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parse/dev/0IywQ8uxJx/0IywQ8uxJx.md

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+# Graph Neural Networks as Gradient Flows
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Dynamical systems minimizing an energy are ubiquitous in geometry and physics.   
+2 We propose a gradient flow framework for GNNs where the equations follow the   
+3 direction of steepest descent of a learnable energy. This approach allows to analyse   
+4 the GNN evolution from a multi-particle perspective as learning attractive and   
+5 repulsive forces in feature space via the positive and negative eigenvalues of a   
+6 symmetric ‘channel-mixing’ matrix. We perform spectral analysis of the solutions   
+7 and conclude that gradient flow graph convolutional models can induce a dynamics   
+8 dominated by the graph high frequencies, which is desirable for heterophilic   
+9 datasets. We also describe structural constraints on common GNN architectures   
+10 allowing to interpret them as gradient flows. We perform thorough ablation studies   
+11 corroborating our theoretical analysis and show competitive performance of simple   
+12 and lightweight models on real-world homophilic and heterophilic datasets.
+
+# 13 1 Introduction and motivations
+
+14 Graph neural networks (GNNs) [38, 20, 21, 36, 7, 15, 27] and in particular their Message Passing   
+15 formulation (MPNN) $\boxed { 1 9 }$ have become the standard ML tool for dealing with different types of   
+16 relations and interactions, ranging from social networks to particle physics and drug design. One   
+17 of the often cited drawbacks of traditional GNN models is their poor ‘explainability’, making it   
+18 hard to know why and how they make certain predictions $\boxed { 1 4 6 } \boxed { 4 7 }$ , and in which situations they   
+19 may work and when they would fail. Limitations of GNNs that have attracted attention are over  
+20 smoothing $\pm \boxed { 2 9 } \boxed { 3 0 } \boxed { 8 }$ , over-squashing and bottlenecks $\boxed { 1 } \boxed { 4 0 } \boxed { }$ , and performance on heterophilic data   
+21 [31, 51, 13, 4, 45] – where adjacent nodes usually have different labels.   
+22 Contributions. We propose a Gradient Flow Framework   
+23 (GRAFF) where the GNN equations follow the direction of steep  
+24 est descent of a learnable energy. Thanks to this framework we can   
+25 (i) interpret GNNs as a multi-particle dynamics where the learned   
+26 parameters determine pairwise attractive and repulsive potentials   
+27 in the feature space. This sheds light on how GNNs can adapt to   
+28 heterophily and explains their performance and the smoothness of   
+29 the prediction. (ii) GRAFF leads to residual convolutional models   
+30 where the channel-mixing W is performed by a shared symmet  
+31 ric bilinear form inducing attraction and repulsion via its positive   
+32 and negative eigenvalues, respectively. We theoretically investi  
+33 gate the interaction of the graph spectrum with the spectrum of the   
+34 channel-mixing, proving that if there is more mass on the negative   
+35 eigenvalues of W, then the dynamics is dominated by the graph  
+36 high frequencies, which could be desirable on heterophilic graphs.   
+37 We also extend results of $\boxed { 2 9 } \boxed { 3 0 } \boxed { 8 }$ by showing that when we drop   
+38 the residual connection intrinsic to the gradient flow framework,   
+39 graph convolutional models always induce a low-frequency dominated dynamics independent of the   
+40 sign and magnitude of the spectrum of the channel-mixing. We also discuss how simple choices   
+41 make common architectures fit GRAFF and conduct thorough ablation studies to corroborate the the  
+42 oretical analysis on the role of the spectrum of W. (iii) We crystallize an instance of our framework   
+43 into a linear, residual, convolutional model that achieves competitive performance on homophilic and   
+44 heterophilic real world graphs whilst being faster than GCN.   
+45 Related work. Our analysis is related to studying GNNs as filters on the graph spectrum [15, 24,   
+46 2, 25] and over-smoothing $\pmb { \bigtriangledown } \bigtriangledown \bigtriangledown \bigtriangledown \bigtriangledown \bigtriangledown \bigtriangledown \bigtriangledown \bigtriangledown \bigtriangledown$ and partly adopts techniques similar to $\textcircled { 1 3 0 } \textcircled { 1 }$ . The key   
+47 difference is that we also consider the spectrum of the ‘channel-mixing’ matrix. The concept of   
+48 gradient flows has been a standard tool in physics and geometry [16], from which they were adopted   
+49 for image processing $\pmb { \mathbb { \mathbb { \mathbb { 2 6 } } } }$ , and recently used in ML $\textcircled { 1 3 5 }$ for the analysis of Transformers [41] – see   
+50 also $\boxed { \boxed { 1 8 } }$ for discussion of loss landscapes. Our continuous-time evolution equations follows the spirit   
+51 of Neural ODES [22, 12, 3] and the study of GNNs as continuous dynamical systems [44, 10, 17, 9].   
+52 Outline. In Section 2, we review the continuous and discrete Dirichlet energy and the associated   
+53 gradient flow framework. We formalize the notion of over-smoothing and low(high)-frequency  
+54 dominated dynamics to investigate GNNs and study the dominant components in their evolution. We   
+55 extend the graph Dirichlet energy to allow for a non-trivial norm for the feature edge-gradient. This   
+56 leads to gradient flow equations that diffuse the features and over-smooth in the limit. Accordingly,   
+57 in Section $^ 3$ we introduce a more general energy with a symmetric channel-mixing matrix W giving   
+58 rise to attractive and repulsive pairwise terms via its positive and negative eigenvalues and show   
+59 that the negative spectrum can induce high-frequency-dominant dynamics. In Section 4 we first   
+60 compare with continuous GNN models and then discretize the equations and provide a ‘recipe’ for   
+61 making standard GNN architectures fit a gradient flow framework. We adapt the spectral analysis to   
+62 discrete-time showing that gradient flow convolutional models can generate a dynamics dominated by   
+63 the high frequencies via the negative eigenvalues of W while this is impossible if we drop the residual   
+64 connection. In Section $. 5$ we corroborate our theoretical analysis on the role of the spectrum of W   
+65 via ablation studies on graphs with varying homophily. Experiments on real world datasets show a   
+66 competitive performance of our model despite its simplicity and reduced number of parameters.
+
+![](images/bbd3047d2cbd0b9bc92189a22e8836ee7d60b4c122913cbc88bde0574145b67a.jpg)  
+Figure 1: GRAFF dynamics: attractive and repulsive forces lead to a non-smoothing process able to separate labels.
+
+# 67 2 Gradient-flow formalism
+
+68 Notations adopted throughout the paper. Let $\mathsf { G } = ( \mathsf { V } , \mathsf { E } )$ be an undirected graph with $n$ nodes.   
+69 We denote by $\textbf { F } \in \mathbb { R } ^ { n \times d }$ the matrix of $d$ -dimensional node features, by $\mathbf { f } _ { i } ~ \in ~ \mathbb { R } ^ { d }$ its $i$ -th row   
+70 (transposed), by $\mathbf { f } ^ { r } \in \mathbb { R } ^ { n }$ its $r$ -th column, and by $\mathrm { v e c } ( \mathbf { F } ) \in \mathbb { R } ^ { n d }$ the vectorization of $\mathbf { F }$ obtained   
+71 by stacking its columns. Given a symmetric matrix $\mathbf { B }$ , we let $\lambda _ { + } ^ { \mathbf { B } }$ , $\lambda _ { - } ^ { \mathbf { B } }$ denote its most positive and   
+72 negative eigenvalues, respectively, and $\rho _ { \mathbf { B } }$ be its spectral radius. If $\mathbf { B } \succeq 0$ , then $\mathrm { g a p } ( \mathbf { B } )$ denotes the   
+73 positive smallest eigenvalue of $\mathbf { B }$ . ${ \dot { f } } ( t )$ denotes the temporal derivative, $\otimes$ is the Kronecker product   
+74 and ‘a.e.’ means almost every w.r.t. Lebesgue measure and usually refers to data in the complement   
+75 of some lower dimensional subspace in $\mathbb { R } ^ { \bar { n } \times d }$ . Proofs and additional results appear in the Appendix.   
+76 Starting point: a geometric parallelism. To motivate a gradient-flow approach for GNNs, we start   
+77 from the continuous case (see Appendix A.1 for details). Consider a smooth map $f : \mathbb { R } ^ { n } \to ( \mathbb { R } ^ { d } , h )$   
+78 with $h$ a constant metric represented by $\mathbf { \overline { { H } } } \succeq 0$ . The Dirichlet energy of $f$ is defined by
+
+$$
+\mathcal { E } ( f , h ) = \frac { 1 } { 2 } \int _ { \mathbb { R } ^ { n } } \| \nabla f \| _ { h } ^ { 2 } d x = \frac { 1 } { 2 } \sum _ { q , r = 1 } ^ { d } \sum _ { j = 1 } ^ { n } \int _ { \mathbb { R } ^ { n } } h _ { q r } \partial _ { j } f ^ { q } \partial _ { j } f ^ { r } ( x ) d x
+$$
+
+79 and measures the ‘smoothness’ of $f$ . A natural approach to find minimizers of $\mathcal { E }$ - called harmonic   
+80 maps - was introduced in $\mathbb { I } \mathbb { 1 6 } ]$ and consists in studying the gradient flow of $\mathcal { E }$ , wherein a given map   
+81 $f ( 0 ) = f _ { 0 }$ is evolved according to $\dot { f } ( t ) = - \nabla _ { f } \mathcal { E } ( f ( t ) )$ . These type of evolution equations have   
+82 historically been the core of variational and $P D E$ -based image processing; in particular, gradient   
+83 flows of the Dirichlet energy were shown $[ \overline { { 2 6 } } ]$ to recover the Perona-Malik nonlinear diffusion $\pmb { \left. 3 2 \right. }$ .   
+84 Motivation: GNNs for node-classification. We wish to extend the gradient flow formalism to node   
+85 classification on graphs. Assume we have a graph $\sf G$ , node-features $\mathbf { F } _ { 0 }$ and labels $\{ y _ { i } \}$ on $\mathsf { V } _ { \mathrm { t r a i n } } \subset \mathsf { V }$   
+86 and that we want to predict the labels on $\mathsf { V } _ { \mathrm { t e s t } } \subset \mathsf { V }$ . A GNN typically evolves the features via some   
+87 parametric rule, ${ \mathrm { G N N } } _ { \theta } ( { \mathsf { G } } , { \mathbf { F } } _ { 0 } )$ , and uses a decoding map for the prediction $y = \psi _ { \mathrm { D E } } ( \mathrm { G N N } _ { \theta } ( \mathsf { G } , \mathbf { F } _ { 0 } ) )$ .   
+88 In graph convolutional models $\mathbb { I m }$ , $\mathrm { G N N } _ { \theta }$ consists of two operations: applying a shared linear   
+89 transformation to the features (‘channel mixing’) and propagating them along the edges of the graph   
+90 (‘diffusion’). Our goal consists in studying when $\mathrm { G N N } _ { \theta }$ is the gradient flow of some parametric class   
+91 of energies $\mathcal { E } _ { \theta } : \mathbb { R } ^ { \bar { n } \times d }  \mathbb { R }$ , which generalize the Dirichlet energy. This means that the parameters   
+92 can be interpreted as ‘finding the right notion of smoothness’ for our task. We evolve the features by   
+93 $\dot { \mathbf { F } } ( t ) = - \bar { \nabla _ { \mathbf { F } } } \mathcal { E } _ { \theta } ( \mathbf { F } ( t ) )$ with prediction $y = \psi _ { \mathrm { D E } } ( \mathbf { F } ( T ) )$ for some optimal time $T$ .   
