plotting Data Generation Can you help with heatmap chart plotting data generation for my heatmap plotting? **Data Example** This plotting data example below is an input data dictionary for the first domain category "1 - Media & Entertainment". ```python HEATMAP_DICT = "heatmap_data": [[80.5, 20.5, 55.5, 90.0, 60.0], [30.5, 75.0, 45.5, 60.0, 80.0], [90.0, 10.5, 80.0, 20.0, 70.5], [80.0, 60.5, 30.0, 90.0, 59.5], [77.5, 80.5, 66.5, 25.5, 95.5]], "heatmap_category": "singular": "rating score", "plural": "rating scores", "x_labels": ["Titanic", "Star Wars", "The Lord of the Rings", "Snow White and the Seven Dwarfs", "The Lion King"], "y_labels": ["2005", "2010", "2015", "2020", "2025"], "x_label": "Movies", "y_label": "Rating Year", "image_title": "Correlation Between Movie Ratings and Rating Year", ``` **Your Task** For the second domain category "2 - Geography & Demography", can you help generate a new plotting data using the same dictionary structure and data format? **Important Requirements** * Please generate the new plotting data specifically for the new domain category. * Each data entry should be unique and should simulate real-world circumstances of the new domain category. * Please be careful about `heatmap_category`, which will be used as how to generally call the heatmap cells. For example, `sentence=f"these heatmap cells represent different HEATMAP_DICT['heatmap_category']['plural']"` will become "these heatmap cells represent different *movie rating scores*". * Please response in the following format: ```python HEATMAP_DICT = <your new data dictionary> ``` HEATMAP_DICT = "heatmap_data": [ [70.5, 82.0, 65.0, 90.0, 75.0], [60.0, 55.0, 85.0, 80.0, 70.0], [55.0, 78.0, 88.0, 60.0, 65.0], [80.0, 68.0, 72.0, 85.0, 90.0], [95.0, 75.0, 60.0, 70.0, 80.0] ], "heatmap_category": "singular": "urbanization index", "plural": "urbanization indices", "x_labels": ["Tokyo", "New York", "Paris", "São Paulo", "Lagos"], "y_labels": ["1990", "2000", "2010", "2020", "2030"], "x_label": "Cities", "y_label": "Year", "image_title": "Urbanization Indices Over Time in Major Global Cities" Figure 20: Example of Plotting Data Generation Prompt. We prompt GPT-4o to generate the source plotting data, which will be used as the input for chart drawing functions. This example is for the heatmap plotting data generation of the second domain category (Tab. 4). C.2 Multi-Level Curriculum Implementing curriculum learning to progressively increase reasoning difficulty across three distinct levels (Fig. 5), each level targets at specific aspects of visual reasoning development (Tab. 5): read, statistics, extrema, sorting, comparison, filtering, thresholding, subset constraints, localization, and relation. The knowledge transfer between levels contributes to the increase of task complexity (§C). 30 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning LEVEL 1: Foundational Single-Operation Reasoning.LEVEL 1 establishes fundamental chart understanding capabilities (reasoning depth:D 1 =1) with single-operation reasoning processes on single-plot charts: SINGLE(x) = f (x)(7) wheref (·)represents basic operations such as direct value reading, simple arithmetic, and elementary comparisons within a single-plot chart. Hereby, the reasoning depth of a CQA data sample, denoted asD l , is defined as the number of nested operations for curriculum level l. Accordingly, the template structure of LEVEL-1 CQA data follow the format: TEMPLATE 1 =Q,R t , B ∗ t T 1 t=1 , A(8) whereQis the question,T 1 ≥1 is the number of reasoning steps,R t is thet-th reasoning step,B ∗ t is the corresponding ground-truth visual grounding with reasoning depthD 1 =1, and A is the final answer. LEVEL 2: Multi-Operation Reasoning.LEVEL 2 introduces compositional reasoning (reasoning depth: D 2 > 1) through nested operations on single charts: SINGLE(x) = F( f (x))(9) whereF(·)represents composite operations applied tof (x), e.g.,F( f (x)) = h(g( f (x))). This level requires models to perform sequential reasoning and visual grounding where each step builds upon previous computations. Consequently, LEVEL 2 templates extend to multi-step reasoning: TEMPLATE 2 =Q,R t , B ∗ t T 2 t=1 , A(10) whereT 2 ≥2 is the number of reasoning steps, each reasoning stepR t progresses through nested operations with reasoning depthD 2 ≥2, andB ∗ t is the corresponding ground-truth visual grounding for the t-th step. LEVEL 3: Complex Multi-Chart ReasoningLEVEL 3 represents the most challenging scenarios that involve complex reasoning (D 3 ≥2) across multiple subplots and chart types: MULTI(x) = MULTI(SINGLE i (x))(11) whereiconforms to 1≤ i ≤ nand 1≤ n ≤ subplotnum. LEVEL 3 templates thereby incorporate multi-step and cross-chart dependencies: TEMPLATE 3 =Q,R t , B ∗ t , C t T 3 t=1 , A(12) whereT 3 ≥3 is the number of reasoning steps, each reasoning stepR t progresses through complex nested operations with reasoning depthD 3 ≥3,B ∗ t is the corresponding ground- truth visual grounding, andC t indicates the chart index for thet-th reasoning step. Specifi- cally, our multi-plot reasoning incorporates both localization(n−1)and relation(n>1) operations across multiple charts. C.3 Fine-Grained Curriculum Tiers We construct our three-level curriculum dataset (Fig. 5) based on reasoning depth and chart complexity (§C.2). According to their fine-grained problem-solving difficulty, we categorize them into five curriculum tiers: •Tier 1: Curriculum Level 1 (D =1). In Tier 1, all CQA data correspond to queries about single-plot chart image input. Reasoning is limited to one depth level, i.e., single-function reasoning (Eq. 7, D 1 ≥ 1), to derive the final answer. •Tier 2: Curriculum Level 2 (D =2). In Tier 2, all CQA data correspond to queries about single-plot chart image input. Reasoning requires two depth levels, i.e., constructed through two nested functions (Eq. 9, D 2 ≥ 2), to derive the final answer. 31 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning •Tier 3: Curriculum Level 2 (D ≥3). In Tier 3, all CQA data correspond to queries about single-plot chart image input, with reasoning that involves three or more depth levels, i.e., constructed through three or more nested functions (Eq. 9,D 2 ≥3), to derive the final answer. •Tier 4: Curriculum Level 3 - Localization (D ≥3). In Tier 4, all CQA data correspond to queries about multi-plot chart image input, with reasoning that involves three or more depth levels, i.e., constructed through three or more nested functions (Eq. 11,D 3 ≥3), to derive the final answer. While different from single-plot charts, multi-plot CQA tasks in Tier 4 involves the precise localization of target subplot(s) that directly yield the answer. •Tier 5: Curriculum Level 3 - Relation (D ≥3). CQA tasks in Tier 5 are similar to the constitution of Tier 4, corresponding to queries about multi-plot charts with reasoning that involves three or more depth levels, i.e., constructed through three or more nested functions (Eq. 11,D 3 ≥3). The key distinction is that, while Tier 4 emphasizes precise localization of target subplot(s), Tier 5 additionally demands the modeling of relations across the identified subplots. C.4 Meta-Learning Supported Curriculum Learning Our curriculum learning design (§3) is reinforced through meta-learning, which provides a principled way to structure both data and task complexity. Specifically, we leverage meta-learning through the following aspects: 1. Domain diversity as meta-tasks. We construct CCQA using 30 domain categories (Tab. 4), where each category contributes one source plotting data, and thus one chart image. This structured diversity provides a wide range of meta-tasks that expose MLLMs to domain-generalizable visual reasoning. 2. Chart-type variability as meta-structures. We employ 7 fundamental chart types (Tab. 4) to visualize the 30 domain datasets. Multiplying 30 plotting datasets by 7 chart types yields 210 unique chart-structure metadata, based on which the entire dataset is system- atically constructed. This ensures that each domain is represented across diverse chart structures, promoting cross-task adaptation. 3.Operator set as meta-functions. To support multi-layer nested reasoning, we define 12 fundamental operators (Tab. 5). These operators serve as compositional primitives for constructing multi-level CQA tasks. By progressively increasing the depth of nesting, we control CQA difficulty level, thereby enabling MLLMs to gradually acquire higher-order reasoning capabilities. 4. Decomposition as learning scaffolds. Following decomposition insights (§A.1), we disentangle each task into visual decomposition (low-level chart components) and reasoning decomposition (singular operations across nested functions). This scaffolding allows MLLMs to incrementally learn fine-grained visual perception and step-wise logical reasoning, supporting the high-level composition of accurate chart understanding. 