Analyzing Transformers in Embedding Space

Analyzing Transformers in Embedding Space Guy Dar11^1start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPT Mor Geva22^2start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPT Ankit Gupta11^1start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPT Jonathan Berant11^1start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPT 11^1start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPTThe Blavatnik School of Computer Science, Tel-Aviv University 22^2start_FLOATSUPERSCRIPT 2 end_FLOATSUPERSCRIPTAllen Institute for Artificial Intelligence guy.dar,joberant@cs.tau.ac.il, morp@allenai.org, ankitgupta.iitkanpur@gmail.com Abstract Understanding Transformer-based models has attracted significant attention, as they lie at the heart of recent technological advances across machine learning. While most interpretability methods rely on running models over inputs, recent work has shown that an input-independent approach, where parameters are interpreted directly without a forward/backward pass is feasible for some Transformer parameters, and for two-layer attention networks. In this work, we present a conceptual framework where all parameters of a trained Transformer are interpreted by projecting them into the embedding space, that is, the space of vocabulary items they operate on. Focusing mostly on GPT-2 for this paper, we provide diverse evidence to support our argument. First, an empirical analysis showing that parameters of both pretrained and fine-tuned models can be interpreted in embedding space. Second, we present two applications of our framework: (a) aligning the parameters of different models that share a vocabulary, and (b) constructing a classifier without training by “translating” the parameters of a fine-tuned classifier to parameters of a different model that was only pretrained. Overall, our findings show that at least in part, we can abstract away model specifics and understand Transformers in the embedding space. 1 Introduction Figure 1: Applications of the embedding space view. Left: interpreting parameters in embedding space. The most active vocabulary items in a feed-forward key (k) and a feed-forward value (v). The most active pairs of vocabulary items in an attention query-key matrix WQKsubscriptQKW_QKWQK and an attention value-output matrix WVOsubscriptVOW_VOWVO (see §2). Center: Aligning the parameters of different BERT instances that share a vocabulary. Right: Zero-shot “stitching”, where representations of a fine-tuned classifier are translated through the embedding space (multiplying by EA⁢EB−1subscriptsuperscriptsubscript1E_AE_B^-1Eitalic_A Eitalic_B- 1) to a pretrained-only model. Transformer-based models [Vaswani et al., 2017] currently dominate Natural Language Processing [Devlin et al., 2018; Radford et al., 2019; Zhang et al., 2022] as well as many other fields of machine learning [Dosovitskiy et al., 2020; Chen et al., 2020; Baevski et al., 2020]. Consequently, understanding their inner workings has been a topic of great interest. Typically, work on interpreting Transformers relies on feeding inputs to the model and analyzing the resulting activations [Adi et al., 2016; Shi et al., 2016; Clark et al., 2019]. Thus, interpretation involves an expensive forward, and sometimes also a backward pass, over multiple inputs. Moreover, such interpretation methods are conditioned on the input and are not guaranteed to generalize to all inputs. In the evolving literature on static interpretation, i.e., without forward or backward passes, Geva et al. [2022b] showed that the value vectors of the Transformer feed-forward module (the second layer of the feed-forward network) can be interpreted by projecting them into the embedding space, i.e., multiplying them by the embedding matrix to obtain a representation over vocabulary items.111We refer to the unique items of the vocabulary as vocabulary items, and to the (possibly duplicate) elements of a tokenized input as tokens. When clear, we might use the term token for vocabulary item. Elhage et al. [2021] have shown that in a 2-layer attention network, weight matrices can be interpreted in the embedding space as well. Unfortunately, their innovative technique could not be extended any further. In this work, we extend and unify the theory and findings of Elhage et al. [2021] and Geva et al. [2022b]. We present a zero-pass, input-independent framework to understand the behavior of Transformers. Concretely, we interpret all weights of a pretrained language model (LM) in embedding space, including both keys and values of the feed-forward module (Geva et al. [2020, 2022b] considered just F values) as well as all attention parameters (Elhage et al. [2021] analyzed simplified architectures up to two layers of attention with no MLPs). Our framework relies on a simple observation. Since Geva et al. [2022b] have shown that one can project hidden states to the embedding space via the embedding matrix, we intuit this can be extended to other parts of the model by projecting to the embedding space and then projecting back by multiplying with a right-inverse of the embedding matrix. Thus, we can recast inner products in the model as inner products in embedding space. Viewing inner products this way, we can interpret such products as interactions between pairs of vocabulary items. This applies to (a) interactions between attention queries and keys as well as to (b) interactions between attention value vectors and the parameters that project them at the output of the attention module. Taking this perspective to the extreme, one can view Transformers as operating implicitly in the embedding space. This entails the existence of a single linear space that depends only on the tokenizer, in which parameters of different Transformers can be compared. Thus, one can use the embedding space to compare and transfer information across different models that share a tokenizer. We provide extensive empirical evidence for the validity of our framework, focusing mainly on GPT-2 medium [Radford et al., 2019]. We use GPT-2 for two reasons. First, we do this for concreteness, as this paper is mainly focused on introducing the new framework and not on analyzing its predictions. Second, and more crucially, unlike many other architectures (such as BERT [Devlin et al., 2018], RoBERTa [Liu et al., 2019], and T5 [Raffel et al., 2019]), the GPT family has a linear language modeling head (LM head) – which is simply the output embedding matrix. All the other architectures’ LM heads are two layer networks that contain non-linearities before the output embedding matrix. Our framework requires a linear language modeling head to work. That being said, we believe in practice this will not be a major obstacle, and we indeed see in the experiments that model alignment works well for BERT in spite of the theoretical difficulties. We leave the non-linearities in the LM head for future work. On the interpretation front (Fig. 1, Left), we provide qualitative and quantitative evidence that Transformer parameters can be interpreted in embedding space. We also show that when fine-tuning GPT-2 on a sentiment analysis task (over movie reviews), projecting changes in parameters into embedding space yields words that characterize sentiment towards movies. Second (Fig. 1, Center), we show that given two distinct instances of BERT pretrained from different random seeds [Sellam et al., 2022], we can align layers of the two instances by casting their weights into the embedding space. We find that indeed layer i of the first instance aligns well to layer i of the second instance, showing the different BERT instances converge to a semantically similar solution. Last (Fig. 1, Right), we take a model fine-tuned on a sentiment analysis task and “transfer” the learned weights to a different model that was only pretrained by going through the embedding spaces of the two models. We show that in 30% of the cases, this procedure, termed stitching, results in a classifier that reaches an impressive accuracy of 70% on the IMDB benchmark [Maas et al., 2011] without any training. Overall, our findings suggest that analyzing Transformers in embedding space is valuable both as an interpretability tool and as a way to relate different models that share a vocabulary and that it opens the door to interpretation methods that operate in embedding space only. Our code is available at https://github.com/guyd1995/embedding-space. 2 Background We now present the main components of the Transformer [Vaswani et al., 2017] relevant to our analysis. We discuss the residual stream view of Transformers, and recapitulate a view of the attention layer parameters as interaction matrices WVOsubscriptVOW_VOWVO and WQKsubscriptQKW_QKWQK [Elhage et al., 2021]. Similar to them, we exclude biases and layer normalization from our analysis. 2.1 Transformer Architecture The Transformer consists of a stack of layers, each including an attention module followed by a Feed-Forward (F) module. All inputs and outputs are sequences of N vectors of dimensionality d. Attention Module takes as input a sequence of representations X∈ℝN×dsuperscriptℝX ^N× dX ∈ blackboard_RN × d, and each layer L is parameterized by four matrices WQ(L),WK(L),WV(L),WO(L)∈ℝd×dsubscriptsuperscriptsubscriptsuperscriptsubscriptsuperscriptsubscriptsuperscriptsuperscriptℝW^(L)_Q,W^(L)_K,W^(L)_V,W^(L)_O ^d× dW( L )Q , W( L )K , W( L )V , W( L )O ∈ blackboard_Rd × d (we henceforth omit the layer superscript for brevity). The input X is projected to produce queries, keys, and values: Qatt=X⁢WQ,Katt=X⁢WK,Vatt=X⁢WVformulae-sequencesubscriptattsubscriptformulae-sequencesubscriptattsubscriptsubscriptattsubscriptQ_att=XW_Q,K_att=XW_K,V_att=XW_VQatt = X Witalic_Q , Katt = X Witalic_K , Vatt = X Witalic_V. Each one of Qatt,Katt,VattsubscriptattsubscriptattsubscriptattQ_att,K_att,V_attQatt , Katt , Vatt is split along the columns to H different heads of dimensionality ℝN×dHsuperscriptℝR^N× dHblackboard_RN × divide start_ARG d end_ARG start_ARG H end_ARG, denoted by Qatti,Katti,VattisuperscriptsubscriptattsuperscriptsubscriptattsuperscriptsubscriptattQ_att^i,K_att^i,V_att^iQattitalic_i , Kattitalic_i , Vattitalic_i respectively. We then compute H attention maps: Ai=softmax⁢(Qatti⁢Katti⁢Td/H+M)∈ℝN×N,superscriptsoftmaxsuperscriptsubscriptattsuperscriptsubscriptattTsuperscriptℝA^i= softmax ( Q_att^iK_att^i % T d/H+M ) ^N× N,Aitalic_i = softmax ( divide start_ARG Qattitalic_i Kattitalic_i T end_ARG start_ARG square-root start_ARG d / H end_ARG end_ARG + M ) ∈ blackboard_RN × N , where M∈ℝN×NsuperscriptℝM ^N× NM ∈ blackboard_RN × N is the attention mask. Each attention map is applied to the corresponding value head as Ai⁢VattisuperscriptsuperscriptsubscriptattA^iV_att^iAitalic_i Vattitalic_i, results are concatenated along columns and projected via WOsubscriptW_OWitalic_O. The input to the module is added via a residual connection, and thus the attention module’s output is: X+⁢[A1⁢Vatt1,…,Ai⁢Vatti,…,AH⁢VattH]⁢WO.superscript1superscriptsubscriptatt1…superscriptsuperscriptsubscriptatt…superscriptsuperscriptsubscriptattsubscript X+Concat [A^1V_att^1,…,A^iV_% att^i,…,A^HV_att^H ]W_O.X + Concat [ A1 Vatt1 , … , Aitalic_i Vattitalic_i , … , Aitalic_H Vattitalic_H ] Witalic_O . (1) F Module is a two-layer neural network, applied to each position independently. Following past terminology [Sukhbaatar et al., 2019; Geva et al., 2020], weights of the first layer are called F keys and weights of the second layer F values. This is an analogy to attention, as the F module too can be expressed as: f⁢(Q⁢KT)⁢VsuperscriptTf(QK T)Vf ( Q KT ) V, where f is the activation function, Q∈ℝN×dsuperscriptℝQ ^N× dQ ∈ blackboard_RN × d is the output of the attention module and the input to the F module, and K,V∈ℝd×dsuperscriptℝsubscriptK,V ^d_f× dK , V ∈ blackboard_Rdff × d are the weights of the first and second layers of the F module. Unlike attention, keys and values are learnable parameters. The output of the F module is added to the output of the attention module to form the output of the layer via a residual connection. The output of the i-th layer is called the i-th hidden state. Embedding Matrix To process sequences of discrete tokens, Transformers use an embedding matrix E∈ℝd×esuperscriptℝE ^d× eE ∈ blackboard_Rd × e that provides a d-dimensional representation to vocabulary items before entering the first Transformer layer. In different architectures, including GPT-2, the same embedding matrix E is often used [Press and Wolf, 2016] to take the output of the last Transformer layer and project it back to the vocabulary dimension, i.e., into the embedding space. In this work, we show how to interpret all the components of the Transformer model in the embedding space. 2.2 The Residual Stream We rely on a useful view of the Transformer through its residual connections popularized by Elhage et al. [2021].222Originally introduced in nostalgebraist [2020]. Specifically, each layer takes a hidden state as input and adds information to the hidden state through its residual connection. Under this view, the hidden state is a residual stream passed along the layers, from which information is read, and to which information is written at each layer. Elhage et al. [2021] and Geva et al. [2022b] observed that the residual stream is often barely updated in the last layers, and thus the final prediction is determined in early layers and the hidden state is mostly passed through the later layers. An exciting consequence of the residual stream view is that we can project hidden states in every layer into embedding space by multiplying the hidden state with the embedding matrix E, treating the hidden state as if it were the output of the last layer. Geva et al. [2022a] used this approach to interpret the prediction of Transformer-based language models, and we follow a similar approach. 