+94 Why a gradient flow? Since $\dot { \mathcal { E } } _ { \theta } ( { \bf F } ( t ) ) = - | | \nabla _ { \bf F } \mathcal { E } _ { \theta } ( { \bf F } ( t ) ) | | ^ { 2 }$ , the energy dissipates along the gradient   
+95 flow. Accordingly, this framework allows to explain the GNN dynamics as flowing the node features   
+96 in the direction of steepest descent of $\mathcal { E } _ { \theta }$ . Indeed, we find that parametrizing an energy leads to   
+97 equations governed by attractive and repulsive forces that can be controlled via the spectrum of   
+98 symmetric ‘channel-mixing’ matrices. This shows that by learning to distribute more mass over the   
+99 negative (positive) eigenvalues of the channel-mixing, gradient flow models can generate dynamics   
+100 dominated by the higher (respectively, lower) graph frequencies and hence tackle different homophily   
+101 scenarios. The gradient flow framework also leads to sharing of the weights across layers (since we   
+102 parametrize the energy rather than the evolution equations, as usually done in GNNs), allowing us to   
+103 reduce the number of parameters without compromising performance (see Table 1).   
+104 Analysis on graphs: preliminaries. Given a connected graph G with self-loops, its adjacency   
+105 matrix A is defined as $a _ { i j } = 1$ if $( i , j ) \in { \mathsf { E } }$ and zero otherwise. We let $\mathbf { D } = \mathrm { d i a g } ( \bar { d } _ { i } )$ be the degree   
+106 matrix and write $\bar { \mathbf { A } } : = \mathbf { D } ^ { - 1 / 2 } \mathbf { A } \mathbf { D } ^ { - 1 / 2 }$ . Let $\mathbf { F } \in \mathbb { R } ^ { n \times d }$ be the matrix representation of a signal. Its   
+107 graph gradient is $( \nabla { \bf F } ) _ { i j } : = { \bf f } _ { j } / \sqrt { d _ { j } } - { \bf f } _ { i } / \sqrt { d _ { i } }$ . We define the Laplacian as $\begin{array} { r } { \Delta : = - \frac { 1 } { 2 } \mathrm { d i v } \bigtriangledown } \end{array}$ (the   
+108 divergence div is the adjoint of $\nabla$ ), represented by $\pmb { \Delta } = \mathbf { I } - \bar { \mathbf { A } } \succeq 0$ . We refer to the eigenvalues of   
+109 $\pmb { \Delta }$ as frequencies: the lowest frequency is always 0 while the highest frequency is $\rho \Delta \stackrel { - } { \leq } 2$ [14]. As   
+110 for the continuum case, the gradient allows to define a (graph) Dirichlet energy as [49]
+
+$$
+\mathcal { E } ^ { \mathrm { D i r } } ( \mathbf { F } ) : = \frac { 1 } { 4 } \sum _ { i } \sum _ { j : ( i , j ) \in \mathsf { E } } | | ( \nabla \mathbf { F } ) _ { i j } | | ^ { 2 } \equiv \frac { 1 } { 4 } \sum _ { ( i , j ) \in \mathsf { E } } | | \frac { \mathbf { f } _ { i } } { \sqrt { d _ { i } } } - \frac { \mathbf { f } _ { j } } { \sqrt { d _ { j } } } | | ^ { 2 } = \frac { 1 } { 2 } \mathrm { t r a c e } ( \mathbf { F } ^ { \top } \Delta \mathbf { F } ) ,
+$$
+
+where the extra 111 $\begin{array} { l } { { \frac { 1 } { 2 } } } \end{array}$ is for convenience. As for manifolds, ${ \mathcal { E } } ^ { \mathrm { { D i r } } }$ measures smoothness. If we stack the columns of 112 $\mathbf { F }$ into $\mathrm { v e c } ( \mathbf { F } ) \in \mathbb { R } ^ { n d }$ , the gradient flow of ${ \mathcal { E } } ^ { \mathrm { { D i r } } }$ yields the heat equation on each channel:
+
+$$
+\operatorname { v e c } ( { \dot { \mathbf { F } } } ( t ) ) = - \nabla _ { \operatorname { v e c } ( \mathbf { F } ) } { \mathcal { E } } ^ { \mathrm { D i r } } ( \operatorname { v e c } ( \mathbf { F } ( t ) ) ) = - ( \mathbf { I } _ { d } \otimes \Delta ) \operatorname { v e c } ( \mathbf { F } ( t ) ) \iff { \dot { \mathbf { f } } } ^ { r } ( t ) = - \Delta \mathbf { f } ^ { r } ( t ) ,
+$$
+
+113 for $1 \leq r \leq d$ . Similarly to $\textcircled { 8 }$ , we rely on ${ \mathcal { E } } ^ { \mathrm { { D i r } } }$ to assess whether a given dynamics $t \mapsto \mathbf { F } ( t )$ is a   
+114 smoothing process. A different choice of Laplacian $\mathbf { L } = \mathbf { D } - \mathbf { A }$ with non-normalized adjacency   
+115 induces the analogous Dirichlet energy $\begin{array} { r } { \mathcal E _ { \mathbf { L } } ^ { \mathrm { D i r } } ( \mathbf { \dot { F } } ) = \frac { 1 } { 2 } \mathrm { t r a c e } ( \mathbf { F } ^ { \top } \mathbf { L } \mathbf { F } ) } \end{array}$ . Throughout this paper, we rely   
+116 on the following definitions (see Appendix A.3 for further equivalent formulations and justifications):   
+117 Definition 2.1. $\dot { \mathbf { F } } ( t ) = \mathrm { G N N } _ { \theta } ( \mathbf { F } ( t ) , t )$ initialized at $\mathbf F ( 0 )$ is smoothing if $\mathcal { E } ^ { \mathrm { D i r } } ( { \bf F } ( t ) ) \leq C + \varphi ( t )$ ,   
+118 with $C$ a constant only depending on $\dot { \mathcal { E } } ^ { \mathrm { D i r } } ( \mathbf { F } ( 0 ) )$ and $\dot { \varphi } ( t ) \leq 0$ . Over-smoothing occurs if either   
+119 $\mathcal { E } ^ { \mathrm { D i r } } ( { \bf F } ( t ) )  0$ or $\mathcal { E } _ { \mathbf { L } } ^ { \mathrm { D i r } } ( \mathbf { F } ( t ) )  \mathrm { 0 }$ Efor $t \to \infty$ .   
+120 Our notion of ‘over-smoothing’ is a relaxed version of the definition in $\pm \mathbb { I } -$ although in the linear   
+121 case one always finds an exponential decay of ${ \mathcal { E } } ^ { \mathrm { D i r } }$ . We note that $\mathcal { E } ^ { \mathrm { D i r } } ( { \bf F } ( t ) )  0$ iff $\Delta \mathbf { f } ^ { r } ( t )  \mathbf { 0 }$ for   
+122 each column $\mathbf { f } ^ { r }$ . As in $\textcircled { 1 3 0 }$ , this corresponds to a loss of separation power along the solution where   
+123 nodes with equal degree become indistinguishable since we converge to $\ker ( \Delta )$ (if we replaced $\pmb { \Delta }$   
+124 with $\mathbf { L }$ then we would not even be able to separate nodes with different degrees in the limit).   
+125 To motivate the next definition, consider $\dot { \mathbf { F } } ( t ) = \bar { \mathbf { A } } \mathbf { F } ( t )$ . Despite $| | \mathbf { F } ( t ) | |$ being unbounded for a.e.   
+126 $\mathbf F ( 0 )$ , the low-frequency components are growing the fastest and indeed ${ \bf F } ( t ) \bar { \ } | | { \bf F } ( t ) | |  { \bf F } _ { \infty }$ s.t.   
+127 $\Delta \mathbf { f } _ { \infty } ^ { r } = \mathbf { 0 }$ for $1 \leq r \leq d$ . We formalize this scenario – including the opposite case of high-frequency   
+128 components being dominant – by studying $\mathcal { E } ^ { \mathrm { D i r } } ( { \bf F } ( t ) / | | { \bf F } ( t ) | | )$ , i.e. the Rayleigh quotient of $\mathbf { I } _ { d } \otimes \Delta$ .   
+129 Definition 2.2. $\dot { \mathbf { F } } ( t ) ~ = ~ \mathrm { G N N } _ { \theta } ( \mathbf { F } ( t ) , t )$ initialized at $\mathbf F ( 0 )$ is Low/High-Frequency-Dominant   
+130 (L/HFD) if $\mathcal { E } ^ { \mathrm { D i r } } ( { \bf F } ( t ) / | | { \bf F } ( t ) | | )  0$ (respectively, $\mathcal { E } ^ { \mathrm { D i r } } ( { \bf F } ( t ) / | | { \bf F } ( t ) | | )  \rho _ { \Delta } / 2 )$ for $t \to \infty$ .   
+131 We report a consequence of Definition $| 2 . 2 |$ and refer to Appendix ${ \bf A } . 3$ for additional details and   
+132 motivations for the characterizations of $\overline { { \mathrm { L F D } } }$ and HFD.   
+133 Lemma 2.3. $\mathrm { G N N } _ { \theta }$ is LFD (HFD) iff for each $t _ { j } ~  ~ \infty$ there exist $t _ { j _ { k } } \ \to \ \infty$ and $\mathbf { F } _ { \infty }$ s.t.   
+134 $\mathbf { F } ( t _ { j _ { k } } ) / | | \mathbf { F } ( t _ { j _ { k } } ) | |  \mathbf { F } _ { \infty }$ and $\pmb { \Delta } \mathbf { f } _ { \infty } ^ { r } = \mathbf { 0 }$ ( $\Delta \mathbf { f } _ { \infty } ^ { r } = \rho \Delta \mathbf { \bar { f } } _ { \infty } ^ { r }$ , respectively).   
+135 If a graph is homophilic, adjacent nodes are likely to share the same label and we expect a smoothing   
+136 or LFD dynamics enhancing the low-frequency components to be successful at node classification   
+137 tasks $\underline { { \lVert 4 3 \rVert } } \dot { \lVert 2 8 \rVert }$ . In the opposite case of heterophily, the high-frequency components might contain more   
+138 relevant information for separating classes $i \boxed { 4 } \boxed { 5 } \rbrack$ – the prototypical example being the eigenvector of   
+139 $\pmb { \Delta }$ associated with largest frequency $\rho _ { \Delta }$ separating a regular bipartite graph. In other words, the class   
+140 of heterophilic graphs contain instances where signals should be sharpened by increasing ${ \mathcal { E } } ^ { \mathrm { { D i r } } }$ rather   
+141 than smoothed out. Accordingly, an ideal framework for learning on graphs must accommodate both   
+142 of these opposite scenarios by being able to induce either an LFD or a HFD dynamics.   