5.Meta-learning for transferability. Beyond dataset construction, our design leverages meta-learning to encourage transferability across chart types, domains, and reason- ing depths. By repeatedly exposing MLLMs to varied meta-tasks with systematically controlled complexity, we enable them to acquire generalizable strategies rather than overfitting to particular chart types or reasoning templates. Our meta-learning implemen- tation strengthens the robustness of curriculum learning by aligning it with principles of adaptation and generalization. Together, these design principles ensure that our curriculum learning is not only systematic but also meta-learnable, allowing MLLMs to progressively integrate visual and reasoning competencies across tasks of increasing complexity. C.5 Quality Control To ensure high data fidelity for both training and evaluation, we enforce rigorous quality control throughout the construction of CCQA: 32 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning First, we implement the meta-learning-driven construction pipeline in which each chart type is defined by human-authored meta-functions. These meta-functions explicitly encode the data generation logic and reasoning structure, with configurable chart attributes such as the number of chart elements, spatial arrangements, structural relationships, shapes and color settings, etc. This design enables precise and controllable synthesis of visual reasoning samples while preserving data diversity across multiple chart visual dimensions, such as colors, shapes, sizes, positions, etc. Prior to inclusion in the dataset, all generated instances are pre-executed to verify functional correctness and internal consistency. Second, we perform manual validation of the visualization metadata, including ground- truth visual grounding masks and the corresponding step-by-step reasoning annotations associated with each metadata instance. In addition to manually verifying the 30 meta images for each type of charts, we randomly sample 10 visual reasoning instances of each category (i.e., 10×30×7 in total) to validate accurate vision-reasoning alignment (Tab. 6). This process ensures that every visual grounding step accurately corresponds in multimodal spaces, and that the resulting reasoning chains are coherent, interpretable, and faithful to the underlying data. Critically, we restrict the role of LLM (GPT-4o) to generating randomized numerical param- eters within predefined domain categories for chart construction, which will only be used as inputs in plotting functions as visual grounded reasoning chain templates. That is, these parameters are deterministically incorporated into plotting functions and template-based reasoning chains, pre-computed in predefined equations prior to data instantiation. As a result, the use of LLMs is ensured to exhibit no effect on the correctness of ground-truth answers, thereby preserving the overall reliability and integrity of the dataset. Human ValidationBarHistogramScatterLineHeatmapPieRadar Meta Image30303030303030 CQA Instance 10101010101010 Table 6: Human Validation on Data Quality D Visual Grounding Strategies We propose three visual grounding strategies — applied, boxed, and cropped (Fig. 3, Tab. 7) — to enable dynamic visual focus navigation throughout the evolution of multi-step reasoning. All three strategies follow the same RVA process where the model generates reasoning steps accompanied by grounded bounding box coordinates, while the “dynamic” nature refers to how the visual focus adaptively changes as the train of thoughts progresses. Each strategy implements a distinct grounding mechanism for directing the model’s visual attention to corresponding image regions of focus while maintaining coherent reasoning flow. On the other hand, these strategies also represent different trade-offs between visual clarity, computational efficiency, and reasoning precision (Tab. 7). D.1 Applied: Grounding Through Dynamic Visual Focus Highlighting Method. The applied grounding strategy directly underlines the predicted regions of focus through semi-transparent yellow highlighting overlays. As the reasoning progresses, the yellow highlighting adaptively shifts to emphasize different visually focused regions mirroring each reasoning step, while preserving full visual context. Specifically, our applied grounding strategy implements visual grounding through semi- transparent highlighting overlays that mask visual focuses: I ′ vis,t = I orig ⊙ (1− κ· M f ocus,t ) + κ· H yellow ⊙ M f ocus,t (13) 33 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Method Low Computation High Precision Full Context No Occlusion Multi- Region Easy Integration Easy Comprehension applied✓✗✓✗✓ boxed✓✗✓ cropped✗✓✗ Table 7: Comparative Analysis of Visual Grounding Strategies. We evaluate each ground- ing strategy across seven key dimensions that are significant for effective visual reasoning. ✓indicates the strategy possesses the advantage, while✗indicates limitation. Boxed ground- ing achieves the best overall balance, applied grounding provides clear interpretability and highlighting with moderate trade-offs, and cropped grounding maximizes precision at the cost of computational efficiency and understanding straightforwardness. whereI orig is the original input image,M f ocus,t is the binary mask derived from the list of bounding boxesB t,i N t i=1 where eachB t,i = [x min ,y min ,x max ,y max ]andN t is the number of bounding boxes at reasoning stept,H yellow is the highlight color (i.e., yellow),κcontrols transparency, and⊙ denotes element-wise multiplication. Examples. Fig. 21 shows an example for multi-step CoT reasoning with applied visual grounding. D.2 Boxed: Grounding Through Dynamic Visual Box Guides Method. The boxed grounding strategy straightforwardly guides visual attention by adding red rectangular borders to the focus regions. These red boxes dynamically re- locate and resize along with the evolution of the reasoning chain, emphasizing current regions of focus through explicit visual boundaries. Particularly, the border guides are generated by drawing rectangular outlines at the specified coordinates: I ′ vis,t = BOX(I orig ,B t,i N t i=1 , C red , τ)(14) whereI orig is the original input image,B t,i N t i=1 is the list of bounding boxes at reasoning steptwhere eachB t,i = [x min ,y min ,x max ,y max ],BOX(·)draws colored rectangular border lines for each specified regions of focus,C red is the border line color (i.e., red), andτis the thickness of border lines. Examples. Fig. 22 shows an example for multi-step CoT reasoning with boxed visual grounding. D.3 Cropped: Grounding Through Dynamic Visual Focus Zooming Method. The cropped grounding strategy localizes corresponding regions of focus by zooming in, extracting and presenting the focused sub-regions as separate zoomed images alongside the full chart. As reasoning evolves, different cropped regions are dynamically generated and presented, enabling detailed examination of the specific components relevant to each reasoning step. Therefore, the cropping operation extracts sub-regions using array indexing based on the bounding box coordinates: I ′ vis,t = I orig ,CROP(I orig , B t,i ) N t i=1 (15) where CROP(I orig , B t,i ) = I orig [y min : y max , x min : x max ](16) 34 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Here,I orig is the original chart image,B t,i N t i=1 is the list of bounding boxes at reasoning stept, and the model processes both the full contextI orig and multiple zoomed crops CROP(I orig , B t,i ) N t i=1 simultaneously. Examples. Fig. 23 shows an example for multi-step CoT reasoning with cropped visual grounding. Applied Visual Grounding What is lowest Annual CO₂ Emissions (Million Metric Tons) among bars with their Annual CO₂ Emissions (Million Metric Tons) higher than 410 but lower than 1200? <think>First, I need to find all the bars with their Annual CO₂ E missions (Million Metric Tons) higher than 410 but lower than 1200.</think> <|box_start|>[[2180, 334, 2522, 656], [60, 344, 1766, 500], [2614, 391, 2783, 461], [60, 539, 1093, 695], [2614, 587, 2767, 657], [2180, 711, 2490, 907], [60, 735, 1154, 891], [2614, 782, 2809, 852], [60, 1125, 1031, 1282], [2614, 1173, 2731, 1243], [60, 1321, 1827, 1477], [2614, 1369, 2841, 1439], [60, 1516, 1256, 1673], [2614, 1564, 2766, 1634], [60, 1712, 1521, 1868], [2614, 1760, 2752, 1830], [60, 1907, 1977, 2064], [1979, 1951, 2092, 2021], [2614, 1955, 2735, 2025]]<|box_end|> <think>Second, I need to compare their Annual CO₂ Emissions (Million Metric Tons) to find the one with the lowest Annual CO₂ E missions (Million Metric Tons).</think><|box_start|>[[2180, 586, 2412, 656], [60, 1125, 1031, 1282], [2614, 1173, 2731, 1243]]<|box_end|> <answer>430</answer> Figure 21: Example of Applied Visual Grounding. The applied visual grounding method directly accentuates the regions of focus through semi-transparent yellow highlighting overlays. 35 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Boxed Visual Grounding What is lowest Annual CO₂ Emissions (Million Metric Tons) among bars with their Annual CO₂ Emissions (Million Metric Tons) higher than 410 but lower than 1200? <think>First, I need to find all the bars with their Annual CO₂ E missions (Million Metric Tons) higher than 410 but lower than 1200.