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A Mechanistic Analysis of Sim-and-Real Co-Training in Generative Robot Policies
Yu Lei, Minghuan Liu, Abhiram Maddukuri, Zhenyu Jiang, Yuke Zhu
Intelligence
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Summary
This paper investigates the mechanisms behind sim-and-real co-training in generative robot policies. The authors identify two primary intrinsic effects: 'structured representation alignment' (a balance between cross-domain alignment and domain discernibility) and the 'importance reweighting effect' (domain-dependent action weighting). Through theoretical analysis and experiments on robotic manipulation tasks, they demonstrate that structured representation alignment is the dominant factor for performance, while the importance reweighting effect plays a secondary, modulatory role. The study provides a unified interpretation of existing co-training techniques and proposes a method to improve performance by explicitly managing these effects.
Entities (5)
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Structured Representation Alignment → influences → downstream performance
confidence 95% · plays a primary role in downstream performance
Importance Reweighted Effect → modulates → Action Weighting
confidence 95% · arises from domain-dependent modulation of action weighting
Co-training → utilizes → Surrogate Data
confidence 95% · combines limited in-domain real-world data with abundant surrogate data
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Abstract
Abstract:Co-training, which combines limited in-domain real-world data with abundant surrogate data such as simulation or cross-embodiment robot data, is widely used for training generative robot policies. Despite its empirical success, the mechanisms that determine when and why co-training is effective remain poorly understood. We investigate the mechanism of sim-and-real co-training through theoretical analysis and empirical study, and identify two intrinsic effects governing performance. The first, \textbf{``structured representation alignment"}, reflects a balance between cross-domain representation alignment and domain discernibility, and plays a primary role in downstream performance. The second, the \textbf{``importance reweighting effect"}, arises from domain-dependent modulation of action weighting and operates at a secondary level. We validate these effects with controlled experiments on a toy model and extensive sim-and-sim and sim-and-real robot manipulation experiments. Our analysis offers a unified interpretation of recent co-training techniques and motivates a simple method that consistently improves upon prior approaches. More broadly, our aim is to examine the inner workings of co-training and to facilitate research in this direction.
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- Source: https://arxiv.org/abs/2604.13645v1
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A Mechanistic Analysis of Sim-and-Real Co-Training in Generative Robot Policies Yu Lei 1 Minghuan Liu 1 Abhiram Maddukuri 1 Zhenyu Jiang 2 Yuke Zhu 1 3 https://science-of-co-training.github.io/ Primary Intrinsic Effect in Co-Tr a i n i n g : Structured Representation Alignment Co-Tr a i n i n g w i t h G e n e r a t i v e R o b o t P o l i c y Robot Action Mixed Batches Limited Real Data Large-scale Sim Data Data Sources Sample Train Real-World Execution Deploy Image Robot State Language Prompt Action Head Encoder Trunk Vision Stem Global AlignmentLocal Alignment Binary Classifier Real or Sim ? Accuracy ≈100% (b) (c) (a) Latent Feature Layer Depth ①Visual Feature Observation Feature from Sim/Real ②Observation Feature Domain DiscernibilityRepresentation Alignment Correlation Practical Improvement State Stem Lang. Stem ① ② Network Module Data Input/Output Figure 1. (a) A workflow example of co-training systems for generative robot policies. We identify Structured Representation Alignment as the main intrinsic effect in co-training: it refers to both representation alignment and domain discernibility, which lay the foundation of action transfer and adaptation. (b) Representative observations: representation alignment can be implicitly learned with appropriate mixing ratios, showing local geometric alignment in shallow layers and global alignment in deep layers (Sec. 4.1). The globally aligned features still preserve discernibility (Sec. 4.2). Representation alignment has a strong positive correlation with success rate. (c) Inspired by these observations, a simple fix of co-training with enhanced structured alignment can improve the success rate by another∼ 20%. Abstract Co-training, which combines limited in-domain real-world data with abundant surrogate data such as simulation or cross-embodiment robot data, is widely used for training generative robot poli- cies. Despite its empirical success, the mecha- nisms that determine when and why co-training is effective remain poorly understood. We inves- tigate the mechanism of sim-and-real co-training through theoretical analysis and empirical study, 1 Department of Computer Science, The University of Texas at Austin 2 Amazon FAR 3 NVIDIA. Correspondence to: Yu Lei <yulei@utexas.edu>. Preprint. April 16, 2026. and identify two intrinsic effects governing per- formance. The first, “structured representation alignment”, reflects a balance between cross- domain representation alignment and domain dis- cernibility, and plays a primary role in down- stream performance. The second, the “impor- tance reweighting effect”, arises from domain- dependent modulation of action weighting and operates at a secondary level. We validate these ef- fects with controlled experiments on a toy model and extensive sim-and-sim and sim-and-real robot manipulation experiments. Our analysis offers a unified interpretation of recent co-training tech- niques and motivates a simple method that con- sistently improves upon prior approaches. More 1 arXiv:2604.13645v1 [cs.RO] 15 Apr 2026 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? broadly, our aim is to examine the inner work- ings of co-training and to facilitate research in this direction. 1. Introduction Data scarcity remains a fundamental bottleneck in robotics, motivating the use of inexpensive and abundant surrogate data such as simulation and cross-embodiment data (Bjorck et al., 2025; Physical Intelligence, 2025). Although these data sources contain rich task-relevant information, they in- troduce substantial domain gaps that make effective knowl- edge transfer difficult in practice. Recently, a simple co- training paradigm — jointly training on in-domain real data and surrogate data with a data mixing ratiow— has demon- strated strong empirical performance across sim-and-real and human-to-robot settings (Cheng et al., 2025; Yuan et al., 2025; Kareer et al., 2025a;b). Despite some work (Wei et al., 2025) providing valuable empirical analysis, co-training remains poorly understood: its internal mechanisms are largely treated as a black box, and the factors that govern its effectiveness are unclear. In this work, we focus on investigating the co-training paradigm as described above, in particular, on sim-and-real data, in diffusion-based models, which are representative in modern generative robot policies. (Pan et al., 2025) We begin with a theoretical analysis that examines the learn- ing objective induced by jointly mixing data from multi- ple domains. This analysis reveals two intrinsic effects that independently influence co-training performance: (1) Structuredrepresentationalignmentis characterized by a two-fold property. On one hand, representations become aligned across domains in a domain-invariant subspace, en- abling the transfer of task-relevant knowledge. On the other hand, representations retain discernibility with respect to domain-specific factors, allowing actions to adapt to the real world rather than being directly copied from surrogate do- mains. This balance is instrumental for effective co-training, as it determines whether adaptive action transfer is possible. (2)Importancereweightingeffectrefers to the domain- dependent logit modulation within action weightings. This effect operates locally in the space conditioned on observa- tions and controls how much each training sample from each domain contributes to an action decision during training. It is determined by the data mixing ratiow, the dataset size |D|and domain gaps. Through controlled toy co-training experiments, we verify the presence of both effects and find that structured representation alignment is the primary factor underlying strong model performance, while the importance reweighting effect plays a modulatory role. These findings motivate the following questions: Do similar effects arise in realistic robot manipulation tasks? How can these insights guide the design of more effective co-training algorithms? In the second part of the paper, we address these ques- tions through comprehensive sim-and-sim and sim-and-real robotic manipulation experiments. In end-to-end co-training systems, the data mixing ratiowis typically the only ex- plicit control variable, yet it simultaneously influences both internal effects. Empirically, we find