+143 Parametric Dirichlet energy: channel-mixing as metric in feature space. In eq. $\boxed { 1 }$ a constant   
+144 nontrivial metric $h$ in $\mathbb { R } ^ { d }$ leads to the mixing of the feature channels. We adapt this idea by considering   
+145 a symmetric positive semi-definite $\mathbf { H } = \bar { \mathbf { W } } ^ { \top } \mathbf { W }$ with $\mathbf { W } \in \mathbb { R } ^ { d \times d }$ and using it to generalize ${ \mathcal { E } } ^ { \mathrm { { D i r } } }$ as
+
+$$
+\mathcal { E } _ { \mathbf { W } } ^ { \mathrm { D i r } } ( \mathbf { F } ) : = \frac { 1 } { 4 } \sum _ { q , r = 1 } ^ { d } \sum _ { i } \sum _ { j : ( i , j ) \in \mathsf { E } } h _ { q r } ( \nabla \mathbf { f } ^ { q } ) _ { i j } ( \nabla \mathbf { f } ^ { r } ) _ { i j } = \frac { 1 } { 4 } \sum _ { ( i , j ) \in \mathsf { E } } | | \mathbf { W } ( \nabla \mathbf { F } ) _ { i j } | | ^ { 2 } .
+$$
+
+146 We note the analogy with eq. $\mathbb { \underline { { \left( 1 \right) } } }$ , where the sum over the nodes replaces the integration over the   
+147 148 domain and the We generally tr $j$ -t $\mathbf { W }$ erivative at some point as learnable weights a $_ { i }$ is replaced by the gradientd study the gradient flow of $\mathcal { E } _ { \mathbf { W } } ^ { \mathrm { D i r } }$ g the edge : $( i , j ) \in \mathsf E$
+
+$$
+\dot { \mathbf { F } } ( t ) = - \nabla _ { \mathbf { F } } \mathcal { E } _ { \mathbf { W } } ^ { \mathrm { D i r } } ( \mathbf { F } ( t ) ) = - \Delta \mathbf { F } ( t ) \mathbf { W } ^ { \top } \mathbf { W } .
+$$
+
+149 We see that eq. $( 5 )$ generalizes eq. $( 3 )$ . Below ‘smoothing’ is intended as in Definition 2.1.
+
+Proposition 2.4. Let 150 $P _ { \mathbf { W } } ^ { \mathrm { k e r } }$ be the projection onto $\ker ( \mathbf { W } ^ { \top } \mathbf { W } )$ . Equation $\textcircled { 5 }$ is smoothing since
+
+$$
+\begin{array} { r } { \mathcal { E } ^ { \mathrm { D i r } } ( \mathbf { F } ( t ) ) \leq e ^ { - 2 t \mathrm { g a p } ( \mathbf { W } ^ { \top } \mathbf { W } ) \mathrm { g a p } ( \Delta ) } | | \mathbf { F } ( 0 ) | | ^ { 2 } + \mathcal { E } ^ { \mathrm { D i r } } ( ( P _ { \mathbf { W } } ^ { \mathrm { k e r } } \otimes \mathbf { I } _ { n } ) \mathrm { v e c } ( \mathbf { F } ( 0 ) ) ) , \quad t \geq 0 . } \end{array}
+$$
+
+In fact 151 $\mathbf { F } ( t )  \mathbf { F } _ { \infty } s . t . \exists \phi _ { \infty } \in \mathbb { R } ^ { d } .$ : for each $i \in \mathsf { V }$ we have $( \mathbf { f } _ { \infty } ) _ { i } = \sqrt { d _ { i } } \phi _ { \infty } + P _ { \mathbf { W } } ^ { \mathrm { k e r } } \mathbf { f } _ { i } ( 0 ) .$
+
+152 Proposition $2 . 4$ implies that no weight matrix $\mathbf { W }$ in eq. $( \bar { 5 } )$ can separate the limit embeddings $\mathbf { F } ( \infty )$   
+153 of nodes with same degree and input features. If W has a trivial kernel, then nodes with same degrees   
+154 converge to the same representation and over-smoothing occurs as per Definition $2 . 1 .$ Differently   
+155 from $\frac { 1 } { \vert 2 9 \vert } \textcircled { 3 0 } \textcircled { 8 }$ , over-smoothing occurs independently of the spectral radius of the ‘channel-mixing’   
+156 if its eigenvalues are positive – even for equations which lead to residual GNNs when discretized   
+157 [12]. According to Proposition $\underline { { \left. \left[ 2 . 4 \right] \right. } }$ we do not expect eq. $\textcircled{5}$ to succeed on heterophilic graphs where   
+158 smoothing processes are generally harmful – this is confirmed in Figure $\bigstar$ (see prod-curve). To   
+159 remedy this problem, we generalize eq. $_ { ( 5 ) }$ to a gradient flow that can be HFD as per Definition 2.2.
+
+# 160 3 A general parametric energy for pairwise interactions
+
+We first rewrite the energy 161 $\mathcal { E } _ { \mathbf { W } } ^ { \mathrm { D i r } }$ in eq. (4) as
+
+$$
+\mathcal { E } _ { \mathbf { W } } ^ { \mathrm { D i r } } ( \mathbf { F } ) = \frac { 1 } { 2 } \sum _ { i } \langle \mathbf { f } _ { i } , \mathbf { W } ^ { \top } \mathbf { W } \mathbf { f } _ { i } \rangle - \frac { 1 } { 2 } \sum _ { i , j } \bar { a } _ { i j } \langle \mathbf { f } _ { i } , \mathbf { W } ^ { \top } \mathbf { W } \mathbf { f } _ { j } \rangle .
+$$
+
+162 We then define a new, more general energy by replacing the occurrences of $\mathbf { W } ^ { \top } \mathbf { W }$ with new symmetric matrices 163 $\boldsymbol { \Omega } , \mathbf { W } \in \breve { \mathbb { R } } ^ { d \times d }$ since we also want to generate repulsive forces:
+
+$$
+\mathcal { E } ^ { \mathrm { t o t } } ( \mathbf { F } ) : = \frac { 1 } { 2 } \sum _ { i } \langle \mathbf { f } _ { i } , \boldsymbol { \Omega } \mathbf { f } _ { i } \rangle - \frac { 1 } { 2 } \sum _ { i , j } \bar { a } _ { i j } \langle \mathbf { f } _ { i } , \mathbf { W } \mathbf { f } _ { j } \rangle \equiv \mathcal { E } _ { \Omega } ^ { \mathrm { e x t } } ( \mathbf { F } ) + \mathcal { E } _ { \mathbf { W } } ^ { \mathrm { p a i r } } ( \mathbf { F } ) ,
+$$
+
+164 with associated gradient flow of the form (see Appendix B)
+
+$$
+\dot { \mathbf { F } } ( t ) = - \nabla _ { \mathbf { F } } \mathcal { E } ^ { \mathrm { t o t } } ( \mathbf { F } ( t ) ) = - \mathbf { F } ( t ) \boldsymbol { \Omega } + \bar { \mathbf { A } } \mathbf { F } ( t ) \mathbf { W } .
+$$
+
+Note that eq.165 $( 8 ) \ : i s$ gradient flow of some energy $\mathbf { F } \mapsto \mathcal { E } ^ { \mathrm { t o t } } ( \mathbf { F } )$ iff both $\pmb { \Omega }$ and W are symmetric.
+
+166 A multi-particle system point of view: attraction vs repulsion. Consider the $d$ -dimensional   
+167 node-features as particles in $\mathbb { R } ^ { d }$ with energy ${ \mathcal { E } } ^ { \mathrm { t o t } }$ . While the term $\mathcal { E } _ { \Omega } ^ { \mathrm { e x t } }$ is independent of the graph   
+168 topology and represents an external field in the feature space, the second term $\mathcal { E } _ { \mathbf { W } } ^ { \mathrm { p a i r } }$ constitutes a   
+169 potential energy, with W a bilinear form determining the pairwise interactions of adjacent node
+
+representations. Given a symmetric 170 $\mathbf { W }$ , we write $\mathbf { W } = \Theta _ { + } ^ { \top } \Theta _ { + } - \Theta _ { - } ^ { \top } \Theta _ { - }$ , by decomposing the spectrum of W in positive and negative values.We can rewrite 171 $\mathcal { E } ^ { \mathrm { t o t } } = \mathcal { E } _ { \Omega - \mathbf { W } } ^ { \mathrm { e x t } } + \mathcal { E } _ { \Theta _ { + } } ^ { \mathrm { D i r } } - \mathcal { E } _ { \Theta _ { - } } ^ { \mathrm { D i r } }$ , i.e.
+
+$$
+\mathcal { E } ^ { \mathrm { t o t } } ( \mathbf { F } ) = \frac { 1 } { 2 } \sum _ { i } \langle \mathbf { f } _ { i } , ( \Omega - \mathbf { W } ) \mathbf { f } _ { i } \rangle + \frac { 1 } { 4 } \sum _ { i , j } \vert \vert \Theta _ { + } ( \nabla \mathbf { F } ) _ { i j } \vert \vert ^ { 2 } - \frac { 1 } { 4 } \sum _ { i , j } \vert \vert \Theta _ { - } ( \nabla \mathbf { F } ) _ { i j } \vert \vert ^ { 2 } .
+$$
+
+172 The gradient flow of ${ \mathcal { E } } ^ { \mathrm { t o t } }$ minimizes $\mathcal { E } _ { \Theta _ { + } } ^ { \mathrm { D i r } }$ and maximizes $\mathcal { E } _ { \Theta _ { - } } ^ { \mathrm { D i r } }$ . The matrix $\mathbf { W }$ encodes repulsive   
+173 pairwise interactions via its negative-definite component $\Theta _ { - }$ which lead to terms $| | \Theta _ { - } ( \nabla \mathbf { F } ) _ { i j } | |$   
+174 increasing along the solution. The latter affords a ‘sharpening’ effect desirable on heterophilic graphs   
+175 where we need to disentangle adjacent node representations and hence ‘magnify’ the edge-gradient.   
+176 Spectral analysis of the channel-mixing. We will now show that eq. $( 8 )$ can lead to a HFD   
+177 dynamics. To this end, we assume that $\mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega } \mathbf { \Omega }$ so that eq. $\textcircled { 8 }$ becomes $\dot { \mathbf { F } } ( t ) = \bar { \mathbf { A } } \mathbf { F } ( t ) \mathbf { W }$ . According   
+178 179 to eq.dyna $\textcircled { 9 }$ the negative eigens as per Definition $2 . 2$ s of W  We let $P _ { \mathbf { W } } ^ { \rho _ { - } }$ to repulsion. We show that the latter can induce HFDbe the orthogonal projection into the eigenspace of   
+180 $\mathbf { \bar { W } } \otimes \bar { \mathbf { A } }$ associated with the eigenvalue $\rho _ { - } : = | \lambda _ { - } ^ { \mathbf { W } } | ( \rho _ { \Delta } - 1 )$ . We define $\epsilon _ { \mathrm { H F D } }$ explicitly in eq. $\textcircled { 2 4 }$ .
+
+Proposition 3.1. I181 $f \rho _ { - } > \lambda _ { + } ^ { \mathbf { W } }$ , then $\dot { \mathbf { F } } ( t ) = \bar { \mathbf { A } } \mathbf { F } ( t ) \mathbf { W }$ is HFD for a.e. $\mathbf F ( 0 )$ : there exists ✏HFD s.t.