</think> <|box_start|>[[2180, 334, 2522, 656], [60, 344, 1766, 500], [2614, 391, 2783, 461], [60, 539, 1093, 695], [2614, 587, 2767, 657], [2180, 711, 2490, 907], [60, 735, 1154, 891], [2614, 782, 2809, 852], [60, 1125, 1031, 1282], [2614, 1173, 2731, 1243], [60, 1321, 1827, 1477], [2614, 1369, 2841, 1439], [60, 1516, 1256, 1673], [2614, 1564, 2766, 1634], [60, 1712, 1521, 1868], [2614, 1760, 2752, 1830], [60, 1907, 1977, 2064], [1979, 1951, 2092, 2021], [2614, 1955, 2735, 2025]]<|box_end|> <think>Second, I need to compare their Annual CO₂ Emissions (Million Metric Tons) to find the one with the lowest Annual CO₂ E missions (Million Metric Tons).</think><|box_start|>[[2180, 586, 2412, 656], [60, 1125, 1031, 1282], [2614, 1173, 2731, 1243]]<|box_end|> <answer>430</answer> Figure 22: Example of Boxed Visual Grounding. The boxed visual grounding method directly accentuates the regions of focus through semi-transparent yellow highlighting overlays. 36 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Cropped Visual Grounding What is lowest Annual CO₂ Emissions (Million Metric Tons) among bars with their Annual CO₂ Emissions (Million Metric Tons) higher than 410 but lower than 1200? <think>First, I need to find all the bars with their Annual CO₂ E missions (Million Metric Tons) higher than 410 but lower than 1200.</think> <|box_start|>[[2180, 334, 2522, 656], [60, 344, 1766, 500], [2614, 391, 2783, 461], [60, 539, 1093, 695], [2614, 587, 2767, 657], [2180, 711, 2490, 907], [60, 735, 1154, 891], [2614, 782, 2809, 852], [60, 1125, 1031, 1282], [2614, 1173, 2731, 1243], [60, 1321, 1827, 1477], [2614, 1369, 2841, 1439], [60, 1516, 1256, 1673], [2614, 1564, 2766, 1634], [60, 1712, 1521, 1868], [2614, 1760, 2752, 1830], [60, 1907, 1977, 2064], [1979, 1951, 2092, 2021], [2614, 1955, 2735, 2025]]<|box_end|> <think>Second, I need to compare their Annual CO₂ Emissions (Million Metric Tons) to find the one with the lowest Annual CO₂ E missions (Million Metric Tons).</think><|box_start|>[[2180, 586, 2412, 656], [60, 1125, 1031, 1282], [2614, 1173, 2731, 1243]]<|box_end|> <answer>430</answer> Figure 23: Example of Cropped Visual Grounding. The cropped visual grounding method directly accentuates the regions of focus through semi-transparent yellow highlighting overlays. E Generation Mode E.1 Mode A Fig. 25 shows an example of generation mode A, where the model directly outputs the answer without intermediate reasoning and visual grounding. E.2 Mode VA Fig. 24 shows an example of generation mode VA, through which the model first generates its intermediate visual grounding via applied grounding method, followed by its final answer. For clarity, the input instructions for VA generation are omitted in the figure. To save space, the user’s intermediate responses are shown as smaller images on the right of each model response, corresponding to the model’s response on the left in each turn of the multi-step interaction. 37 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning E.3 Mode RA Fig. 26 shows an example of generation mode RA, through which the model first generates its intermediate CoT reasoning, followed by its final answer. The model is prompted to produce its CoT reasoning and final answer in a single-turn manner. For clarity, the input instructions of CoT reasoning and answering for RA mode are omitted in the figure. E.4 Mode RVA Fig. 27 shows an example of generation mode RVA, where the model first produces its intermediate reasoning with visual grounding, followed by its final answer. Similar to Fig. 24, for clarity, the input instructions for RVA mode generation are omitted in the figure. To save space, the user’s intermediate responses are shown as smaller images on the right of each model response, corresponding to the model’s response on the left in each turn of the multi-step RVA reasoning. Mode VA What is the highest Plant Height (Meters) among plants that have their Plant Height (Meters) above the average Plant Height (Meters) of all plants? <|box_start|>[[2541, 125, 2900, 288], [554, 578, 776, 705], [2719, 1122, 2954, 1288], [1651, 1213, 1872, 1363], [1119, 1304, 1437, 1444], [765, 1394, 1063, 1527], [2533, 1551, 2629, 2193], [2638, 1554, 2899, 2190], [677, 1576, 877, 1706], [291, 1757, 475, 1876], [519, 1848, 719, 1974], [783, 2029, 1069, 2162]]<|box_end|> <answer>16</answer> <|box_start|>[[2541, 125, 2900, 288], [2719, 1122, 2954, 1288], [1651, 1213, 1872, 1363], [2533, 1803, 2629, 1941], [2638, 1806, 2899, 1938], [2534, 2054, 2628, 2130], [2639, 2057, 2775, 2127]]<|box_end|> <|box_start|>[[2719, 1122, 2954, 1288], [2534, 1803, 2628, 1879], [2639, 1806, 2773, 1876]]<|box_end|> <|box_start|>[[2541, 125, 2900, 288], [554, 578, 776, 705], [2719, 1122, 2954, 1288], [1651, 1213, 1872, 1363], [1119, 1304, 1437, 1444], [765, 1394, 1063, 1527], [2533, 1551, 2629, 2193], [2638, 1554, 2899, 2190], [677, 1576, 877, 1706], [291, 1757, 475, 1876], [519, 1848, 719, 1974], [783, 2029, 1069, 2162]]<|box_end|> Figure 24: Example of Generation Mode VA. A CQA example resolved through generation mode VA. 38 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Mode A What is the highest Plant Height (Meters) among plants that have their Plant Height (Meters) above the average Plant Height (Meters) of all plants? <answer>16</answer> Figure 25: Example of Generation Mode A. A CQA example resolved through generation mode A. Mode RA What is the highest Plant Height (Meters) among plants that have their Plant Height (Meters) above the average Plant Height (Meters) of all plants? <think> First, I need to read the Plant Height (Meters) of each plant in this chart. Second, I need to compute the average Plant Height (Meters) of all plants. Third, I need to find all the plants whose Plant Height (Meters) is higher than 7.23. Fourth, I need to identify the plant with the highest Plant Height (Meters) among these plants. </think> <answer>16</answer> Figure 26: Example of Generation Mode RA. A CQA example resolved through generation mode RA. 39 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Mode RVA What is the highest Plant Height (Meters) among plants that have their Plant Height (Meters) above the average Plant Height (Meters) of all plants? <think>First, I need to read the Plant Height (Meters) of each plant in this chart.</think> <|box_start|>[[2541, 125, 2900, 288], [554, 578, 776, 705], [2719, 1122, 2954, 1288], [1651, 1213, 1872, 1363], [1119, 1304, 1437, 1444], [765, 1394, 1063, 1527], [2533, 1551, 2629, 2193], [2638, 1554, 2899, 2190], [677, 1576, 877, 1706], [291, 1757, 475, 1876], [519, 1848, 719, 1974], [783, 2029, 1069, 2162]]<|box_end|> <answer>16</answer> <think>Third, I need to find all the plants whose Plant Height (Meters) is higher than 7.23.</think><|box_start|>[[2541, 125, 2900, 288], [2719, 1122, 2954, 1288], [1651, 1213, 1872, 1363], [2533, 1803, 2629, 1941], [2638, 1806, 2899, 1938], [2534, 2054, 2628, 2130], [2639, 2057, 2775, 2127]]<|box_end|> <think>Fourth, I need to identify the plant with the highest Plant Height (Meters) among these plants.</think><|box_start|>[[2719, 1122, 2954, 1288], [2534, 1803, 2628, 1879], [2639, 1806, 2773, 1876]]<|box_end|> <think>Second, I need to compute the average Plant Height (Meters) of all plants.</think> <|box_start|> [[2541, 125, 2900, 288], [554, 578, 776, 705], [2719, 1122, 2954, 1288], [1651, 1213, 1872, 1363], [1119, 1304, 1437, 1444], [765, 1394, 1063, 1527], [2533, 1551, 2629, 2193], [2638, 1554, 2899, 2190], [677, 1576, 877, 1706], [291, 1757, 475, 1876], [519, 1848, 719, 1974], [783, 2029, 1069, 2162]]<|box_end|> Figure 27: Example of Generation Mode RVA. A CQA example resolved through genera- tion mode RVA. F Evaluation Metrics To elaborate more details in §5.2, our evaluation incorporates multiple complementary metrics to assess different aspects of model performance. F.1 Evaluation of Answers In pursuit of accurate evaluation on multimodal datasets that contain both multi-choice and free-form responses, we compute answer accuracy by comparing model outputs with their corresponding ground-truth answers. Aiming for more comprehensive assessment, we employ two complementary evaluation approaches: MLLM-as-judge for semantic understanding and rule-based evaluation for systematic accuracy measurement. The overall accuracy score for each dataset is calculated as the mean accuracy across all test samples. 