2.3 WQKsubscriptQKW_QKWQK and WVOsubscriptVOW_VOWVO Following Elhage et al. [2021], we describe the attention module in terms of interaction matrices WQKsubscriptQKW_QKWQK and WVOsubscriptVOW_VOWVO which will be later used in our mathematical derivation. The computation of the attention module (§2.1) can be re-interpreted as follows. The attention projection matrices WQ,WK,WVsubscriptQsubscriptKsubscriptVW_Q,W_K,W_VWQ , WK , WV can be split along the column axis to H equal parts denoted by WQi,WKi,WVi∈ℝd×dHsuperscriptsubscriptQsuperscriptsubscriptKsuperscriptsubscriptVsuperscriptℝW_Q^i,W_K^i,W_V^i ^d× % dHWQitalic_i , WKitalic_i , WVitalic_i ∈ blackboard_Rd × divide start_ARG d end_ARG start_ARG H end_ARG for 1≤i≤H11≤ i≤ H1 ≤ i ≤ H. Similarly, the attention output matrix WOsubscriptOW_OWO can be split along the row axis into H heads, WOi∈ℝdH×dsuperscriptsubscriptOsuperscriptℝW_O^i dH× dWOitalic_i ∈ blackboard_Rdivide start_ARG d end_ARG start_ARG H end_ARG × d. We define the interaction matrices as WQKi:=WQi⁢WKi⁢T∈ℝd×d,assignsuperscriptsubscriptQKsuperscriptsubscriptQsuperscriptsubscriptKTsuperscriptℝ W_QK^i:=W_Q^iW_K^i T% ^d× d,WQKitalic_i := WQitalic_i WKitalic_i T ∈ blackboard_Rd × d , WVOi:=WVi⁢WOi∈ℝd×d.assignsuperscriptsubscriptVOsuperscriptsubscriptVsuperscriptsubscriptOsuperscriptℝ W_VO^i:=W_V^iW_O^i % ^d× d.WVOitalic_i := WVitalic_i WOitalic_i ∈ blackboard_Rd × d . Importantly, WQKi,WVOisuperscriptsubscriptQKsuperscriptsubscriptVOW_QK^i,W_VO^iWQKitalic_i , WVOitalic_i are input-independent. Intuitively, WQKsubscriptQKW_QKWQK encodes the amount of attention between pairs of tokens. Similarly, in WVOisuperscriptsubscriptVOW_VO^iWVOitalic_i, the matrices WVsubscriptVW_VWV and WOsubscriptOW_OWO can be viewed as a transition matrix that determines how attending to certain tokens affects the subsequent hidden state. We can restate the attention equations in terms of the interaction matrices. Recall (Eq. 1) that the output of the i’th head of the attention module is Ai⁢VattisuperscriptsuperscriptsubscriptattA^iV_att^iAitalic_i Vattitalic_i and the final output of the attention module is (without the residual connection): ⁢[A1⁢Vatt1,…,Ai⁢Vatti,…,AH⁢VattH]⁢WO=superscript1superscriptsubscriptatt1…superscriptsuperscriptsubscriptatt…superscriptsuperscriptsubscriptattsubscriptOabsent [A^1V_att^1,...,A^iV_% att^i,...,A^HV_att^H ]W_O=Concat [ A1 Vatt1 , … , Aitalic_i Vattitalic_i , … , Aitalic_H Vattitalic_H ] WO = (2) ∑i=1HAi⁢(X⁢WVi)⁢WOi=∑i=1HAi⁢X⁢WVOi.superscriptsubscript1superscriptsuperscriptsubscriptVsuperscriptsubscriptOsuperscriptsubscript1superscriptsuperscriptsubscriptVO _i=1^HA^i(XW_V^i)W_O^i= _i=1% ^HA^iXW_VO^i.∑i = 1H Aitalic_i ( X WVitalic_i ) WOitalic_i = ∑i = 1H Aitalic_i X WVOitalic_i . Similarly, the attention map AisuperscriptA^iAitalic_i at the i’th head in terms of WQKsubscriptQKW_QKWQK is (softmax is done row-wise): Ai=softmax⁢((X⁢WQi)⁢(X⁢WKi)Td/H+M)superscriptsoftmaxsuperscriptsubscriptQsuperscriptsuperscriptsubscriptKT A^i= softmax ( (XW_Q^i)(XW_% K^i) T d/H+M )Aitalic_i = softmax ( divide start_ARG ( X WQitalic_i ) ( X WKitalic_i )T end_ARG start_ARG square-root start_ARG d / H end_ARG end_ARG + M ) (3) =softmax⁢(X⁢(WQKi)⁢XTd/H+M).absentsoftmaxsuperscriptsubscriptQKsuperscriptT = softmax ( X(W_QK^i)X T% d/H+M ).= softmax ( divide start_ARG X ( WQKitalic_i ) XT end_ARG start_ARG square-root start_ARG d / H end_ARG end_ARG + M ) . 3 Parameter Projection In this section, we propose that Transformer parameters can be projected into embedding space for interpretation purposes. We empirically support our framework’s predictions in §4-§5. Given a matrix A∈ℝN×dsuperscriptℝA ^N× dA ∈ blackboard_RN × d, we can project it into embedding space by multiplying by the embedding matrix E as A^=A⁢E∈ℝN×e^superscriptℝ A=AE ^N× eover start_ARG A end_ARG = A E ∈ blackboard_RN × e. Let E′ be a right-inverse of E, that is, E⁢E′=I∈ℝd×dsuperscript′ℝE =I ^d× dE E′ = I ∈ blackboard_Rd × d.333E′ exists if d≤ed≤ ed ≤ e and E is full-rank. We can reconstruct the original matrix with E′ as A=A⁢(E⁢E′)=A^⁢E′^superscript′A=A(E )= AE A = A ( E E′ ) = over start_ARG A end_ARG E′. We will use this simple identity to reinterpret the model’s operation in embedding space. To simplify our analysis we ignore LayerNorm and biases. This has been justified in prior work [Elhage et al., 2021]. Briefly, LayerNorm can be ignored because normalization changes only magnitudes and not the direction of the update. At the end of this section, we discuss why in practice we choose to use E′=ETsuperscript′TE =E TE′ = ET instead of a seemingly more appropriate right inverse, such as the pseudo-inverse [Moore, 1920; Bjerhammar, 1951; Penrose, 1955]. In this section, we derive our framework and summarize its predictions in Table 1. Attention Module Recall that WVOi:=WVi⁢WOi∈ℝd×dassignsuperscriptsubscriptVOsuperscriptsubscriptVsuperscriptsubscriptOsuperscriptℝW_VO^i:=W_V^iW_O^i ^d× dWVOitalic_i := WVitalic_i WOitalic_i ∈ blackboard_Rd × d is the interaction matrix between attention values and the output projection matrix for attention head i. By definition, the output of each head is: Ai⁢X⁢WVOi=Ai⁢X^⁢E′⁢WVOisuperscriptsuperscriptsubscriptVOsuperscript^superscript′subscriptVOA^iXW_VO^i=A^i XE W_VO^iAitalic_i X WVOitalic_i = Aitalic_i over start_ARG X end_ARG E′ WVOitalic_i. Since the output of the attention module is added to the residual stream, we can assume according to the residual stream view that it is meaningful to project it to the embedding space, similar to F values. Thus, we expect the sequence of N e-dimensional vectors (Ai⁢X⁢WVOi)⁢E=Ai⁢X^⁢(E′⁢WVOi⁢E)superscriptsuperscriptsubscriptVOsuperscript^superscript′subscriptVO(A^iXW_VO^i)E=A^i X(E W_VO^iE)( Aitalic_i X WVOitalic_i ) E = Aitalic_i over start_ARG X end_ARG ( E′ WVOitalic_i E ) to be interpretable. Importantly, the role of AisuperscriptA^iAitalic_i is just to mix the representations of the updated N input vectors. This is similar to the F module, where F values (the parameters of the second layer) are projected into embedding space, and F keys (parameters of the first layer) determine the coefficients for mixing them. Hence, we can assume that the interpretable components are in the term X^⁢(E′⁢WVOi⁢E)^superscript′subscriptVO X(E W_VO^iE)over start_ARG X end_ARG ( E′ WVOitalic_i E ). Zooming in on this operation, we see that it takes the previous hidden state in the embedding space (X^ Xover start_ARG X end_ARG) and produces an output in the embedding space which will be incorporated into the next hidden state through the residual stream. Thus, E′⁢WVOi⁢Esuperscript′subscriptVOE W_VO^iE′ WVOitalic_i E is a transition matrix that takes a representation of the embedding space and outputs a new representation in the same space. Similarly, the matrix WQKisuperscriptsubscriptQKW_QK^iWQKitalic_i can be viewed as a bilinear map (Eq. 2.3). To interpret it in embedding space, we perform the following operation with E′: X⁢WQKi⁢XT=(X⁢E⁢E′)⁢WQKi⁢(X⁢E⁢E′)T=superscriptsubscriptQKsuperscriptTsuperscript′subscriptQKsuperscriptsuperscript′Tabsent XW_QK^iX T=(XEE )W_QK^i% (XEE ) T=X WQKitalic_i XT = ( X E E′ ) WQKitalic_i ( X E E′ )T = (X⁢E)⁢E′⁢WQKi⁢E′⁣T⁢(X⁢E)T=X^⁢(E′⁢WQKi⁢E′⁣T)⁢X^T.superscript′subscriptQKsuperscript′TsuperscriptT^superscript′subscriptQKsuperscript′Tsuperscript^T (XE)E W_QK^iE T(XE)^% T= X(E W_QK^iE T) X^% T.( X E ) E′ WQKitalic_i E′ T ( X E )T = over start_ARG X end_ARG ( E′ WQKitalic_i E′ T ) over start_ARG X end_ARGT . Therefore, the interaction between tokens at different positions is determined by an e×e× e × e matrix that expresses the interaction between pairs of vocabulary items. F Module Geva et al. [2022b] showed that F value vectors V∈ℝd×dsuperscriptℝsubscriptV ^d_f× dV ∈ blackboard_Rdff × d are meaningful when projected into embedding space, i.e., for a F value vector v∈ℝdsuperscriptℝv ^dv ∈ blackboard_Rd, v⁢E∈ℝesuperscriptℝvE ^ev E ∈ blackboard_Re is interpretable (see §2.1). In vectorized form, the rows of V⁢E∈ℝd×esuperscriptℝsubscriptVE ^d_f× eV E ∈ blackboard_Rdff × e are interpretable. On the other hand, the keys K of the F layer are multiplied on the left by the output of the attention module, which are the queries of the F layer. Denoting the output of the attention module by Q, we can write this product as Q⁢KT=Q^⁢E′⁢KT=Q^⁢(K⁢E′⁣T)TsuperscriptT^superscript′T^superscriptsuperscript′TTQK T= QE K T= Q(KE T% ) TQ KT = over start_ARG Q end_ARG E′ KT = over start_ARG Q end_ARG ( K E′ T )T. Because Q is a hidden state, we assume according to the residual stream view that Q^ Qover start_ARG Q end_ARG is interpretable in embedding space. When multiplying Q^ Qover start_ARG Q end_ARG by K⁢E′⁣Tsuperscript′TKE TK E′ T, we are capturing the interaction in embedding space between each query and key, and thus expect K⁢E′⁣Tsuperscript′TKE TK E′ T to be interpretable in embedding space as well. Overall, F keys and values are intimately connected – the i-th key controls the coefficient of the i-th value, so we expect their interpretation to be related. While not central to this work, we empirically show that key-value pairs in the F module are similar in embedding space in Appendix B.1. Symbol Projection Approximate Projection F values v v⁢EvEv E v⁢EvEv E F keys k k⁢E′⁣Tsuperscript′TkE Tk E′ T k⁢EkEk E Attention query-key WQKisuperscriptsubscriptQKW_QK^iWQKitalic_i E′⁢WQKi⁢E′⁣Tsuperscript′subscriptQKsuperscript′TE W_QK^iE TE′ WQKitalic_i E′ T ET⁢WQKi⁢EsuperscriptTsuperscriptsubscriptQKE TW_QK^iEET WQKitalic_i E Attention value-output WVOisuperscriptsubscriptVOW_VO^iWVOitalic_i E′⁢WVOi⁢Esuperscript′subscriptVOE W_VO^iE′ WVOitalic_i E ET⁢WVOi⁢EsuperscriptTsuperscriptsubscriptVOE TW_VO^iEET WVOitalic_i E Attention value subheads WVi,jsuperscriptsubscriptVW_V^i,jWVitalic_i , j WVi,j⁢E′⁣TsuperscriptsubscriptVsuperscript′TW_V^i,jE TWVitalic_i , j E′ T WVi,j⁢EsuperscriptsubscriptVW_V^i,jEWVitalic_i , j E Attention output subheads WOi,jsuperscriptsubscriptOW_O^i,jWOitalic_i , j WOi,j⁢EsuperscriptsubscriptOW_O^i,jEWOitalic_i , j E WOi,j⁢EsuperscriptsubscriptOW_O^i,jEWOitalic_i , j E Attention query subheads WQi,jsuperscriptsubscriptQW_Q^i,jWQitalic_i , j WQi,j⁢E′⁣TsuperscriptsubscriptQsuperscript′TW_Q^i,jE TWQitalic_i , j E′ T WQi,j⁢EsuperscriptsubscriptQW_Q^i,jEWQitalic_i , j E Attention key subheads WKi,jsuperscriptsubscriptKW_K^i,jWKitalic_i , j WKi,j⁢E′⁣TsuperscriptsubscriptKsuperscript′TW_K^i,jE TWKitalic_i , j E′ T WKi,j⁢EsuperscriptsubscriptKW_K^i,jEWKitalic_i , j E Table 1: A summary of our approach for projecting Transformer components into embedding space. The ‘Approximate Projection’ shows the projection we use in practice where E′=ETsuperscript′TE =E TE′ = ET. Subheads Another way to interpret the matrices WVOisuperscriptsubscriptVOW_VO^iWVOitalic_i and WQKisuperscriptsubscriptQKW_QK^iWQKitalic_i is through the subhead view. We use the following identity: A⁢B=∑j=1bA:,j⁢Bj,:superscriptsubscript1subscript:subscript:AB= _j=1^bA_:,jB_j,:A B = ∑j = 1b A: , j Bitalic_j , :, which holds for arbitrary matrices A∈ℝa×b,B∈ℝb×cformulae-sequencesuperscriptℝsuperscriptℝA ^a× b,B ^b× cA ∈ blackboard_Ra × b , B ∈ blackboard_Rb × c, where A:,j∈ℝa×1subscript:superscriptℝ1A_:,j ^a× 1A: , j ∈ blackboard_Ra × 1 are the columns of the matrix A and Bj,:∈ℝ1×csubscript:superscriptℝ1B_j,: ^1× cBitalic_j , : ∈ blackboard_R1 × c are the rows of the matrix B. Thus, we can decompose WVOisuperscriptsubscriptVOW_VO^iWVOitalic_i and WQKisuperscriptsubscriptQKW_QK^iWQKitalic_i into a sum of dH dHdivide start_ARG d end_ARG start_ARG H end_ARG rank-1 matrices: WVOi=∑j=1dHWVi,j⁢WOi,j,WQKi=∑j=1dHWQi,j⁢WKi,jT.formulae-sequencesuperscriptsubscriptVOsuperscriptsubscript1superscriptsubscriptVsuperscriptsubscriptOsuperscriptsubscriptQKsuperscriptsubscript1superscriptsubscriptQsuperscriptsuperscriptsubscriptKT W_VO^i= _j=1 dHW_V^i,jW_% O^i,j, W_QK^i= _j=1 dHW_Q^% i,jW_K^i,j T.WVOitalic_i = ∑j = 1divide start_ARG d end_ARG start_ARG H end_ARG WVitalic_i , j WOitalic_i , j , WQKitalic_i = ∑j = 1divide start_ARG d end_ARG start_ARG H end_ARG WQitalic_i , j WKitalic_i , jT . where WQi,j,WKi,j,WVi,j∈ℝd×1superscriptsubscriptQsuperscriptsubscriptKsuperscriptsubscriptVsuperscriptℝ1W_Q^i,j,W_K^i,j,W_V^i,j ^d× 1WQitalic_i , j , WKitalic_i , j , WVitalic_i , j ∈ blackboard_Rd × 1 are columns of WQi,WKi,WVisuperscriptsubscriptQsuperscriptsubscriptKsuperscriptsubscriptVW_Q^i,W_K^i,W_V^iWQitalic_i , WKitalic_i , WVitalic_i respectively, and WOi,j∈ℝ1×dsuperscriptsubscriptOsuperscriptℝ1W_O^i,j ^1× dWOitalic_i , j ∈ blackboard_R1 × d are the rows of WOisuperscriptsubscriptOW_O^iWOitalic_i. We call these vectors subheads. This view is useful since it allows us to interpret subheads directly by multiplying them with the embedding matrix E. Moreover, it shows a parallel between interaction matrices in the attention module and the F module. Just like the F module includes key-value pairs as described above, for a given head, its interaction matrices are a sum of interactions between pairs of subheads (indexed by j), which are likely to be related in embedding space. We show this is indeed empirically the case for pairs of subheads in Appendix B.1. Choosing E′=Esuperscriptnormal-′superscriptE =E TE′ = ET In practice, we do not use an exact right inverse (e.g. the pseudo-inverse). We use the transpose of the embedding matrix E′=ETsuperscript′TE =E TE′ = ET instead. The reason pseudo-inverse doesn’t work is that for interpretation we apply a top-k operation after projecting to embedding space (since it is impractical for humans to read through a sorted list of 50⁢K5050K50 K tokens). So, we only keep the list of the vocabulary items that have the k largest logits, for manageable values of k. In Appendix A, we explore the exact requirements for E′ to interact well with top-k. We show that the top k entries of a vector projected with the pseudo-inverse do not represent the entire vector well in embedding space. We define keep-k robust invertibility to quantify this. It turns out that empirically ETsuperscriptTE TET is a decent keep-k robust inverse for E in the case of GPT-2 medium (and similar models) for plausible values of k. We refer the reader to Appendix A for details. To give intuition as to why ETsuperscriptTE TET works in practice, we switch to a different perspective, useful in its own right. Consider the F keys for example – they are multiplied on the left by the hidden states. In this section, we suggested to re-cast this as hT⁢K=(hT⁢E)⁢(E′⁢K)superscriptℎsuperscriptℎsuperscript′h^TK=(h^TE)(E K)hitalic_T K = ( hitalic_T E ) ( E′ K ). Our justification was that the hidden state is interpretable in the embedding space. A related perspective (dominant in previous works too; e.g. Mickus et al. [2022]) is thinking of the hidden state as an aggregation of interpretable updates to the residual stream. That is, schematically, h=∑i=1kαi⁢riℎsuperscriptsubscript1subscriptsubscripth= _i=1^k _ir_ih = ∑i = 1k αitalic_i ritalic_i, where αisubscript _iαitalic_i are scalars and risubscriptr_iritalic_i are vectors corresponding to specific concepts in the embedding space (we roughly think of a concept as a list of tokens related to a single topic). Inner product is often used as a similarity metric between two vectors. If the similarity between a column KisubscriptK_iKitalic_i and hℎh is large, the corresponding i-th output coordinate will be large. Then we can think of K as a detector of concepts where each neuron (column in K) lights up if a certain concept is “present” (or a superposition of concepts) in the inner state. To understand which concepts each detector column encodes we see which tokens it responds to. Doing this for all (input) token embeddings and packaging the inner products into a vector of scores is equivalent to simply multiplying by ETsuperscriptTE TET on the left (where E is the input embedding in this case, but for GPT-2 they are the same). A similar argument can be made for the interaction matrices as well. For example for WVOsubscriptVOW_VOWVO, to understand if a token embedding eisubscripte_ieitalic_i maps to a ejsubscripte_jeitalic_j under a certain head, we apply the matrix to eisubscripte_ieitalic_i, getting eiT⁢WVOsuperscriptsubscriptsubscriptVOe_i^TW_VOeitalic_iitalic_T WVO and use the inner product as a similarity metric and get the score eiT⁢WVO⁢ejsuperscriptsubscriptsubscriptVOsubscripte_i^TW_VOe_jeitalic_iitalic_T WVO eitalic_j. 4 Interpretability Experiments Figure 2: Left: Average RksubscriptR_kRitalic_k score (k=100100k=100k = 100) across tokens per layer for activated parameter vectors against both the aligned hidden state h^^ℎ hover start_ARG h end_ARG at the output of the layer and a randomly sampled hidden state h^randsubscript^ℎrand h_randover start_ARG h end_ARGrand. Parameters are F keys (top-left), F values (top-right), attention values (bottom-left), and attention outputs (bottom-right). In this section, we provide empirical evidence for the viability of our approach as a tool for interpreting Transformer parameters. For our experiments, we use Huggingface Transformers (Wolf et al. [2020]; License: Apache-2.0). 