+
+$$
+\mathcal { E } ^ { \mathrm { D i r } } ( \mathbf { F } ( t ) ) = e ^ { 2 t \rho _ { - } } \left( \frac { \rho _ { \Delta } } { 2 } | | P _ { \mathbf { W } } ^ { \rho - } \mathbf { F } ( 0 ) | | ^ { 2 } + \mathcal { O } ( e ^ { - 2 t \epsilon _ { \mathrm { H F D } } } ) \right) , \quad t \geq 0 ,
+$$
+
+and 182 $\mathbf { F } ( t ) / | | \mathbf { F } ( t ) | |$ converges to $\mathbf { F } _ { \infty } \in \mathbb { R } ^ { n \times d }$ such that $\pmb { \Delta } \mathbf { f } _ { \infty } ^ { r } = \rho \pmb { \Delta } \mathbf { f } _ { \infty } ^ { r }$ , for $1 \leq r \leq d $
+
+183 Proposition $\boxed { 3 . 1 }$ shows that if enough mass of the spectrum of the ‘channel-mixing’ is distributed over   
+184 the negative eigenvalues, then the evolution is dominated by the graph high frequencies. This analysis   
+185 is made possible in our gradient flow framework where W must be symmetric. The HFD dynamics   
+186 induced by negative eigenvalues of $\mathbf { W }$ is confirmed in Figure $\bigtriangledown$ (neg-prod-curve in the bottom chart).
+
+187 A more general energy. Equations with a source term may have better expressive power [44, 11, 39]. 188 In our framework this means adding an extra energy term of the form $\mathcal { E } _ { \tilde { \mathbf { W } } } ^ { \mathrm { s o u r c e } } ( \mathbf { F } ) : = \beta \langle \mathbf { F } , \mathbf { F } ( 0 ) \tilde { \mathbf { W } } \rangle$ to eq.189 $\textcircled { 7 }$ with some learnable $\beta$ and $\tilde { \mathbf { W } }$ . This leads to the following gradient flow:
+
+$$
+\dot { \mathbf { F } } ( t ) = - \mathbf { F } ( t ) \pmb { \Omega } + \bar { \mathbf { A } } \mathbf { F } ( t ) \mathbf { W } - \beta \mathbf { F } ( 0 ) \tilde { \mathbf { W } } .
+$$
+
+190 We also observe that one could replace the fixed matrix $\bar { \mathbf A }$ with a more general symmetric graph   
+191 vector field $\pmb { A }$ satisfying $A _ { i j } = 0$ if $( i , j ) \notin \mathsf E$ , although in this work we focus on the case $\pmb { A } = \bar { \mathbf { A } }$   
+192 We also note that when $\Omega = \mathbf { W }$ , then eq. $\textcircled { 8 }$ becomes $\dot { \mathbf { F } } ( t ) = - \Delta \mathbf { F } ( t ) \mathbf { W }$ . We perform a spectral   
+193 analysis of this case in Appendix B.2.   
+194 Non-linear activations. In Appendix $\boxed { \mathbf { B } . 3 }$ we discuss non-linear gradient flow equations. Here   
+195 we study what happens if the gradient flow in eq. $\underline { { \sqrt { 1 0 } } }$ is activated pointwise by $\sigma : \mathbb { R }  \mathbb { R }$ . We   
+196 show that although we are no longer a gradient flow, the learnable multi-particle energy ${ \mathcal { E } } ^ { \mathrm { t o t } }$ is still   
+197 decreasing along the solution, meaning that the interpretation of the channel-mixing $\mathbf { W }$ inducing   
+198 attraction and repulsion via its positive and negative eigenvalues respectively is preserved.   
+199 Proposition 3.2. Consider a non-linear map $\sigma : \mathbb { R }  \mathbb { R }$ such that the function $x \mapsto x \sigma ( x ) \geq 0 .$ . If   
+200 $t \mapsto \mathbf { F } ( t )$ solves the equation
+
+$$
+\dot { \mathbf { F } } ( t ) = \sigma \left( - \mathbf { F } ( t ) \Omega + \bar { \mathbf { A } } \mathbf { F } ( t ) \mathbf { W } - \beta \mathbf { F } ( 0 ) \tilde { \mathbf { W } } \right) ,
+$$
+
+201 where $\sigma$ acts elementwise, then
+
+$$
+\frac { d { \mathcal E } ^ { \mathrm { t o t } } ( { \bf F } ( t ) ) } { d t } \le 0 .
+$$
+
+202 A proof of this result and more details and discussion are reported in Appendix E. We emphasize   
+203 here that differently from previous results about behaviour of ReLU wrt $\hat { \mathcal { E } } ^ { \mathrm { D i r } }$ [30, 8], we deal with a   
+204 much more general energy that can also induce repulsion and a more general family of activation   
+205 functions (that include ReLU, tanh, arctan and many others).
+
+# 206 4 Comparison with GNNs
+
+207 In this Section, we study standard GNN models from the perspective of our gradient flow framework.
+
+209 Continuous GNN models replace layers with continuous time. In contrast with Proposition 3.1,   
+210 we show that three main linearized continuous GNN models are either smoothing or LFD as   
+211 per Definition $2 . 2 .$ The linearized PDE- $\mathrm { G C N } _ { D }$ model $\pmb { \mathbb { I } 7 }$ corresponds to choosing $\beta = 0$ and   
+212 $\mathbf { \hat { \Omega } } \mathbf { \tilde { \Omega } } = \mathbf { W } = \mathbf { K } ( t ) \mathbf { \bar { \Omega } } ^ { \top } \mathbf { K } ( t )$ in eq. $( 1 0 )$ , for some time-dependent family $\dot { t } \mapsto \mathbf { K } ( t ) \in \mathbb { R } ^ { d \times \breve { d } }$ :
+
+$$
+\dot { \mathbf { F } } _ { \mathrm { P D E - G C N _ { D } } } ( t ) = - \Delta \mathbf { F } ( t ) \mathbf { K } ( t ) ^ { \top } \mathbf { K } ( t ) .
+$$
+
+The CGNN model [44] can be derived from eq.213 $( 1 0 )$ by setting $\boldsymbol { \Omega } = \mathbf { I } - \boldsymbol { \tilde { \Omega } } , \mathbf { W } = \mathbf { \tilde { W } } = \mathbf { I } , \beta = 1$ :
+
+$$
+\dot { \mathbf { F } } _ { \mathrm { C G N N } } ( t ) = - \Delta \mathbf { F } ( t ) + \mathbf { F } ( t ) \tilde { \Omega } + \mathbf { F } ( 0 ) .
+$$
+
+214 Finally, in linearized GRAND $\boxed { 1 0 }$ a row-stochastic matrix $\pmb { A } ( \mathbf { F } ( 0 ) )$ is learned from the encoding   
+215 via an attention mechanism and we have
+
+$$
+\dot { \mathbf { F } } _ { \mathrm { G R A N D } } ( t ) = - \pmb { \Delta } _ { \mathrm { R W } } \mathbf { F } ( t ) = - ( \mathbf { I } - \pmb { A } ( \mathbf { F } ( 0 ) ) ) \mathbf { F } ( t ) .
+$$
+
+216 We note that if $\pmb { A }$ is not symmetric, then GRAND is not a gradient flow.
+
+217 Proposition 4.1. $\mathrm { P D E } - \mathrm { G C N } _ { D }$ , CGNN and GRAND satisfy the following:
+
+(i) $\mathrm { P D E } - \mathrm { G C N } _ { D }$ is a smoothing model: $\dot { \mathcal { E } } ^ { \mathrm { D i r } } ( \mathbf { F } _ { \mathrm { P D E - G C N } _ { D } } ( t ) ) \leq 0 .$ . (ii) For a.e. $\mathbf F ( 0 )$ it holds: CGNN is never HFD and $i f$ we remove the source term, then $\mathcal { E } ^ { \mathrm { D i r } } ( \mathbf { F } _ { \mathrm { C G N N } } ( t ) / | | \mathbf { F } _ { \mathrm { C G N N } } ( t ) | | ) \le e ^ { - \mathrm { g a p } ( \Delta ) t }$ . (iii) If G is connected, $\mathbf { F } _ { \mathrm { G R A N D } } ( t )  \mu$ as $t \to \infty$ , with $\mu ^ { r } = \mathrm { m e a n } ( \mathbf { f } ^ { r } ( 0 ) ) , 1 \leq r \leq d .$
+
+222 By (ii) the source-free CGNN-evolution is LFD independent of $\tilde { \Omega }$ . Moreover, by (iii), over-smoothing   
+223 occurs for GRAND as per Definition $\therefore 2 . 1 .$ On the other hand, Proposition $\boxed { 3 . 1 }$ shows that the negative   
+224 eigenvalues of W can make the source-free gradient flow in eq. $\textcircled { 8 }$ HFD. Experiments in Section 5   
+225 confirm that the gradient flow model outperforms CGNN and GRAND on heterophilic graphs.
+
+# 226 4.2 Discrete case
+
+227 We now describe a discrete version of our gradient flow model and compare it to ‘discrete’ GNNs   
+228 where discrete time steps correspond to different layers. In the spirit of $\lVert \overline { { 1 2 } } \rVert$ , we use explicit Euler   
+229 scheme with step size $\tau \leq 1$ to solve eq. $\bigstar \bigstar$ and set $\tilde { \mathbf { W } } = \mathbf { I }$ . In the gradient flow framework we   
+230 parametrize the energy rather than the actual equations, which leads to symmetric channel-mixing   
+231 matrices $\pmb { \Omega }$ , $\mathbf { W } \in \mathbb { R } ^ { d \times d }$ that are shared across the layers. Since the matrices are square, an encoding   
+232 block $\psi _ { \mathrm { E N } } : \mathbb { R } ^ { n \times p }  \mathbb { R } ^ { n \times d }$ is used to process input features $\mathbf { F } _ { 0 } \in \mathbb { R } ^ { n \times p }$ and generally reduce the   
+233 hidden dimension from $p$ to $d$ . Moreover, the iterations inherently lead to a residual architecture   
+234 because of the explicit Euler discretization:
+
+$$
+\mathbf { F } ( t + \tau ) = \mathbf { F } ( t ) + \tau \left( - \mathbf { F } ( t ) \Omega + \bar { \mathbf { A } } \mathbf { F } ( t ) \mathbf { W } + \beta \mathbf { F } ( 0 ) \right) , \quad \mathbf { F } ( 0 ) = \psi _ { \mathrm { E N } } ( \mathbf { F } _ { 0 } ) ,
+$$
+
+235 with prediction $y = \psi _ { \mathrm { D E } } ( \mathbf { F } ( T ) )$ produced by a decoder $\psi _ { \mathrm { D E } } : \mathbb { R } ^ { n \times d }  \mathbb { R } ^ { n \times k }$ , where $k$ is the   
+236 number of label classes and $T$ integration time of the form $T = m \tau$ , so that $m \in \mathbb { N }$ represents the   
+237 number of layers. Although eq. $\underline { { \tilde { ( 1 1 ) } } }$ is linear, we can include non-linear activations in $\psi _ { \mathrm { E N } } , \psi _ { \mathrm { D E } }$   
+238 making the entire model generally non-linear. We emphasize two important points:
+
+• Since the framework is residual, even if the message-passing is linear, this is not equivalent to collapsing the dynamics into a single layer with diffusion matrix $\bar { \mathbf { A } } ^ { m }$ , with $m$ the number of layers, see eq. $\boxed { 2 7 }$ in the appendix where we derive the expansion of the solution. • We could also activate the equations pointwise and maintain the physics interpretation thanks to Proposition $\cdot 3 . 2$ to gain greater expressive power. In the following though, we mainly stick to the linear discrete gradient flow unless otherwise stated.