40 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning MLLM-Based Answer Evaluation. For MLLM-based answer evaluation, we employ GPT-4.1-mini as the judge, guided by the prompt shown in Fig. 28. Each model response undergoes MLLM-as-judge evaluation to extract the model answer content, ensuring con- sistent comparison with ground truth. The judge performs a True-or-False assessment by evaluating whether the model response semantically matches the ground truth, accounting for variations in phrasing and presentation while maintaining semantic equivalence. Rule-Based Answer Evaluation.To mitigate potential biases introduced by using MLLMs as judges (§5.2), we complement the MLLM-as-judge approach with a systematic rule- based evaluation (Algorithm 1). This rule-based method assesses answer accuracy through predefined parsing and judgment rules, incorporating both strict and relaxed error tolerance through four range criteria: absolute accuracy (acc@0.0) and three progressively relaxed thresholds (acc@0.05, acc@0.1, acc@0.2). Algorithm 1 Rule-Based Answer Evaluation with Tolerance Ranges Require:Ground truth answergt, predicted answerpred, choicesC(optional), tolerance ranges R =0.0, 0.05, 0.1, 0.2 Ensure: Accuracy scores acc@r for each r ∈ R 1: gt← CLEAN(gt), pred← CLEAN( pred) 2: answer ty pe← DETECTTYPE(gt) 3: if gt =∅ and pred =∅ then 4:return acc@r = 1.0 for all r ∈ R 5: end if 6: if answer ty pe = ”multi-choice” then 7:gtlist← PARSECHOICES(gt, C) 8:predlist← PARSECHOICES( pred, C) 9:if gt list = predlist then 10:return acc@r = 1.0 for all r ∈ R 11:else 12:matchrate← |gtlist∩ predlist| |gtlist| 13:for r ∈ R do 14:acc@r ← 1[match rate≥ r] 15:end for 16:end if 17: else if answerty pe∈int, float then 18:gtnum← EXTRACTNUMBER(gt) 19:pred num← EXTRACT( pred) 20:acc@0.0← EXTRACTNUMBER( pred num, gtnum) 21:for r ∈0.05, 0.1, 0.2 do 22:lower ← gt num× (1− r) 23:u p per ← gtnum× (1 + r) 24:acc@r ← 1[lower ≤ prednum≤ u p per] 25:end for 26: else 27:exact match← GRADEANSWER( pred.lower(), gt.lower()) 28:substring match← 1[|gt|> 5 and gt.lower()∈ pred.lower()] 29:acc@r ← max(exactmatch, substringmatch) for all r ∈ R 30: end if 31: return acc@r for all r ∈ R F.2 Evaluation of Reasoning To comprehensively evaluate model reasoning, we implement both micro- and macro-level assessments (§5.2). Our micro-level evaluation relies on five metrics (Eq. 17 - 21), providing the semantic similarity assessment of model reasoning. The final score (acc@mic) is the average across all five metrics. At the macro level (acc@mac), we leverage GPT-4.1-mini as the judge, which rates the quality of model reasoning on a 0−10 scale based on three criteria: (1) visual understanding and grounding, (2) logical coherence and multimodal integration, and (3) alignment with ground-truth reasoning. While micro-level evaluation focuses on fine-grained 41 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning similarity between ground-truth and model reasoning, macro evaluation provides a holistic judgment of reasoning quality through MLLM-as-judge (Fig. 29). LLM As Judge For Answer Evaluation Given a Question, a Model Response, and its Ground-truth Answer, determine whether the Model's reponse is correct. The Question can be either multi-choice or free-form, please treat them separately: * 1. **Multi-Choice Question:** The Model Response is correct only if it **exactly matches** the correct choice letter (e.g., "A", "B", "C", or "D") after necessary normalization. Follow these instructions carefully: (1) If the Model Response is a number (e.g., "2", "3", etc.), map it to the corresponding option letter based on its order in the Question (e.g., 1 → A, 2 → B, etc.). (2) Ignore irrelevant text, explanations, or format differences. Extract the core predicted answer. (3) Compare the final normalized response with the Ground-truth Answer letter. * 2. **Free-Form Question:** The Model Response is correct if it fully aligns with the meaning and key information of the Ground-truth Answer. Respond with True if the Model Response is correct and False otherwise. Question: question Model Response: model_answer Ground-truth Answer: gt_answer Response Format: * Analysis: First extract the model's answer, then explain the comparison. Please enclose your analysis in: <analysis>your analysis here</analysis> * Extracted Model Answer: Please ensure the format of the extracted answer is consistent with the format of the ground-truth answer, but the content of the extracted answer should be exactly what the model responsed. Please enclose the extracted model answer in: <model>extracted model answer</model> * Judge: Return "True" only for exact matches, otherwise "False". Please enclose your True-or-False judge in <judge>True or False</judge> <analysis>...</analysis> <model>...</model> <judge>...</judge> Figure 28: LLM-As-Judge For Answer Evaluation. We employ GPT-4.1-mini as the judge to assess model answer accuracy using the prompt shown in this figure. LLM As Judge For Reasoning Evaluation You are an expert evaluator tasked with assessing the quality of reasoning chains in Chart Question Answering (CQA) tasks. Given an Image, a Question, a Model's Reasoning, and the Ground-truth Reasoning, evaluate ONLY the Model's reasoning process, NOT the final answer. **Evaluation Criteria:** 1. **Visual Understanding and Grounding (0-10)**: Does the reasoning demonstrate accurate perception and interpretation of visual elements in the image? Are visual observations correctly identified, described, and used as evidence? Check for: - Accurate identification of chart elements, components, labels, values, legends, and spatial relationships - Correct interpretation of visual context and relevant details - Appropriate grounding of reasoning in observable visual evidence - Absence of hallucinated or misperceived visual elements 2. **Logical Coherence and Multimodal Integration (0-10)**: Does the reasoning logically connect visual observations with the question requirements? Are the steps coherent and does each conclusion follow from visual evidence and prior reasoning? Evaluate: - Logical flow from visual observations to conclusions - Appropriate integration of visual and textual information - Sound inferential steps that bridge perception and reasoning - Absence of logical fallacies or non-sequiturs 3. **Alignment with Ground-Truth Reasoning (0-10)**: How well does the model's reasoning align with the provided ground-truth reasoning in terms of approach, key insights, and logical structure? Evaluate: - Similarity in problem-solving approach and methodology - Coverage of the same key reasoning steps and insights as the ground-truth - Alignment in the logical progression and chain of thought - Whether the model identifies and addresses the same critical visual elements as the ground-truth - Overall coherence with the expected reasoning pattern, even if expressed differently **Input:** * Image: [The image will be provided to you for reference] * Question: question * Model Reasoning: model_reasoning * Ground-truth Reasoning: gt_reasoning **Instructions:** - Carefully examine the provided image and understand its contents - Compare the Model Reasoning against both the image content and Ground-truth Reasoning - Focus exclusively on the reasoning process, NOT the final answer - Be objective and consistent in your scoring across different types of visual content - Use the full 0-10 scale for each criterion, and each score should be an integer, NOT float - Implement your evaluation by strictly following the three criteria above **Response Format:** <analysis> Provide a detailed analysis of the Model Reasoning across all three criteria: 1. Visual Understanding and Grounding: [Assess accuracy of visual perception, object identification, and use of visual evidence] 2. Logical Coherence and Multimodal Integration: [Evaluate logical flow and integration of visual and textual information] 3. Alignment with Ground-Truth Reasoning: [Compare the model's approach and insights with ground-truth reasoning, assessing similarity in methodology and coverage of key points] Compare with ground-truth reasoning and justify your scores based on visual accuracy, reasoning quality, and alignment with expected reasoning patterns. </analysis> <judge>score1, score2, score3</judge> Where score1 = Visual Understanding and Grounding, score2 = Logical Coherence and Multimodal Integration, score3 = Alignment with Ground-Truth Reasoning **Response Format:** * Analysis: Explain x. Please enclose your analysis in: <analysis>your analysis here</analysis> * Judge: Return three scores respectively for three criteria, separated with comma. Please enclose your judge scores in <judge>score1, score2, score3</judge> <analysis>...</analysis> <judge>...</judge> Figure 29: LLM-as-Judge For Reasoning Evaluation. We employ GPT-4.1-mini as the judge to evaluate model reasoning using the prompt shown in this figure. Prompt is restricted to smaller sizes to save space. 