4.1 Parameter Interpretation Examples Attention Module We take GPT-2 medium (345M parameters; Radford et al. [2019]) and manually analyze its parameters. GPT-2 medium has a total of 384 attention heads (24 layers and 16 heads per layer). We take the embedded transition matrices E′⁢WVOi⁢Esuperscript′subscriptVOE W_VO^iE′ WVOitalic_i E for all heads and examine the top-k pairs of vocabulary items. As there are only 384 heads, we manually choose a few heads and present the top-k pairs in Appendix C.1 (k=5050k=50k = 50). We observe that different heads capture different types of relations between pairs of vocabulary items including word parts, heads that focus on gender, geography, orthography, particular part-of-speech tags, and various semantic topics. In Appendix C.2 we perform a similar analysis for WQKsubscriptQKW_QKWQK. We supplement this analysis with a few examples from GPT-2 base and large (117M, 762M parameters – respectively) as proof of concept, similarly presenting interpretable patterns. A technical note: WVOsubscriptVOW_VOWVO operates on row vectors, which means it operates in a “transposed” way to standard intuition – which places inputs on the left side and outputs on the right side. It does not affect the theory, but when visualizing the top-k tuples, we take the transpose of the projection (E′⁢WVOi⁢E)Tsuperscriptsuperscript′subscriptVOT(E W_VO^iE) T( E′ WVOitalic_i E )T to get the “natural” format (input token, output token). Without the transpose, we would get the same tuples, but in the format (output token, input token). Equivalently, in the terminology of linear algebra, it can be seen as a linear transformation that we represent in the basis of row vectors and we transform to the basis of column vectors, which is the standard one. F Module Appendix C.3 provides examples of key-value pairs from the F modules of GPT-2 medium. We show random pairs (k,v)(k,v)( k , v ) from the set of those pairs such that when looking at the top-100 vocabulary items for k and v, at least 15% overlap. Such pairs account for approximately 5% of all key-value pairs. The examples show how key-value pairs often revolve around similar topics such as media, months, organs, etc. We again include additional examples from GPT-2 base and large. Knowledge Lookup Last, we show we can use embeddings to locate F values (or keys) related to a particular topic. We take a few vocabulary items related to a certain topic, e.g., [‘cm’, ‘kg’, ‘inches’], average their embeddings,444We subtract the average embedding μ from E before averaging, which improves interpretability. and rank all F values (or keys) based on their dot-product with the average. Appendix C.4 shows a few examples of F values found with this method that are related to programming, measurements, and animals. 4.2 Hidden State and Parameters One merit of zero-pass interpretation is that it does not require running inputs through the model. Feeding inputs might be expensive and non-exhaustive. In this section and in this section only, we run a forward pass over inputs and examine if the embedding space representations of dynamically computed hidden states are “similar” to the representations of the activated static parameter vectors. Due to the small number of examples we run over, the overall GPU usage is still negligible. A technical side note: we use GPT-2, which applies LayerNorm to the Transformer output before projecting it to the embedding space with E. Thus, conservatively, LayerNorm should be considered as part of the projection operation. Empirically, however, we observe that projecting parameters directly without LayerNorm works well, which simplifies our analysis in §3. Unlike parameters, we apply LayerNorm to hidden states before projection to embedding space to improve interpretability. This nuance was also present in the code of Geva et al. [2022a]. Experimental Design We use GPT-2 medium and run it over 60 examples from IMDB (25,000 train, 25,000 test examples; Maas et al. [2011]).555Note that IMDB was designed for sentiment analysis and we use it here as a general-purpose corpus. This provides us with a dynamically-computed hidden state hℎh for every token and at the output of every layer. For the projection h^∈ℝe^ℎsuperscriptℝ h ^eover start_ARG h end_ARG ∈ blackboard_Re of each such hidden state, we take the projections of the m most active parameter vectors x^ii=1msuperscriptsubscriptsubscript^1\ x_i\_i=1^m over start_ARG x end_ARGi i = 1m in the layer that computed hℎh and check if they cover the dominant vocabulary items of h^^ℎ hover start_ARG h end_ARG in embedding space. Specifically, let top-k⁢(w⁢E)top-k top-k(wE)top-k ( w E ) be the k vocabulary items with the largest logits in embedding space for a vector w∈ℝdsuperscriptℝw ^dw ∈ blackboard_Rd. We compute: Rk⁢(x^1,…,x^m,h^)=|top-k⁢(h^)∩⋃i=1mtop-k⁢(x^i)|k,subscriptsubscript^1…subscript^^ℎtop-k^ℎsuperscriptsubscript1top-ksubscript^R_k( x_1,..., x_m, h)= | top-k( h)∩% _i=1^m top-k( x_i)|k,Ritalic_k ( over start_ARG x end_ARG1 , … , over start_ARG x end_ARGm , over start_ARG h end_ARG ) = divide start_ARG | top-k ( over start_ARG h end_ARG ) ∩ ⋃i = 1m top-k ( over start_ARG x end_ARGi ) | end_ARG start_ARG k end_ARG , to capture if activated parameter vectors cover the main vocabulary items corresponding to the hidden state. We find the m most active parameter vectors separately for F keys (K), F values (V), attention value subheads (WVsubscriptVW_VWV) (see §3), and attention output subheads (WOsubscriptOW_OWO), where the activation of each parameter vector is determined by the vector’s “coefficient” as follows. For a F key-value pair (k,v)(k,v)( k , v ) the coefficient is σ⁢(qT⁢k)superscriptTσ(q Tk)σ ( qT k ), where q∈ℝdsuperscriptℝq ^dq ∈ blackboard_Rd is an input to the F module, and σ is the F non-linearity. For attention, value-output subhead pairs (v,o)(v,o)( v , o ) the coefficient is xT⁢vsuperscriptTx TvxT v, where x is the input to this component (for attention head i, the input is one of the rows of Ai⁢XsuperscriptA^iXAitalic_i X, see Eq. 2.3). Results and Discussion Figure 2 presents the RksubscriptR_kRitalic_k score averaged across tokens per layer. As a baseline, we compare RksubscriptR_kRitalic_k of the activated vectors x^ii=1msuperscriptsubscriptsubscript^1\ x_i\_i=1^m over start_ARG x end_ARGi i = 1m of the correctly-aligned hidden state h^^ℎ hover start_ARG h end_ARG at the output of the relevant layer (blue bars) against the RksubscriptR_kRitalic_k when randomly sampling h^randsubscript^ℎrand h_randover start_ARG h end_ARGrand from all the hidden states (orange bars). We conclude that representations in embedding space induced by activated parameter vector mirror, at least to some extent, the representations of the hidden states themselves. Appendix §B.2 shows a variant of this experiment, where we compare activated parameters throughout GPT-2 medium’s layers to the last hidden state, which produces the logits used for prediction. 4.3 Interpretation of Fine-tuned Models We now show that we can interpret the changes a model goes through during fine-tuning through the lens of embedding space. We fine-tune the top-3 layers of the 12-layer GPT-2 base (117M parameters) with a sequence classification head on IMDB sentiment analysis (binary classification) and compute the difference between the original parameters and the fine-tuned model. We then project the difference of parameter vectors into embedding space and test if the change is interpretable w.r.t. sentiment analysis. Appendix D shows examples of projected differences randomly sampled from the fine-tuned layers. Frequently, the difference or its negation is projected to nouns, adjectives, and adverbs that express sentiment for a movie, such as ‘amazing’, ‘masterpiece’, ‘incompetence’, etc. This shows that the differences are indeed projected into vocabulary items that characterize movie reviews’ sentiments. This behavior is present across WQ,WK,WV,KsubscriptQsubscriptKsubscriptVW_Q,W_K,W_V,KWQ , WK , WV , K, but not V and WOsubscriptOW_OWO, which curiously are the parameters added to the residual stream and not the ones that react to the input directly. 5 Aligning Models in Embedding Space The assumption Transformers operate in embedding space leads to an exciting possibility – we can relate different models to one another so long as they share the vocabulary and tokenizer. In §5.1, we show that we can align the layers of BERT models trained with different random seeds. In §5.2, we show the embedding space can be leveraged to “stitch” the parameters of a fine-tuned model to a model that was not fine-tuned. 5.1 Layer Alignment Figure 3: Left: Aligning in embedding space the layers of two different BERT models initialized from different random seeds for all parameter groups. Layers that have the same index tend to align with one another. Right: Alignment in feature space leads to unintelligible patterns. Experimental Design Taking our approach to the extreme, the embedding space is a universal space, which depends only on the tokenizer, in which Transformer parameters and hidden states reside. Thus, we can align parameter vectors from different models in this space and compare them even if they come from different models, as long as they share a vocabulary. To demonstrate this, we use MultiBERTs ([Sellam et al., 2022]; License: Apache-2.0), which contains 25 different instantiations of BERT-base (110M parameters) initialized from different random seeds.666Estimated compute costs: around 1728 TPU-hours for each pre-training run, and around 208 GPU-hours plus 8 TPU-hours for associated fine-tuning experiments. We take parameters from two MultiBERT seeds and compute the correlation between their projections to embedding space. For example, let VA,VBsubscriptsubscriptV_A,V_BVitalic_A , Vitalic_B be the F values of models A and B. We can project the values into embedding space: VA⁢EA,VB⁢EBsubscriptsubscriptsubscriptsubscriptV_AE_A,V_BE_BVitalic_A Eitalic_A , Vitalic_B Eitalic_B, where EA,EBsubscriptsubscriptE_A,E_BEitalic_A , Eitalic_B are the respective embedding matrices, and compute Pearson correlation between projected values. This produces a similarity matrix ~∈ℝ|VA|×|VB|~superscriptℝsubscriptsubscript S ^|V_A|×|V_B|over~ start_ARG S end_ARG ∈ blackboard_R| Vitalic_A | × | Vitalic_B |, where each entry is the correlation coefficient between projected values from the two models. We bin ~~ Sover~ start_ARG S end_ARG by layer pairs and average the absolute value of the scores in each bin (different models might encode the same information in different directions, so we use absolute value) to produce a matrix ∈ℝL×LsuperscriptℝS ^L× LS ∈ blackboard_RL × L, where L is the number of layers – that is, the average (absolute) correlation between vectors that come from layer ℓAsubscriptℓ _Aℓitalic_A in model A and layer ℓBsubscriptℓ _Bℓitalic_B in Model B is registered in entry (ℓA,ℓB)subscriptℓsubscriptℓ( _A, _B)( ℓitalic_A , ℓitalic_B ) of SS. Last, to obtain a one-to-one layer alignment, we use the Hungarian algorithm [Kuhn, 1955], which assigns exactly one layer from the first model to a layer from the second model. The algorithm’s objective is to maximize, given a similarity matrix SS, the sum of scores of the chosen pairs, such that each index in one model is matched with exactly one index in the other. We repeat this for all parameter groups (WQ,WK,WV,WO,KsubscriptQsubscriptKsubscriptVsubscriptOW_Q,W_K,W_V,W_O,KWQ , WK , WV , WO , K). Results and Discussion Figure 3 (left) shows the resulting alignment. Clearly, parameters from a certain layer in model A tend to align to the same layer in model B across all parameter groups. This suggests that different layers from different models that were trained separately (but with the same training objective and data) serve a similar function. As further evidence, we show that if not projected, the matching appears absolutely random in Figure §3 (right). We show the same results for other seed pairs as well in Appendix B.3. 5.2 Zero-shot Stitching Model stitching [Lenc and Vedaldi, 2015; Csiszárik et al., 2021; Bansal et al., 2021] is a relatively under-explored feature of neural networks, particularly in NLP. The idea is that different models, even with different architectures, can learn representations that can be aligned through a linear transformation, termed stitching. Representations correspond to hidden states, and thus one can learn a transformation matrix from one model’s hidden states to an equivalent hidden state in the other model. Here, we show that going through embedding space one can align the hidden states of two models, i.e., stitch, without training. Given two models, we want to find a linear stitching transformation to align their representation spaces. According to our theory, given a hidden state v∈ℝd1superscriptℝsubscript1v ^d_1v ∈ blackboard_Rd1 from model A, we can project it to the embedding space as v⁢EAsubscriptvE_Av Eitalic_A, where EAsubscriptE_AEitalic_A is its embedding matrix. Then, we can re-project to the feature space of model B, with EB+∈ℝe×d2superscriptsubscriptsuperscriptℝsubscript2E_B^+ ^e× d_2Eitalic_B+ ∈ blackboard_Re × d2, where EB+superscriptsubscriptE_B^+Eitalic_B+ is the Penrose-Moore pseudo-inverse of the embedding matrix EBsubscriptE_BEitalic_B.777Since we are not interested in interpretation we use an exact right-inverse and not the transpose. This transformation can be expressed as multiplication with the kernel KA⁢B:=EA⁢EB+∈ℝd1×d2assignsubscriptsubscriptsuperscriptsubscriptsuperscriptℝsubscript1subscript2K_AB:=E_AE_B^+ ^d_1× d_2Kitalic_A B := Eitalic_A Eitalic_B+ ∈ blackboard_Rd1 × d2. We employ the above approach to take representations of a fine-tuned classifier, A, and stitch them on top of a model B that was only pretrained, to obtain a new classifier based on B. Figure 4: Accuracy on the IMDB evaluation set. We ran stitching randomly 11 times and obtained 3 models with higher than random accuracy when stitching over top layers. Dashed red line indicates random performance. Experimental Design We use the 24-layer GPT-2 medium as model A and 12-layer GPT-2 base model trained in §4.3 as model B. We fine-tune the last three layers of model B on IMDB, as explained in §4.3. Stitching is simple and is performed as follows. Given the sequence of N hidden states HAℓ∈ℝN×d1superscriptsubscriptℓsuperscriptℝsubscript1H_A ^N× d_1Hitalic_Aroman_ℓ ∈ blackboard_RN × d1 at the output of layer ℓ ℓ of model A (ℓ ℓ is a hyperparameter), we apply the stitching layer, which multiplies the hidden states with the kernel, computing HAℓ⁢KA⁢BsuperscriptsubscriptℓsubscriptH_A K_ABHitalic_Aroman_ℓ Kitalic_A B. This results in hidden states HB∈ℝN×d2subscriptsuperscriptℝsubscript2H_B ^N× d_2Hitalic_B ∈ blackboard_RN × d2, used as input to the three fine-tuned layers from B. Results and Discussion Stitching produces models with accuracies that are higher than random on IMDB evaluation set, but not consistently. Figure 4 shows the accuracy of stitched models against the layer index from model A over which stitching is performed. Out of 11 random seeds, three models obtained accuracy that is significantly higher than the baseline 50% accuracy, reaching an accuracy of roughly 70%, when stitching is done over the top layers. 