+
+Are discrete GNNs gradient flows? Given a (learned) symmetric graph vector field 245 $\ b { A } \in \mathbb { R } ^ { n \times n }$ 246 satisfying $A _ { i j } = 0$ if $( i , j ) \notin \mathsf E$ , consider a family of linear GNNs with shared weights of the form
+
+$$
+\mathbf { F } ( t + 1 ) = \mathbf { F } ( t ) \Omega + A \mathbf { F } ( t ) \mathbf { W } + \beta \mathbf { F } ( 0 ) { \tilde { \mathbf { W } } } , \quad 0 \leq t \leq T .
+$$
+
+247 Symmetry is the key requirement to interpret GNNs in eq. $( 1 2 )$ in a gradient flow framework.
+
+Lemma 4.2. Equation (12) is the unit step size discrete gradient flow of 248 $\mathcal { E } _ { \mathbf { I } - \Omega } ^ { \mathrm { e x t } } + \mathcal { E } _ { A , \mathbf { W } } ^ { \mathrm { p a i r } } - \mathcal { E } _ { \tilde { \mathbf { W } } } ^ { \mathrm { s o u r c e } }$ with 249 $\mathcal { E } _ { A , \mathbf { W } } ^ { \mathrm { { p a i r } } }$ defined by replacing $\bar { \mathbf A }$ with $\pmb { A }$ in eq. $\bigstar$ , iff $\pmb { \Omega }$ and W are symmetric.
+
+250 Lemma $\boxed { 4 . 2 }$ provides a recipe for making standard architectures into a gradient flow, with symmetry   
+251 being the key requirement. When eq. ${ \overset { \smile } { ( 1 2 ) } }$ is a gradient flow, the underlying GNN dynamics is   
+252 equivalent to minimizing a multi-particle energy by learning attractive and repulsive directions in   
+253 feature space as discussed in Section $\textcircled { 3 }$ In Appendix C.2, we show how Lemma $\boxed { 4 . 2 }$ covers linear   
+254 versions of GCN [27, 43], GAT $\pmb { \bigtriangledown }$ , GraphSAGE $\pmb { \bigtriangledown } 3 \mathbf { h }$ and GCNII [11] to name a few.   
+255 Over-smoothing analysis in discrete setting. By Proposition $3 . 1$ we know that the continuous   
+256 version of eq. $( 1 \bar { 1 ^ { \cdot } } )$ can be HFD thanks to the negative eigenvalues of W. The next result represents a   
+257 258 discrete counterpart of Propmodels can be HFD. Below $P _ { \mathbf { W } } ^ { \rho _ { - } }$ onis $\boxed { 3 . 1 }$ and shows that residual, symmetrized graph convolutionalprojection into the eigenspace associated with the eigenvalue   
+259 $\rho _ { - } : = | \lambda _ { - } ^ { \mathbf { W } } | ( \rho _ { \Delta } - 1 )$ and we report the explicit value of $\delta _ { \mathrm { H F D } }$ in eq. $\boxed { 2 8 }$ in Appendix $\boxed { \mathsf { C . 3 } }$ We let:
+
+$$
+\lambda _ { + } ^ { \mathbf { W } } ( \rho _ { \Delta } - 1 ) ) ^ { - 1 } < | \lambda _ { - } ^ { \mathbf { W } } | < 2 ( \tau ( 2 - \rho _ { \Delta } ) ) ^ { - 1 } .
+$$
+
+Theorem 4.3. Given 260 $\mathbf { F } ( t + \tau ) = \mathbf { F } ( t ) + \tau \bar { \mathbf { A } } \mathbf { F } ( t ) \mathbf { W }$ , with W symmetric, if eq. (13) holds then
+
+$$
+\mathscr { E } ^ { \mathrm { D i r } } ( { \bf F } ( m \tau ) ) = ( 1 + \tau \rho _ { - } ) ^ { 2 m } \left( \frac { \rho _ { \Delta } } { 2 } | | P _ { \bf W } ^ { \rho _ { - } } { \bf F } ( 0 ) | | ^ { 2 } + \mathcal { O } \left( \left( \frac { 1 + \tau \delta _ { \mathrm { H F D } } } { 1 + \tau \rho _ { - } } \right) ^ { 2 m } \right) \right) , \quad \delta _ { \mathrm { H F D } } < \rho _ { - } ,
+$$
+
+261 hence the dynamics is HFD for a.e. $\mathbf F ( 0 )$ and in fact $\mathbf { F } ( m \tau ) / | | \mathbf { F } ( m \tau ) | |  \mathbf { F } _ { \infty }$ s.t. $\pmb { \Delta } \mathbf { f } _ { \infty } ^ { r } = \rho \pmb { \Delta } \mathbf { f } _ { \infty } ^ { r }$   
+262 Conversely, $i f G$ is not bipartite, then for a.e. $\mathbf F ( 0 )$ the system $\mathbf { F } ( t + \tau ) = \tau \bar { \mathbf { A } } \mathbf { F } ( t ) \bar { \mathbf { W } }$ , with W   
+263 symmetric, is LFD independent of the spectrum of $\mathbf { W }$ .   
+264 Theorem $\boxed { 4 . 3 }$ shows that linear discrete gradient flows can be HFD due to the negative eigenvalues of   
+265 W. This differs from statements that standard GCNs act as low-pass filters and thus over-smooth in   
+266 the limit. Indeed, in these cases the spectrum of $\mathbf { W }$ is generally ignored $\textcircled { 4 3 } \textcircled { 1 1 }$ or required to be   
+267 sufficiently small in terms of singular value decomposition $\overline { { \lVert 2 9 \rVert 3 0 } } , \overline { { \ 8 } } \rVert$ when no residual connection   
+268 is present. On the other hand, Theorem $\mathbf { \delta } _ { . 4 . 3 }$ emphasizes that the spectrum of W plays a key role to   
+269 enhance the high frequencies when enough mass is distributed over the negative eigenvalues provided   
+270 that a residual connection exists – this is confirmed by the neg-prod-curve in Figure 2.   
+271 The residual connection from a spectral perspective. Given a sufficiently small step-size so   
+272 that the right hand side of inequality 13 is satisfied, $\mathbf { F } ( t + \tau ) = \mathbf { F } ( t ) + \tau \bar { \mathbf { A } } \mathbf { F } ( \dot { t } ) \mathbf { W }$ is HFD for a.e.   
+273 $\mathbf F ( 0 )$ if $| \lambda _ { - } ^ { \mathbf { \tilde { W } } } | ( \rho _ { \Delta } - 1 ) > \lambda _ { + } ^ { \mathbf { W } }$ , i.e. ‘there is more mass’ in the negative spectrum of W than in the   
+274 positive one. This means that differently from $\pm \varTheta \left| 3 0 \right| \bigotimes 1$ , there is no requirement on the minimal   
+275 magnitude of the spectral radius of W coming from the graph topology as long as $\lambda _ { + } ^ { \mathbf { w } }$ is small   
+276 enough. Conversely, without a residual term, the dynamics is LFD for a.e. ${ \bf \ddot { F } } ( 0 )$ independently of the   
+277 sign and magnitude of the eigenvalues of W. This is also confirmed by the GCN-curve in Figure 2.   
+278 Over-smoothing vs LFD. We highlight how in general a linear GCN equation as $\mathbf { F } ( t + \tau ) =$   
+279 $\tau \bar { \mathbf { A } } \mathbf { F } ( t ) \mathbf { W }$ may avoid over-smoothing in the sense of Definition $2 . 1 .$ meaning that $\mathcal { E } ^ { \mathrm { D i r } } ( { \bf F } ( t ) )  \infty$   
+280 as soon as there exist $\lambda _ { i } ^ { \pmb { \Delta } } \in ( 0 , 1 )$ and the spectral radius of W is large enough. However, this   
+281 will not lead to over-separation since the dominating term is the lowest frequency one: in other   
+282 words, once we re-set the scale right as per the normalization in Theorem $4 . 3 _ { \cdot }$ we encounter loss of   
+283 separability even with large (and possibly negative) spectrum of W.
+
+# 284 5 Experiments
+
+285 In this section we evaluate the gradient flow framework (GRAFF). We corroborate the spectral   
+286 analysis using synthetic data with controllable homophily. We confirm that having negative (positive)   
+287 eigenvalues of the channel-mixing W are essential in heterophilic (homophilic) scenarios where the   
+288 gradient flow should align with HFD (LFD) respectively. We show that the gradient flow in eq. $( 1 1 )$   
+289 – a linear, residual, symmetric graph convolutional model – achieves competitive performance on   
+290 heterophilic datasets.   
+291 Methodology. We crystallize GRAFF in the model presented in eq. $( 1 1 )$ with $\psi _ { \mathrm { E N } } , \psi _ { \mathrm { D E } }$ im  
+292 plemented as single linear layers or MLPs, and we set $\pmb { \Omega }$ to be diagonal. For the real-world   
+293 experiments we consider diagonally-dominant (DD), diagonal (D) and time-dependent choices   
+294 for the structure of $\mathbf { W }$ that offer explicit control over its spectrum. In the (DD)-case, we consider   
+295 a $\mathbf { W } ^ { 0 } \in \mathbb { R } ^ { d \times d }$ symmetric with zero diagonal and $\mathbf { w } \in \mathbb { R } ^ { d }$ defined by $\begin{array} { r } { \mathbf { w } _ { \alpha } = q _ { \alpha } \sum _ { \beta } \lvert \mathbf { W } _ { \alpha \beta } ^ { 0 } \rvert + r _ { \alpha } } \end{array}$   
+296 and set $\mathbf { W } = \mathrm { d i a g } ( \mathbf { w } ) + \mathbf { W } ^ { 0 }$ . Due to the Gershgorin Theorem the eigenvalues of W belong to   
+297 $\begin{array} { r } { [ \mathbf { w } _ { \alpha } - \sum _ { \beta } \vert \mathbf { W } _ { \alpha \beta } ^ { 0 } \vert , \dot { \mathbf { w } } _ { \alpha } + \sum _ { \beta } \vert \mathbf { W } _ { \alpha \beta } ^ { 0 } \vert ] } \end{array}$ , so the model ‘can’ easily re-distribute mass in the spectrum of   
+298 W via $q _ { \alpha } , r _ { \alpha }$ . This generalizes the decomposition of $\mathbf { W }$ in $\pmb { \mathbb { \ m } }$ providing a justification in terms of   
+299 its spectrum and turns out to be more efficient w.r.t. the hidden dimension $d$ as shown in Figure $\boxed { 4 }$ in   
+300 the Appendix. For (D) we take $\mathbf { W }$ to be diagonal, with entries sampled $\boldsymbol { \mathcal { U } } [ - 1 , 1 ]$ and fixed – i.e., we   
+301 do not train over W – and only learn $\psi _ { \mathrm { E N } }$ , $\psi _ { \mathrm { D E } }$ . We also include a time-dependent model where $\mathbf { W } _ { t }$   
+302 varies across layers. To investigate the role of the spectrum of $\mathbf { W }$ on synthetic graphs, we construct   
+303 three additional variants: $\mathbf { W } = \mathbf { W } ^ { \prime } + \mathbf { W } ^ { \prime } ^ { \top }$ , $\mathbf { W } = \pm \mathbf { W } ^ { \prime \top } \mathbf { W } ^ { \prime }$ named sum, prod and neg-prod   
+304 respectively where prod (neg-prod) variants have only non-negative (non-positive) eigenvalues.   