that structured rep- resentation alignment, in both local and global space, can emerge implicitly within an appropriate range of mixing ra- tios (which we refer to as “balanced mixing ratios”), and that its strength exhibits a moderate-to-strong correlation with task success. At the same time, preserving domain discerni- bility is necessary for effective action adaptation to the real world; when this property is lost, performance even shows a negative correlation with representation alignment. These observations provide a unified perspective for understand- ing existing co-training techniques. We benchmark three recent representative co-training techniques on our tasks — optimal transport-based feature regularization (Cheng et al., 2025; Punamiya et al., 2025), adversarial discriminative do- main adaptation (Cai et al., 2025; Yuan et al., 2025), and classifier-free guidance (Wei et al., 2025). We observe that each method primarily emphasizes only one aspect of struc- tured representation alignment, which often leads to unsta- ble or marginal improvements. Motivated by this analysis, we propose a simple combination of co-training techniques that jointly promotes alignment while preserving domain discernibility, and that also offers a more controllable inter- face for knowledge transfer during inference. This approach yields consistent and substantial improvements over prior methods. In summary, our contributions are as follows. •We systematically identify, for the first time, the working mechanisms of co-training through theoretical analysis and experimental support. •We find that structured representation alignment can be learned implicitly, and validate the effects and require- ments of both alignment and discernibility through com- prehensive robotic manipulation experiments. •We benchmark representative co-training techniques through the lens of our analysis, which inspires us to intro- duce a simple approach that stably improves performance, and opens the door to new algorithm designs. 2. Theoretical Analysis of Co-Training Generative robot policies use generative modeling archi- tectures, such as diffusion/flow models, and autoregressive transformers, as parameterizations of the mapping from ob- servation to action. Given their popularity and adoption in industry, our analysis throughout this paper focuses on 2 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? the most popular policy form — diffusion/flow matching policy (Chi et al., 2025). In the following, we use “dif- fusion” as an umbrella term for both diffusion and flow matching (Liu et al., 2022) models, as they are equivalent under our analysis (Gao et al., 2025). We start with illus- trating why structured representation alignment matters in co-training with diffusion policy (Sec. 2.1); then, we provide an analysis of the importance reweighting effect (Sec. 2.2). 2.1. Structured Representation Alignment Training diffusion policy corresponds to jointly learning a feature encoderf φ :O →Zto project observations into a latent space, and a policy modelπ θ :Z →Athat maps the learned representations to the action space. Formally, given limited robotic datasetD T = (o i ,a i ) N i=1 in the target domain, and abundant datasetD S =(o j ,a j ) M j=1 from a source domain (M ≫ N), we train a diffusion policy model (Chi et al., 2025) with mixing ratiow. This gives us the learning objective as: L w (t;φ,θ) := w·L D T + (1− w)·L D S (1) whereL D = E (o i ,a i )∼D,ε∈N(0,I d ) [||ε−ε θ (a t ,t,o)|| 2 2 ]. We can prove that there exists an analytical optimal solution. In this paper, we adopt the score parameterization (proof provided in Appendix B.1): s ∗ w (a t ,t,o) = ˆw t · s ∗ t (a t ,t,o) + ˆw s · s ∗ s (a t ,t,o)(2) where ˆw t = w· p t (a t ,f φ (o)) w· p t (a t ,f φ (o)) + (1− w)· p s (a t ,f φ (o)) p k (a t ,z) = 1 |D k | X i∼D k p(a t |a 0 i )· K(z,z i ), k ∈t,s. (3) HereK(·,·)is a kernel measuring how closely the current observation matches the observations in the dataset. So the behavior of the empirical optimal score function de- pends heavily on the learned observation representations z = f φ (o). Based on the degree of representation alignment between the source and the target domain induced byf φ , we hypothesize three different scenarios in co-training: 1) Disjoint: observation representations of source and target domains are located in totally different clusters. During inference in the target domain, withp s (a t ,z) ≈ 0, the dy- namic weightˆw t stays near1. So the policy ignores data from the source domain, thus no positive transfer from source to target will occur. 2) Structured aligned: the pol- icy learns task-relevant, domain-invariant representations while retaining sufficient domain-specific information, such that observation representations of source and target do- mains are close but not collapsed. In this case, the action prediction will be effectively guided by neighbors in the source domain but dominated by data from the target do- main. This informs our definition of structured representa- tion alignment at the beginning. 3) Overlapping: Although the observation representations of the source and target do- mains are fully aligned, the corresponding actions differ due to domain gaps. As a result, the policy prediction is unaware of the actual environment and instead exhibits a bimodal distribution over source and target actions, leading to negative transfer. 2.2. Importance Reweighting Effect Mixing ratio directly provides additional modulation in this transfer process. Given any specific observationo, Eq.(3) will degrade to: ˆw t := wp t (a t ) wp t (a t ) + (1− w)p s (a t ) (4) wherep t (a t ) = 1 N P N i:o i =o p(a t |a 0 = a i ),p s (a t ) = 1 M P M j:o j =o p(a t |a 0 = a j ) . At a largetduring inference, as data are greatly perturbed by noise,p t (a t ) ≈ p s (a t ), the model will approximate a global average between two domains; at a smallert,p(a t )will be concentrated on one domain with the existence of domain gaps, so the model pre- diction will converge to one specific domain withˆw t ≈ 1. At any timestept, asa t are usually distributed as Gaussians around the training sample, for each data point we define: r k (a t ,t) := ||a t −α t a k || σ t √ d . We can have a closer look at the merged score function with further simplifications: s ∗ w (a t ,t) = N X i t g i t s ∗ i t + M X i s g i s s ∗ i s g i k = Softmax(ln(w k )− r 2 k (a t ,t)∗ d/2), k ∈t,s w t = w/N,w s = (1− w)/M (5) wheres ∗ i is the optimal score towards each action data (derivation provided in Appendix B.2). It reshapes the ac- tion sampling distribution to learn by reweighting the score functions in two domains during training as shown below. a real a sim g it ⋅S it g is ⋅S is S w 1 S w 2 noisy action a t clean action a 0 denoising process Figure 2. Importance reweighting reshapes the learned action distribution via reweighting score functions during training time. 3 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? 0.00 0.25 0.50 0.75 1.00 z=0 0.0 0.5 1.0 1.5 2.0 z=0 0.0 2.5 5.0 7.5 10.0 z=0 w = 0 (source-only) w = 0.01w = 0.1 w = 1 (target-only) Overlapping Structured Aligned Disjoint Source ManifoldSource SamplesTarget ManifoldTarget SamplesPrediction Figure 3. Visualization of controlled toy example. We co-train∼30 and∼3000 samples from target and source domain. Vertically in each column, the differences of prediction samples showcase the impact of representation alignment; horizontally in each row, the differences showcase the shift of importance reweighting effect controlled by mixing ratio w. In a special case, we can further have the following relation of the relative weight ratio: g i t g i s ∝F( w 1− w , M N ,|a i t − a i s |) (6) The amplitude of this modulation is influenced by bothw, the dataset size and domain gaps. A detailed characteriza- tion about this property is provided in Appendix B.3. Based on the analysis above, we can summarize as fol- lows:The effectiveness ofco-trainingwithdiffusionpolicy ismainlydecidedbytwointrinsiceffects:(1)structured representationalignment;(2)theimportancereweighting effect.With the above theoretical analysis, we are now ready to find empirical evidence that supports our insight. 