6 Related Work Interpreting Transformers is a broad area of research that has attracted much attention in recent years. A large body of work has focused on analyzing hidden representations, mostly through probing [Adi et al., 2016; Shi et al., 2016; Tenney et al., 2019; Rogers et al., 2020]. Voita et al. [2019a] used statistical tools to analyze the evolution of hidden representations throughout layers. Recently, Mickus et al. [2022] proposed to decompose the hidden representations into the contributions of different Transformer components. Unlike these works, we interpret parameters rather than the hidden representations. Another substantial effort has been to interpret specific network components. Previous work analyzed single neurons [Dalvi et al., 2018; Durrani et al., 2020], attention heads [Clark et al., 2019; Voita et al., 2019b], and feedforward values [Geva et al., 2020; Dai et al., 2021; Elhage et al., 2022]. While these works mostly rely on input-dependent neuron activations, we inspect “static” model parameters, and provide a comprehensive view of all Transformer components. Our work is most related to efforts to interpret specific groups of Transformer parameters. Cammarata et al. [2020] made observations about the interpretability of weights of neural networks. Elhage et al. [2021] analyzed 2-layer attention networks. We extend their analysis to multi-layer pre-trained Transformer models. Geva et al. [2020, 2022a, 2022b] interpreted feedforward values in embedding space. We coalesce these lines of work and offer a unified interpretation framework for Transformers in embedding space. 7 Discussion While our work has limitations (see §8), we think the benefits of our work overshadow its limitations. We provide a simple approach and a new set of tools to interpret Transformer models and compare them. The realm of input-independent interpretation methods is still nascent and it might provide a fresh perspective on the internals of the Transformer, one that allows to glance intrinsic properties of specific parameters, disentangling their dependence on the input. Moreover, many models are prohibitively large for practitioners to run. Our method requires only a fraction of the compute and memory requirements, and allows interpreting a single parameter in isolation. Importantly, our framework allows us to view parameters from different models as residents of a canonical embedding space, where they can be compared in model-agnostic fashion. This has interesting implications. We demonstrate two consequences of this observation (model alignment and stitching) and argue future work can yield many more use cases. 8 Limitations Our work has a few limitations that we care to highlight. First, it focuses on interpreting models through the vocabulary lens. While we have shown evidence for this, it does not preclude other factors from being involved. Second, we used E′=ETsuperscript′TE =E TE′ = ET, but future research may find variants of E that improve performance. Additionally, most of the work focused on GPT-2. This is due to shortcomings in the current state of our framework, as well as for clear presentation. We believe non-linearities in language modeling are resolvable, as is indicated in the experiment with BERT. In terms of potential bias in the framework, some parameters might consider terms related to each due to stereotypes learned from the corpus. References Adi et al. [2016] Y. Adi, E. Kermany, Y. Belinkov, O. 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Appendix A Rethinking Interpretation Figure 5: Each row represents a model in the following order from top to bottom: GPT-2 base, GPT-2 medium, GPT-2 large. Left: The keep-k inverse scores for three distributions: normal distribution, hidden states, and F values, for k∈10,50,100,200,300,5001050100200300500k∈\10,50,100,200,300,500\k ∈ 10 , 50 , 100 , 200 , 300 , 500 . Right: for k∈10,50,100,200,300,5001050100200300500k∈\10,50,100,200,300,500\k ∈ 10 , 50 , 100 , 200 , 300 , 500 . The process of interpreting a vector v in Geva et al. [2022b] proceeds in two steps: first the projection of the vector to the embedding space (v⁢EvEv E); then, we use the list of the tokens that were assigned the largest values in the projected vector, i.e.: top-k⁢(v⁢E)top-k top-k(vE)top-k ( v E ), as the interpretation of the projected vector. This is reasonable since (a) the most activated coordinates contribute the most when added to the residual stream, and (b) this matches how we eventually decode: we project to the embedding space and consider the top-1 token (or one of the few top tokens, when using beam search). In this work, we interpret inner products and matrix multiplications in the embedding space: given two vectors x,y∈ℝdsuperscriptℝx,y ^dx , y ∈ blackboard_Rd, their inner product xT⁢ysuperscriptTx TyxT y can be considered in the embedding space by multiplying with E and then by one of its right inverses (e.g., its pseudo-inverse E+superscriptE^+E+ [Moore, 1920, Bjerhammar, 1951, Penrose, 1955]): xT⁢y=xT⁢E⁢E+⁢y=(xT⁢E)⁢(E+⁢y)superscriptTsuperscriptTsuperscriptsuperscriptTsuperscriptx Ty=x TE^+y=(x TE)(E^+y)xT y = xT E E+ y = ( xT E ) ( E+ y ). Assume x⁢ExEx E is interpretable in the embedding space, crudely meaning that it represents logits over vocabulary items. We expect y, which interacts with x, to also be interpretable in the embedding space. Consequently, we would like to take E+⁢ysuperscriptE^+yE+ y to be the projection of y. However, this projection does not take into account the subsequent interpretation using top-k. The projected vector E+⁢ysuperscriptE^+yE+ y might be harder to interpret in terms of its most activated tokens. To alleviate this problem, we need a different “inverse” matrix E′ that works well when considering the top-k operation. Formally, we want an E′ with the following “robustness” guarantee: keep-k⁢(xT⁢E)⁢keep-k⁢(E′⁢y)≈xT⁢ykeep-ksuperscriptTkeep-ksuperscript′T keep-k(x TE) keep-k(E y)≈ x % Tykeep-k ( xT E ) keep-k ( E′ y ) ≈ xT y, where keep-k⁢(v)keep-k keep-k(v)keep-k ( v ) is equal to v for coordinates whose absolute value is in the top-k, and zero elsewhere. This is a stronger notion of inverse – not only is E⁢E′≈Isuperscript′E ≈ IE E′ ≈ I, but even when truncating the vector in the embedding space we can still reconstruct it with E′. We claim that ETsuperscriptTE TET is a decent instantiation of E′ and provide some empirical evidence. While a substantive line of work [Ethayarajh, 2019, Gao et al., 2019, Wang et al., 2020, Rudman et al., 2021] has shown that embedding matrices are not isotropic (an isotropic matrix E has to satisfy E⁢ET=α⁢IsuperscriptTEE T=α IE ET = α I for some scalar α), we show that it is isotropic enough to make ETsuperscriptTE TET a legitimate compromise. We randomly sample 300 vectors drawn from the normal distribution ⁢(0,1)01N(0,1)N ( 0 , 1 ), and compute for every pair x,yx,yx , y the cosine similarity between xT⁢ysuperscriptTx TyxT y and keep-k⁢(xT⁢E)⁢keep-k⁢(E′⁢y)keep-ksuperscriptTkeep-ksuperscript′ keep-k(x TE) keep-k(E y)keep-k ( xT E ) keep-k ( E′ y ) for k=10001000k=1000k = 1000, and then average over all pairs. We repeat this for E′∈E+,ETsuperscript′superscriptTE ∈\E^+,E T\E′ ∈ E+ , ET and obtain a score of 0.100.100.100.10 for E+superscriptE^+E+, and 0.830.830.830.83 for ETsuperscriptTE TET, showing the ETsuperscriptTE TET is better under when using top-k. More globally, we compare E′∈E+,ETsuperscript′superscriptTE ∈\E^+,E T\E′ ∈ E+ , ET for k∈10,50,100,200,300,5001050100200300500k∈\10,50,100,200,300,500\k ∈ 10 , 50 , 100 , 200 , 300 , 500 with three distributions: - x,yx,yx , y drawn from the normal ⁢(0,1)01N(0,1)N ( 0 , 1 ) distribution - x,yx,yx , y chosen randomly from the F values - x,yx,yx , y drawn from hidden states along Transformer computations. In Figure 5 we show the results, where dashed lines represent E+superscriptE^+E+ and solid lines represent ETsuperscriptTE TET. The middle row shows the plots for GPT-2 medium, which is the main concern of this paper. For small values of k (which are more appropriate for interpretation), ETsuperscriptTE TET is superior to E+superscriptE^+E+ across all distributions. Interestingly, the hidden state distribution is the only distribution where E+superscriptE^+E+ has similar performance to ETsuperscriptTE TET. Curiously, when looking at higher values of k the trend is reversed (k=512,1024,2048,4096,10000,15000,20000,3000051210242048409610000150002000030000k=\512,1024,2048,4096,10000,15000,20000,30000\k = 512 , 1024 , 2048 , 4096 , 10000 , 15000 , 20000 , 30000 ) - see Figure 5 (Right). This settles the deviation from findings showing embedding matrices are not isotropic, as we see that indeed as k grows, ETsuperscriptTE TET becomes an increasingly bad approximate right-inverse of the embedding matrix. The only distribution that keeps high performance with ETsuperscriptTE TET is the hidden state distribution, which is an interesting direction for future investigation. For completeness, we provide the same analysis for GPT-2 base and large in Figure 5. We can see that GPT-2 base gives similar conclusions. GPT-2 large, however, seems to show a violent zigzag movement for E+superscriptE^+E+ but for most values it seems to be superior to ETsuperscriptTE TET. It is however probably best to use ETsuperscriptTE TET since it is more predictable. This zigzag behavior is very counter-intuitive and we leave it for future work to decipher. Appendix B Additional Material B.1 Corresponding Parameter Pairs are Related Figure 6: Average Simk⁢(x^,y^)subscriptSim^ Sim_k( x, y)Simk ( over start_ARG x end_ARG , over start_ARG y end_ARG ) for k=100100k=100k = 100 by layer, where blue is when matching pairs are aligned, and orange is when pairs are shuffled within the layer. Top Left: F keys and F values. Top Right: The subheads of WOsubscriptW_OWitalic_O and WVsubscriptW_VWitalic_V. Bottom: The subheads of WQsubscriptW_QWitalic_Q and WKsubscriptW_KWitalic_K. We define the following metric applying on vectors after projecting them into the embedding space: Simk⁢(x^,y^)=|top-k⁢(x^)∩top-k⁢(y^)||top-k⁢(x^)∪top-k⁢(y^)|subscriptSim^^top-k^top-k^top-k^top-k Sim_k( x, y)= | top-k( x)∩ % top-k( y)|| top-k( x)∪ top-k( y)|Simk ( over start_ARG x end_ARG , over start_ARG y end_ARG ) = divide start_ARG | top-k ( over start_ARG x end_ARG ) ∩ top-k ( over start_ARG y end_ARG ) | end_ARG start_ARG | top-k ( over start_ARG x end_ARG ) ∪ top-k ( over start_ARG y end_ARG ) | end_ARG where top-k⁢(v)top-k top-k(v)top-k ( v ) is the set of k top activated indices in the vector v (which correspond to tokens in the embedding space). This metric is the Jaccard index [Jaccard, 1912] applied to the top-k tokens from each vector. In Figure 6, Left, we demonstrate that F key vectors and their corresponding value vectors are more similar (in embedding space) than two random key and value vectors. In Figure 6, Right, we show a similar result for attention value and output vectors. In Figure 6, Bottom, the same analysis is done for attention query and key vectors. This shows that there is a much higher-than-chance relation between corresponding F keys and values (and the same for attention values and outputs). B.2 Final Prediction and Parameters We show that the final prediction of the model is correlated in embedding space with the most activated parameters from each layer. This implies that these objects are germane to the analysis of the final prediction in the embedding space, which in turn suggests that the embedding space is a viable choice for interpreting these vectors. Figure 7 shows that just like §4.2, correspondence is better when hidden states are not randomized, suggesting their parameter interpretations have an impact on the final prediction. Figure 7: Left: Average RksubscriptR_kRitalic_k score (k=100100k=100k = 100) across tokens per layer for activated parameter vectors against both the aligned hidden state h^^ℎ hover start_ARG h end_ARG at the output of the final layer and a randomly sampled hidden state h^randsubscript^ℎrand h_randover start_ARG h end_ARGrand. Parameters are F keys (top-left), F values (top-right), attention values (bottom-left), and attention outputs (bottom-right). B.3 Parameter Alignment Plots for Additional Model Pairs Alignment in embedding space of layers of pairs of BERT models trained with different random seeds for additional model pairs. Seed 1 VS Seed 2 Seed 2 VS Seed 3 Seed 3 VS Seed 4 Seed 4 VS Seed 5 Appendix C Example Cases C.1 WVOsubscriptVOW_VOWVO Matrices Below we show output-value pairs from different heads of GPT-2 medium. For each head, we show the 50 pairs with the largest values in the e×e× e × e transition matrix. There are 384 attention heads in GPT-2 medium from which we manually choose a subset. Throughout the section some lists are marked with asterisks indicating the way this particular list was created: * - pairs of the form (x,x)(x,x)( x , x ) were excluded from the list ** - pairs where both items are present in the corpus (we use IMDB training set). Along with GPT-2 medium, we also provide a few examples from GPT-2 base and GPT-2 large. C.1.1 Low-Level Language Modeling GPT-2 Medium - Layer 21 Head 7* ⬇ (’NF’, ’FN’), (’Ram’, ’ Ramos’), (’Hug’, ’ Hughes’), (’gran’, ’GR’), (’FN’, ’NF’), (’CLA’, ’CL’), (’McC’, ’ McCain’), (’Marsh’, ’ Marshall’), (’ Hughes’, ’Hug’), (’Tan’, ’ Tanner’), (’nih’, ’NH’), (’NRS’, ’NR’), (’ Bowman’, ’Bow’), (’ Marshall’, ’Marsh’), (’Jac’, ’ Jacobs’), (’Hay’, ’ Hayes’), (’ Hayes’, ’Hay’), (’McC’, ’ McCorm’), (’NI’, ’NR’), (’ sidx’, ’ Dawson’), (’ Tanner’, ’Tan’), (’gra’, ’GR’), (’JA’, ’jac’), (’zos’, ’zo’), (’NI’, ’NF’), (’McC’, ’ McCull’), (’ Jacobs’, ’Jac’), (’ Beetle’, ’ Beet’), (’GF’, ’FG’), (’jas’, ’ja’), (’Wil’, ’ Wilkinson’), (’ Ramos’, ’Ram’), (’GRE’, ’GR’), (’ NF’, ’FN’), (’ McCorm’, ’McC’), (’Scar’, ’ Scarborough’), (’ Baal’, ’Ba’), (’FP’, ’FG’), (’FH’, ’FN’), (’ Garfield’, ’Gar’), (’jas’, ’jac’), (’nuts’, ’nut’), (’WI’, ’ Wis’), (’ Vaughn’, ’ Vaughan’), (’FP’, ’PF’), (’RNA’, ’RN’), (’ Jacobs’, ’jac’), (’FM’, ’FN’), (’ Knox’, ’Kn’), (’NI’, ’nic’) lstlisting GPT-2 Medium - Layer 19 Head 13 (first letter/consonant of the word and last token of the word) lstlisting[backgroundcolor= white] (’ R’, ’senal’), # arsenal (’senal’, ’R’), (’ G’, ’vernment’), # government (’ Madness’, ’ M’), (’ M’, ’ Mayhem’), (’ W’, ’nesday’), # wednesday (’vernment’, ’G’), (’M’, ’ Madness’), (’ N’, ’lace’), # necklace (’nesday’, ’W’), (’Rs’, ’senal’), (’ g’, ’vernment’), (’ N’, ’farious’), # nefarious (’eneg’, ’ C’), (’ r’, ’senal’), (’ F’, ’ruary’), # february (’senal’, ’RIC’), (’ R’, ’ondo’), (’ N’, ’ Mandela’), # nelson (’ Mayhem’, ’M’), (’ RD’, ’senal’), (’ C’, ’estine’), (’Gs’, ’vernment’), (’RF’, ’senal’), (’ N’, ’esis’), (’ N’, ’Reviewed’), (’ C’, ’arette’), # cigarette (’rome’, ’ N’), (’ N’, ’theless’), # nonetheless (’lace’, ’N’), (’ H’, ’DEN’), (’ V’, ’ versa’), (’ P’, ’bably’), # probably (’vernment’, ’GF’), (’g’, ’vernment’), (’GP’, ’vernment’), (’ C’, ’ornia’), # california (’ilipp’, ’ F’), (’ N’, ’umbered’), (’ C’, ’arettes’), (’RS’, ’senal’), (’ N’, ’onsense’), (’RD’, ’senal’), (’RAL’, ’senal’), (’ F’, ’uci’), (’R’, ’ondo’), (’ RI’, ’senal’), (’ H’, ’iday’), # holiday (’senal’, ’ Rx’), (’ F’, ’odor’) GPT-2 Medium - Layer 20 Head 9 ⬇ (’On’, ’ behalf’), (’ On’, ’ behalf’), (’ on’, ’ behalf’), (’during’, ’ periods’), (’within’, ’ bounds’), (’ inside’, ’ envelope’), (’outside’, ’door’), (’inside’, ’ envelope’), (’ Under’, ’ regime’), (’ during’, ’ periods’), (’ LIKE’, ’lihood’), (’ on’, ’ occasions’), (’Under’, ’ regime’), (’inside’, ’door’), (’during’, ’period’), (’Like’, ’lihood’), (’ During’, ’ periods’), (’Inside’, ’ envelope’), (’for’, ’ sake’), (’ inside’, ’ doors’), (’ under’, ’ regime’), (’ ON’, ’ behalf’), (’for’, ’ purposes’), (’On’, ’ occasions’), (’inside’, ’ doors’), (’ on’, ’ basis’), (’ Under’, ’ regimes’), (’outside’, ’doors’), (’inside’, ’ Osc’), (’During’, ’ periods’), (’ inside’, ’door’), (’ UNDER’, ’ regime’), (’ under’, ’ regimes’), (’Under’, ’ regimes’), (’inside’, ’doors’), (’inside’, ’zx’), (’during’, ’ period’), (’inside’, ’ascript’), (’Inside’, ’door’), (’ On’, ’ occasions’), (’BuyableInstoreAndOnline’, ’ysc’), (’ Inside’, ’ envelope’), (’during’, ’ pauses’), (’under’, ’ regime’), (’ on’, ’ occasion’), (’outside’, ’ doors’), (’ UNDER’, ’ banner’), (’within’, ’ envelope’), (’ here’, ’abouts’), (’during’, ’ duration’) GPT-2 Base - Layer 10 Head 11** ⬇ (’ sources’, ’ources’) (’ repertoire’, ’ reperto’) (’ tales’, ’ stories’) (’ stories’, ’ tales’) (’ journals’, ’ magazines’) (’stories’, ’ tales’) (’ journal’, ’ journals’) (’ magazines’, ’Magazine’) (’ magazines’, ’ newspapers’) (’ reperto’, ’ repertoire’) (’ cameras’, ’ Camer’) (’ source’, ’ sources’) (’ newspapers’, ’ magazines’) (’ position’, ’ positions’) (’ tale’, ’ tales’) (’ positions’, ’ position’) (’ obstacles’, ’ hurdles’) (’ chores’, ’ tasks’) (’ journals’, ’ papers’) (’ role’, ’ roles’) (’ hurdles’, ’ obstacles’) (’ journals’, ’ journal’) (’ windows’, ’ doors’) (’ ceiling’, ’ ceilings’) (’ loophole’, ’ loopholes’) (’ Sources’, ’ources’) (’source’, ’ sources’) (’ documentaries’, ’ films’) (’ microphone’, ’ microphones’) (’ cameras’, ’ camera’) (’Journal’, ’ journals’) (’ restrooms’, ’ bathrooms’) (’ tasks’, ’ chores’) (’ perspectives’, ’ viewpoints’) (’ shelf’, ’ shelves’) (’ rooms’, ’ bedrooms’) (’ hurdle’, ’ hurdles’) (’ barriers’, ’ fences’) (’ magazines’, ’ journals’) (’ journals’, ’Magazine’) (’ sources’, ’ source’) (’ manuals’, ’ textbooks’) (’ story’, ’ stories’) (’ labs’, ’ laboratories’) (’ tales’, ’ Stories’) (’ chores’, ’ duties’) (’ roles’, ’ role’) (’ ceilings’, ’ walls’) (’ microphones’, ’ microphone’) (’ pathway’, ’ pathways’) GPT-2 Large - Layer 27 Head 6 ⬇ (’ where’, ’upon’), (’where’, ’upon’), (’with’, ’ regard’), (’with’, ’ regards’), (’ with’, ’ regards’), (’ Where’, ’upon’), (’ Like’, ’lihood’), (’of’, ’ course’), (’ with’, ’ regard’), (’ LIKE’, ’lihood’), (’Where’, ’upon’), (’from’, ’ afar’), (’with’, ’stood’), (’ FROM’, ’ afar’), (’ like’, ’lihood’), (’ WHERE’, ’upon’), (’Like’, ’lihood’), (’ with’, ’stood’), (’ of’, ’ course’), (’of’, ’course’), (’Of’, ’ course’), (’ from’, ’ afar’), (’ WITH’, ’ regard’), (’ where’, ’abouts’), (’with’, ’ impunity’), (’ WITH’, ’ regards’), (’With’, ’stood’), (’for’, ’ purposes’), (’with’, ’ respect’), (’ With’, ’stood’), (’like’, ’lihood’), (’ Of’, ’ course’), (’With’, ’ regard’), (’ With’, ’ regard’), (’where’, ’abouts’), (’ WITH’, ’stood’), (’With’, ’ regards’), (’ OF’, ’ course’), (’ From’, ’ afar’), (’ with’, ’ impunity’), (’ With’, ’ regards’), (’ with’, ’ respect’), (’From’, ’ afar’), (’with’, ’standing’), (’ on’, ’ behalf’), (’ by’, ’products’), (’ for’, ’ purposes’), (’ or’, ’acle’), (’for’, ’ sake’), (’ with’, ’standing’) C.1.2 Gender GPT-2 Medium - Layer 18 Head 1 ⬇ (’women’, ’ Marie’), (’ actresses’, ’ Marie’), (’women’, ’ Anne’), (’Women’, ’ Anne’), (’woman’, ’ Marie’), (’Women’, ’ Marie’), (’woman’, ’ Anne’), (’Woman’, ’ Marie’), (’ actresses’, ’ Anne’), (’ heroine’, ’ Marie’), (’Women’, ’Jane’), (’ heroine’, ’ Anne’), (’women’, ’Jane’), (’Women’, ’ actresses’), (’Woman’, ’ Anne’), (’Women’, ’ Esther’), (’women’, ’ Esther’), (’girls’, ’ Marie’), (’Mrs’, ’ Anne’), (’ actress’, ’ Marie’), (’women’, ’ actresses’), (’Woman’, ’Jane’), (’ girls’, ’ Marie’), (’ actresses’, ’Jane’), (’Woman’, ’Anne’), (’Girls’, ’ Marie’), (’women’, ’Anne’), (’Girls’, ’ Anne’), (’Woman’, ’ actresses’), (’ Women’, ’ Marie’), (’ Women’, ’ Anne’), (’ girls’, ’ Anne’), (’girl’, ’ Anne’), (’Women’, ’Anne’), (’Woman’, ’Women’), (’girls’, ’ Anne’), (’ actresses’, ’Anne’), (’women’, ’ Michelle’), (’ Actress’, ’ Marie’), (’girl’, ’ Marie’), (’ Feminist’, ’ Anne’), (’ women’, ’ Marie’), (’Women’, ’ Devi’), (’Women’, ’ Elizabeth’), (’ actress’, ’ Anne’), (’Mrs’, ’Anne’), (’answered’, ’Answer’), (’woman’, ’Anne’), (’Woman’, ’maid’), (’women’, ’Marie’) GPT-2 Large - Layer 27 Head 12 ⬇ (’ herself’, ’ Marie’), (’ hers’, ’ Marie’), (’she’, ’ Marie’), (’ she’, ’ Marie’), (’ her’, ’ Marie’), (’She’, ’ Marie’), (’ hers’, ’Maria’), (’ actresses’, ’ actresses’), (’ herself’, ’Maria’), (’ her’, ’Maria’), (’ herself’, ’ Anne’), (’She’, ’Maria’), (’ hers’, ’ Louise’), (’ herself’, ’ Louise’), (’ hers’, ’ Anne’), (’ hers’, ’pher’), (’she’, ’Maria’), (’ actress’, ’ actresses’), (’ herself’, ’ Isabel’), (’ herself’, ’pher’), (’ she’, ’Maria’), (’ SHE’, ’ Marie’), (’ herself’, ’ Gloria’), (’ herself’, ’ Amanda’), (’ Ivanka’, ’ Ivanka’), (’ her’, ’ Louise’), (’ herself’, ’ Kate’), (’ her’, ’pher’), (’ her’, ’ Anne’), (’ she’, ’pher’), (’she’, ’ Louise’), (’ herself’, ’Kate’), (’ she’, ’ Louise’), (’ she’, ’ Anne’), (’ She’, ’ Marie’), (’she’, ’ Gloria’), (’She’, ’ Louise’), (’ hers’, ’ Gloria’), (’ herself’, ’ Diana’), (’She’, ’ Gloria’), (’she’, ’ Anne’), (’she’, ’pher’), (’Her’, ’ Marie’), (’ she’, ’ Gloria’), (’ Paleo’, ’ Paleo’), (’ hers’, ’ Diana’) GPT-2 Base - Layer 9 Head 7** ⬇ (’ her’, ’ herself’) (’She’, ’ herself’) (’ she’, ’ herself’) (’she’, ’ herself’) (’Her’, ’ herself’) (’ She’, ’ herself’) (’ SHE’, ’ herself’) (’their’, ’ themselves’) (’ hers’, ’ herself’) (’Their’, ’ themselves’) (’ Her’, ’ herself’) (’ Their’, ’ themselves’) (’ THEIR’, ’ themselves’) (’ HER’, ’ herself’) (’ their’, ’ themselves’) (’They’, ’ themselves’) (’His’, ’ himself’) (’ herself’, ’erest’) (’they’, ’ themselves’) (’his’, ’ himself’) (’Their’, ’selves’) (’ They’, ’ themselves’) (’ herself’, ’ Louise’) (’their’, ’selves’) (’her’, ’ herself’) (’ his’, ’ himself’) (’ herself’, ’ Marie’) (’He’, ’ himself’) (’She’, ’ Louise’) (’ they’, ’ themselves’) (’their’, ’chairs’) (’ herself’, ’ dow’) (’ herself’, ’eva’) (’ THEY’, ’ themselves’) (’ herself’, ’ Mae’) (’ His’, ’ himself’) (’clinton’, ’enegger’) (’She’, ’erest’) (’ her’, ’ Louise’) (’ herself’, ’ Devi’) (’ Their’, ’selves’) (’Their’, ’chairs’) (’ Himself’, ’enegger’) (’ she’, ’ Louise’) (’ herself’, ’ Anne’) (’Its’, ’ itself’) (’ her’, ’erest’) (’ herself’, ’ Christina’) (’she’, ’erest’) (’their’, ’ selves’) C.1.3 Geography GPT-2 Base - Layer 11 Head 2** ⬇ (’ Halifax’, ’ Scotia’) (’Saudi’, ’ Arabia’) (’ Nova’, ’ Scotia’) (’ Tamil’, ’ Nadu’) (’ Finnish’, ’onen’) (’ Saudi’, ’ Arabia’) (’Pitt’, ’sburgh’) (’Dutch’, ’ijk’) (’ Schwartz’, ’enegger’) (’ Afghans’, ’ Kabul’) (’ Icelandic’, ’sson’) (’ Finland’, ’onen’) (’Pitt’, ’enegger’) (’ Czech’, ’oslov’) (’ Manitoba’, ’ Winnipeg’) (’ Malaysian’, ’ Lumpur’) (’ Swedish’, ’borg’) (’ Saskatchewan’, ’ Sask’) (’ Chennai’, ’ Nadu’) (’ Argentine’, ’ Aires’) (’ Iceland’, ’ Icelandic’) (’ Swedish’, ’sson’) (’ Tasman’, ’ Nadu’) (’Houston’, ’ Astros’) (’Colorado’, ’ Springs’) (’ Kuala’, ’ Lumpur’) (’Tai’, ’pport’) (’Houston’, ’ Dynamo’) (’ Manitoba’, ’Marginal’) (’ Afghan’, ’ Kabul’) (’ Buenos’, ’ Aires’) (’ Alberta’, ’ Calgary’) (’ Stockholm’, ’sson’) (’ Sweden’, ’borg’) (’Brazil’, ’ Paulo’) (’ Iceland’, ’sson’) (’ Winnipeg’, ’ Manitoba’) (’ Sweden’, ’sson’) (’ Carolina’, ’ Hurricanes’) (’ Dutch’, ’ijk’) (’ Swed’, ’borg’) (’ Aki’, ’pport’) (’ Winnipeg’, ’Marginal’) (’ Argentine’, ’ pes’) (’ Halifax’, ’imore’) (’ Brisbane’, ’enegger’) (’ Melbourne’, ’ Nadu’) (’ Adelaide’, ’ Nadu’) (’ Cambod’, ’ Nguyen’) (’ Vietnamese’, ’ Nguyen’) GPT-2 Medium - Layer 16 Head 6* ⬇ (’ Chennai’, ’ Mumbai’), (’India’, ’ Mumbai’), (’ Mumbai’, ’ Chennai’), (’ Queensland’, ’ Tasmania’), (’India’, ’ Rahul’), (’India’, ’ Gujar’), (’ Chennai’, ’ Bangalore’), (’England’, ’Scotland’), (’ Chennai’, ’ Kerala’), (’ Delhi’, ’ Mumbai’), (’Britain’, ’Scotland’), (’ Bangalore’, ’ Mumbai’), (’Pakistan’, ’India’), (’Scotland’, ’Ireland’), (’ Mumbai’, ’ Bangalore’), (’ Bangalore’, ’ Chennai’), (’ Aadhaar’, ’ Gujar’), (’ Mumbai’, ’ Maharashtra’), (’ Maharashtra’, ’ Gujarat’), (’ Gujarat’, ’ Gujar’), (’Australian’, ’Australia’), (’India’, ’ Gujarat’), (’ Rahul’, ’ Gujar’), (’ Maharashtra’, ’ Mumbai’), (’Britain’, ’England’), (’India’, ’ Chennai’), (’ Mumbai’, ’ Bombay’), (’ Tamil’, ’ Kerala’), (’ Hindi’, ’ Mumbai’), (’ Tasmania’, ’ Tasman’), (’ Mumbai’, ’India’), (’ Hindi’, ’ Gujar’), (’ Maharashtra’, ’ Gujar’), (’ Australians’, ’Austral’), (’ Maharashtra’, ’ Kerala’), (’India’, ’ Bangalore’), (’India’, ’ Kerala’), (’India’, ’ Bombay’), (’Australia’, ’Austral’), (’ Aadhaar’, ’India’), (’ Sharma’, ’ Mumbai’), (’Australian’, ’Austral’), (’ Mumbai’, ’ Kerala’), (’Scotland’, ’England’), (’ Mumbai’, ’ Gujar’), (’ Rahul’, ’ Mumbai’), (’ Queensland’, ’ Tasman’), (’ Tamil’, ’ Chennai’), (’ Gujarat’, ’ Maharashtra’), (’India’, ’ Modi’) GPT-2 Medium - Layer 16 Head 2* ⬇ (’Austral’, ’ Australians’), (’Australia’, ’Austral’), (’ Canberra’, ’Austral’), (’Austral’, ’ Canberra’), (’ Winnipeg’, ’ Edmonton’), (’Australian’, ’Austral’), (’ Alberta’, ’ Edmonton’), (’Australia’, ’ Australians’), (’ Australians’, ’Austral’), (’Ukraine’, ’ovych’), (’ Quebec’, ’ Canad’), (’Australian’, ’ Australians’), (’ Winnipeg’, ’ Manitoba’), (’ Manitoba’, ’ Winnipeg’), (’Canadian’, ’Canada’), (’Moscow’, ’ Bulgar’), (’ Manitoba’, ’ Edmonton’), (’berra’, ’Austral’), (’Austral’, ’Australian’), (’ Ukrainians’, ’ovych’), (’Canada’, ’ Canadians’), (’ Canberra’, ’ Australians’), (’Canada’, ’Canadian’), (’ Yanukovych’, ’ovych’), (’Canada’, ’ Trudeau’), (’ Dmitry’, ’ Bulgar’), (’ Australia’, ’Austral’), (’ Mulcair’, ’ Canad’), (’berra’, ’ Canberra’), (’Turkish’, ’oglu’), (’udeau’, ’Canada’), (’ Edmonton’, ’ Oilers’), (’Australia’, ’ Canberra’), (’Canada’, ’ Edmonton’), (’ Edmonton’, ’ Calgary’), (’ Alberta’, ’ Calgary’), (’udeau’, ’ Trudeau’), (’ Calgary’, ’ Edmonton’), (’Canadian’, ’ Trudeau’), (’Australian’, ’ Canberra’), (’ Vancouver’, ’ Canucks’), (’Australia’, ’Australian’), (’ Vancouver’, ’ Fraser’), (’Canadian’, ’ Edmonton’), (’Austral’, ’elaide’), (’Tex’, ’ Braz’), (’Canada’, ’ RCMP’), (’Moscow’, ’sov’), (’Russia’, ’ Bulgar’), (’ Canadians’, ’Canada’) GPT-2 Medium - Layer 21 Head 12* ⬇ (’ Indonesian’, ’ Indones’), (’ Vietnamese’, ’ Nguyen’), (’ Indonesian’, ’ Jakarta’), (’ Indonesian’, ’ Indonesia’), (’Turkish’, ’oglu’), (’ Indonesia’, ’ Indones’), (’ Jakarta’, ’ Indones’), (’ Korean’, ’ Koreans’), (’ Turkish’, ’oglu’), (’ Taiwan’, ’ Taiwanese’), (’ Thai’, ’ Nguyen’), (’ Brazilian’, ’Brazil’), (’ Indones’, ’ Indonesia’), (’Tai’, ’ Taiwanese’), (’ Istanbul’, ’oglu’), (’ Indones’, ’ Indonesian’), (’ Indones’, ’ Jakarta’), (’ Laos’, ’ Nguyen’), (’ Slovenia’, ’ Sloven’), (’ Koreans’, ’ Korean’), (’ Cambod’, ’ Nguyen’), (’Italy’, ’zzi’), (’ Taiwanese’, ’Tai’), (’ Indonesia’, ’ Jakarta’), (’ Indonesia’, ’ Indonesian’), (’ Bulgarian’, ’ Bulgaria’), (’ Iceland’, ’ Icelandic’), (’ Korea’, ’ Koreans’), (’Brazil’, ’ Brazilian’), (’ Bulgarian’, ’ Bulgar’), (’ Malaysian’, ’ Malays’), (’ Ankara’, ’oglu’), (’ Bulgaria’, ’ Bulgarian’), (’ Malays’, ’ Indones’), (’ Taiwanese’, ’ Tai’), (’Turkey’, ’oglu’), (’Brazil’, ’ Janeiro’), (’Italian’, ’zzi’), (’ Kuala’, ’ Malays’), (’Japanese’, ’ Fuk’), (’ Jakarta’, ’ Indonesian’), (’ Taiwanese’, ’ Taiwan’), (’ Erdogan’, ’oglu’), (’ Viet’, ’ Nguyen’), (’ Philippine’, ’ Filipino’), (’ Jakarta’, ’ Indonesia’), (’ Koreans’, ’ Jong’), (’ Filipino’, ’ Duterte’), (’ Azerbaijan’, ’ Azerbai’), (’ Bulgar’, ’ Bulgarian’) GPT-2 Large - Layer 23 Head 5 ⬇ (’Canada’, ’ Trudeau’), (’ Canadians’, ’ Trudeau’), (’Canadian’, ’ Trudeau’), (’ Queensland’, ’ Tasman’), (’ Tasman’, ’ Tasman’), (’ Canada’, ’ Trudeau’), (’ Canberra’, ’ Canberra’), (’ Winnipeg’, ’ Winnipeg’), (’ Canberra’, ’ Tasman’), (’Canadian’, ’Canada’), (’ Canadian’, ’ Trudeau’), (’ Brisbane’, ’ Brisbane’), (’ Quebec’, ’ Trudeau’), (’Canadian’, ’ Canadian’), (’ Brisbane’, ’ Tasman’), (’ Tasmania’, ’ Tasman’), (’Canadian’, ’ Canadians’), (’ RCMP’, ’ Trudeau’), (’ Manitoba’, ’ Trudeau’), (’ Queensland’, ’ Brisbane’), (’ Queensland’, ’ Canberra’), (’Canada’, ’ Saskatchewan’), (’Canadian’, ’ Saskatchewan’), (’Canada’, ’ Canadian’), (’ RCMP’, ’ Saskatchewan’), (’ Canberra’, ’ Brisbane’), (’ Canadians’, ’Canada’), (’ Winnipeg’, ’ Trudeau’), (’Canadian’, ’ Canada’), (’Canada’, ’ Canadians’), (’Australian’, ’ Canberra’), (’ Melbourne’, ’ Canberra’), (’ RCMP’, ’ Canad’), (’ Canadians’, ’ Canadians’), (’CBC’, ’ Trudeau’), (’ Canadian’, ’ Canadian’), (’Canadian’, ’ Winnipeg’), (’ Australians’, ’ Canberra’), (’ Quebec’, ’Canada’), (’ Canadian’, ’Canada’), (’ NSW’, ’ Canberra’), (’Toronto’, ’ Canad’), (’Canada’, ’Canada’), (’ NSW’, ’ Tasman’), (’ RCMP’, ’ RCMP’), (’ Canadian’, ’ Canadians’), (’ Saskatchewan’, ’ Saskatchewan’), (’ Canadians’, ’ Saskatchewan’), (’Canadian’, ’ Canad’), (’ Ottawa’, ’ Winnipeg’) C.1.4 British Spelling GPT-2 Medium - Layer 19 Head 4 ⬇ (’ realise’, ’ Whilst’), (’ Whilst’, ’ Whilst’), (’ realised’, ’ Whilst’), (’ organise’, ’ Whilst’), (’ recognise’, ’ Whilst’), (’ civilisation’, ’ Whilst’), (’ organisation’, ’ Whilst’), (’ whilst’, ’ Whilst’), (’ organising’, ’ Whilst’), (’ organised’, ’ Whilst’), (’ organis’, ’ Whilst’), (’ util’, ’ Whilst’), (’ apologise’, ’ Whilst’), (’ emphas’, ’ Whilst’), (’ analyse’, ’ Whilst’), (’ organisations’, ’ Whilst’), (’ recognised’, ’ Whilst’), (’ flavours’, ’ Whilst’), (’ colour’, ’ Whilst’), (’colour’, ’ Whilst’), (’ Nasa’, ’ Whilst’), (’ Nato’, ’ Whilst’), (’ analys’, ’ Whilst’), (’ flavour’, ’ Whilst’), (’ colourful’, ’ Whilst’), (’ colours’, ’ Whilst’), (’ realise’, ’ organising’), (’ behavioural’, ’ Whilst’), (’ coloured’, ’ Whilst’), (’ learnt’, ’ Whilst’), (’ favourable’, ’ Whilst’), (’isation’, ’ Whilst’), (’ programmes’, ’ Whilst’), (’ realise’, ’ organis’), (’ authorised’, ’ Whilst’), (’ practise’, ’ Whilst’), (’ criticised’, ’ Whilst’), (’ organisers’, ’ Whilst’), (’ organise’, ’ organising’), (’ analysed’, ’ Whilst’), (’ programme’, ’ Whilst’), (’ behaviours’, ’ Whilst’), (’ humour’, ’ Whilst’), (’isations’, ’ Whilst’), (’ tyres’, ’ Whilst’), (’ aluminium’, ’ Whilst’), (’ realise’, ’ organised’), (’ favour’, ’ Whilst’), (’ ageing’, ’ Whilst’), (’ organise’, ’ organis’) C.1.5 Related Words GPT-2 Medium - Layer 13 Head 8* ⬇ (’ miraculous’, ’ mirac’), (’ miracle’, ’ mirac’), (’ nuance’, ’ nuanced’), (’ smarter’, ’Better’), (’ healthier’, ’ equitable’), (’ liberated’, ’ liberating’), (’ untouched’, ’ unaffected’), (’ unbiased’, ’ equitable’), (’failed’, ’ inconsistent’), (’ liberated’, ’ emanc’), (’ humane’, ’ equitable’), (’ liberating’, ’ liberated’), (’failed’, ’ incompatible’), (’ miracles’, ’ mirac’), (’ peacefully’, ’ consensual’), (’ unconditional’, ’ uncond’), (’ unexpectedly’, ’ unexpected’), (’ untouched’, ’ unconditional’), (’ healthier’, ’Better’), (’ unexpected’, ’ unexpectedly’), (’ peacefully’, ’ graceful’), (’ emancipation’, ’ emanc’), (’ seamlessly’, ’ effortlessly’), (’ peacefully’, ’ honorable’), (’ uncond’, ’ unconditional’), (’ excuses’, ’ rubbish’), (’ liberating’, ’ emanc’), (’ peacefully’, ’ equitable’), (’ gracious’, ’ Feather’), (’ liberated’, ’ emancipation’), (’ nuances’, ’ nuanced’), (’ avoids’, ’icable’), (’ freeing’, ’ liberated’), (’ freeing’, ’ liberating’), (’ lousy’, ’ inconsistent’), (’failed’, ’ lousy’), (’ unaffected’, ’ unconditional’), (’ivable’, ’ equitable’), (’Honest’, ’ equitable’), (’ principled’, ’erning’), (’surv’, ’ survival’), (’ lackluster’, ’ocre’), (’ liberating’, ’ equitable’), (’Instead’, ’Bah’), (’ inappropriate’, ’ incompatible’), (’ emanc’, ’ emancipation’), (’ unaffected’, ’ unchanged’), (’ peaceful’, ’ peacefully’), (’ safer’, ’ equitable’), (’ uninterrupted’, ’ unconditional’) GPT-2 Medium - Layer 12 Head 14* ⬇ (’ died’, ’ perished’), (’ dies’, ’ perished’), (’ testifying’, ’ testify’), (’ interven’, ’ intervened’), (’ advising’, ’ advises’), (’ disband’, ’ disbanded’), (’ perished’, ’lost’), (’ perished’, ’ died’), (’ applaud’, ’ applauded’), (’ dictate’, ’ dictates’), (’ prevailed’, ’ prev’), (’ advising’, ’ advise’), (’thood’, ’shed’), (’orsi’, ’Reviewed’), (’ perished’, ’ dies’), (’ publishes’, ’published’), (’ prevail’, ’ prevailed’), (’ dies’, ’ died’), (’ testifying’, ’ testified’), (’ testify’, ’ testifying’), (’ governs’, ’ dictates’), (’ complicity’, ’ complicit’), (’ dictate’, ’ dictated’), (’CHO’, ’enough’), (’independence’, ’ skelet’), (’ prescribe’, ’ Recomm’), (’ perished’, ’essential’), (’CHO’, ’noticed’), (’ approving’, ’avorable’), (’ perished’, ’ perish’), (’ oversee’, ’ overseeing’), (’shed’, ’ skelet’), (’chart’, ’EY’), (’ overseeing’, ’ presiding’), (’pees’, ’ fundament’), (’appro’, ’ sanction’), (’ prevailed’, ’ prevail’), (’ regulates’, ’ governs’), (’shed’, ’tails’), (’chart’, ’ Period’), (’hower’, ’lihood’), (’ prevail’, ’ prev’), (’helps’, ’ aids’), (’ dict’, ’ dictated’), (’ dictates’, ’ dictated’), (’itta’, ’ Dise’), (’CHO’, ’REC’), (’ORTS’, ’exclusive’), (’helps’, ’ Helpful’), (’ciples’, ’bart’) GPT-2 Medium - Layer 14 Head 1* ⬇ (’ incorrectly’, ’ misunderstand’), (’ properly’, ’ Proper’), (’ incorrectly’, ’ inaccur’), (’ wrongly’, ’ misunderstand’), (’ incorrectly’, ’ misinterpret’), (’ incorrectly’, ’ incorrect’), (’ incorrectly’, ’ mistakes’), (’ incorrectly’, ’ misunderstanding’), (’ properly’, ’ proper’), (’ incorrectly’, ’fail’), (’ incorrectly’, ’ faulty’), (’ incorrectly’, ’ misrepresent’), (’ fails’, ’ failing’), (’ incorrectly’, ’ inaccurate’), (’ incorrectly’, ’ errors’), (’ Worse’, ’ harmful’), (’ wrong’, ’ misunderstand’), (’ improperly’, ’ misunderstand’), (’ incorrectly’, ’wrong’), (’ incorrectly’, ’ harmful’), (’ incorrectly’, ’ mistake’), (’ incorrectly’, ’ mis’), (’ fails’, ’fail’), (’ Worse’, ’ detrimental’), (’ properly’, ’ rightful’), (’ inappropriately’, ’ misunderstand’), (’ unnecessarily’, ’ harmful’), (’ unnecessarily’, ’ neglect’), (’ properly’, ’ correctly’), (’ Worse’, ’ Worst’), (’ fails’, ’ failure’), (’ adequately’, ’ satisfactory’), (’ incorrectly’, ’ defective’), (’ mistakenly’, ’ misunderstand’), (’ Worse’, ’ harming’), (’ incorrectly’, ’ mishand’), (’ adequately’, ’adequ’), (’ incorrectly’, ’ misuse’), (’ fails’, ’Failure’), (’ Worse’, ’ hurts’), (’wrong’, ’ misunderstand’), (’ incorrectly’, ’ mistakenly’), (’ fails’, ’ failures’), (’ adequately’, ’ adequate’), (’ correctly’, ’ properly’), (’ Worse’, ’ hurting’), (’ correctly’, ’ Proper’), (’ fails’, ’ fail’), (’ incorrectly’, ’ mistaken’), (’ adversely’, ’ harming’) GPT-2 Large - Layer 24 Head 9 ⬇ (’ interviewer’, ’ interviewer’), (’ lectures’, ’ lectures’), (’ lecture’, ’ lecture’), (’ interview’, ’Interview’), (’ interview’, ’ interview’), (’ interview’, ’ interviewer’), (’ interviewing’, ’ interviewing’), (’ magazine’, ’ magazine’), (’ Reviews’, ’ Reviews’), (’ reviewer’, ’ reviewer’), (’ reviewers’, ’ reviewers’), (’ lectures’, ’ lecture’), (’ testers’, ’ testers’), (’ editors’, ’ editors’), (’ interviewer’, ’ interview’), (’ Interview’, ’Interview’), (’ interviewer’, ’Interview’), (’Interview’, ’Interview’), (’ lecture’, ’ lectures’), (’ interviewing’, ’ interviewer’), (’ journal’, ’ journal’), (’ interviewer’, ’ interviewing’), (’ blogs’, ’ blogs’), (’ editorial’, ’ editorial’), (’ tests’, ’ tests’), (’ presentations’, ’ presentations’), (’ Editorial’, ’ Editorial’), (’ interview’, ’ Interview’), (’ reviewer’, ’ reviewers’), (’ interviews’, ’Interview’), (’ interview’, ’ interviewing’), (’ interviewer’, ’ Interview’), (’ interviews’, ’ interview’), (’ Interview’, ’ Interview’), (’ interviewing’, ’Interview’), (’Interview’, ’ interviewer’), (’ testifying’, ’ testifying’), (’ reviewers’, ’ reviewer’), (’ blogging’, ’ blogging’), (’ broadcast’, ’ broadcast’), (’ Interview’, ’ interviewer’), (’ magazine’, ’ magazines’), (’ editorial’, ’ Editorial’), (’ interview’, ’ interviews’), (’ interviewing’, ’ interview’), (’ Interview’, ’ interview’), (’ interviews’, ’ interviews’), (’ tests’, ’tests’), (’ interviews’, ’ interviewing’), (’Interview’, ’ interview’) lstlisting GPT-2 Medium - Layer 14 Head 13 * lstlisting[backgroundcolor= white] (’ editorial’, ’ editors’), (’ broadcasting’, ’ broadcasters’), (’ broadcasts’, ’ broadcasting’), (’ broadcasts’, ’ broadcast’), (’ broadcasters’, ’ Broadcasting’), (’ Editorial’, ’ editors’), (’ broadcast’, ’ broadcasters’), (’ broadcast’, ’ Broadcasting’), (’ lecture’, ’ lectures’), (’ broadcasting’, ’ Broadcast’), (’ broadcaster’, ’ broadcasters’), (’ broadcasts’, ’ broadcasters’), (’ publishing’, ’ Publishers’), (’ broadcast’, ’ broadcasting’), (’ Broadcasting’, ’ broadcasters’), (’ Publishing’, ’ Publishers’), (’ lectures’, ’ lecture’), (’ editorial’, ’ Editors’), (’ broadcasting’, ’ broadcast’), (’ broadcasts’, ’ Broadcasting’), (’ broadcasters’, ’ broadcasting’), (’ journalistic’, ’ journalism’), (’Journal’, ’reports’), (’ Broadcasting’, ’ Broadcast’), (’Publisher’, ’ Publishers’), (’ Broadcasting’, ’azeera’), (’Journal’, ’Reporting’), (’ journalism’, ’ journalistic’), (’ broadcaster’, ’ Broadcasting’), (’ broadcaster’, ’ broadcasting’), (’ broadcasting’, ’ broadcaster’), (’ publication’, ’ editors’), (’journal’, ’ journalism’), (’Journal’, ’ Journalists’), (’ documentaries’, ’ documentary’), (’ filmed’, ’ filming’), (’ publishing’, ’ publishers’), (’Journal’, ’ journalism’), (’ broadcasts’, ’ Broadcast’), (’ broadcasters’, ’ broadcast’), (’Journal’, ’ articles’), (’reports’, ’ reporting’), (’ manuscript’, ’ manuscripts’), (’ publishing’, ’ publish’), (’ broadcasters’, ’azeera’), (’ publication’, ’ Publishers’), (’ publications’, ’ Publishers’), (’ Newsp’, ’ newspapers’), (’ broadcasters’, ’ Broadcast’), (’Journal’, ’ Readers’) C.2 Query-Key Matrices GPT-2 Large - Layer 19 Head 7** ⬇ (’ tonight’, ’Friday’), (’ Copyright’, ’Returns’), (’TM’, ’review’), (’ Weekend’, ’Preview’), (’ tonight’, ’Thursday’), (’ recently’, ’Closure’), (’ Copyright’, ’Contents’), (’ Copyright’, ’Wisconsin’), (’ Copyright’, ’Methods’), (’ tonight’, ’Sunday’), (’ tomorrow’, ’ postpone’), (’ tomorrow’, ’ tonight’), (’ recently’, ’acerb’), (’ Copyright’, ’Rated’), (’ myself’, ’ my’), (’ Copyright’, ’Cop’), (’ Wednesday’, ’Closure’), (’ Billion’, ’ 1935’), (’ tonight’, ’Saturday’), (’ tonight’, ’ celebr’), (’ tomorrow’, ’ postponed’), (’ Copyright’, ’Show’), (’ Wednesday’, ’Friday’), (’ Copyright’, ’Earn’), (’ Billion’, ’ 1934’), (’ Eric’, ’Larry’), (’ 2015’, ’Released’), (’ Copyright’, ’Rat’), (’ tomorrow’, ’ postp’), (’ 2017’, ’Latest’), (’ previous’, ’obin’), (’ controversial’, ’Priv’), (’ recently’, ’ nightly’), (’Base’, ’ LV’), (’ recently’, ’Project’), (’ historically’, ’ globalization’), (’ recently’, ’ vulner’), (’ tonight’, ’Wednesday’), (’ Copyright’, ’Abstract’), (’ Tuesday’, ’Friday’), (’ Anthony’, ’Born’), (’ Budget’, ’Premium’), (’ tonight’, ’Welcome’), (’yle’, ’lite’), (’ Wednesday’, ’Latest’), (’ Latest’, ’show’), (’ B’, ’ pione’), (’ Copyright’, ’cop’), (’ Pablo’, ’ Dia’), (’ recent’, ’Latest’) GPT-2 Medium - Layer 22 Head 1 ⬇ (’ usual’, ’ usual’), (’ occasional’, ’ occasional’), (’ aforementioned’, ’ aforementioned’), (’ general’, ’ usual’), (’ usual’, ’ slightest’), (’agn’, ’ealous’), (’ traditional’, ’ usual’), (’ free’, ’amina’), (’ major’, ’ major’), (’ frequent’, ’ occasional’), (’ generous’, ’ generous’), (’ free’, ’lam’), (’ regular’, ’ usual’), (’ standard’, ’ usual’), (’ main’, ’ usual’), (’ complete’, ’ Finished’), (’ main’, ’liest’), (’ traditional’, ’ traditional’), (’ latest’, ’ aforementioned’), (’ current’, ’ aforementioned’), (’ normal’, ’ usual’), (’ dominant’, ’ dominant’), (’ free’, ’ministic’), (’ brief’, ’ brief’), (’ biggest’, ’liest’), (’usual’, ’ usual’), (’ rash’, ’ rash’), (’ regular’, ’ occasional’), (’ specialized’, ’ specialized’), (’ free’, ’iosis’), (’ free’, ’hero’), (’ specialty’, ’ specialty’), (’ general’, ’iosis’), (’ nearby’, ’ nearby’), (’ best’, ’liest’), (’ officially’, ’ formal’), (’ immediate’, ’mediate’), (’ special’, ’ ultimate’), (’ free’, ’otropic’), (’ rigorous’, ’ comparative’), (’ actual’, ’ slightest’), (’ complete’, ’ comparative’), (’ typical’, ’ usual’), (’ modern’, ’ modern’), (’ best’, ’ smartest’), (’ free’, ’ free’), (’ highest’, ’ widest’), (’ specialist’, ’ specialist’), (’ appropriate’, ’ slightest’), (’ usual’, ’liest’) GPT-2 Large - Layer 20 Head 13 ** ⬇ (’ outdoors’, ’ outdoors’), (’ outdoor’, ’ outdoors’), (’ Gre’, ’burg’), (’ healing’, ’ healing’), (’ indoor’, ’ outdoors’), (’ Hemp’, ’burg’), (’ Ticket’, ’ Ticket’), (’ accommodations’, ’ accommodations’), (’eco’, ’aco’), (’prem’, ’otti’), (’ Candy’, ’cott’), (’ decorative’, ’ ornament’), (’yan’, ’ava’), (’ deadlines’, ’ schedule’), (’ Lor’, ’ian’), (’ architectural’, ’ ornament’), (’ Ratings’, ’ Ratings’), (’ Bod’, ’za’), (’ exotic’, ’ exotic’), (’ food’, ’ baths’), (’ Marketplace’, ’ Marketplace’), (’ heal’, ’ healing’), (’ Ex’, ’ilus’), (’ indoors’, ’ outdoors’), (’ therm’, ’ therm’), (’ bleach’, ’ coated’), (’ Sod’, ’opol’), (’ District’, ’ Metropolitan’), (’ Anonymous’, ’ Rebell’), (’ Corn’, ’burg’), (’ indoor’, ’ indoors’), (’ R’, ’vale’), (’rom’, ’otti’), (’ ratings’, ’ Ratings’), (’ attendance’, ’ attendance’), (’ destinations’, ’ destinations’), (’ VIDEOS’, ’ VIDEOS’), (’yan’, ’opol’), (’ Suffolk’, ’ville’), (’ retali’, ’ against’), (’mos’, ’oli’), (’ pacing’, ’ pacing’), (’ Spectrum’, ’ QC’), (’ Il’, ’ian’), (’ archived’, ’ archived’), (’ Pledge’, ’ Pledge’), (’alg’, ’otti’), (’ Freedom’, ’USA’), (’anto’, ’ero’), (’ decorative’, ’ decoration’) GPT-2 Medium - Layer 0 Head 9 ⬇ (’59’, ’27’), (’212’, ’39’), (’212’, ’38’), (’217’, ’39’), (’37’, ’27’), (’59’, ’26’), (’54’, ’88’), (’156’, ’39’), (’212’, ’79’), (’59’, ’28’), (’57’, ’27’), (’212’, ’57’), (’156’, ’29’), (’36’, ’27’), (’217’, ’79’), (’59’, ’38’), (’63’, ’27’), (’72’, ’39’), (’57’, ’26’), (’57’, ’34’), (’59’, ’34’), (’156’, ’27’), (’91’, ’27’), (’156’, ’38’), (’63’, ’26’), (’59’, ’25’), (’138’, ’27’), (’217’, ’38’), (’72’, ’27’), (’54’, ’27’), (’36’, ’29’), (’72’, ’26’), (’307’, ’39’), (’37’, ’26’), (’217’, ’57’), (’37’, ’29’), (’54’, ’38’), (’59’, ’29’), (’37’, ’28’), (’307’, ’38’), (’57’, ’29’), (’63’, ’29’), (’71’, ’27’), (’138’, ’78’), (’59’, ’88’), (’89’, ’27’), (’561’, ’79’), (’212’, ’29’), (’183’, ’27’), (’54’, ’29’) GPT-2 Medium - Layer 17 Head 6* ⬇ (’ legally’, ’ legal’), (’ legal’, ’ sentencing’), (’ legal’, ’ arbitration’), (’ boycot’, ’ boycott’), (’ legal’, ’ criminal’), (’ legal’, ’ Judicial’), (’ legal’, ’ rulings’), (’ judicial’, ’ sentencing’), (’ marketing’, ’ advertising’), (’ legal’, ’ confidential’), (’ protesting’, ’ protest’), (’ recruited’, ’ recruit’), (’ recruited’, ’ recruits’), (’ judicial’, ’ criminal’), (’ legal’, ’ exemptions’), (’ demographics’, ’ demographic’), (’ boycott’, ’ boycot’), (’ sentencing’, ’ criminal’), (’ recruitment’, ’ recruits’), (’ recruitment’, ’ recruit’), (’ Constitutional’, ’ sentencing’), (’ Legal’, ’ sentencing’), (’ constitutional’, ’ sentencing’), (’ legal’, ’ subpoena’), (’ injury’, ’ injuries’), (’ FOIA’, ’ confidential’), (’ legal’, ’ licenses’), (’ donation’, ’ donations’), (’ disclosure’, ’ confidential’), (’ negotiation’, ’ negotiating’), (’ Judicial’, ’ legal’), (’ legally’, ’ criminal’), (’ legally’, ’ confidential’), (’ legal’, ’ jur’), (’ legal’, ’ enforcement’), (’ legal’, ’ lawyers’), (’ legally’, ’ enforcement’), (’ recruitment’, ’ recruiting’), (’ recruiting’, ’ recruit’), (’ criminal’, ’ sentencing’), (’ legal’, ’ attorneys’), (’ negotiations’, ’ negotiating’), (’ legally’, ’ arbitration’), (’ recruited’, ’ recruiting’), (’ legally’, ’ exemptions’), (’ legal’, ’ judicial’), (’ voting’, ’ Vote’), (’ negotiated’, ’ negotiating’), (’ legislative’, ’ veto’), (’ funding’, ’ funded’) GPT-2 Medium - Layer 17 Head 7 ⬇ (’tar’, ’idia’), (’ [...]’, ’..."’), (’ lecture’, ’ lectures’), (’ Congress’, ’ senate’), (’ staff’, ’ staffers’), (’ Scholarship’, ’ collegiate’), (’ executive’, ’ overseeing’), (’ Scholarship’, ’ academic’), (’ academ’, ’ academic’), (’."’, ’..."’), (’ [’, ’..."’), (’";’, ’..."’), (’ Memorial’, ’priv’), (’ festival’, ’conference’), (’crew’, ’ supervisors’), (’ certification’, ’ grading’), (’ scholarship’, ’ academic’), (’ rumored’, ’ Academic’), (’ Congress’, ’ delegated’), (’ staff’, ’ technicians’), (’Plex’, ’ CONS’), (’ congress’, ’ senate’), (’ university’, ’ tenure’), (’ Congress’, ’ appointed’), (’ Congress’, ’ duly’), (’ investigative’, ’ investig’), (’ legislative’, ’ senate’), (’ademic’, ’ academic’), (’bench’, ’ academic’), (’ scholarship’, ’ tenure’), (’ campus’, ’ campuses’), (’ staff’, ’ Facilities’), (’ Editorial’, ’mn’), (’ clinic’, ’ laboratory’), (’ crew’, ’ crews’), (’ Scholarship’, ’ academ’), (’ staff’, ’ staffer’), (’icken’, ’oles’), (’?"’, ’..."’), (’ Executive’, ’ overseeing’), (’ academic’, ’ academ’), (’ Congress’, ’atra’), (’aroo’, ’anny’), (’ academic’, ’ academia’), (’ Congress’, ’ Amendments’), (’ academic’, ’ academics’), (’student’, ’ academic’), (’ committee’, ’ convened’), (’",’, ’..."’), (’ove’, ’idia’) GPT-2 Medium - Layer 16 Head 13 ⬇ (’ sugg’, ’ hindsight’), (’ sugg’, ’ anecdotal’), (’ unsuccessfully’, ’ hindsight’), (’didn’, ’ hindsight’), (’orously’, ’staking’), (’illions’, ’uries’), (’until’, ’era’), (’ lobbied’, ’ hindsight’), (’ incorrectly’, ’ incorrect’), (’ hesitate’, ’ hindsight’), (’ECA’, ’ hindsight’), (’ regret’, ’ regrets’), (’inventoryQuantity’, ’imore’), (’consider’, ’ anecdotal’), (’ errone’, ’ incorrect’), (’ someday’, ’ eventual’), (’illions’, ’Murray’), (’ recently’, ’recent’), (’ Learned’, ’ hindsight’), (’before’, ’ hindsight’), (’ lately’, ’ealous’), (’upon’, ’rity’), (’ja’, ’ hindsight’), (’ regretted’, ’ regrets’), (’ unsuccessfully’, ’udging’), (’ lately’, ’dated’), (’ sugg’, ’ anecd’), (’ inform’, ’imore’), (’ lately’, ’recent’), (’ anecd’, ’ anecdotal’), (’orously’, ’ hindsight’), (’ postwar’, ’ Era’), (’ lately’, ’ recent’), (’ skept’, ’ cynicism’), (’ sugg’, ’informed’), (’ unsuccessfully’, ’ealous’), (’ebin’, ’ hindsight’), (’ underest’, ’ overest’), (’ Jinn’, ’ hindsight’), (’ someday’, ’2019’), (’ recently’, ’turned’), (’ sugg’, ’ retrospect’), (’ unsuccessfully’, ’didn’), (’ unsuccessfully’, ’gged’), (’ mistakenly’, ’ incorrect’), (’assment’, ’)</’), (’ja’, ’didn’), (’illions’, ’ hindsight’), (’ sugg’, ’ testimony’), (’jri’, ’ hindsight’) GPT-2 Medium - Layer 12 Head 9 ⬇ (’ PST’, ’ usual’), (’etimes’, ’ foreseeable’), (’uld’, ’uld’), (’ Der’, ’ Mankind’), (’ statewide’, ’ yearly’), (’ guarantees’, ’ guarantees’), (’ Flynn’, ’ Logged’), (’borne’, ’ foreseeable’), (’ contiguous’, ’ contiguous’), (’ exceptions’, ’ exceptions’), (’ redist’, ’ costly’), (’ downstream’, ’ day’), (’ ours’, ’ modern’), (’ foreseeable’, ’ foreseeable’), (’ Posted’, ’ Posted’), (’ anecdotal’, ’ anecdotal’), (’ moot’, ’ costly’), (’ successor’, ’ successor’), (’ any’, ’ ANY’), (’ generational’, ’ modern’), (’ temporarily’, ’ costly’), (’ overall’, ’ overall’), (’ effective’, ’ incentiv’), (’ future’, ’ tomorrow’), (’ ANY’, ’ lifetime’), (’ dispatch’, ’ dispatch’), (’ legally’, ’ WARRANT’), (’ guarantees’, ’ incentiv’), (’ listed’, ’ deductible’), (’ CST’, ’ foreseeable’), (’ anywhere’, ’ any’), (’ guaranteed’, ’ incentiv’), (’ successors’, ’ successor’), (’ weekends’, ’ day’), (’iquid’, ’ expensive’), (’ Trib’, ’ foreseeable’), (’ phased’, ’ modern’), (’ constitutionally’, ’ foreseeable’), (’ any’, ’ anybody’), (’ anywhere’, ’ ANY’), (’ veto’, ’ precedent’), (’ veto’, ’ recourse’), (’ hopefully’, ’ hopefully’), (’ potentially’, ’ potentially’), (’ ANY’, ’ ANY’), (’ substantive’, ’ noteworthy’), (’morrow’, ’ day’), (’ancial’, ’ expensive’), (’listed’, ’ breastfeeding’), (’ holiday’, ’ holidays’) GPT-2 Medium - Layer 11 Head 10 ⬇ (’ Journalism’, ’ acron’), (’ democracies’, ’ governments’), (’/-’, ’verty’), (’ legislatures’, ’ governments’), (’ocracy’, ’ hegemony’), (’osi’, ’ RAND’), (’ Organizations’, ’ organisations’), (’ellectual’, ’ institutional’), (’ Journalists’, ’ acron’), (’eworks’, ’ sponsors’), (’ Inqu’, ’ reviewer’), (’ocracy’, ’ diversity’), (’ careers’, ’ Contributions’), (’gency’, ’\\-’), (’ellectual’, ’ exceptions’), (’ Profession’, ’ specializing’), (’online’, ’ Online’), (’ Publications’, ’ authorised’), (’Online’, ’ Online’), (’ sidx’, ’ Lazarus’), (’eworks’, ’ Networks’), (’ Groups’, ’ organisations’), (’ Governments’, ’ governments’), (’ democracies’, ’ nowadays’), (’ psychiat’, ’ Mechdragon’), (’ educ’, ’ Contributions’), (’ Ratings’, ’ organisations’), (’vernment’, ’spons’), (’..."’, ’),"’), (’ Caucas’, ’ commodity’), (’ dictators’, ’ governments’), (’istration’, ’ sponsor’), (’iquette’, ’ acron’), (’ Announce’, ’ answ’), (’ Journalism’, ’ empowering’), (’Media’, ’ bureaucr’), (’ Discrimination’, ’ organizations’), (’ Journalism’, ’Online’), (’FAQ’, ’sites’), (’ antitrust’, ’ Governments’), (’..."’, ’..."’), (’Questions’, ’ acron’), (’rities’, ’ organisations’), (’ Editorial’, ’ institutional’), (’ tabl’, ’ acron’), (’ antitrust’, ’ governments’), (’ Journalism’, ’ Everyday’), (’icter’, ’ Lieberman’), (’ defect’, ’SPONSORED’), (’ Journalists’, ’ organisations’) GPT-2 Medium - Layer 22 Head 5 (names and parts of names seem to attend to each other here) ⬇ (’ Smith’, ’ovich’), (’ Jones’, ’ovich’), (’ Jones’, ’Jones’), (’ Smith’, ’Williams’), (’ Rogers’, ’opoulos’), (’Jones’, ’ovich’), (’ Jones’, ’inez’), (’ug’, ’ Ezek’), (’ Moore’, ’ovich’), (’orn’, ’roit’), (’van’, ’actionDate’), (’ Jones’, ’inelli’), (’ Edwards’, ’opoulos’), (’ Jones’, ’ Lyons’), (’Williams’, ’opoulos’), (’Moore’, ’ovich’), (’ Rodriguez’, ’hoff’), (’ North’, ’ suburbs’), (’ Smith’, ’chio’), (’Smith’, ’ovich’), (’ Smith’, ’opoulos’), (’Mc’, ’opoulos’), (’Johnson’, ’utt’), (’ Jones’, ’opoulos’), (’Ross’, ’Downloadha’), (’pet’, ’ilage’), (’ Everett’, ’ Prairie’), (’ Cass’, ’isma’), (’ Jones’, ’zynski’), (’Jones’, ’Jones’), (’ McCl’, ’elman’), (’ Smith’, ’Jones’), (’ Simmons’, ’opoulos’), (’ Smith’, ’brown’), (’ Mc’, ’opoulos’), (’ Jones’, ’utt’), (’ Richards’, ’Davis’), (’ Johnson’, ’utt’), (’ Ross’, ’bred’), (’ McG’, ’opoulos’), (’ Stevens’, ’stadt’), (’ra’, ’abouts’), (’ Johnson’, ’hoff’), (’ North’, ’ Peninsula’), (’ Smith’, ’Smith’), (’Jones’, ’inez’), (’ Hernandez’, ’hoff’), (’ Lucas’, ’Nor’), (’ Agu’, ’hoff’), (’Jones’, ’utt’) GPT-2 Medium - Layer 19 Head 12 ⬇ (’ 2015’, ’ADVERTISEMENT’), (’ 2014’, ’2014’), (’ 2015’, ’2014’), (’ 2015’, ’Present’), (’ 2013’, ’2014’), (’ 2017’, ’ADVERTISEMENT’), (’ 2016’, ’ADVERTISEMENT’), (’itor’, ’ Banner’), (’2015’, ’ Bulletin’), (’2012’, ’ Bulletin’), (’2014’, ’ Bulletin’), (’ Airl’, ’Stream’), (’2016’, ’ Bulletin’), (’ 2016’, ’2014’), (’2017’, ’ Bulletin’), (’ 2013’, ’ 2014’), (’ 2012’, ’2014’), (’ stadiums’, ’ventions’), (’ 2015’, ’ Bulletin’), (’2013’, ’ Bulletin’), (’ 2017’, ’2014’), (’ 2011’, ’ 2011’), (’ 2014’, ’ 2014’), (’ 2011’, ’ 2009’), (’ mile’, ’eming’), (’ 2013’, ’ADVERTISEMENT’), (’ 2014’, ’2015’), (’ 2014’, ’Present’), (’ 2011’, ’2014’), (’ 2011’, ’2009’), (’ 2015’, ’ 2014’), (’ 2013’, ’ Bulletin’), (’ 2015’, ’2015’), (’ 2011’, ’ 2003’), (’ 2011’, ’ 2010’), (’ 2017’, ’Documents’), (’2017’, ’iaries’), (’ 2013’, ’2015’), (’2017’, ’Trend’), (’ 2011’, ’2011’), (’ 2016’, ’Present’), (’ 2011’, ’ 2014’), (’ years’, ’years’), (’Plug’, ’Stream’), (’ 2014’, ’ADVERTISEMENT’), (’2015’, ’Present’), (’ 2018’, ’thora’), (’ 2017’, ’thora’), (’ 2012’, ’ 2011’), (’ 2012’, ’ 2014’) C.3 Feedforward Keys and Values Key-value pairs, (ki,vi)subscriptsubscript(k_i,v_i)( kitalic_i , vitalic_i ), where at least 15% of the top-k vocabulary items overlap, with k=100100k=100k = 100. We follow our forerunner’s convention of calling the index of the value in the layer “dimension” (Dim). Here again we use two asterisks (**) to represent lists where we discarded tokens outside the corpus vocabulary. GPT-2 Medium - Layer 0 Dim 116 ⬇ #annels #Els #netflix #osi telev #mpeg #tv #vous #avi #iane #flix transmitter Television Sinclair #outube Streaming #channel #channel Vid mosqu #Channel broadcaster documentaries airs #videos Broadcasting Hulu broadcasts channels streams #levision channels DVDs broadcasters broadcasts broadcasting #azeera #RAFT MPEG #oded televised htt aired transmissions broadcasters playback Streaming Instruction viewership nic #TV Sirius Kodi viewership ITV radio #ovies #achers channel channel GPT-2 Medium - Layer 3 Dim 2711 ⬇ purposes purposes sake sake purpose reasons reasons purpose convenience ages reason reason Seasons #ummies #Plex #going Reasons foreseeable #ummies Reasons #asons #reason #lation #pur #alsh Developers #agos #akers #ACY transl STATS Reason #itas consideration ages #purpose #purpose beginners #=[ awhile #gencies Pur Millennium #benefit Brewers #atel Festival #tun EVENT pur #payment Ages #=- preservation #printf Metatron beginners um Expo #KEN GPT-2 Medium - Layer 4 Dim 621 ⬇ #ovie headlined newspapers pestic television dime editorial describ #journal Afric broadcasters broadcasts #Journal #(’ publication #umbnails Newsweek #adish Zeit #uggest columnist splash Editorial #ZX newsletter objectionable cartoon #article #eport Bucc telev #London radio reprint headlined #azine #ribune Giov BBC #ender reprint headline sitcom #oops reprinted #articles broadcast snipp tabloid Ajax documentaries marqu journalist #(" TV #otos headline mast news #idem GPT-2 Medium - Layer 7 Dim 72 ⬇ sessions session dinners sessions #cation #cation session #iesta dinner Booth #eteria screenings Dinner booked #Session #rogram rehears vacation baths baths Lunch #pleasant #hops meetings visits #Session Session greet #session #athon meetings Sessions chatting boarding lunch rituals chats booking festivities Grape boarding #miah #workshop #session #rooms Pars #tests simulated seated Dispatch visit Extras appointments toile #vu Evening #rations showers #luaj abroad GPT-2 Medium - Layer 10 Dim 8 ⬇ Miy Tai #imaru #jin Gong Jin Jinn Makoto Xia #etsu Makoto Shin Kuro Hai Shin Fuj #Tai Dai Yamato Miy Tai #iku Ichigo Yun #Shin Ryu #atsu Shu Haku Hua Chun Suzuki #ku Yang Qing Xia Tsuk #Shin Hua #iru Jiang Yu Nanto #yu manga Chang Yosh Nan yen Qian Osaka #hao Qian Fuk #uku Chun #iku Yong Yue #Tai GPT-2 Medium - Layer 11 Dim 2 ⬇ progressing toward #Progress towards #progress Pace #osponsors progression #oppable #inness advancement onward progress canon Progress #progress #senal pace #venge #peed queue advancement #pun advancing progression progressing #wagon ladder advancing path #cknowled honoring #Goal ranks momentum standings #zag goal #hop #grand pursuits momentum #encing #ometer #Improve timetable STEP nearing #chini quest standings spiral #eway trajectory #chie progress #ibling accelerating Esports escal GPT-2 Medium - Layer 15 Dim 4057 ⬇ EDITION copies versions Version copies #edition version #Version Version version edition #download editions download reprint versions #edition #Download EDIT copy Edition #release reproduce #version originals release #edited #copy VERS VERS #Versions #pub #Publisher Download reprodu #released #uploads editions playthrough edition Printed reprint reproduction Release #Reviewed #Available copy #published #Version #Published paperback EDITION preview print surv #Quantity #Download #available circulate RELEASE GPT-2 Medium - Layer 16 Dim 41 ⬇ #duino alarm #Battery alarms Morse signal alarms circuit GPIO GPIO LEDs timers batteries voltage #toggle signals signal circuitry circuitry electrical #PsyNetMessage circuits alarm LEDs autop standby signalling signalling #volt signaling volt lights signals Idle voltage triggers LED batteries electrom Morse timers LED malfunction #LED amplifier button radios Signal wiring timer #Alert wiring signaling buzz #Clock disconnect arming Arduino Arduino triggered GPT-2 Medium - Layer 17 Dim 23 ⬇ responsibility responsibility Responsibility respons responsibilities responsibilities #ipolar Responsibility #responsible oversee duties #respons #respons duties superv supervision supervision superv #abwe stewards Adin chore respons oversight oversee oversees entrusted responsible overseeing #responsible helicop handling presided handles overseen overseeing #dyl chores responsible manage #ADRA managing reins duty #accompan Respons chores charge oversees reins supervised handle blame oversaw oversaw CONTROL #archment RESP RESP tasks GPT-2 Medium - Layer 19 Dim 29 ⬇ subconscious thoughts thoughts thought #brain Thoughts #Brain minds memories mind OCD thinking flashbacks #thought brainstorm imagination Anxiety Thinking #mind Thought fantas imagin amygdala thinker impuls #thinking Thinking #mind #Memory memories Thoughts #think dreams imagining #ocamp impulses #Psych fantasies #mares think mentally urges #mental desires mind dreams #thinking delusions #Mind subconscious #dream emotions psyche imag prefrontal #dream PTSD conscience Memories visions GPT-2 Medium - Layer 20 Dim 65 ⬇ exercises volleyball #Sport tennis #athlon sports Exercise sport #ournaments #basketball volleyball Tennis Recre soccer Mahjong golf #basketball playground exercise Golf bowling athletics skating #athlon spar athletic skiing rugby gymn amusement #sports gymn drills sled #Training #Sport tournaments cricket sled Soccer Volunte amuse skate Activities golf recreational #Pract Ski dunk activities #hower basketball athletics #games sport skating Solitaire hockey #BALL #sports GPT-2 Medium - Layer 21 Dim 86 ⬇ IDs number identifiers #number surname #Number surn Number identifier NUM initials numbers #Registered Numbers NAME #Numbers #names address pseudonym #address #codes #Num nomine #NUM names addresses username Address #IDs identifier ID #Address registration #num #76561 ID #soDeliveryDate numbering #ADRA IDs CLSID #ID numbering identifiers #ername identification #address numer addresses digits codes #numbered #Names numerical regist Ident name numeric Names Identification GPT-2 Medium - Layer 21 Dim 400 ⬇ #July Oct July Feb #February Sept #January Dec #Feb Jan November Nov #October Aug January #Oct Feb May October #Nov #September Apr September March #June April #Sept #Sept February June #November #Aug #April October April #Feb June July #December December August Sep #March November Sept #Jan December #May Aug August March Jul #August Jun #Aug September #wcs January Apr February GPT-2 Medium - Layer 23 Dim 166 ⬇ #k #k #ks #K #kish #ks #K #KS #kat k #kus #kt #KS K #ked #kr #kr #kl #kB #kish #kan #kos #kw #king #ket #ked #king #kie #kb #KB #kos #k #kHz #kowski #k #KR #kick #KING #kers #KT #kowski #K #KB #KC #krit #kw #KING #kb #kt #Ka #ksh #krit #kie #KN #ky #kar #KY #kh #ku #ket GPT-2 Medium - Layer 23 Dim 907 ⬇ hands hand hand #Hand #hands Hand #hand #hand fingers hands #feet Hands fingertips fist claws #hands paw finger paws handed metab thumb palms fingers fingert foot #Hand #handed fists paw wrists handing levers #finger thumbs #hander tentacles fingertips feet claw limb fingert slider #Foot #handed Stick #dimension arm jaws #Accessory skelet #fing lapt Foot ankles index