42 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Micro-Evaluation: Reasoning Similarity.We employ five metrics to measure the semantic similarity between ground-truth and model reasoning, including ROUGE-L (Eq. 17), BLEU (Eq. 18), METEOR (Eq. 20), BERTSCORE (Eq. 19), and COSINE SIMILARITY (Eq. 21). ROUGE = ROUGE-L = 2· P lcs · R lcs P lcs + R lcs (17) BLEU = BLEU-4 = BP· exp 4 ∑ n=1 w n log p n ! (18) where P lcs and R lcs are precision and recall of longest common subsequences. BERTSCORE = 1 |S p | ∑ s i ∈S p max s j ∈S g v T i v j |v i ||v j | (19) whereS p andS g are the sets of predicted and ground-truth reasoning tokens respectively, and v i , v j are their corresponding BERT contextual embeddings. METEOR = (1 + η 1 )· P r · R r η 1 · P r + R r · 1− η 2 · c u m η 3 (20) COSINE = e p · e g |e p ||e g | (21) whereP r andR r are precision and recall of reasoning tokens,u m is the number of matched unigrams,cis the number of chunks, and we defineη 1 =0.9,η 2 =0.5, andη 3 =3 as hyperparameters controlling the weight of recall, penalty magnitude, and penalty sharpness, respectively; and e p and e g are the embedding vectors of predicted and ground-truth reasoning steps respectively. Macro-Evaluation: Reasoning Quality. The quality of model reasoning is evaluated through MLLM-as-judge assessment. Specifically, we employ GPT-4.1-mini as the judge, guided by the prompt shown in Fig. 29, to assign a quality score on a 0−10 scale based on three criteria, including visual understanding and grounding, logical coherence and multimodal integration, and alignment with ground-truth reasoning. The final quality score for each dataset is calculated as the mean score across all test samples. •Criterion 1: Visual Understanding and Grounding. Reasoning accuracy in identifying, interpreting, and grounding reasoning in visual elements, meanwhile without introducing hallucinated details •Criterion 2: Logical Coherence and Multimodal Integration. Logical progression throughout the entire reasoning chain, with appropriate integration of multimodal infor- mation. •Criterion 3: Alignment with Ground-Truth Reasoning. Consistency with ground-truth reasoning, especially in terms of problem-solving approach, key insights, and logical structure, even if expressed differently. F.3 Evaluation of Visual Grounding As introduced in (§5.2), we employ two IoU variants, CIOU (Eq. 22) and GIOU (Eq. 23), as the primary evaluation metrics for visual grounding assessment. CIOU = IOU− ρ 2 (c p , c g ) d 2 (22) GIOU = IOU− | A c − A u | A c (23) whereρ 2 (c p ,c g )is the squared distance between predicted and ground-truth centroids,dis the diagonal of the enclosing box, A c is the enclosing area, and A u is the union area. 43 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning F.4 Evaluation Mode Furnishing models with the capabilities to reason through dynamic visual grounding, we employ different generation modes (§5.1) to support comparable evaluations: •Mode A: Answer-Only mode where MLLMs are prompted to directly generate the final answer. •Mode RA: Reason-Answer mode where MLLMs first go through the intermediate reason- ing process, followed by the final answer. •Mode VA: Vision-Answer mode where MLLMs first generate their visual grounding coordinates, followed by the final answer. •Mode RVA: Reason-Vision-Answer mode where MLLMs first go through the reasoning process with dynamic visual grounding, and then generate the final answer. G In-Depth Analysis G.1 Implementation Details. We train each model for 3 epochs with an initial learning ratelr =1e−4 usingcosinesched- uler. The ratio of training and validation is set totrain:val=9:1. Employing two NVIDIA 80G H100 GPUs, our model training is powered by LoRA for memory efficiency. For hyper- parameter settings, Stage I supervises visual grounding only (λ V =1.0,λ R =0.0,λ A =0.0), and Stage I supervises reasoning, grounding, and answer jointly (Eq. 6). Implementing different generation modes, reasoning (R) is enclosed within<think></think>, visual grounding (V) is enclosed within<|boxstart|><|boxend|>for explicit visual grounding while represented as<GROUND>in implicit visual ground (§G.5), and the final answer (A) is enclosed within<answer></answer>. G.2 Grounding Method & Computation Cost Employing zoom-in visual grounding, the cropped grounding method requires substantially larger memory at the same resolution. To mitigate this cost, we reduce the training resolu- tion of cropped grounding to 128×128, thereby maintaining a comparable computational overhead. Despite the resolution degradation, reasoning with cropped visual grounding achieves notably higher accuracy than the baseline (up to 4.72% improvement on CCQA) and performs competitively with the other two grounding methods (Tab. 2). These results highlight the effectiveness of zoom-in visual enhancement, albeit at the expense of increased computational cost when aiming for higher performance. MethodResolutionD max T max SFTRLInference Applied448× 448452× 80GB ≥ 8× 80GB1× 80GB Boxed448× 448452× 80GB ≥ 8× 80GB1× 80GB Cropped 128× 128452× 80GB ≥ 8× 80GB1× 80GB Table 8: The Computation & Configuration Of Different Grounding Method. This table summarizes the visual computation requirements and parameter configuration. G.3 The Role of Visual Grounding: From Extrinsic Assistance To Intrinsic Abilities A critical finding from our experiments reveals the fundamental distinction between the utility of multi-step visual reasoning during training versus inference (Tab. 11). While incorporating visual grounding in the training process significantly enhances models’ in- trinsic visual reasoning capabilities, directly applying the same multi-step approach during inference can paradoxically degrade performance due to error accumulation (Fig. H.2 & §H.1). 44 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Training Benefits of Visual Grounding. Our curriculum learning approach with visual grounding supervision effectively teaches models to develop stronger intrinsic represen- tations for chart understanding. By learning to align reasoning steps with visual focuses during training, models internalize the ability to focus on relevant image components, leading to improved performance even when generating direct answers without explicit visual grounding steps. Inference Challenges with Multi-Turn Visual Grounding. On the other hand, when models are required to explicitly generate visual grounding coordinates during inference (i.e., Mode RVA), performance degrades in comparison with direct answer generation (Mode A) and reasoning without grounding (Mode RA). This degradation stems from two primary factors: (1) Cumulative grounding errors: Inaccurate bounding box predictions in early reasoning steps propagate and compound errors in subsequent steps; (2) Reasoning- grounding misalignment: Discrepancies between intended visual focus and actual predicted coordinates lead to reasoning based on incorrect visual regions. Power of Intrinsic Visual Reasoning Capabilities. Results in Tab. 11 demonstrate that visual grounding serves as an effective training signal rather than an inference mechanism. Our curriculum learning with visual supervision enables models to learn better intrinsic visual-textual alignments, which manifest as improved performance in direct answer gener- ation Tab. 2. However, explicitly requiring visual grounding during inference introduces additional complexity and error sources that outweigh the potential benefits. Nevertheless, compared with baselines, our finetuned models manage to achieve remarkably higher performance in not only Mode A, but also Modes RA, VA, and RVA. G.4 Visual Grounded Reasoning via Reinforcement Learning To further validate the generality of CURV, we extend our two-stage training framework to reinforcement learning (RL), reformulating training objectives as specialized reward signals that directly incentivize accurate visual grounding, faithful reasoning, and correct answer generation. Stage I: Visual Grounding.In Stage I, we define visual grounding rewardR V that evalu- ates the geometric quality of the predicted visual focusV t against the ground-truth focus region V ∗ t using CIoU (Eq. 22) and GIoU (Eq. 23): R V (V t , V ∗ t ) = λ CIoU R CIoU (V t , V ∗ t ) + λ GIoU R GIoU (V t , V ∗ t ),(24) where λ CIoU = λ GIoU = 0.5. As such, Stage I training objective becomes: R (S1) = T ∑ t=1 R V (V t , V ∗ t ).(25) Stage I: Interleaved Visual Reasoning Reward.In Stage I, the training objective extends to the full interleaved visual grounded reasoning chain with a composite reward: R (S2) = λ R T ∑ t=1 R R (R t , R ∗ t ) + λ V T ∑ t=1 R V (V t , V ∗ t ) + λ A R A (A, A ∗ ) + λ F R F ,(26) whereλ R ,λ V ,λ A , andλ F are weighting coefficients balancing the four reward components. Each component is defined as follows: Reasoning rewardR R (R t ,R ∗ t )measures the semantic similarity between the predicted reasoning stepR t and its ground-truth counterpartR ∗ t as a rubric-based aggregate of ROUGE-L (Eq. 17), BERTSCORE (Eq. 19), and COSINE SIMILARITY (Eq. 21): SIMILARITY = λ rouge · ROUGE + λ bert · BERT + λ cosine · COSINE(27) 45 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Model Level 1Level 2Level 3 @M@0.0@0.05@0.1@0.2@M@0.0@0.05@0.1@0.2@M@0.0@0.05@0.1@0.2 Baseline Baseline45.2543.5251.5456.2561.5422.7522.8631.1437.2145.3616.1816.0021.8925.8231.71 Ours (RL) Stage I53.6451.7157.4361.3665.0024.6125.3935.2141.0451.5716.2118.8625.2529.3233.96 Stage I+I55.1453.2361.3767.3870.5426.2528.4640.3548.0557.9419.0722.5327.4331.4335.58 Ours (SFT) Stage I54.2151.2159.5065.0068.5025.8627.1439.1847.2555.1816.8617.3223.9328.6133.11 Stage I+I60.7157.2163.7169.5773.7930.1129.6141.8248.8757.4617.9818.2124.2529.0734.50 Ours (SFT + RL) Stage I55.8653.6460.7966.4369.2927.9628.7940.7548.6457.2520.6421.3926.7930.2534.93 Stage I+I62.2959.7965.8672.2173.7929.3629.7442.5750.3959.3922.4323.3929.2532.4336.89 Table 9: CURV with Reinforcement Learning on CCQA. Employing Qwen2.5-VL-3B as the base model, we evaluate the performance of its baseline and finetuned versions on CCQA across three curriculum levels via five accuracy evaluation metricsacc@XwhereXis MLLM (abbreviated as@ M) or ranges (§ 5.2). All settings use CURV@Applied throughout training and inference. whereλ rouge ,λ bert , andλ cosine are weighting coefficients respectively set to 0.3, 0.4, 0.3 during Stage I training. Grounding