3. Controlled Toy Example Generally, the two effects interact during end-to-end co- training and jointly influence the learning dynamics. To disentangle and understand their individual contributions to co-training, we designed a pilot toy experiment. In this simplified setting, we seek to learn a policy modelπ θ with a pre-trained feature encoder that defines the input distribution p(x), where each input dimension corresponds to a principal direction in the latent space. We adopt a 4-layer Multi-Layer Perceptron (MLP) as the diffusion model architecture. Experiment design. The policy modelπ θ is co-trained to learn the mapping π(y|x) : R 3 → R 2 . We manually define two manifolds,M S andM T , with different distributions, corresponding to the intrinsic data distributions of the source and target domains, respectively. Then, we sample paired data points from these two manifoldsD S =(x i ,y i ) N S ∼ M S andD T = (x i ,y i ) N T ∼ M T , whereN S ≫ N T andD T is sampled partially as shown in Fig. 3. This design simulates the common case in which target domain data is usually sparser, limited, and less diverse than source data. We align the two manifolds along two principal directions but vary the distance between them along the remaining one to create different representation alignment scenarios. The results are shown in Fig. 3. Finding 1: The toy model behavior aligns with our the- oretical analysis. As expected, under three different rep- resentation scenarios described in Sec. 2.1, the co-trained model exhibits distinct behaviors: 1) In disjoint, the pre- diction remains close to that of the model trained solely on target-domain data, where the model can easily distinguish between two domains but fails to transfer knowledge, i.e., the learned mapping, from the source domain. Due to the limited amount of data, the model tends to memorize each data point (He et al., 2025), failing to interpolate within the training distribution and extrapolate beyond it. 2) In structured alignment, this setting represents a sweet spot, where the model achieves a balance between representation alignment and domain discernibility. As a result, the output distribution is reconstructed with high fidelity. 3) In overlap- ping, the model predictions become randomly distributed 4 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? between the source and target domains. Here, the model cannot effectively distinguish between the two domains and instead treats them as identical, preventing meaningful adap- tation of transferred knowledge. On the other hand, the data mixing ratiowexerts an independent but secondary influence on this capability via importance reweighting, as discussed in Sec. 2.2. Specifically, it adjusts the relative amplitude of transferred knowledges ∗ s and target-domain adaptations ∗ t . As illustrated by the horizontal comparison in Fig. 3, whenwis relatively small (e.g., the second col- umn from the left), the output becomes noisier due to the increased contribution of source-domain data during the early denoising steps. In addition, we observe an intriguing phenomenon: with appropriate co-training settings, the model can reasonably make predictions in the out-of-distribution (OOD) region, indicating OOD generalization capability. Notably, this capability does not arise from simply copying knowledge from the source domain; rather, it emerges from preserving the distribution shift in the learned representations, which is crucial for accurate OOD prediction. Finding 2: Structured representation alignment is a dom- inant driver of strong model performance. Since we have the ground-truth mapping, we provide a quantitative measure usingL2loss in Fig. 4. The overall importance reweighting effect is constrained by the underlying represen- tation alignment. That is, changing the mixing ratio alone cannot compensate for poorly aligned representations, nor can it induce OOD generalization in the absence of sufficient alignment, e.g., the red and blue curves where the mixing ratio nearly has no effect on the final performance. Based on this, we conduct an ANOVA-style variance decomposition analysis (Fisher & Fisher, 1971) on these two factors. We find that changes in structured representation alignment ex- plain around50%of the loss variance, while the importance reweighting effect of the mixing ratio accounts for only 20%. In this sense, structured representation alignment is the primary determinant of model behavior, whilewserves as a modulation factor that fine-tunes the balance between source and target domain knowledge. By drawing an analogy from the toy example to sim-and- real co-training, we hypothesize a similar underlying mech- anism: structured representation alignment enables effective knowledge transfer from simulation, while maintaining suffi- cient domain discernibility to adapt actions to the real world. A key question, however, is whether this mechanism can be empirically observed in practical sim-and-real settings, rather than remaining a purely conceptual intuition. Moreover, Finding 2 raises a deeper question: can structured representation alignment emerge in end-to-end co-training, given that the data mixing ratiowis the only explicit control variable? To answer these questions, we conduct extensive 0.0010.010.050.10.30.50.70.90.99 Mixing Ratio w 10 0 10 1 10 2 10 3 L2 Loss (log) L2 Loss Δz Δz=0.01 Δz=0.10 Δz=0.50 Δz=1.00 Δz=3.00 Δz=5.00 Δz=10.00 Δz=20.00 Δz=50.00 Δz w interaction 0.0 0.1 0.2 0.3 0.4 0.5 Variance Explained ANOVA Decomposition Figure 4. L2 loss with sweeping mixing ratio and deltazdis- tance, and ANOVA variance decomposition. When representa- tions overlap (red line) or are disjoint (blue line), co-trained models are insensitive to mixing ratios. The importance reweighting effect (blue bar) can only explain 20% of the performance variance. experiments on real-world robotic manipulation tasks. 4.Sim-and-Real Co-Training for Manipulation To find out more empirical evidence to validate our hy- pothesis in robot manipulation, in particular, sim-and-real co-training, we design a set of sim-and-sim and sim-and-real co-training experiments on manipulation tasks (Fig. 5). The sim-and-sim experiments are designed to explicitly control the domain gaps between source and target domains to en- sure our observations are consistent across different domain gaps. Across all experiments, we adopt a transformer-based diffusion model (Chi et al., 2025) with ResNet18 (He et al., 2016) as the vision backbone and train end-to-end. Task suites. We choose three manipulation tasks from robo- suite (Zhu et al., 2020):NutAssembly,MugHang, and MugCleanup.NutAssemblyandMugHangrequire more precise control than common pick-and-place tasks, as it includes dense object interactions. And there are more rotation motions in the actions ofMugHangdemonstrations. In addition, the model needs relatively long-horizon reason- ing and execution to succeed inMugCleanup. These tasks represent many key challenges in robot manipulation. Environment setup. For sim-and-real experiments, fol- lowing the recipe in Maddukuri et al. (2025), we calibrate the camera pose and intrinsics to minimize camera align- ment differences between simulation and the real world. For sim-and-sim experiments, we utilize the same source-sim environments and create a second target-sim environment with domain gaps. Domain gaps categorization. Simulation and the real- world data contain domain shifts from various aspects. To identify the different effects of co-training across them, we decompose the gaps along two dimensions — visual appear- ance and environment physics. We manually introduce these 5 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? source-simphysics-onlyvisual-onlyvisual-physicstarget-real (a) NutAssembly (b) MugCleanup (c) MugHang Figure 5. Visualizations of the designed sim-and-sim and sim-and-real tasks. In the physics-only setting, we vary the physical parameters of objects, including mass, friction, and size, while keeping the appearance similar. Across all target environments, we tune the robot controller configurations to match those in the real world. gaps and construct three sim-and-sim co-training settings, i.e., visual-only, physics-only and visual-physics. Data preparation. For the target domains, we collect 50 human demonstrations for each task. For the source domains, we use MimicGen (Mandlekar et al., 2023) to further synthesize∼3000 trajectories based on 50 human demonstrations. Following Wei et al. (2025), we de- finew n = |D r | |D r |+|D s | as the natural mixing ratio, where |D r |and|D s |are the sizes of real-world and simulation datasets, respectively. This is equivalent to concatenating the sim and real datasets. For experiments in this section, we co-train policies by sweeping a set of mixing ratios w ∈0, 0.005,w n = 0.016, 0.1, 0.3, 0.5, 0.8, 1. 4.1. Observations on Representation Alignment Representation alignment can be learned implicitly in end-to-end co-training. Our experiments began with vi- sualizing the latent embeddings of simulation and real- world observation features across different layers using UMAP (McInnes et al., 2018) at different mixing ratios. Specifically, we look into the features after the vision stem and the final-layer output embeddings of encoder trunk f φ , which include other modality information such as pro- prioception and language. What is surprising is that, in a certain range of mixing ratios, visual features exhibit local geometry alignment sharing very similar geomet- ric structures, while the observation features show repre- sentation alignment in global space, as shown in Fig. 6. This can inform us about how the representation alignment evolves through the networks. We further quantify the lo- cal and global representation alignment using the Gromov- Wasserstein distance (M ́ emoli, 2011) and Wasserstein dis- tance (R ̈ uschendorf, 1985), respectively. By varying the data mixing ratio, we observe a clear correlation: smaller distances between real and simulation features correspond to more similar latent geometries and stronger alignment as shown in Fig. 6 (Full visualizations are available in Ap- pendix D.2). This trend holds consistently across both sim- to-real and sim-to-sim experiments. These