weap toe foot #auntlet GPT-2 Large - Layer 25 Dim 2685** ⬇ #manager engineering #Engineers Marketing chemist #engineering humanities Communications sciences #communications anthropology anthropology lingu Engineering #engineering lingu psychologist psychology Coordinator neurolog Analyst Economics #iologist designer accountant sociology strategist communications #ographer marketing curator pharmac Engineers sciences archae economics Designer Accounting Editing #econom biologist chemist #ologist merch psychologists pharm theolog economist Marketing architect #Manager engineer Architects Architect sociology #technical engineer architects physicist logistics GPT-2 Large - Layer 21 Dim 3419** ⬇ #overty impoverished #wana poverty poverty poorest #Saharan poorer poorest Yemen Poverty families malnutrition Poverty Senegal marginalized impoverished refugees #poor subsistence Gujar displaced homelessness hardship Homeless refugee #heid households Ramadan migrant #Palest disadvantaged poorer Sudan Rahman oppressed #amily socioeconomic illiter peasant Mahmoud homeless Haitian poor #advertisement Ethiopian #hya Kaf #African Rw wealthier #poor Africans Af caste rural homeless #fam Hait needy GPT-2 Large - Layer 25 Dim 2442** ⬇ Tracker tracking gau Tracker charts tracker tracker Tracking #Measure quant measurement #Stats measuring gau #Tracker GPS gauge Track tracking estimating Tracking tally #Monitor #ometers #chart tracked Meter calculate #HUD calculating #ometers measurement surve gauge #Stats estimation #Statistics monitoring calculate #stats Measure #tracking quant track #asuring measuring Calculator Monitoring #ometer #Detailed calculator #ometer Monitoring estim #Maps stats pione charts timet timet GPT-2 Base - Layer 9 Dim 1776 ⬇ radios cable antennas modem radio wireless modem WiFi voltage wired transformer broadband Ethernet Ethernet telev radios #Radio power electricity radio loudspe Cable kW Wireless #radio telephone broadband network volt signal microphones Networks telecommunications networks cable electricity Telephone wifi amplifier #levision wifi coax broadcasting transmit transistor transmitter Radio TV wireless Network LTE television watts transmission microwave router telephone cables amps amplifier GPT-2 Base - Layer 9 Dim 2771 ⬇ arous increase freeing increasing incent accelerating stimulate allev induce exped discourage enhanced inducing aggrav mitigating enhance stimulating inhib emanc improving alleviate infl empowering #oint preventing alien #ufact alter #HCR enabling influencing incre handc indu disadvant #Impro #roying intens arresting improve allev easing weaken elevate depri encouraging dissu accelerate impede enlarg convol energ encouraging accent #xiety acceler #akening depri lowering elong GPT-2 Base - Layer 1 Dim 2931 ⬇ evening week #shows evening night night #sets morning #lav afternoon afternoon month #/+ #’s Night #naissance Loll #genre Kinnikuman semester Weekend #ched morning #ague #enna weekend Saturday latest Sunday #cher week #EST Blossom #icter #Night happens #atto day #vertising happened #spr #essim #Sunday Masquerade #morning #ished #Thursday sounded Week #ching Panc pesky Evening #chy #allery trope #ADVERTISEMENT #feature #Street #fy GPT-2 Base - Layer 0 Dim 1194 ⬇ Pay receipts #Pay depos refund Deposit police deduct #pay #milo #paying #igree #Tax #eln debit levied PayPal deposit ATM #enforcement cops endot tax #soType ID paperwork #payment deposits payment loopholes checkout waivers #police receipt agents waive DMV loophole application arresting card commissioner applications Forms office transporter arrested Dupl #paid confisc pay Clapper #tax #ventures RCMP #Tax PAY whistleblowers APPLIC #ADRA GPT-2 Base - Layer 9 Dim 2771 ⬇ flaws flaws lurking weaknesses failings dangers vulnerabilities scams inaccur shortcomings scams pitfalls shortcomings injust flawed faults glitches flawed pitfalls abuses inconsistencies imperfect rigged lurking biases wrongdoing deficiencies corruption weaknesses inaccur discrepancies inadequ hypocrisy fraud rigging inequ deceptive weakness misinformation scam #urities hazards lur problematic imperfect hoax regress danger #abase failings #errors problems #lived injustice abuses plagiar misinterpret plag suspic deceptive C.4 Knowledge Lookup Given a few seed embeddings of vocabulary items we find related F values by taking a product of the average embeddings with F values. Seed vectors: ["python", "java", "javascript"] Layer 14 Dim 1215 (ranked 3rd) ⬇ filesystem debugging Windows HTTP configure Python debug config Linux Java configuration cache Unix lib runtime kernel plugins virtual FreeBSD hash plugin header file server PHP GNU headers Apache initialization Mozilla Seed vectors: ["cm", "kg", "inches"] Layer 20 Dim 2917 (ranked 1st) ⬇ percent years hours minutes million seconds inches months miles weeks pounds #% kilometers ounces kilograms grams kilometres metres centimeters thousand days km yards Years meters #million acres kg #years inch lstlisting Seed vectors: |["horse", "dog", "lion"]| \\ Layer 21 Dim 3262 (ranked 2nd) lstlisting[backgroundcolor= white] animal animals Animal dogs horse wildlife Animals birds horses dog mammal bird mammals predator beasts Wildlife species #Animal #animal Dogs fish rabbits deer elephants wolves pets veterinary canine beast predators reptiles rodent primates hunting livestock creature rabbit rept elephant creatures human hunters hunter shark Rept cattle wolf Humane tiger lizard lstlisting Appendix D Sentiment Analysis Fine-Tuning Vector Examples This section contains abusive language Classification Head Parameters Below we show the finetuning vector of the classifier weight. “POSITIVE” designates the vector corresponding to the label “POSITIVE”, and similarly for “NEGATIVE”. ⬇ POSITIVE NEGATIVE ----------- ------------ #yssey bullshit #knit lame #etts crap passions incompetent #etooth inco #iscover bland pioneers incompetence #emaker idiots Pione crappy #raft shitty #uala idiot prosper pointless #izons retarded #encers worse #joy garbage cherish CGI loves FUCK #accompan Nope strengthens useless #nect shit comr mediocre honoured poorly insepar stupid embraces inept battled lousy #Together fuck intrig sloppy #jong Worse friendships Worst #anta meaningless In the following sub-sections, we sample 4 difference vectors per each parameter group (F keys, F values; attention query, key, value, and output subheads), and each one of the fine-tuned layers (layers 9-11). We present the ones that seemed to contain relevant patterns upon manual inspection. We also report the number of “good” vectors among the four sampled vectors for each layer and parameter group. F Keys Layer 9 4 out of 4 ⬇ diff -diff --------------- ------------ reperto wrong congratulations unreasonable Citation horribly thanks inept Recording worst rejo egregious Profile #wrong Tradition unfair canopy worse #ilion atro extracts stupid descendant egreg #cele bad enthusiasts terribly :-) ineffective #photo nonsensical awaits awful believer #worst #IDA incompetence welcomes #icably ⬇ diff -diff ------------ ------------ incompetence #knit bullshit #Together crap Together useless versatile pointless #Discover incompetent richness idiots #iscover incompet forefront garbage inspiring meaningless pioneering stupid #accompan crappy unparalleled shitty #Explore nonexistent powerfully worthless #"," Worse #love lame admired worse #uala inco innovative ineffective enjoyed Layer 10 4 out of 4 ⬇ diff -diff ------------------ ------------- isEnabled wonderfully guiActiveUnfocu... beautifully #igate cinem waivers cinematic expires wonderful expire amazing reimb Absolutely expired storytelling #rollment fantastic #Desktop Definitely prepaid unforgettable #verning comedy #andum movie reimbursement comedic Advisory hilarious permitted #movie #pta #Amazing issuance scenes Priebus Amazing #iannopoulos enjoyable ⬇ diff -diff ------------- ------------- #Leaks loving quotas love #RNA loved subsidy lovers #?’" wonderful Penalty lover #iannopoulos nostalgic #>] alot discredited beautiful #conduct amazing #pta great waivers passionate Authorization admire #admin passion HHS lovely arbitrarily loves #arantine unforgettable #ERC proud memorandum inspiration #Federal #love Layer 11 4 out of 4 ⬇ diff -diff --------------- ----------- #SpaceEngineers love nuisance definitely #erous always #aband wonderful Brist loved racket wonderfully Penalty cherish bystand loves #iannopoulos truly Citiz enjoy Codec really courier #olkien #>] beautifully #termination #love incapac great #interstitial LOVE fugitive never breaching adore targ loving thug amazing ⬇ diff -diff ------------ ------------ #knit bullshit passions crap #accompan idiots #ossom goddamn #Explore stupid welcomes shitty pioneering shit forefront garbage embraces fuck pioneers incompetence intertw crappy #izons bogus #iscover useless unparalleled idiot evolving #shit Together pointless vibrant stupidity prosper fucking strengthens nonsense #Together FUCK F Values Layer 9 0 out of 4 Layer 10 0 out of 4 Layer 11 0 out of 4 WQsubscriptQW_QWQ Subheads Layer 9 3 out of 4 ⬇ diff -diff ------------ ----------- bullshit strengthens bogus Also faux #helps spurious adjusts nonsense #ignt nonsensical evolves inept helps crap grew junk grows shitty #cliffe fake recognizes incompetence #assadors crappy regulates phony flourished sloppy improves dummy welcomes mediocre embraces lame gathers outrage greets inco prepares ⬇ diff -diff ---------- ------------ alot Provision kinda coerc amazing Marketable definitely contingency pretty #Dispatch tho seiz hilarious #verning VERY #iannopoulos really #Reporting lol #unicip wonderful Fiscal thats issuance dont provision pics #Mobil doesnt #etooth underrated policymakers funny credential REALLY Penalty #love #activation alright #Officials Layer 10 4 out of 4 ⬇ diff -diff ------------- ------------ love Worse unforgettable Nope beautiful #Instead loved Instead #love #Unless loving incompetence amazing incapable #joy Unless inspiring #failed passion incompet adventure incompetent loves ineffective excitement #Fuck joy #Wr LOVE inept together spurious memories #Failure wonderful worthless enjoyment obfusc themes inadequate ⬇ diff -diff --------- ----------- crap #egu bullshit #etooth shit #verning :( #ounces lol #accompan stupid coh filler #assadors shitty #pherd fucking #acio pointless #uchs idiots strengthens anyways #reprene nonsense Scotia anyway #rocal crappy reciprocal stupidity Newly fuck fost #shit #ospons anymore #onductor Nope governs Layer 11 3 out of 4 ⬇ diff -diff ------------- ------------------ #also meaningless #knit incompetence helps inco strengthens pointless :) incompetent broaden Worse #ossom inept incorporates nonsensical #Learn coward incorporate unint #"," obfusc enjoy excuses enjoyed panicked complementary useless #etts bullshit enhances stupid integrates incompet #ospons incomprehensibl... differs stupidity #arger lifeless ⬇ diff -diff ------------- --------------- amazing #iannopoulos beautifully expired love ABE wonderful Yiannopoulos wonderfully liability unforgettable #SpaceEngineers beautiful #isance loving Politico #love waivers #beaut #utterstock enjoyable excise #Beaut #Stack inspiring phantom fantastic PubMed defin #ilk incredible impunity memorable ineligible greatness Coulter amazingly issuance timeless IDs WKsubscriptKW_KWK Subheads Layer 9 3 out of 4 ⬇ diff -diff ------------- ----------- Then any Instead #ady Unfortunately #imate Why #cussion Sometimes #ze Secondly appreci #Then #raq But currently Luckily #kers Anyway #apixel And active Suddenly significant Thankfully #ade Eventually #imal Somehow specific Fortunately #ability Meanwhile anyone What #ker Obviously #unction Because reap ⬇ diff -diff ----------- --------- bullshit #avorite anyway #ilyn crap #xtap anyways #insula unless #cedented nonsense #aternal #falls #lyak fuck #rieve #. #uana fallacy #accompan #tics #ashtra #punk #icer damned #andum #fuck Mehran stupidity #andise shit #racuse commercials #assadors because #Chel despite rall movies #abella Layer 10 2 out of 4 ⬇ diff -diff -------- --------- #sup #etting Amazing #liness #airs #ktop awesome #ulkan Bless #enthal Loving #enance my #yre #OTHER #eeds #BW omission #perfect #reys #-) #lihood amazing #esian #adult #holes perfect syndrome welcome grievance Rated offenders #Amazing #wig #anch #hole FANT #creen #anche #pmwiki Layer 11 2 out of 4 ⬇ diff -diff -------------- ----------- #ly #say storytelling actionGroup sounding prefers spectacle #ittees #ness #reon #hearted presumably cinematic waivers #est #aucuses portrayal #Phase quality #racuse paced #arge combination #hers juxtap #sup representation #later mixture expired #!!!!! stricter filmmaking #onds enough #RELATED thing #rollment rendition #orders WVsubscriptVW_VWV Subheads Layer 9 4 out of 4 ⬇ diff -diff -------- ------------- crap jointly shit #verning bullshit #pora fucking #rocal idiots #raft fuck #etooth goddamn #estead stupid #ilitation FUCK #ourse #fuck migr shitty #ourses damn #iership #shit Pione lol #iscover fuckin pioneering nonsense #egu crappy #ivities kinda neighbourhood Fuck pioneer idiot nurt ⬇ diff -diff --------- -------------- anime #rade kinda #jamin stuff #ounces shit #pherd lol Unable tho #pta realism Roche damn Payments :) Gupta fucking #odan alot #uez movie #adr funny #ideon anyways #Secure enjoyable #raught crap Bei comedy sovere genre unsuccessfully anyway #moil fun #Register Layer 10 4 out of 4 ⬇ diff -diff ----------- --------- #"]," crap #verning stupid #etooth shit #"," fucking Browse fuck #Register shitty #Lago bullshit #raft crappy #egu idiots jointly horrible #iership stupidity strengthens kinda Scotia goddamn #ounces awful #uania mediocre #iann pathetic workspace #fuck seiz damn Payments FUCK #Learn damned ⬇ diff -diff ------------ ------------- bullshit Pione crap pioneers stupid pioneering nonsense complementary incompetence #knit idiots #Learn shit #accompan stupidity pioneer pointless invaluable inco #ossom retarded #Together idiot Browse vomit versatile lame welcomes meaningless #"," goddamn admired nonsensical jointly garbage Sharing #shit Together useless #Discover Layer 11 4 out of 4 ⬇ diff -diff ------------ --------- crap #rocal fucking #verning bullshit #etooth fuck #uania goddamn caches shit Browse #fuck #"," stupidity #imentary pathetic exerc spoiler #Lago stupid #"]," inept #cium blah #enges FUCK #ysis awful quarterly shitty #iscover trope Scotia Godd #resso inco #appings incompetence jointly ⬇ diff -diff ------------ ------------- Worse #knit bullshit pioneers Nope pioneering crap inspiring incompetence #iscover idiots complementary incompetent pioneer stupid #ossom incompet passionate pointless passions inco journeys Stupid unique meaningless embraces nonsense admired lame forefront idiot richness worse invaluable #Fuck prosper whining vibrant nonsensical enriched WOsubscriptOW_OWO Subheads Layer 9 0 out of 4 Layer 10 0 out of 4 Layer 11 0 out of 4