+305 Complexity and number of parameters. If we treat the number of layers as a constant, the discrete   
+306 gradient flow scales as $\mathcal { O } ( | \bar { \mathsf { V } } | p d + | \mathsf { E } | d ^ { 2 } )$ , where $p$ and $d$ are input feature and hidden dimension   
+307 respectively, with $p \geq d$ usually. Note that GCN has complexity ${ \dot { \mathcal { O } } } ( | \mathsf { E } | p d )$ and in fact our model is   
+308 faster than GCN as confirmed in Figure $5$ in Appendix $\dot { \mathbf { D } _ { \cdot } }$ Since $\psi _ { \mathrm { E N } } , \psi _ { \mathrm { D E } }$ are single linear layers   
+309 (MLPs), we can bound the number of parameters by $p d \stackrel {  } { + } d ^ { 2 } + 3 d + d k$ , with $k$ the number of label   
+310 classes, in the (DD)-variant while in the (D)-variant we have $p d + 3 d + d k$ . Further ablation studies   
+311 appear in Figure 4 in the Appendix showing that (DD) outperforms sum and GCN – especially in the   
+312 lower hidden dimension regime – on real-world benchmarks with varying homophily.   
+313 Synthetic experiments and ablation studies.   
+314 To investigate our claims in a controlled environ  
+315 ment we use the synthetic Cora dataset of $\textcircled { 5 1 }$ Ap  
+316 pendix G]. Graphs are generated for target levels   
+317 of homophily via preferential attachment – see   
+318 Appendix $\mathbf { D } \dot { . } \dot { 3 }$ for details. Figure $2 \cdot$ confirms the   
+319 spectral analysis and offers a better understanding   
+320 in terms of performance and smoothness of the   
+321 predictions. Each curve – except GCN – repre  
+322 sents one version of W as in ‘methodology’ and   
+323 we implement eq. $^ { ( 1 1 ) }$ with $\beta = 0$ , $\pmb { \Omega } = \mathbf { 0 }$ . Fig  
+324 ure $\bigstar$ (top) reports the test accuracy vs true label   
+325 homophily. Neg-prod is better than prod on low  
+326 homophily and viceversa on high-homophily. This   
+327 confirms Proposition $\underline { { \boldsymbol { \left. 3 . 1 \right. } } }$ where we have shown   
+328 that the gradient flow can lead to a HFD dy  
+329 namics – that are generally desirable with low  
+330 homophily – through the negative eigenvalues of   
+331 W. Conversely, the prod configuration (where we   
+332 have an attraction-only dynamics) struggles in low  
+333 homophily scenarios even though a residual connection is present. Both prod and neg-prod are   
+334 ‘extreme’ choices and serve the purpose of highlighting that by turning off one side of the spectrum   
+335 this could be the more damaging depending on the underlying homophily. In general though ‘neutral’   
+336 variants like sum and (DD) are indeed more flexible and better performing. In fact, (DD) outperforms   
+337 GCN especially in low-homophily scenarios, confirming Theorem 4.3 where we have shown that   
+338 without a residual connection convolutional models are LFD – and hence more sensitive to underlying   
+339 homophily – irrespectively of the spectrum of W. This is further confirmed in Figure 3.   
+340 In Figure 2 (bottom) we compute the homophily of the prediction (cross) for a given method and we   
+341 compare with the homophily (circle) of the prediction read from the encoding (i.e. graph-agnostic).   
+342 The homophily here is a proxy to assess whether the evolution is smoothing, the goal being explaining   
+343 the smoothness of the prediction via the spectrum of W as per our theoretical analysis. For neg-prod   
+344 the homophily after the evolution is lower than that of the encoding, supporting the analysis that   
+345 negative eigenvalues of W enhance high-frequencies. The opposite behaviour occurs in the case of   
+346 prod and explains that in the low-homophily regime prod is under-performant due to the prediction   
+347 being smoother than the true homophily. (DD) and sum variants adapt better to the true homophily.   
+348 We note how the encoding compensates when the dynamics can only either attract or repulse (i.e. the   
+349 spectrum of W has a sign) by decreasing or increasing the initial homophily respectively.   
+350 Real world experiments. We test GRAFF against a range of datasets with varying homophily   
+351 [37, 33, 31] (see Appendix $\textcircled { \mathbf { D . 4 } }$ for additional details). We use results provided in [45, Table 1],   
+352 which includes standard baselines as GCN $\lVert 2 7 \rVert$ , GraphSAGE $\lVert 2 3 \rVert$ , GAT [42], PairNorm [48] and   
+353 recent models tailored towards the heterophilic setting (GGCN [45], Geom-GCN [31], H2GCN   
+354 [51] and GPRGNN $\pmb { \mathbb { I 3 } } \mathbf { I }$ . For Sheaf $[ \sqrt { 5 } ]$ , a recent top-performer on heterophilic datasets, we took   
+355 the best performing variant (out of six provided) for each dataset. We also include continuous   
+356 baselines CGNN $\checkmark$ and GRAND $\boxed { 1 0 }$ to provide empirical evidence for Proposition 4.1. Splits   
+357 taken from $\textcircled { 3 1 }$ are used in all the comparisons. The GRAFF model discussed in ‘methodology’   
+358 is a very simple architecture with shared parameters across layers and run-time smaller than GCN   
+359 and more recent models like GGCN designed for heterophilic graphs (see Figure 5 in the Appendix).   
+360 Nevertheless, it achieves competitive results on all datasets, performing on par or better than more   
+361 complex recent models. Moreover, comparison with the ‘time-dependent’ (DD) variant confirms   
+362 that by sharing weights across layers we do not lose performance. We note that on heterophilic   
+363 graphs short integration time is usually needed due to the topology being harmful and the negative   
+364 eigenvalues of W leading to exponential behaviour (see Appendix D).
+
+![](images/0d7e8be689840459aa1ffe8642b06bbea77c3f7ad6f1af46e1567c0e0253e831.jpg)  
+Figure 2: Experiments on synthetic datasets with controlled homophily.
+
+Table 1: Results on heterophilic and homophilic datasets   
+
+<table><tr><td>Hom level #Nodes #Edges</td><td>Texas 0.11 183 295</td><td>Wisconsin 0.21 251 466</td><td>Cornell 0.30 183 280</td><td>Film 0.22 7,600 26.752</td><td>Squirrel 0.22 5,201 198.493</td><td>Chameleon 0.23 2.277 31,421</td><td>Citeseer 0.74 3,327 4.676</td><td>Pubmed 0.80 18,717 44,327</td><td>Cora 0.81 2,708 5,278</td></tr><tr><td>#Classes</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>7</td><td>3</td><td>6</td></tr><tr><td>GGCN</td><td>84.86 ± 4.55</td><td>86.86 ±3.29</td><td>85.68 ± 6.63</td><td>37.54 ± 1.56</td><td>55.17 ± 1.58</td><td>71.14 ± 1.84</td><td>77.14 ± 1.45</td><td>89.15 ± 0.37</td><td>87.95 ± 1.05</td></tr><tr><td>GPRGNN</td><td>78.38 ± 4.36</td><td>82.94 ± 4.21</td><td>80.27 ± 8.11</td><td>34.63 ± 1.22</td><td>31.61 ± 1.24</td><td>46.58 ± 1.71</td><td>77.13 ± 1.67</td><td>87.54 ± 0.38</td><td>87.95 ± 1.18</td></tr><tr><td>H2GCN</td><td>84.86 ± 7.23</td><td>87.65 ± 4.98</td><td>82.70 ± 5.28</td><td>35.70 ± 1.00</td><td>36.48 ± 1.86</td><td>60.11 ± 2.15</td><td>77.11 ± 1.57</td><td>89.49 ± 0.38</td><td>87.87 ± 1.20</td></tr><tr><td>GCNII</td><td>77.57 ± 3.83</td><td>80.39 ± 3.40</td><td>77.86 ± 3.79</td><td>37.44 ± 1.30</td><td>38.47 ± 1.58</td><td>63.86 ± 3.04</td><td>77.33 ± 1.48</td><td>90.15 ± 0.43</td><td>88.37 ± 1.25</td></tr><tr><td>Geom-GCN</td><td>66.76 ± 2.72</td><td>64.51 ± 3.66</td><td>60.54 ± 3.67</td><td>31.59 ± 1.15</td><td>38.15 ± 0.92</td><td>60.00 ± 2.81</td><td>78.02 ± 1.15</td><td>89.95 ± 0.47</td><td>85.35 ± 1.57</td></tr><tr><td>PairNorm</td><td>60.27 ± 4.34</td><td>48.43 ± 6.14</td><td>58.92 ± 3.15</td><td>27.40 ± 1.24</td><td>50.44 ± 2.04</td><td>62.74 ± 2.82</td><td>73.59 ± 1.47</td><td>87.53 ± 0.44</td><td>85.79 ± 1.01</td></tr><tr><td>GraphSAGE</td><td>82.43 ± 6.14</td><td>81.18 ± 5.56</td><td>75.95 ± 5.01</td><td>34.23 ± 0.99</td><td>41.61 ± 0.74</td><td>58.73 ± 1.68</td><td>76.04 ± 1.30</td><td>88.45 ± 0.50</td><td>86.90 ± 1.04</td></tr><tr><td>GCN</td><td>55.14 ± 5.16</td><td>51.76 ± 3.06</td><td>60.54 ± 5.30</td><td>27.32 ± 1.10</td><td>53.43 ± 2.01</td><td>64.82 ± 2.24</td><td>76.50 ± 1.36</td><td>88.42 ± 0.50</td><td>86.98 ± 1.27</td></tr><tr><td>GAT</td><td>52.16 ± 6.63</td><td>49.41 ± 4.09</td><td>61.89 ± 5.05</td><td>27.44 ± 0.89</td><td>40.72 ± 1.55</td><td>60.26 ± 2.50</td><td>76.55 ± 1.23</td><td>87.30 ± 1.10</td><td>86.33 ± 0.48</td></tr><tr><td>MLP CGNN</td><td>80.81 ± 4.75</td><td>85.29 ± 3.31 74.31 ± 7.26</td><td>81.89 ± 6.40 66.22 ± 7.69</td><td>36.53 ± 0.70 35.95 ± 0.86</td><td>28.77 ± 1.56</td><td>46.21 ± 2.99</td><td>74.02 ± 1.90</td><td>75.69 ± 2.00 87.70 ± 0.49</td><td>87.16 ± 0.37 87.10 ± 1.35</td></tr><tr><td>GRAND</td><td>71.35 ± 4.05 75.68 ± 7.25</td><td>79.41 ± 3.64</td><td>82.16 ± 7.09</td><td>35.62 ± 1.01</td><td>29.24 ± 1.09 40.05 ± 1.50</td><td>46.89 ± 1.66 54.67 ± 2.54</td><td>76.91 ± 1.81 76.46 ± 1.77</td><td>89.02 ± 0.51</td><td>87.36 ± 0.96</td></tr><tr><td>Sheaf (max)</td><td>85.95 ± 5.51</td><td>89.41 ± 4.74</td><td>84.86 ± 4.71</td><td>37.81 ± 1.15</td><td>56.34 ± 1.32</td><td>68.04 ± 1.58</td><td>76.70 ± 1.57</td><td>89.49 ± 0.40</td><td>86.90 ± 1.13</td></tr><tr><td>GRAFF (DD)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>GRAFF (D)</td><td>88.38 ± 4.53</td><td>87.45 ± 2.94</td><td>83.24 ± 6.49</td><td>36.09 ± 0.81</td><td>54.52 ± 1.37</td><td>71.08 ± 1.75</td><td>76.92 ± 1.70</td><td>88.95 ± 0.52 90.04 ± 0.41</td><td>87.61 ± 0.97</td></tr><tr><td></td><td>88.11 ± 5.57</td><td>88.83 ± 3.29</td><td>84.05 ± 6.10</td><td>37.11 ± 1.08</td><td>47.36 ± 1.89</td><td>66.78 ± 1.28</td><td>77.30 ± 1.85</td><td></td><td>88.01 ± 1.03</td></tr><tr><td>GRAFF-timedep (DD)</td><td>87.03 ± 4.49</td><td>87.06 ± 4.04</td><td>82.16 ± 7.07</td><td>35.93 ± 1.23</td><td>53.97 ± 1.45</td><td>69.56 ± 1.20</td><td>76.59 ± 1.53</td><td>88.26 ± 0.41</td><td>87.38 ± 1.05</td></tr></table>
+
+# 365 6 Conclusions
+
+366 In this work, we developed a framework for GNNs where the evolution can be interpreted as   
+367 minimizing a multi-particle learnable energy. This translates into studying the interaction between   
+368 the spectrum of the graph and the spectrum of the ‘channel-mixing’ leading to a better understanding   
+369 of when and why the induced dynamics is low (high) frequency dominated. From a theoretical   
+370 perspective, we refined existing asymptotic analysis of GNNs to account for the role of the spectrum of   
+371 the channel-mixing as well. From a practical perspective, our framework allows for ‘educated’ choices   
+372 resulting in a simple convolutional model that achieves competitive performance on homophilic   
+373 and heterophilic benchmarks while being faster than GCN. Our results refute the folklore of graph   
+374 convolutional models being too simple for heterophilic benchmarks.   