results suggest that co-training remains sensitive to the data mixing ratiow because adjusting it simultaneously and substantially alters primary intrinsic effect—representation alignment itself. In other words, the mixing ratio does not merely re-weight source and target data contributions, but also implicitly re- shapes the learned representation space. Similar phenomena have also been observed in Kareer et al. (2025b), where the alignment emerges from scaling pre-training data. Representation alignment positively correlates with model performance. We compute the correlation between the log-transformed Wasserstein distance obtained above and the corresponding success rate across different settings. For each checkpoint, we evaluate the policy over 200 trials (sim) and 30 trials (real) for computing the mean success rate. We report both Pearson’s correlation coefficient, which captures linear associations, and Spearman’s rank correla- tion coefficient, which is robust to non-linear but monotonic relationships. As shown in Fig. 7, in all settings except the physics-only condition in sim-and-sim co-training, both Pearson and Spearman correlation coefficients fall in the range of0.6 ∼ 0.8, with p-values< 0.04. These results indicate a moderate-to-strong positive association between representation alignment and model performance. In some cases, one of the correlation coefficients (Pearson or Spear- 6 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? 2 4 6 8 10 12 -2 0 2 4 6 8 0 2 4 6 8 10 12 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 -2 0 2 4 6 8 10 0 2 4 6 8 10 12 -2 0 2 4 6 8 10 12 -2 0 2 4 6 8 2 3 4 5 6 7 8 9 10 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 -4 -2 0 2 4 6 8 10 0 2 4 6 8 10 12 -4 -2 0 2 4 6 8 10 12 -4 -2 0 2 4 6 8 10 -2 0 2 4 6 -4 -2 0 2 4 6 8 10 12 0 2 4 6 8 10 2 4 6 8 10 -2 0 2 4 6 8 10 12 14 -2 0 2 4 6 8 10 12 2 4 6 8 10 12 -2 0 2 4 6 8 10 2 4 6 8 10 2 3 4 5 6 7 8 9 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 12.5 0 2 4 6 8 2 3 4 5 6 7 8 9 -2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 0 2 4 6 8 10 12 14 -1 0 1 2 3 4 5 -2 0 2 4 6 8 10 12 -2 0 2 4 6 8 10 12 14 1 2 3 4 5 6 7 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 -2 0 2 4 6 8 10 6 7 8 9 10 11 12 NutAssembly MugCleanup MugHang Visual features Observation features f φ Visual features Observation features f φ Co-Training Real-Only RealSim Figure 6. UMAP visualization of latent features. We visualize the features after the vision stem and after the encoder trunkf φ . Red and blue dots represent the real and simulation features, respectively. We show the results for a specific mixing ratio in this figure. More details and results are provided in the Appendix D.2. man) is lower (e.g.,∼ 0.4), suggesting that the relationship may be non-linear or only partially monotonic rather than strictly linear. Importantly, this overall pattern is consis- tently observed across all three tasks. Suppressing representation alignment results in perfor- mance degradation. To further verify the causal effect of representation alignment, we conduct a minimal ablation that explicitly encourages representation separation in vis- phys sim-and-sim setting. Inspired by adversarial domain adaptation (Tzeng et al., 2017), we keep the domain classi- fier operating on the learned representation, but intention- ally remove the gradient reversal layer, thereby promoting domain-discriminative features instead of domain-invariant ones. The performance across 3 tasks drops consistently. 4.2. Observations on Domain Discernibility Although representations align in low-dimensional space, they are easily discernible by shallow neural networks. We perform a simple linear probing study by training a 2- layer MLP for binary-domain classification on the encoder trunk’s output embeddings. Surprisingly, even if the repre- sentations seem to be aligned well in low-dimensional space, a simple MLP can easily achieve∼ 100%success rate on validation sets in all settings. This suggests that represen- tations are in the partially aligned scenario, and co-training policies do retain the domain-specific information. Discernibility is indispensable for actions to adapt to the target domain. We report the success rate of each task in each setting in Fig. 8. We can find that in the four sim-and-sim settings, the success rate of the physics-only policy is even lower than the vis-phys policy on the task of NutAssemblyMugCleanupMugHangAvg co-training0.850.440.270.52 +discrimination0.790.4150.2150.47 Table 1. Performance degrades after adding penalty for repre- sentation alignment. The experiments are run with w = 0.1. 7 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? -1.5-1.0-0.50.00.51.01.52.02.5 Normalized WD -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Normalized SR Pearson r = -0.76 (p=7e-05) Spearman ρ = -0.65 (p=0.001) Visual-only -2.5-2.0-1.5-1.0-0.50.00.51.01.5 Normalized WD Pearson r = 0.57 (p=0.007) Spearman ρ = 0.49 (p=0.03) Physics-only -2-1012 Normalized WD Pearson r = -0.70 (p=0.0004) Spearman ρ = -0.85 (p=1e-06) Visual-Physics -1.0-0.50.00.51.01.52.02.5 Normalized WD Pearson r = -0.44 (p=0.04) Spearman ρ = -0.55 (p=0.01) Target-Real Figure 7. Correlation between representation alignment and success rate. We group the results of different tasks in each setting and compute the Spearman and Pearson correlation coefficients after normalization. Blue, yellow, and green-series dots represent the same task in Fig. 8. Gray dash line indicates the fitted curve. 0.0050.0160.1000.3000.5000.8001.000 Mixing ratio w 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 Success rate NutAssembly visual-only physics-only visual-physics 0.0050.0160.1000.3000.5000.8001.000 Mixing ratio w 0.2 0.3 0.4 0.5 0.6 Success rate MugCleanup visual-only physics-only visual-physics 0.0050.0160.1000.3000.5000.8001.000 Mixing ratio w 0.10 0.15 0.20 0.25 0.30 0.35 Success rate MugHang visual-only physics-only visual-physics Figure 8. Success rate of sim-and-sim co-training with different domain gaps. Each data point is computed over three policy checkpoints, and each checkpoint is evaluated for 200 trials. The best performance is consistently achieved in the range of(0.016, 0.3). NutAssemblyandMugCleanup. As we largely change the object’s physical parameters while keeping its visual appearance similar, it is harder for co-trained policies to dis- tinguish between the two environments. Interestingly, from Fig. 7, we observe that the correlation between representa- tion alignment and model performance in the physics-only policy can even become negative, suggesting that blind rep- resentation alignment can be harmful. 5. A Unified View of Co-Training Methods Although a wide range of co-training techniques has been proposed, it often remains unclear why these methods yield performance gains in some settings while failing in others. In this section, we revisit three representative co-training approaches through the lens of our findings and show that their empirical behavior can largely be explained by how they balance representation alignment and domain discerni- bility. Specifically, we categorize existing methods based on whether they primarily promote cross-domain alignment or preserve domain-specific information. 5.1. Prior Co-Training Methods Optimal transport (OT) (Courty et al., 2016)-based meth- ods aim to align simulation and real-world data by explicitly matching their representation distributions, either in latent or trajectory space. Recent work (Punamiya et al., 2025; Cheng et al., 2025) formulated co-training as a joint opti- mal transport problem, in which samples from simulation and real domains are softly coupled to minimize a global discrepancy: min φ,θ L w (φ,θ) + λ· L OT (D r ,D s ).(7) L OT is usually computed using the Wasserstein distance between two domains. Under our hypothesis, such methods strongly encourage representation overlap across domains, effectively pushing simulation and real observations into a shared latent space. We implement OT-regularized co- training as in Cheng et al. (2025), and the only difference is that we drop the offline data pairing sampler. Adversarial domain adaptation (ADDA) methods (Tzeng et al., 2017; Cai et al., 2025) similarly seek domain-invariant representations by training a discriminator to distinguish between simulation and real data, while simultaneously learning an encoder that attempts to fool the discriminator: min φ,θ L w (φ,θ) + λ· L disc (D r ,D s ).(8) L disc can be implemented simply using binary cross-entropy loss. From the perspective of our hypothesis, adversarial 8 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? 