rewardR V (V t ,V ∗ t )evaluates geometric alignment of the predicted visual focus at each reasoning step, as defined in Eq. 24. Answer rewardR A (A,A ∗ )is a binary exact-match signal that returns 1 if the predicted final answer A matches the ground-truth A ∗ and 0 otherwise. Format rewardR F is a binary compliance signal that returns 1 if the model output adheres to the required structured format, ensuring output parsability throughout training. G.5 Explicit vs. Implicit Visual Grounded Reasoning We introduce two designs for integrating visual grounding into the multi-step reasoning framework: explicit and implicit visual grounded reasoning. Both approaches are based on our two-stage training framework CURV (Fig. 3), but differ fundamentally in how the MLLM is encouraged to attend to chart regions across reasoning steps. G.5.1 Explicit Visual Grounding In explicit visual grounded reasoning, the MLLM is trained end-to-end to produce multi-step reasoning with dynamically changed visual focuses. Concretely, in Stage I, given the chart imageIand questionQ, the model learns to dynamically ground logical reasoning in corresponding visually focused regions by predicting bounding boxesV t for each reasoning stepR t acrosststeps of reasoning (Eqs. 3-4). Each predicted bounding box is represented as a normalized coordinate tuple[x min ,y min ,x max ,y max ]over the image canvas, and the grounding lossL V directly supervises the model’s spatial predictionsV t against ground- truth focus regionsV ∗ t . In Stage I, the model transitions to interleaved visual grounded reasoning (RVA). At each stept, the bounding boxV t is applied to the original chart image Ithrough one of our visual grounding strategies (§D) to construct the grounded visual stateI t , which is then provided as an additional visual input at the next reasoning step to implement theV t ′ →I t ′ mapping (Eq. 5). Note that during training, we use ground-truthV ∗ t as intermediate visual augmentation; while in inference, the model continues its reasoning based on its last-step V t . Under this design, the MLLM jointly learns to directly articulate its visual focus as explicit coordinate predictions and dynamically augment its reasoning by linking eachR t with corresponding visually focused regions. The training objective in Eq. 6 jointly supervises reasoning, grounding, and final answer quality, withλ R ,λ V , andλ A balancing the three components. The key characteristic of explicit visual grounded reasoning is that the MLLM 46 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning internalizes both roles on its own, inherently encouraged to dynamically shift attention alongside reasoning. G.5.2 Implicit Visual Grounding In implicit visual grounded reasoning design, visual grounding is performed by a dedicated grounder module that operates alongside the MLLM, implicitly affecting MLLM reasoning. Grounder Module.The grounder is a lightweight grounding headG φ that operates on the model’s internal representations to produce patch-level attention maps over the chart image. Specifically, it takes as input (1) the visual patch embeddings E v ∈R N v ×d , extracted from the frozen vision encoder withN v patches and hidden dimensiond, and (2) the decoder hidden state h t ∈R d at a designated<GROUND>token position appended after each reasoning step R t . As such, the grounder computes a patch-level attention map: a t = g φ ( E v , h t ) ∈∆ N v −1 (28) where∆ N v −1 denotes the(N v −1)-simplex (i.e., the output is a distribution over visual patches). Internally,g φ applies self-attention over E v to capture contextual relationships among patches, followed by separate projection networks for the visual and decoder rep- resentations, and a scaled dot-product operation to compute attention logits. The training objective forg φ combines KL divergence (Eq. 30), binary cross-entropy (Eq. 31), Dice loss (Eq. 32), and cross-entropy (Eq. 33), all computed against ground-truth binary focus masks M ∗ t ∈0, 1 N v : L V (V t , V ∗ t ) = λ KL L KL + λ BCE L BCE + λ Dice L Dice + λ CE L CE (29) where each term supervises a distinct aspect of the predicted attention map a t against the ground-truth binary focus mask M ∗ t : L KL = N v ∑ i=1 ̃ M ∗ t,i log ̃ M ∗ t,i a t,i + ε (30) L BCE =− 1 N v N v ∑ i=1 M ∗ t,i log(a t,i + ε) + (1− M ∗ t,i ) log(1− a t,i + ε) (31) L Dice = 1− 2 ∑ N v i=1 a t,i M ∗ t,i ∑ N v i=1 a t,i + ∑ N v i=1 M ∗ t,i + ε (32) L CE =− log a t, arg max i M ∗ t,i (33) where ̃ M ∗ t = M ∗ t /( ∑ i M ∗ t,i + ε) is the row-normalized ground-truth mask treated as a target probability distribution for the KL term,εis a small constant for numerical stability, and a t,i denotes the predicted attention weight for thei-th visual patch.L KL penalizes distributional divergence between the predicted attention map and the normalized ground-truth mask; L BCE enforces patch-level binary classification independently across all patches;L Dice optimizes region overlap between the predicted and ground-truth focus regions; andL CE treats the patch with the highest ground-truth activation as the target class, encouraging the model to concentrate probability mass on the most salient region. The loss weightsλ KL , λ BCE , λ Dice , and λ CE are set to 0.25 uniformly in our experiments. Stage I: Grounder Training. In Stage I, the MLLM parametersθare kept frozen while the grounderG φ is trained to predict accurate visual focuses for each reasoning step. The decoder hidden state h t at<GROUND>position is extracted and passed toG φ together with E v . The training objective follows Eq. 4, withL V computed as in Eq. 29. 47 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Stage I: MLLM Training with Visual Grounded Feedback. In Stage I, the grounder G φ is frozen and the MLLM parametersθare finetuned on the full interleaved reasoning objective. The model is trained to produce reasoning chains of the form: R 1 |z step 1 < GROUND>R 2 |z step 2 < GROUND> · R t |z step t < GROUND>A |z answer < GROUND>(34) where<GROUND>tokens serve as anchors at which the grounder produces attention maps a t T t=1 , which are used to construct grounded visual statesI t T t=1 through our visual grounding strategies (§D). However, as the MLLM generates reasoning stepsR t without explicit coordinate predictions, the visual grounding signal remains implicit. As such, the model is encouraged to internally shift its internal attention focus at each reasoning step, revealing less effective visual augmentation than explicit visual grounded reasoning (§5.4). This performance gap also highlights differences between visual grounded reasoning tasks that leverage learn-to-focus dynamics to enhance reasoning and visual grounding tasks aimed at accurately localizing visual focus. G.6 Curriculum Learning: Effectiveness of CL in Chart Understanding Figure 30: Effects of Curriculum Learning. Using Qwen2.5-VL-7B as the base model, we compare its per- formance across different training settings. To validate the effectiveness of curriculum learning, we further compare CURV with the baseline model (Qwen2.5-VL-7B) in both untrained and non-curriculum training settings. As shown in Fig. 30, although training with- out curriculum learning improves performance on CCQA, it con- sistently underperforms as com- pared to CURV trained with ei- ther Stage I or Stage I+I. This gap becomes more pronounced as task complexity increases (i.e., levels 1→3), with diminishing performance gains observed in the non-curriculum setting. These results highlight the effectiveness of curriculum learning in guiding the model to learn from fundamental chart elements and gradually develop intrinsic visual reasoning capabili- ties. This structured learning process not only improves the model’s adaptability to more complex chart understanding tasks but also enhances its generalization to out-of-domain multimodal reasoning. G.7 Two-Stage Learning: Strength of Training Curriculum Figure 31: Strength of Curriculum Training. Using Qwen2.5-VL-7B as the base model, we compare its per- formance across different training settings. In addition to data curriculum (§G.6), CURV also implements the two-stage curriculum training paradigm from internalizing to en- hancing MLLM visual reasoning capabilities (§3). In Stage I, MLLM learns vision-reasoning alignment through decoupled visual ground- ing, establishing explicit associ- ations between visual evidence and reasoning logics. In Stage I, MLLM is further trained to per- form interleaved visual reasoning, leveraging the grounding capabil- ities acquired in Stage I to support 48 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning Method Level 1Level 2Level 3 acc@Macc@0.0acc@Macc@0.0acc@Macc@0.0 Base Models Qwen2.5-VL-7B54.2150.7928.6828.8219.0119.38 Qwen3-VL-4B48.0747.2826.8626.6120.6820.04 Chart Specialist ChartGemma (Masry et al., 2024)28.7121.3612.468.257.718.86 Tool-Use Agents Thyme (Zhang et al., 2025c)56.5755.5724.3924.9618.0718.54 DeepEyes (Zheng et al., 2025)58.4357.2130.0429.2920.7520.29 Ours CURV (Qwen3-VL-4B)64.2163.7934.8634.5723.8223.25 CURV (Qwen2.5-VL-7B)69.8666.7940.2138.6426.1124.25 Table 10: Comparison Against Chart Specialist & Tool-Use Agents on CCQA. We