+375 Limitations and future works. We limited our attention to a constant bilinear form W, which   
+376 might be excessively rigid. It is possible to derive non-constant alternatives that are aware of the   
+377 features or the position in the graph. The main challenge amounts to matching the requirement for   
+378 local ‘heterogeneity’ with efficiency: we reserve this question for future work. Our analysis is also a   
+379 first step into studying the interaction of the graph and ‘channel-mixing’ spectra; we did not explore   
+380 other dynamics that are neither LFD nor HFD as per our definitions. The energy formulation points   
+381 to new models more ‘physics’ inspired; this will be explored in future work.   
+382 Societal impact. Our work sheds light on the actual dynamics of GNNs and could hence improve   
+383 their understanding, which is crucial for assessing their impact on large-scale applications. We also   
+384 show that instances of our framework achieve competitive performance on heterophilic data despite   
+385 being faster than GCN, providing evidence for efficient methods with reduced footprint.
+
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+
+# 523 Checklist
+
+524 The checklist follows the references. Please read the checklist guidelines carefully for information on   
+525 how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or   
+526 [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing   
+527 the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes] See Section ??.   
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   
+• Did you include the license to the code and datasets? [N/A]
+
+532 Please do not modify the questions and only use the provided macros for your answers. Note that the   
+533 Checklist section does not count towards the page limit. In your paper, please delete this instructions   
+534 block and only keep the Checklist section heading above along with the questions/answers below.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] , in Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] in the Societal impact paragraph in Section 6.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] in Appendix A Appendix B and Appendix C.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Code and README in SM, dataloaders in code   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Splits and hyperparameters provided in code zip   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] Standard deviations are stated in results table   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] in appendix D
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] datasets and standard libraries cited in appendix D   
+(b) Did you mention the license of the assets? [Yes] industry standard libraries and benchmark datasets were used in accordance with licences   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] code provided in SM zip   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] no personal data is contained within benchmarking datasets
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
+
+572 (b) Did you describe any potential participant risks, with links to Institutional Review   
+573 Board (IRB) approvals, if applicable? [N/A]   
+574 (c) Did you include the estimated hourly wage paid to participants and the total amount   
+575 spent on participant compensation? [N/A]

parse/dev/17RDXeF-skZ/17RDXeF-skZ.md

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+# DOING FAST ADAPTATION FAST: CONDITIONALLY INDEPENDENT DEEP ENSEMBLES FOR DISTRIBUTION SHIFTS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Classifiers in a diverse ensemble capture distinct predictive signals, which is valuable for datasets containing multiple strongly predictive signals. Performing fast adaptation at test time allows us to generalize to distributions where certain signals are no longer predictive, or to avoid relying on sensitive or protected attributes. However, ensemble learning is often expensive, even more so when we need to enforce diversity constraints between the high-dimensional representations of the classifiers. Instead, we propose an efficient and fast method for learning ensemble diversity. We minimize conditional mutual information of the output distributions between classifiers, a quantity which can be cheaply and exactly computed from empirical data. The resulting ensemble contains individually strong predictors that are only dependent because they predict the label. We demonstrate the efficacy of our method on shortcut learning tasks. Performing fast adaptation on our ensemble selects shortcut-invariant models that generalize well to test distributions where the shortcuts are uncorrelated with the label.
+
+# 1 INTRODUCTION
+
+Some of the strongest scientific theories are supported by multiple sources of evidence, a principle described by 19th century philosopher William Whewell as “consilience”. Evolution is one such example, having been firmly corroborated by fields ranging from paleontology to genetics. In many real-world applications of machine learning, datasets can similarly contain multiple predictive signals that explain the label well. In these settings, a standard model typically learns from a combination of predictive features (Ross et al., 2018; Kirichenko et al., 2022). Such a model will fail to generalize to distribution shifts that break the correlation between certain signals and the label (Hovy & Søgaard, 2015; Hashimoto et al., 2018; Puli et al., 2022).
+
+This shortcoming can be addressed by learning a diverse set or ensemble of classifiers. Such methods typically exploit some notion of independence to learn multiple classifiers that rely on different predictive signals. We can then perform fast adaptation, using a small amount of out-of-distribution (OOD) validation data to select the model that generalizes best. Learning diversity is also beneficial in and of itself: these classifiers are empirically shown to be more human-interpretable than if we were to fit a single model (Ross et al., 2018), possibly because they learn disentangled representations that correspond to natural factors of variation (Shu et al., 2019).
+
+The key challenge is quantifying the right notion of diversity. Existing work has exploited concepts like input gradient or parameter orthogonality as a proxy for statistical independence (Teney et al., 2021; Xu et al., 2021). To tackle OOD generalization, which fundamentally requires additional assumptions or data beyond the observed training data (Bareinboim et al., 2022; Scholkopf et al., ¨ 2021), previous work have also assumed access to unlabelled test data and measured disagreement on those examples (Lee et al., 2022; Pagliardini et al., 2022). However, these objectives or assumptions are often prohibitive or unrealistic in real-world settings. For example, group-balanced test data is not always obtainable, e.g. when deploying a pneumonia model to multiple new hospitals whose patient profiles may change over time. Another costly example is enforcing input gradient orthogonality on high-dimensional covariates like images or text, where it can be challenging to avoid learning from orthogonal covariates of the same underlying feature, such as neighboring pixels.
+
+To avoid the pitfalls of operating in high-dimensional input or parameter space, a promising line of work instead adopts the information-theoretic perspective and tackles the problem as representation learning. These approaches apply the information bottleneck method and minimize mutual information between the representations learnt by each classifier. Such an objective forces the classifiers to rely on distinctly meaningful features for prediction. Most notably, Pace et al. (2020) and Rame & Cord (2021) minimize mutual information between the classifier representations conditioned on the label. Since any pair of predictors cannot both be accurate while remaining unconditionally independent, the extra conditioning prevents learning weak classifiers. The resulting ensemble contains accurate classifiers that nevertheless rely on distinct predictive signals. The only core assumption is that the underlying predictive signals are themselves conditionally independent.
+
+These approaches are conceptually appealing but practically challenging. Mutual information between high-dimensional representations is intractable and must be approximated, either via variational (e.g. Fischer, 2020) or contrastive (e.g. Oord et al., 2018) bounds. Furthermore, such approximations are computationally expensive, a problem that is compounded in the ensemble setting where we wish to train multiple classifiers speedily.
+
+We seek to learn ensemble diversity fast and effectively. Our key insight is that it suffices to enforce conditional independence on the output distributions of the classifiers. Our first contribution is proposing conditional mutual information (CMI) between output distributions as the regularizing objective. Assuming conditionally independent predictive signals, enforcing CMI between output distributions also guarantees that the ensemble where separate predictive signals are learnt by separate classifiers is a minimizing solution. Since the output distribution is categorical, CMI can be cheaply and exactly computed from empirical data. In addition, our method avoids using additional sources of data that cannot be found in many real-world domains, such as unlabelled test data or “group” labels for each predictive signal in the dataset. We only permit a small amount of validation data from the test distribution for (1) hyperparameter tuning and (2) selecting the final predictor from our ensemble. We dub our approach as Conditionally Independent Deep Ensembles (CoDE).
+
+Our second contribution is evaluating CoDE on benchmark datasets for shortcut learning (Geirhos et al., 2020). Shortcuts are signals that are (i) highly but spuriously correlated to the label in the training distribution, possibly due to biases in data collection or other systematic pre-processing errors (Torralba & Efros, 2011), and (ii) preferentially learnt by a neural network, possibly due to simplicity biases (Shah et al., 2020) or architectural biases (e.g. convolutional neural networks (CNNs) relying on texture over shape (Baker et al., 2018)). An empirical risk minimizing (ERM) model will rely on shortcuts and fail to generalize to test distributions where they are no longer correlated to the label. This is a natural application for our method as the core assumption of conditional independence applies to many such datasets — for example, in natural images, the foreground is typically the label and is thus conditionally independent from the background (shortcut). We show that CoDE effectively recovers an ensemble where the shortcut features and the true signal are learnt by separate classifiers.
+
+# 2 PRELIMINARIES: SETUP AND NOTATION
+
+In Section 3, we will fully motivate the assumptions behind our model of the data-generating process (DGP). However, we describe it here first to establish key terminology and concepts.