01234 0.70 0.75 0.80 0.85 0.90 0.95 Success rate NutAssembly 01234 0.2 0.3 0.4 0.5 0.6 MugCleanup 01234 0.15 0.20 0.25 0.30 0.35 0.40 MugHang 01234 0.40 0.45 0.50 0.55 0.60 0.65 Average Co-Training+OT+ADDA+CFG+CFG-ADDA UnbalancedBalanced Figure 9. Average performance on sim-and-sim experiments. We can find that alignment-based methods (+OT/+ADDA) get improvement in the balanced mixing ratio group, while discernibility-based methods (+CFG) get improvement mainly in the unbalanced mixing ratio group. More detailed results are available in the Appendix. D.1 Method Success Rate Avg NutAssemblyMugCleanupMugHang Real-only11/308/306/308.6/30 Mixing Ratio w0.0160.10.30.0160.10.30.0160.10.3- Co-Training17/3011/3016/3016/309/307/308/3013/307/3015.3/30 + OT15/3017/3011/308/3015/3015/3011/309/304/3014.3/30 + ADDA 13/3013/3015/306/3014/3011/3010/3014/307/3014.3/30 + CFG 15/3014/3011/306/3017/3014/308/3014/3010/3015.3/30 + CFG-ADDA(λ = 0)20/3017/3011/3015/3019/3014/3015/3017/3013/3018.6/30 + CFG-ADDA(λ =−0.5)23/3015/3018/3011/3022/3017/3018/3015/308/3021/30 Table 2. Real-world policy performance under different co-training strategies. alignment also prioritizes cross-domain overlap, but does so implicitly through representation indistinguishability rather than explicit distribution matching. We implement it in the same way as Tzeng et al. (2017). Classifier-free guidance (CFG) (Ho & Salimans, 2022) introduces a distinct mechanism for co-training by interpo- lating between conditioned and unconditioned policies at inference time. Rather than enforcing representation align- ment during training, CFG modulates the influence of real- derived signals via a guidance scale. Under our hypothesis, CFG preserves domain discernibility by maintaining sepa- rate conditional pathways, while still enabling controlled knowledge transfer from simulation: ̃s θ (a,o,c,t) = (1+λ)·s θ (a,o,c,t)−λ·s θ (a,o, ∅,t). (9) We implement it by concatenating a one-hot embeddingc as environment labels to the observation features after the vision encoder, settingλto0as recommended in Wei et al. (2025). CFG-ADDA: a simple combination. Viewed through our explanation framework, existing co-training methods pri- marily differ in how they trade off representation alignment and domain discernibility. OT and ADDA-based methods emphasize alignment, which can be beneficial when domain discrepancies are small but may lead to negative transfer when discrepancies are large. In contrast, classifier-free guidance preserves domain awareness while allowing flex- ible information sharing. This unified perspective clari- fies the strengths and limitations of prior approaches and motivates our proposed combination strategy, which ex- plicitly balances these two competing objectives. We sim- ply combine the techniques of CFG and ADDA, named as CFG-ADDA. Specifically, we attach one-hot embeddings as environment labels to enable domain guidance, while en- couraging the remaining representation dimensions to align through an adversarial discriminator. Training details are provided in Appendix C. With this explicit disentanglement of domain-invariant and domain-specific features, we want to point out a new per- spective towards the score interpolation coefficientλ— as only the environment labels are dropped ins θ (a,o, ∅,t), it actually represents the average log-probability gradient direction in all domains. Soλcan be viewed as a more flex- ible control variable to transfer the “averaged knowledge” during inference as opposed to transferring during training by importance reweighting effect. We setλ = −0.5for CFG-ADDA as default. 5.2. Experiments and Analysis Sim-and-Sim Experiments. We implement the above techniques on top of our co-training model and conduct visual-physics sim-and-sim co-training experiments. The 9 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? -2.0-1.5-1.0-0.50.00.51.01.52.0 λ 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 Success Rate Success rate vs λ Real-only Co-Train CFG CFG-ADDA (ours) Figure 10. Performance of different co-training methods on MugCleanup withw = 0.1in sim-and-sim settings. The plots are similar for all tasks and balanced mixing ratios. results are shown in Fig. 9. We group the data mixing ratios into two regimes: balanced and unbalanced mix- ing. Performances with balanced mixing ratios are con- sistently better than with unbalanced mixing ratios. Under balanced mixing, where simulation and real data are present in comparable proportions, alignment-oriented methods (OT and ADDA) consistently improve performance across tasks. This indicates that representation alignment effectively fa- cilitates cross-domain knowledge transfer when both do- mains are sufficiently observed during training. In con- trast, under unbalanced mixing, alignment-only methods exhibit pronounced performance degradation, particularly onMugCleanupandMugHang. This behavior suggests that when one domain dominates the training data, enforc- ing strong alignment biases the learned representation to- ward suboptimal invariances, thereby hindering real-world adaptation. CFG, which explicitly preserves domain in- formation, demonstrates greater robustness in this regime, but its peak performance remains limited. Notably, CFG- ADDA achieves strong performance across both regimes. By combining adversarial alignment with explicit domain conditioning, it leverages transferable structure from sim- ulation under balanced mixing while maintaining domain discernibility under unbalanced mixing. Sim-and-Real Experiments. Since balanced mixing ratios are the main choice for effective co-training, we conduct real-world evaluations only in this regime. The observations in the Table 2 are similar to sim-and-sim settings. We even find more stable and substantial improvement with our pro- posed method in the real world, achieving∼ 74%success rate on these challenging tasks. Ablations on Guidance Scale. In contrast to using only the positive values, we sweepλin the interval of(−2, 2)with both CFG and CFG-ADDA. As shown in Fig. 10, our pro- posed method consistently outperforms CFG with different guidance scales. Besides, both methods exhibit improve- ment withλ =−0.5. So instead of settingλ > 0to amplify the action gaps in a traditional way, we advocate setting λ < 0to actively transfer knowledge from the surrogate domains during inference. These results support our findings that effective sim-and- real co-training requires both representation alignment for transfer and domain discernibility for adaptive behavior. 6. Discussions and Future Work To understand how co-training works, we present a system- atic study that combines theoretical analysis with extensive empirical validation, yielding a unified explanatory frame- work. Within this framework, we identify two intrinsic effects underlying effective co-training: structured represen- tation alignment and importance reweighting. The effec- tiveness of structured representation alignment requires a careful balance between two competing objectives: aligning representations along domain-invariant dimensions to en- able transfer, while preserving domain-specific dimensions to maintain adaptability. This perspective unifies several ex- isting co-training methods and highlights the effectiveness of a simple combination strategy. We further identify the effects of mixing ratios and dataset size, which can help narrow the search space for future large-scale co-training experiments. A guideline is provided in Appendix D.5. Overall, we hope that this work sheds light on the mecha- nisms behind co-training and informs the design of more principled, robust co-training algorithms. Limitations and Future Work. First, our empirical study primarily focuses on sim-to-sim and sim-to-real co-training settings. Although we observe qualitatively similar trends in other co-training scenarios, such as human–robot co- training, validating the generality of our findings across a broader range of domains remains an important direction for future work. Second, our analysis concentrates on the end effects of the two identified mechanisms, without ex- plicitly characterizing their interaction during the dynamic learning process — particularly how the mixing ratio shapes representation learning over the course of training. In addi- tion, we do not investigate the potential impact of practical factors such as limited batch sizes. Third, we study the rela- tive relationships between representations, rather than their intrinsic structure. That is, we do not directly characterize what representations the model ultimately learns. Under- standing the nature of these representations, especially those that generalize across domains, may provide further insights into the emergence of structured representation alignment. Finally, while our study is grounded in imitation learning, the co-training paradigm is broadly applicable to other learn- ing settings, including world modeling and reinforcement learning. We hope this work encourages further exploration 10 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? across diverse domains, ultimately contributing to a deeper understanding and more effective use of co-training. Acknowledgment We would like to thank the Texas Advanced Computing Cen- ter (TACC) for its valuable support of computing resources. 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PMLR, 2023. 