evaluate CURV against a newer base model (Qwen3-VL-4B), a chart-specialist baseline (Chart- Gemma), and tool-augmented agents (Thyme, DeepEyes) across three curriculum levels. step-by-step reasoning over visual inputs. To validate the effectiveness of this design, we compare the performance among Qwen2.5-VL-7B, CURV@Stage I, CURV@Stage I, and CURV@Stage I+I (Fig. 31). Results show that CURV@Stage I+I consistently outperforms both single-stage variants, indicating the advantage of combining decoupled alignment with interleaved reasoning via two-stage curriculum training. In contrast, CURV@Stage I alone improves performance on level 1 simple CQA while degrades on harder single-chart and multi-chart understanding tasks. This suggests that decoupled grounding, while effective for learning basic visual-reasoning associations, is insufficient for supporting compositional reasoning that requires integrating multiple pieces of visual evidence. Conversely, CURV@Stage I alone fails to fully realize these benefits, implying that interleaved reasoning without well-established grounding leads to weaker visual-reasoning alignment. Collectively, these findings highlight that the strength of CURV lies in its staged curriculum training design: Stage I builds foundational visual grounding ability, while Stage I enables the model to operationalize this grounding ability into coherent, step-by-step visual reason- ing. Their combination results in more robust and generalizable visual reasoning capacity across varying task complexities. G.8 Foundational Learning: Trade-off between Robustness & Adaptability Fig. 9 reveals a critical finding that strongly validates our curriculum learning design (§4). While training on level 1 alone provides solid foundational performance (↑35.08% on level 1), progressive training on levels 1+2 demonstrates the optimal learning accumulation, achieving the best overall performance across all difficulty levels (↑15.65% on level 1,↑11.53% on level 2,↑7.10% on level 3). However, extending training to include level 3 unfolds a concerning trade-off between robustness and adaptability: While notably improves complex reasoning performance (↑12.82% on level 3), it significantly degrades foundational reasoning abilities, only higher than baseline by↑ 5.93% and↑ 0.03% on levels 1 and 2, respectively. G.9 Broader Baselines: Comparison Against Chart Specialist & Tool-Use Agents We broaden our comparison along three axes: a newer base modelQwen3-VL-4B(Yang et al., 2025), a chart-specialist baselineChartGemma(Masry et al., 2024), and tool-augmented agents that invoke external visual toolsThyme(Zhang et al., 2025c) andDeepEyes(Zheng et al., 2025). All methods are evaluated on our CCQA test set across the three curriculum levels 49 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning using both acc@M and the absolute accuracy acc@0.0 (§5). As shown in Tab. 10, CURV is generalizable to newer backbones, improving over its base model (Qwen3-VL-4B) by up to ↑16.14%. This validates that the improvements from CURV are not tied to a specific base model, but instead reflect a transferable strengthening of intrinsic visual-grounded reasoning capabilities. Moreover, CURV also outperforms chart specialist and tool-augmented agents built on the same Qwen2.5-VL-7B backbone, providing further evidence to the effectiveness and robustness of CURV in the face of varying chart complexity levels. H Multi-Step Reasoning With Dynamic Visual Grounding ModelSize CCQA Level 1Level 2Level 3 ReasoningGroundingAnswerReasoningGroundingAnswerReasoningGroundingAnswer acc@macacc@micmIoUacc@macacc@macacc@micmIoUacc@macacc@macacc@micmIoUacc@mac Baselines GPT-4o-53.1751.0222.0350.0053.3253.6013.0124.0050.2046.8611.1421.50 Qwen2.5-VL 3B44.9039.0737.1236.0038.7240.0829.7314.0032.1735.4727.4312.00 7B48.5341.7048.1745.0040.4940.6832.2521.5035.8238.1032.2617.50 Ours Applied (Qwen2.5-VL) 3B50.5053.6347.1750.0045.7945.6640.5919.0041.9538.7634.6015.00 7B56.1354.0157.5552.0049.8747.1745.7223.0046.0538.8339.9019.50 Boxed (Qwen2.5-VL) 3B50.2349.9045.5546.0040.7042.7438.1114.5036.9836.9733.3713.50 7B51.2748.2050.8849.0042.9743.4443.0922.5041.3337.4534.5218.00 Cropped (Qwen2.5-VL) 3B 42.1746.6851.2239.0040.5242.6945.8214.5039.6335.8634.2713.00 7B50.3350.7951.3347.0044.9743.7550.1024.0042.4338.5139.1919.00 Table 11: Performance Evaluation for RVA Mode Inference. Employing the set of eval- uation metrics (§ 5.2), we assess model reasonnig, visual grounding, and final answer, respectively. H.1 Challenges In Multi-Step Visual Grounding We leverage CCQA, randomly selecting 500 samples to evaluate model inference through RVA mode (§5.1). Training MLLMs with explicit reasoning and visual grounding as intermediate outputs effectively enhances model’s intrinsic visual reasoning capabilities (§5.3). This step-by-step visual reasoning guides the model to decompose complex tasks into structured reasoning chains through dynamic attention grounding. With intermediate grounding naturally supporting more coherent reasoning trajectories, this in turn enhances model’s ability to establish interleaved thinking-perception correspondences. Aligning with human visual reasoning, decomposed reasoning chains effectively help models to develop and strengthen their intrinsic visual reasoning capabilities. Different from learning, during inference, human visual reasoning is rather a composed process that interleaves logical reasoning with visual comprehension, while compositing all intermediate steps into a coherent chain of thought. In contrast, inference in RVA exposes the fragility of step-wise generation: once an intermediate step is flawed, whether by incorrect calculation or inaccurate visual comprehension, the error propagates through the chain, breaking the balance between perception and reasoning that eventually leads to incorrect final answers (§H.1). Therefore, it can be an effective way of learning, while may not be as useful in inference. Tab. 11 summarizes the evaluation results of RVA inference. Fintuned models achieve noticeable improvements across reasoning (up to 10.23% absolute gain), grounding (up to 9.38% absolute gain), and answering (up to 14% absolute gain). Beyond these numerical results, qualitative inspection (§H.3) reveals distinct behavioral patterns where training with step-by-step visual grounding encourages systematic reasoning chains with sharper object localization, showcasing stronger alignment with human-like reasoning trajectories. Despite these improvements, however, the answering performance remains lower than that of CURV when using the same base model and grounding method (Tab. 2). This indicates that, while RVA training can effectively enhance intrinsic visual reasoning capabilities, RVA inference magnifies the vulnerability to intermediate error accumulation. Build upon our discussions above, we present the challenge for MLLMs in CQA through mode RVA. Although RVA training notably enhances model’s intrinsic visual reasoning capabilities, RVA inference reveals critical limitations. During RVA inference, cumulative 50 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning intermediate errors can cascade through the reasoning chain, ultimately compromising the model’s ability to produce correct answers. For example, Fig. 32 shows a CQA example that is responded to by GPT-4o and Qwen2.5-VL-7B, respectively. Both GPT-4o and Qwen2.5- VL-7B fail to generate the correct answer due to their disrupted reasoning chains that evolve through their incorrect visual grounding. Question Among bars whose Starting Salary ($ Thousand) are above the mean Starting Salary ($ Thousand) of all bars, what is the lowest Starting Salary ($ Thousand)? <think>Now, I need to compare their Starting Salary ($ Thousand) to find the one with the lowest Starting Salary ($ Thousand).</think> <|box_start|>[[536, 120, 707, 372], [345, 294, 534, 370], [244, 377, 528, 651]]<|box_end|> <answer>68.50</answer> <think>First, I need to read their Starting Salary ($ Thousand).</think> <|box_start|> [[536, 120, 707, 372], [1328, 229, 1855, 1922], [345, 294, 534, 370], [244, 377, 528, 651], [1303, 893, 1855, 1922], [1253, 1951, 1855, 2185], [1126, 2045, 1241, 2121], [1076, 2128, 1225, 2185]]<|box_end|> GPT-4o <think>Next, I will identify the bars whose Starting Salary ($ Thousand) are above the mean Starting Salary ($ Thousand) calculated in the previous step.</think><|box_start|>[[70, 20, 150, 80], [220, 20, 300, 80], [450, 20, 530, 80], [600, 20, 680, 80]]<|box_end|> <answer>The lowest Starting Salary ($ Thousand) among the bars whose Starting Salary ($ Thousand) are above the mean is 78.60</answer> <think> First, I will calculate the mean Starting Salary ($ Thousand) of all bars.</think><|box_start|>[[0, 0, 800, 600]<|box_end|> Qwen2.5-VL-7B <think>Finally, I will determine the lowest Starting Salary ($ Thousand) among the identified bars.