13 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? Contents 1 Introduction2 2 Theoretical Analysis of Co-Training2 2.1Structured Representation Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 2.2Importance Reweighting Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 3 Controlled Toy Example4 4 Sim-and-Real Co-Training for Manipulation5 4.1Observations on Representation Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 4.2Observations on Domain Discernibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 5 A Unified View of Co-Training Methods8 5.1Prior Co-Training Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 5.2Experiments and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 6 Discussions and Future Work10 A Related Work15 A.1 Co-Training for Robot Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 A.2 Representation Learning in Domain Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15 B Theoretical Details15 B.1 Demonstration of Empirical Optimal Score Function in Co-Training . . . . . . . . . . . . . . . . . . . .15 B.2 Derivation from Gaussian Distribution to Softmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 B.3 Empirical Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 C Training Details18 D Additional Experiments Results19 D.1 Detailed Results of Sim-and-sim Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 D.2 Details on Representation Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 D.3 Robustness to Different Policy Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19 D.4 Comparison to “Simulation Pre-training+Real Fine-tine” . . . . . . . . . . . . . . . . . . . . . . . . . .20 D.5 A Formal Statement of Mixing Ratio Selection Guideline . . . . . . . . . . . . . . . . . . . . . . . . . .20 D.6 Three Regimes of “Representation” in Co-Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 14 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? A. Related Work A.1. Co-Training for Robot Learning Generally, we can define any learning system that utilizes heterogeneous data as co-training. To mitigate the gap between limited in-domain robot data and large-scale surrogate data or even multimodal resources, co-training has been employed in numerous studies. Based on the types of large-scale surrogate data, we can coarsely categorize current work into the following three intersecting types: sim-and-real co-training, cross-embodiment co-training, and non-robot data co-training. Sim-and-Real Co-Training. Using simulation as a surrogate data source is a promising way for co-training. The main domain gaps include visual and physics gaps, with the physics gap being much more challenging. But with the advancement of high-fidelity physics simulators (Mittal et al., 2025; Nasiriany et al., 2024; Zakka et al., 2025) and automated data generation tools (Mandlekar et al., 2023; Jiang et al., 2025; Lin et al., 2025), high-quality and massive robot trajectories can be obtained easily. Many works have demonstrated the effectiveness of sim-and-real co-training on challenging manipulation tasks with small diffusion policy models (Maddukuri et al., 2025; Wei et al., 2025; Cheng et al., 2025) and even large Vision-Language-Action(VLA) models (Bjorck et al., 2025; Yu et al., 2025). There are also works (Barreiros et al., 2025; Jain et al., 2025) utilizing relatively large-scale real-world data and a small amount of simulation data for co-training, so that the performance evaluated in simulation can effectively reflect the performance in the real world. Cross-Embodiment Co-Training. A large amount of work (Doshi et al., 2024; Yang et al., 2024; O’Neill et al., 2024; Physical Intelligence, 2025; Yuan et al., 2025) has explored using cross-embodiment robot data for co-training, where a single policy is trained with multiple embodiments with a unified architecture and action representation. The main domain gaps include the visual appearance and the embodiment-physics gap. Human data is a special case of them, which can be directly treated as another embodiment. These works show that diverse robot pretraining can produce transferable internal representations (Kareer et al., 2025b). Some works (Cai et al., 2025; Kareer et al., 2025a; Punamiya et al., 2025) utilize representation alignment regularization, such as optimal transport and adversarial discriminator. Another line of work (Xu et al., 2025; Lepert et al., 2025b;a) forces input data distribution alignment via image editing. Non-Robot Data Co-Training. Internet-scale multimodal data without action labels is another valuable co-training resource. Numerous works (Physical Intelligence, 2025; Lee et al., 2025; Zawalski et al., 2024; Zitkovich et al., 2023; Lin et al., 2026) have adopted VL datasets, which contain rich commonsense knowledge, planning and spatial information for co-training in VLAs. These works have demonstrated that co-training with VLM data can transfer knowledge from other modalities. Also, a number of works try to utilize videos for policy co-training. Some works (Kareer et al., 2025a; Lepert et al., 2025a; Luo et al., 2025) explicitly extract action labels but with the problem of accuracy; some works (Ye et al., 2024; Zhou et al., 2024) explored latent action representations by encoding changes between video frames; other works (Li et al., 2025; Zhu et al., 2025; Liang et al., 2025) propose to co-train the action- and actionless video data within a unified architecture. A.2. Representation Learning in Domain Adaptation Co-training can also be viewed as semi-supervised or unsupervised domain adaptation. A central theme in domain adaptation is learning representations that enable knowledge transfer across domains while mitigating distribution shift. Early theoretical work formalized this goal by relating target error to source error and representation-level domain discrepancy, motivating the pursuit of domain-invariant features through shared embeddings or feature transformations (Ben-David et al., 2010; Mansour et al., 2009). Building on this foundation, many practical approaches explicitly align representations across domains, including discrepancy-based methods that minimize statistical distances such as maximum mean discrepancy (MMD) (Long et al., 2015), OT-based alignment (Courty et al., 2017; Damodaran et al., 2018), and adversarial domain adaptation methods that encourage indistinguishability (Ganin et al., 2016; Tzeng et al., 2017). B. Theoretical Details B.1. Demonstration of Empirical Optimal Score Function in Co-Training Diffusion models define a forward corruption process that maps data samples to a simple reference distribution over timesteps t∈ (0, 1). In the case of Gaussian perturbations, this process can be written as x t = α t x 0 + σ t ε , 15 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? whereε∼N(0,I d ). Models are trained to learn the underlying score field of this process, which is then used at inference to generate samples via reverse-time denoising. Learning Objective. Given training dataD =x i N i=1 , under a Gaussian probability path, the learning objective can be written in the denoising score matching form: L origin (t) := E x i ∼D,ε∈N(0,I d ) [||ε− ε θ (x t ,t)|| 2 2 ] .(10) There exists an analytical optimal solution for Eq. (10): s ∗ (x t ,t) =−ε ∗ (x t ,t)/σ t = 1 σ 2 t [α t E[x 0 |x t ]− x t ] .(11) And we have E[x 0 |x t ] = P N i=1 p(x t |x 0 = x i )· x i P N i=j p(x t |x 0 = x j ) (12) where p(x t |x 0 i ) =N(x t ;α t x i ,σ 2 t I d ). Then suppose we have source and target datasets asD T = x i N i=1 andD S = x j M j=1 , co-train a diffusion model with mixing ratio w, this gives us the training objective as: L w (t) := w·L D T + (1− w)·L D S (13) Similarly, we can get the analytical optimal score function as: s ∗ w (x t ,t) = ˆw t · s ∗ t (x t ,t) + ˆw s · s ∗ s (x t ,t)(14) where ˆw t := wp t (x t ) wp t (x t ) + (1− w)p s (x t ) (15) Proof.To findf(B)that minimizesL(f) = wE t [∥A−f(B)∥ 2 ]+(1−w)E s [∥A−f(B)∥ 2 ], we define a mixture probability density p w (a,b) = wp t (a,b) + (1− w)p s (a,b). By expressing the expectations as integrals, we have: L(f) = Z ∥a− f(b)∥ 2 (wp t (a,b) + (1− w)p s (a,b)) dadb = Z ∥a− f(b)∥ 2 p w (a,b)dadb = E w [∥A− f(B)∥ 2 ]. It is a standard property of Hilbert spaces of random variables that the MSE is minimized by the conditional expectation E w [A| B]: f ∗ (b) = Z ap w (a| b)da = R ap w (a,b)da p w (b) = R a (wp t (a,b) + (1− w)p s (a,b)) da wp t (b) + (1− w)p s (b) . As R ap i (a,b)da = E i [A| B = b]p i (b) for i∈t,s, we obtain the optimal solution: f ∗ (B) = wp t (B)E t [A| B] + (1− w)p s (B)E s [A| B] wp t (B) + (1− w)p s (B) . This shows that the co-trained optimal predictor is a weighted combination of the optimal predictors in each domain. B.2. Derivation from Gaussian Distribution to Softmax Starting from Eq. 14 proved above: s ∗ w (a t ,t) = ˆw t · s ∗ t (a t ,t) + ˆw s · s ∗ s (a t ,t)(16) 16 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? where the mixing weight ˆw t is given by: ˆw t := wp t (a t ) wp t (a t ) + (1− w)p s (a t ) (17) Combining Eq. 12 and expanding p t (a t ) , we can have: s ∗ w (a t ,t) = N+M X k=1 w k p(a t |a k ) P j w j p(a t |a j ) · α t a k − a t σ 2 t = N+M X k=1 w k p(a t |a k ) P j w j p(a t |a j ) · s ∗ k (a t ,t)(18) We recognize the fractional term as the posterior weightg k , which can be computed via the Softmax function using the distance metric r k = ||a t −α t a k || σ t √ d : g k = Softmax(ln(w k )− r 2 k ∗ d/2)(19) By further simplification, we obtain the final simplified form: s ∗ w (a t ,t) = α t σ 2 t · ( N X i t g i t a i t + M X i s g i s a i s )− a t σ 2 t = N X i t g i t s ∗ i t + M X i s g i s s ∗ i s (20) 0.00.20.40.60.81.0 w 0.0 0.2 0.4 0.6 0.8 1.0 g r N/M = 0.01 N/M = 0.05 N/M = 0.1 N/M = 0.3 N/M = 0.5 N/M = 0.7 N/M = 0.9 0.00.20.40.60.81.0 w 0.0 0.2 0.4 0.6 0.8 1.0 g s N/M = 0.01 N/M = 0.05 N/M = 0.1 N/M = 0.3 N/M = 0.5 N/M = 0.7 N/M = 0.9 w n w q Figure 11. N/M scaling B.3. Empirical Estimation [Fact 1. Gaussian concentration in high-dimensional space]: for a standard Gaussian vectorε ∼ N(0,I d ), we have ||ε|| = √ d(1 +O(1))with high probability. So for each component inp r (x t ),p s (x t ), it concentrates on a set of thin spherical shells centered at α t x i with radius σ t √ d. [Empirical evidence 1. Data points separation]: for most of the timestepstthat are not too large, in high-dimensional data space, these shells are non-overlapping asσ t √ d≪ min i,j ||α t x i − α t x j ||. We show this in the action-chunking space of diffusion policy as shown in Fig. 12. These make the softmax weights extremely imbalanced, which is nearly biased towards the nearest training data. 17 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? 0.00.20.40.60.81.0 x 0.0 0.2 0.4 0.6 0.8 1.0 Bhattacharyya coefficient Overlap measurement Real-Sim Real-Real Figure 12. Empirical measurement about training sample overlapping using Bhattacharyya coefficient. We compute it in the same way as in Song et al. (2025), but include the distances between observation features as we are modeling the conditional distribution. [A Special Case Study]: Let’s assume an ideal case where the action chunks are uniformly distributed across different states on the trajectories, so the ratio of the number of trajectories between simulation and real is similar to the ratio of the number of action chunks of different states between simulation and real. A special case is that the nearest data points inD r are similarly close to the nearest data points inD s , then we can have: s ∗ w (a t ,t)≈ α t σ 2 t w/N· exp(−r 2 r /2) w/N· exp(−r 2 r /2) + (1− w)/M· exp(−r 2 s /2) · x r + (1− w)/M· exp(−r 2 s /2) w/N· exp(−r 2 r /2) + (1− w)/M· exp(−r 2 s ) · x s (21) Then we have g r g s = 1−w N w N · w 1−w · exp( r 2 s −r 2 r 2 ) = 1−w N w N · w 1−w · exp( |a t −(a r +a s )/2|·|a r −a s | 2σ 2 t d ), also we can draw the relation betweeng r ,g s andw. We can find that as the ratio of sim/real increases, the choice ofwshould be more robust, which aligns with prior observations from Wei et al. (2025). And g r (w n ) = g s (w n ) = 1/2, which aligns with our intuition. Since we find that the curve will be extremely steep for smallN/M, we posit that the balanced mixing ratio should make good use of both real and sim, but place more weight on real. Take the intersection point betweeng r ,g s with diagonal g = 1− wandg = w, the coordinate of intersection point ing r is(w q , 1− w q )wherew q = √ q √ q+1 andq = N M . So the relative domain weight g r g s should be between(1, q M N ), and balanced mixing ratio should be between(w n ,w q ≈ q N M ) whenN ≫ M. With this principle, this range is around(0.016, 0.13)in our experiments which also aligns with our experimental observations. C. Training Details The original images are captured by the camera at a resolution of 720×1280. During preprocessing, the images are downsampled to 90×106, followed by random cropping to 84×84 during training and center cropping during testing. The policy takes stacked history images and robot proprioceptive inputs (joint and gripper positions) as input, and outputs 7-DOF target joint positions along with the gripper action. The overall training procedure of CFG-ADDA is summarized in Algo. 1. We implement the discriminator as a simple 3-layer MLP. Across all of our experiments, we use a batch size of 256, dropping probabilityp = 0.2and a weighting coefficient λ = 0.1. To ensure the discriminator can learn in a compact feature space and also provide effective reverse gradients to the policy, we only start to train the discriminator and compute the discriminative loss after a warming step of 5000. The implementation of CFG and ADDA can be seen as only keeping the corresponding computation part in Algo. 1. 18 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? Algorithm 1 : CFG-ADDA 1: Input: Source datasetD s , target datasetD t , mixing ratio w, randomly initialized f φ ,π θ and discriminator Disc μ . 2: for iterations t = 1 to T do 3:Sample data(o i t ,a i t ) with size N · w fromD t and(o i s ,a i s ) with size N · (1− w) fromD s 4:Create one-hot embeddings as environment labelsc =c i t ∪c i s 5:Randomly set environment labels as ∅ with probability p 6:Compute features z i = f φ (o i ) 7:Compute discriminator lossL disc =−E z i [logDisc μ (z i s )]− E z i [log(1− Disc μ (z i t ))] 8:Simply concatenate z i and c i , compute behavior cloning lossL w (z i ,c i ,a i ) 9:Update f φ ,π θ ,Disc μ with gradient ofL w (φ,θ) + λ·L disc (μ) 10: end for 0.0050.0160.10.30.50.8 Mixing Ratio 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 Success Rate Nut Assembly sim2sim Co-Training +OT +ADDA +CFG +CFG-ADDA (a) NutAssembly 0.0050.0160.10.30.50.8 Mixing Ratio 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 Success Rate Mug Cleanup sim2sim Co-Training +OT +ADDA +CFG +CFG-ADDA (b) MugCleanup 0.0050.0160.10.30.50.8 Mixing Ratio 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 Success Rate Mug Hang sim2sim Co-Training +OT +ADDA +CFG +CFG-ADDA (c) MugHang Figure 13. Detailed results of sim-and-sim evaluations. D. Additional Experiments Results D.1. Detailed Results of Sim-and-sim Evaluation We provide the detailed sim-and-sim evaluation results of comparing different co-training techniques. In Fig. 9, we categorize0.016, 0.1, 0.3as balanced mixing ratio while the others as unbalanced mixing ratio. Compared to the previous techniques which only emphasizes one side of structured representation alignment, our simple combination brings more stable improvement across different mixing ratios. D.2. Details on Representation Alignment Details of Measurement. We use UMAP dimension reduction for qualitative visualizations, and measure the Gromov- Wasserstein distance (M ́ emoli, 2011) and the Wasserstein distance (R ̈ uschendorf, 1985) to show local geometric similarity and global distributional distance for quantitative comparison. We randomly sample a large batch (1024) of data from each domain and normalize the features within each domain independently before quantitative measurement. We assume the down-sampled distribution can represent the distribution of the complete dataset. To compare the Gromov-Wasserstein distance across multiple policies, we normalize the cost matrix by the pooled mean of each domain, thereby controlling for differences in value amplitudes. More Feature Visualizations.As shown in Fig. 17, the changing trends of the two distances with mixing ratios are similar across different tasks. Whenwis in the range of balanced mixing ratio (roughly betweenw n and0.5), the representations of simulation and real-world observations share more similar geometric structures and exhibit greater overlap. D.3. Robustness to Different Policy Architectures As our main experiments are all conducted on encoder-decoder transformer-based diffusion policy, to further support the robustness of our findings to different architectures, we do another series of experiments with the same setting on two other architectures — decoder-only transformer-based model and CNN-based U-Net diffusion policy. The results in Fig. 14 show 19 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? 00.0050.0160.10.30.50.81.0 Mixing ratio w 0.0 0.2 0.4 0.6 0.8 1.0 Success rate NutAssembly Decoder-only U-net 00.0050.0160.10.30.50.81.0 Mixing ratio w 0.0 0.1 0.2 0.3 0.4 Success rate MugCleanup Decoder-only U-net 00.0050.0160.10.30.50.81.0 Mixing ratio w 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Success rate MugHang Decoder-only U-net Figure 14. Balanced mixing ratios are robust to different policy architectures. Each data point is computed over three policy checkpoints, and each checkpoint is evaluated for 200 trials. The best performance is consistently achieved in the range of(0.016, 0.3). that the best performance consistently lies in a narrow range ([0.016, 0.3]), indicating robustness beyond the architecture used in the main paper. D.4. Comparison to “Simulation Pre-training+Real Fine-tine” Although some prior work shows that it is generally less effective than co-training, as this still remains as an important baseline, we include a direct comparison here. We pre-train with simulation data only for∼ 130k steps and fine-tune with real data for∼ 120ksteps. We evaluate all checkpoints along the training and report the highest success rate. Our result is consistent with prior findings. TaskPretrain+FTCo-training NutAssembly0.770.925 MugCleanup0.2150.495 MugHang0.2250.26 Table 3. Performance comparison with Pre-train+FT. D.5. A Formal Statement of Mixing Ratio Selection Guideline The main experiments in the paper rely a heuristic defined mixing ratio grids. Based on the above analysis, we provide a guideline for selecting mixing ratios for co-training, which we hope will be useful to the community. We conduct some additional experiments to further support the effectiveness of our guideline. We co-train with 3 sets of different dataset sizes – 10:3000, 50:500, and 50:100 – and sweep the mixing ratios. 