</think><|box_start|>[[70, 20, 150, 80]] <|box_end|> Figure 32: Challenge of Inference in Mode RVA. Tested on GPT-4o and Qwen2.5-VL, this example illustrates the challenge MLLMs face in performing inference in RVA. H.2 Inference Failure Fig. 35 illustrates examples of model inference failures in RVA. In both cases, the model fails to properly ground its reasoning in the chart, leading to inaccurate extraction and misinterpretation of visual information. Arising in early reasoning steps, these visual comprehension inaccuracies can propagate through the reasoning chain, ultimately resulting in incorrect question answering. H.3 Inference Success Example 1 - Mode A: Figures 33 & 34 exhibit examples on chart question answering in mode A, where the baseline Qwen2.5-VL-7B fails to generate the correct answer, while CURV (Qwen2.5-VL-7B) finetuned through applied grounding succeeds. Fig. 33 (a) is a simple value reading problem (D =1), where the baseline model fails to localize the exact queried chart component. Fig. 33 (b) consists of two nested functions (D =2), where the baseline model fails to localize the queried bar in the given subset of countries. Fig. 33 (c) further enhance the CQA complexity (D =3), involving three nested functions across reasoning, visual grounding, and interleaved calculation that the baseline model fails to correctly response. Different from Fig. 33 that query about a single chart, each CQA sample in Fig. 34 involves multiple charts that significantly complicates question answering. The baseline model fails in Fig. 33 (a) (D =3) as it requires the localization of the exact chart subplot, the required subset, as well as the Y-axis value reading. Fig. 33 (b) increases the 51 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning CQA difficulty (D =4) by including not only accurate localization of chart components, but also extremia comparison of both bar values and spatial positions. Fig. 33 (c) (D =5) presents further enhanced complexity by involving relations across different charts. This relational chart understanding making the problem solving more challenging, unveiling the significance of accurate visual reasoning in tackling complex CQA tasks. Example 2 - Mode RVA: Fig. 36 presents two examples of successful inference in RVA mode. The bar chart example on the left shows reasoning with accurate visual grounding. The heatmap example on the right shows a case where the grounding is not exact but falls close to the regions of focus, also leading to the correct answer. I Limitations & Future Work In this work, we propose CURV (§3), a curriculum learning framework that develops intrinsic visual grounded reasoning capabilities in MLLMs by reformulating chart question answering as multi-step visual grounded reasoning with dynamic spatial attention. To support model learning, we introduce CCQA (§4), a three-level curriculum dataset with scalable synthetic generation across diverse chart types and reasoning patterns. Results demonstrate that CURV demonstrates notable and consistent improvements across chart understanding and out-of-domain multimodal reasoning benchmarks (§5.3). Nevertheless, we acknowledge a few limitations that we aim to investigate further in our future work. First, CCQA encompasses seven common chart types, while less conventional visualization forms, such as Sankey diagrams, treemaps, and geographic maps, are not currently covered, potentially limiting the applicability of CCQA to future work with more specialized training scenarios. Second, our framework and dataset focus exclusively on English-language charts and questions, which may restrict generalizability to multilingual contexts where chart-based communication is equally prevalent. In future work, we aim to address these limitations by extending CCQA to a broader range of chart types and multilingual settings. Beyond these, we also aim to explore agentic chart understanding, where models can better leverage external knowledge, tools, and multi-agent collaboration to complement the intrinsic visual reasoning capabilities established in this work. 52 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning What is the Annual Visitors (Millions) of the museum positioned as the bottom bar among all the museums in this chart? Please round to two decimal places. Qwen2.5-VL-7B <answer>6.10</answer> CCQA - Curriculum Level 1 ( ) Ours - Applied (Qwen2.5-VL-7B) <answer>8.50</answer> Which country corresponds to the rightmost bar among 'Japan, Singapore, Australia, Italy, Iceland'? CCQA - Curriculum Level 2 ( ) Qwen2.5-VL-7B <answer>Australia</answer> Ours - Applied (Qwen2.5-VL-7B) <answer>Iceland</answer> Which country has the lowest Gold Medals among countries that have their Gold Medals above the average Gold Medals of all bars? CCQA - Curriculum Level 2 ( ) Qwen2.5-VL-7B <answer>Germany</answer> Ours - Applied (Qwen2.5-VL-7B) <answer>Australia</answer> (a)(b)(c) Figure 33: Success Examples On Curriculum Levels 1-2. This figure shows three examples on curriculum levels 1-2 of CCQA, where the baseline Qwen2.5-VL-7B fails while our CURV using applied grounding succeeds. According to the bar chart, what is the Gold Medals of the country corresponding to the rightmost bar among all the bars that have their 'Country' labels among 'USA, Japan, UK, Australia, Netherlands'? Qwen2.5-VL-7B <answer>17</answer> CCQA - Curriculum Level 3 - Localization ( ) Ours - Applied (Qwen2.5-VL-7B) <answer>12</answer> According to the bar chart, among the bars that are located as the 1st, 3rd, 4th, 6th, 7th species from left to right, what is the Maximum Height (meters) of the species represented as the rightmost bar? Please round to two decimal places. CCQA - Curriculum Level 3 - Localization ( ) Qwen2.5-VL-7B <answer>92.6</answer> Ours - Applied (Qwen2.5-VL-7B) <answer>86.20</answer> What is the Beneficiaries Served of the organization that has the highest Annual Donations Received ($ Millions) among all the organizations with their Annual Donations Received ($ Millions) less than 800? CCQA - Curriculum Level 3 - Relation ( ) Qwen2.5-VL-7B <answer>150000</answer> Ours - Applied (Qwen2.5-VL-7B) <answer>180000</answer> (a)(b)(c) Figure 34: Success Examples On Curriculum Level 3. This figure shows three examples on the curriculum level 3 of CCQA, where the baseline Qwen2.5-VL-7B fails while our CURV using applied grounding succeeds. 53 CURV: Enhancing Chart Understanding Through Curriculum Visual Grounded Reasoning What is the average Annual Launches of all data points in this line chart? Please round to two decimal places. RVA Mode - Failure Ours - Applied (Qwen2.5-VL-7B) <think>First, I need to read the Annual Launches of each data point in this chart.</think><|box_start|>[[51, 267, 474, 901], [493, 351, 655, 468], [...], [2580, 2933, 2704, 3024]]<|box_end|> Turn 1 <think>Second, I need to calculate the mean Annual Launches of all data points</think><|box_start|>[[51, 267, 474, 901], [493, 351, 655, 468], [...], [2580, 2933, 2704, 3024]]<|box_end|> Turn 2 <answer>12.33</answer> Turn 3 If given that the structure of the urban feature 'Residential Units' is 920, what is the mean structure of all urban features in this chart? Please round to two decimal places. RVA Mode - Failure Ours - Applied (Qwen2.5-VL-7B) <think>First, I need to find the percentage of Residential Units in the pie chart.</think><|box_start|>[[354, 377, 783, 459], [30, 532, 437, 1323], [0, 1326, 67, 1451], [1441, 1326, 1755, 1451]]<|box_end|> Turn 1 <think>Mean Calculation: Residential Units: 920, Commercial Buildings: 730, Civic Infrastructure: 410, Green Spaces: 560, Cultural Sites: 195. total structure = 920 + 730 + 410 + 560 + 195 = 2915. The mean structure = 2915 / 5 = 583.00</think><|box_start|>[[354, 377, 783, 459], [30, 532, 437, 1323], [0, 1326, 67, 1451], [1441, 1326, 1755, 1451], [1105, 1797, 1386, 1879], [890, 1948, 1124, 2120]]<|box_end|> Turn 2 <answer>583.00</answer> Turn 3 Ground-truth Answer: 563.00 Ground-truth Answer: 13.62 Figure 35: Failure Examples Through Mode RVA. This figure shows two examples of RVA inference on CCQA, where the model fails to give correct answers. Among all the bars shown in this chart, what is the Annual Commuter Rail Passengers (Millions) of the city represented as the rightmost bar? RVA Mode - Success Ours - Applied (Qwen2.5-VL-7B) <think>First, I need to find the rightmost bar in this chart.</think> <|box_start|>[[2529, 707, 2856, 777], [2628, 786, 2831, 2013], [2573, 2016, 2762, 2242]]<|box_end|> Turn 1 <think>Second, I need to read its Annual Commuter Rail Passengers (Millions).</think><|box_start|>[[2529, 707, 2856, 777], [2628, 786, 2831, 2013], [2573, 2016, 2762, 2242]]<|box_end|> Turn 2 <answer>395</answer> Turn 3 What rating score corresponds to 2025 and The Lion King? RVA Mode - Success Ours - Applied (Qwen2.5-VL-7B) <think>First, I need to locate the cell at the intersection of 2025 (Rating Year) and The Lion King (Movies).</think> <|box_start|>[[489, 1712, 961, 2091]] <|box_end|> Turn 1 <think>Moved to the cell at the intersection of 2025 (Rating Year) and The Lion King (Movies). I need to read the rating score at this position. </think></think><|box_start|>[[489, 1712, 961, 2091]]<|box_end|> Turn 2 <answer>95.50</answer> Turn 3 Figure 36: Success Examples Through Mode RVA. This figure shows three examples of RVA inference on CCQA, where the baseline Qwen2.5-VL-7B fails while our CURV using applied grounding succeeds. 54

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