00.0010.0030.010.060.10.51 Mixing ratio w 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 Success rate 10:3000 w n w q 00.050.10.20.30.50.81 Mixing ratio w 0.0 0.2 0.4 0.6 0.8 Success rate 50:500 w n w q 00.050.10.330.50.80.91 Mixing ratio w 0.0 0.2 0.4 0.6 0.8 Success rate 50:100 w n w q Figure 15. Experiments on different dataset sizes. Each data point is computed over three policy checkpoints, and each checkpoint is evaluated for 200 trials. The best performance is consistently achieved in the range of (w n ,w q ), which supports our guideline below. 20 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? Algorithm 2 Guideline for Co-Training Mixing Ratio Selection Require: Source dataset size N , target dataset size M with M > N Require: Optional desired target contribution q (e.g., q = 0.8) Ensure: A narrowed search range (w n ,w q ) for the mixing ratio 1: Compute the natural mixing ratio w n = N N + M . 2: Use w n as the lower bound of the search range. 3: if M/N > 5 then 4:Set the upper bound as w q = r N M . 5: else 6:Set a desired target contribution ratio q (e.g., q = 0.8). 7:Compute the upper bound as w q = N ∗ q (1− q)∗ M + N ∗ q . 8:Optionally, set the upper bound empirically to 0.5, which is empirically enough. 9: end if 10: Adjust w n ,w q upward if the source-target domain gap is large. 11: Consider domain gaps from visual appearance, physics, and embodiment. 12: Since no formal estimator is assumed, apply this adjustment accordingly. 13: Perform a simple search for the final mixing ratio within (w n , w q ). D.6. Three Regimes of “Representation” in Co-Training In Sec. 2.1 of the main paper, we point out three different hypothetical scenarios of representation in co-training – overlapping, structured aligned, and disjoint. As mentioned above, we use two coupled and measurable properties to characterize the concept of “structured representation alignment”. So it can be mathematically defined as follows: SRA(p s ,p t ) = (M align ,D disc ) whereM align = W(p s (z),p t (z))andD disc = 1 2 P k∈s,t E z∼p k (z) [max a t p k (a t | z)]. We design another set of exper- iments to show the existence of these three regimes: We introduce an additional and complementary control knob via discriminator regularization, which directly modulates domain discernibility. Specifically, we train models with discriminator loss weights of0, 0.05, 0.5, and within each setting sweep the mixing ratio to vary alignment. This allows us to populate a substantially broader region of the (alignment, discernibility) space. Empirically, these settings correspond to different regimes: (1) No discriminator (0): representations remain weakly aligned (high WD) but highly distinguishable (high discernibility), corresponding to the disjoint regime (and partially covering the boundary toward structured alignment). (2) Moderate discriminator (0.05): representations become better aligned (lower WD) while preserving domain-specific structure (high discernibility), corresponding to the desired structured aligned regime. (3) Strong discriminator (0.5): representations are strongly aligned (low WD), but domain-specific information is suppressed (low discernibility), corresponding to the overlapping regime. Within each regime, varying the mixing ratio further modulates the degree of alignment, producing a set of points spanning different Wasserstein distances. We provide the corresponding 2D visualization (including Wasserstein distance) in Fig. 16. We observe: - In the overlapping regime, the correlation between performance and WD is reversed, consistent with observations in physics-only settings. - In the disjoint regime, the correlation follows the standard co-training trend (better alignment improves performance). - In the structured aligned regime, the correlation remains positive but weaker, indicating a distinct intermediate behavior. Taken together, these regimes exhibit a non-monotonic relationship, resulting in a characteristic inverted-U (or U-shaped) 21 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? curve when alignment and discernibility are jointly considered. This unifies the three regimes as different regions of a single underlying mechanism. -2.0-1.5-1.0-0.50.00.51.0 Normalized WD -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 Normalized SR Pearson r = 0.61 (p=0.007) Spearman ρ = 0.42 (p=0.08) Overlapping -2-1012 Normalized WD Pearson r = -0.31 (p=0.2) Spearman ρ = -0.31 (p=0.2) Structured Aligned -2-1012 Normalized WD Pearson r = -0.70 (p=0.0004) Spearman ρ = -0.85 (p=1e-06) Disjoint (a) Three regimes exhibit U-shape correlation pattern. -2-1012 -2 -1 0 1 2 Normalized SR r=0.61 ρ=0.42 -2-1012 r=-0.31 ρ=-0.31 -2-1012 Normalized WD r=-0.70 ρ=-0.85 Wasserstein Distance High Low Discernibility Low High Overlapping Structured aligned Disjoint Not Feasible Area NutAssemblyMugCleanupMugHang (b) Three regimes on two-axises – representation alignment and domain discernibility. Figure 16. Existence of three regimes in co-training. 22 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 w = 0 5 0 5 10 15 2 0 2 4 6 8 10 12 14 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 w = 0.005 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 4 2 0 2 4 6 8 10 12 2 4 6 8 10 12 w = 0.016 2 0 2 4 6 8 10 12 2 0 2 4 6 8 2 3 4 5 6 7 8 9 10 w = 0.1 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 2 0 2 4 6 8 2 4 6 8 10 w = 0.3 2 0 2 4 6 8 10 12 14 2 0 2 4 6 8 2 4 6 8 10 w = 0.5 2 0 2 4 6 8 10 12 2 0 2 4 6 8 2 4 6 8 w = 0.8 2 4 6 8 10 12 2 0 2 4 6 8 0 2 4 6 8 10 12 w = 1 Increased mixing ratios RealSim (a) NutAssembly. 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 0 2 4 6 8 10 12 0 2 4 6 8 10 w = 0 5 0 5 10 15 0 2 4 6 8 10 12 14 0 2 4 6 8 10 w = 0.005 2 0 2 4 6 8 10 12 14 2 0 2 4 6 8 10 12 2 4 6 8 10 12 w = 0.016 4 2 0 2 4 6 8 10 12 2 0 2 4 6 8 10 1 2 3 4 5 6 7 w = 0.1 2 0 2 4 6 8 10 12 2 0 2 4 6 8 10 0 2 4 6 8 10 w = 0.3 2 0 2 4 6 8 10 12 2 4 6 8 10 12 2 0 2 4 6 8 w = 0.5 0 2 4 6 8 10 12 14 2 0 2 4 6 8 10 12 14 2 0 2 4 6 8 10 w = 0.8 4 2 0 2 4 6 8 10 12 4 2 0 2 4 6 8 10 2 0 2 4 6 w = 1 Increased mixing ratios RealSim (b) MugCleanup. 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 4 2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 w = 0 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 5 0 5 10 15 4 2 0 2 4 6 8 10 12 w = 0.005 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 4 2 0 2 4 6 8 10 12 14 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 w = 0.016 2 0 2 4 6 8 10 12 2 0 2 4 6 8 10 12 14 1 2 3 4 5 6 7 w = 0.1 2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 4 5 6 7 8 9 w = 0.3 2 0 2 4 6 8 10 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 w = 0.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 5 0 5 10 15 4 2 0 2 4 6 8 10 w = 0.8 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 0 2 4 6 8 2 3 4 5 6 7 8 9 w = 1 Increased mixing ratios RealSim (c) MugHang. Figure 17. More UMAP visualizations of the observation representations. 23 How Does Sim-and-Real Co-Training Work in Generative Robot Policies? 4 2 0 2 4 6 8 10 12 14 2 0 2 4 6 8 2 4 6 8 10 w = 0 5 0 5 10 15 4 2 0 2 4 6 8 10 12 2 0 2 4 6 w = 0.005 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 2 0 2 4 6 8 10 3 4 5 6 7 8 9 10 w = 0.016 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 2 0 2 4 6 8 10 0 2 4 6 8 w = 0.1 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 4 2 0 2 4 6 8 10 0 2 4 6 8 10 12 w = 0.3 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 2 0 2 4 6 8 10 12 14 3 4 5 6 7 8 9 w = 0.5 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 2 0 2 4 6 8 10 2 4 6 8 10 w = 0.8 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 2 0 2 4 6 8 10 0 2 4 6 8 10 12 w = 1 Increased mixing ratios RealSim (a) NutAssembly. 0 2 4 6 8 10 12 14 2 0 2 4 6 8 10 1 2 3 4 5 6 7 8 9 w = 0 2 0 2 4 6 8 10 2 0 2 4 6 8 10 12 0 2 4 6 8 10 12 w = 0.005 4 2 0 2 4 6 8 10 0 2 4 6 8 10 0 1 2 3 4 5 6 7 w = 0.016 4 2 0 2 4 6 8 10 0 2 4 6 8 10 2 3 4 5 6 7 8 w = 0.1 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 2 3 4 5 6 7 8 9 2 4 6 8 10 w = 0.3 0 2 4 6 8 10 12 14 2 0 2 4 6 2 3 4 5 6 7 8 w = 0.5 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 w = 0.8 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 2 4 6 8 10 0 2 4 6 8 10 w = 1 Increased mixing ratios RealSim (b) MugCleanup. 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 2 0 2 4 6 8 10 2 3 4 5 6 7 8 9 w = 0 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 2 4 6 8 10 2 0 2 4 6 8 10 w = 0.005 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 2 4 6 8 10 12 4 5 6 7 8 9 w = 0.016 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 2 0 2 4 6 8 10 6 7 8 9 10 11 12 w = 0.1 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 2 0 2 4 6 8 10 2 1 0 1 2 3 w = 0.3 5.0 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 0 2 4 6 8 10 12 14 7 8 9 10 11 12 w = 0.5 4 2 0 2 4 6 8 10 4 6 8 10 12 4 5 6 7 8 9 10 11 12 w = 0.8 2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 0 2 4 6 8 10 12 14 1 0 1 2 3 4 5 w = 1 Increased mixing ratios RealSim (c) MugHang. Figure 18. More UMAP visualizations of the deeper layer representations. 24