+
+Data-Generating Process Let $\mathbf { z }$ denote the set of latent factors that generate the set of observed features $\mathbf { x } \in \mathbb { R } ^ { P }$ . Let $y \in \{ 0 , 1 , \dotsc , K - 1 \}$ denote the label. The data $p _ { e } ( \mathbf { x } , y , \mathbf { z } )$ is generated from a family of distributions indexed by $e$ , the environment. We only consider: (i) a single training environment $\mathbf { \boldsymbol { e } } _ { \mathbf { \lambda } } = t { \boldsymbol { r } } _ { \mathbf { \lambda } } $ ), from which we have access to i.i.d. labelled training examples $D _ { t r } \ = \ \{ { \bf x } _ { i } , y _ { i } \} _ { i = 1 } ^ { N }$ , and (ii) a test environment $\mathit { \Pi } _ { \mathrm { ~ e ~ } } = \mathit { \Pi } _ { t e }$ ), from which we draw unlabelled test examples that our model should perform well on. We also allow access to a small set of labelled validation data $D _ { v a l } = \{ \mathbf { x } _ { i } , y _ { i } \} _ { i = 1 } ^ { N ^ { \prime } }$ from the test environment, which is used only for hyperparameter tuning and ensembling (i.e. constructing the final model from the set of learnt classifiers).
+
+We make the following assumptions on the DGP:
+
+(i) all label information is encoded by $\mathbf { z }$ , i.e. $p _ { e } ( y | \mathbf { x } , \mathbf { z } ) = p _ { e } ( y | \mathbf { z } )$ for all $e$ (ii) $p _ { e } ( \mathbf { x } | \mathbf { z } ) = p ( \mathbf { x } | \mathbf { z } )$ is invariant across all $e$
+
+(iii) $p _ { e } ( { \bf z } ) > 0$ for all $e$ and $\mathbf { z }$   
+(iv) $p _ { e } ( y ) > 0$ for all $e$ and $y$   
+(v) [Latent Conditional Independence] $z _ { i } \perp \perp z _ { j } \mid y$ for all $e$ and $i , j$
+
+Based on these assumptions, we can factorize $p _ { e } ( \mathbf { x } , y , \mathbf { z } )$ as:
+
+$$
+p _ { e } ( \mathbf { z } , \mathbf { x } , y ) = p _ { e } ( y ) \left( \prod _ { i = 1 } ^ { L } p _ { e } ( z _ { i } | y ) \right) p ( \mathbf { x } | \mathbf { z } )
+$$
+
+Example: ColoredMNIST As introduced in Arjovsky et al. (2019), $y$ is a binary label which determines color $( z _ { 1 } \in \{ \mathrm { r e d } , \mathrm { g r e e n } \} )$ with probability $p _ { c }$ and digit $( z _ { 2 } \in \{ 0 \ – 4 , 5 \ – 9 \} )$ ) with probability $p _ { d }$ . $p _ { c }$ and $p _ { d }$ are independently chosen. In the training distribution, $p _ { c } = 0 . 2 5$ and $p _ { d } = 0 . 1$ , as such, an ERM model will primarily learn from color. $p _ { c }$ and $p _ { d }$ can be arbitrary in the test distribution.
+
+Example: Waterbirds As introduced in Sagawa et al. (2019), $y$ is a binary label determining if the image represents a water or land bird. It perfectly determines the foreground $( z _ { 1 } \in \ \left\{ \begin{array} { l l } { \end{array} } \right.$ water bird, land bird}) and is highly but spuriously correlated to the background $( z _ { 2 } \in$ {water, land}) in the training distribution. An ERM model will learn from background features.
+
+Group Robustness When $\mathbf { z }$ is discrete, each possible value that $\mathbf { z }$ can take is known as a group. Due to the spurious correlations created by $p _ { t r } ( z _ { i } | y )$ , groups that are highly represented in the training set are called “majority groups”, and poorly-represented groups are “minority groups”. Group robustness refers to the goal of generalizing well on all groups and is one natural way of evaluating if a model has been learning shortcuts. For example, both ColoredMNIST and Waterbirds admits four groups formed by the Cartesian product of $z _ { 1 }$ and $z _ { 2 }$ .
+
+Ensembles and Fast Adaptation A classifier $f ( \mathbf { x } ) : = p _ { \theta } ( y | \mathbf { x } )$ is parametrized by $\theta$ and outputs class probabilities. We will use $\hat { y } : = p _ { \theta } ( y )$ to denote the unconditional output distribution. We use the term “ensemble” loosely to refer to a set of or sequentially. (Section 4 clarifies the relations $M$ classifiers  to tradition $\{ f _ { m } \} _ { m = 1 } ^ { M }$ that can be learnt joinle methods.) After all $M$ classifiers are learnt, the final model $\theta ^ { * }$ is selected using validation data $D _ { v a l }$ :
+
+$$
+\theta ^ { * } = \arg \operatorname* { m i n } _ { \theta _ { m } , m \in \{ 1 , \dots , M \} } \frac { 1 } { N ^ { \prime } } \sum _ { i = 1 } ^ { N ^ { \prime } } \log p _ { \theta _ { m } } ( y _ { i } | \mathbf { x } _ { i } )
+$$
+
+This process is referred to as fast adaptation.
+
+# 3 CONDITIONALLY INDEPENDENT DEEP ENSEMBLES
+
+To motivate our approach and the assumptions made in (1), we first define what it means to learn a diverse ensemble and explain why conditional independence is a sound measure of diversity.
+
+# 3.1 DIVERSITY AS CONDITIONAL INDEPENDENCE
+
+Diverse classifiers utilize separate predictive signals, intuitively, they predict the “same things for different reasons” (Rame & Cord, 2021). Our setup in Section 2 formalizes this notion of “different reasons” by explicitly defining the latent variable $\mathbf { z }$ , which models the total underlying set of predictive signals that relate $\mathbf { x }$ to $y$ . A classifier that learns a mapping from $\mathbf { x }$ to $y$ can then be interpreted as implicitly inferring $\mathbf { z }$ from $\mathbf { x }$ and learning a mapping from $\mathbf { z }$ to $y$ . We can thus define diverse classifiers that rely on separate predictive signals as learning from separate dimensions or subspaces of $\mathbf { z }$ .
+
+To formalize the idea that a classifier $f$ learns using only a subspace of $\mathbf { z }$ , one naive approach might be to define $f$ as relying only on the subspace ${ \mathbf z } _ { [ a ] }$ if and only if (some distribution computed from) $f$ is independent of its complement ${ \mathbf z } \backslash { \mathbf z } _ { [ a ] }$ . This definition is convenient as it suggests that the appropriate objective to learn a diverse ensemble is simply to enforce statistical independence between the classifiers. This follows because two classifiers that rely on overlapping subspaces of $\mathbf { z }$ will necessarily be dependent.
+
+However, the definition above assumes that distinct predictive signals (i.e. subspaces of $\mathbf { z }$ ) are themselves unconditionally independent. This is not always true when a dataset contains multiple strongly predictive signals. Dimensions of $\mathbf { z }$ can be dependent by virtue of their correlation to $y$ Classifiers that learn from such signals will similarly be dependent. Shortcut learning is precisely a problem because meaningful and spurious features are highly correlated in the training environment.
+
+This conundrum can be resolved by establishing independence of the latent factors with conditioning on $y$ . Doing so is equivalent to assuming that upon knowing the true label, observing one set of features yields no additional information about other features. This is usually a realistic assumption to make. As the Waterbirds example in Section 2 shows, backgrounds and foregrounds are often conditionally independent in the test distributions we care about. This motivates our assumption (v) of latent conditional independence in Section 2, where the individual factors $z _ { i }$ are conditionally independent given $y$ . We formalize this notion of “diversity as conditional independence” below.
+
+Definition 3.1. Let $\mathbf { z } _ { [ a ] } : = ( z _ { a _ { 1 } } , \ldots , z _ { a _ { l } } )$ denote some subspace of $\mathbf { z }$ . Let $\hat { h } ( f )$ denote some distribution computed from $f$ . We say $f$ is invariant to ${ \mathbf z } _ { [ a ] }$ if $\hat { h } \perp \perp ( z _ { a _ { 1 } } , \ldots , z _ { a _ { l } } ) | y$ . Let $\mathbf { z } _ { [ i ] }$ be the maximal subset of $\mathbf { z }$ that $f$ is invariant to. Then $f$ is said to rely on $\mathbf { z } _ { - [ i ] } : = \mathbf { z } \backslash \mathbf { z } _ { [ i ] }$ for prediction.
+
+Definition 3.2. Let $f$ and $f ^ { \prime }$ be a pair of classifiers that rely on $\mathbf { z } _ { [ i ] }$ and $\mathbf { z } _ { [ i ^ { \prime } ] }$ respectively. $f$ and $f ^ { \prime }$ are said to be diverse if $\mathbf { z } _ { [ i ] } \bigcap _ { . . } \mathbf { z } _ { [ i ^ { \prime } ] } = \emptyset$ . An ensemble $\{ f _ { m } \} _ { m = 1 } ^ { M }$ is diverse if every pair of classifiers $f _ { j } , f _ { k }$ in the ensemble are diverse.
+
+It follows immediately from Definition 3.2 that diverse classifiers must themselves be conditionally independent, i.e. $\hat { h } _ { i } \perp \perp \hat { h } _ { j } | y$ . Our training objective for learning a diverse ensemble should therefore enforce conditional independence on all pairs of classifiers:
+
+$$
+\begin{array} { l } { \displaystyle \arg \underset { \theta _ { 1 } , \dots , \theta _ { M } } { \operatorname* { m a x } } \sum _ { i = 1 } ^ { N } \sum _ { m = 1 } ^ { M } \log p _ { \theta _ { m } } ( y _ { i } | \mathbf { x } _ { i } ) } \\ { \displaystyle \mathrm { s u b j e c t ~ t o } \hat { h } _ { s } \perp \hat { h } _ { t } \vert y \qquad \forall s , t } \end{array}
+$$
+
+We can interpret (3) as follows: the main objective guarantees that the learnt ensemble contains individually strong predictors, whereas the constraint guarantees that each predictor is uninformative of the others when conditioned on the label. Put together, (3) learns classifiers that rely on conditionally independent subspaces of $\mathbf { z }$ and thus provide no additional information about each other. As is typical in machine learning (Krogh & Hertz, 1991; Deb, 2014), we optimize an unconstrained analogue of (3) by expressing the constraint as a regularization term.
+
+# 3.2 ENFORCING CONDITIONAL INDEPENDENCE VIA OUTPUT DISTRIBUTIONS
+
+It remains for us to decide on the distribution $\hat { h }$ that we constrain, as well as the (unconstrained) regularization objective from (3). These choices are crucial in many ways. Since independence with respect to $\hat { h }$ underpins the notions of invariance and diversity in Definitions 3.1 and 3.2, it must be informative about the underlying predictive signals that a classifier is relying on. Furthermore, $\hat { h }$ and the regularization objective must be tractable.
+
+Earlier work such as Pace et al. (2020) and Rame & Cord (2021) choose $\hat { h }$ to be the representations learnt by the classifiers, e.g. by constructing $f = f _ { l } \circ f _ { e }$ as a deep encoder network $f _ { e }$ that is attached to a linear classifier $f _ { l }$ and letting $\hat { h } \ = \ f _ { e } ( \mathbf { x } )$ . As the regularization objective for conditional independence, Rame & Cord (2021) compute pairwise conditional mutual information $\mathcal { C M T } ( f _ { e , s } , f _ { e , t } )$ whereas Pace et al. (2020) compute total correlation $\mathcal { T C } ( f _ { e , 1 } , \dots , f _ { e , M } )$ . Since the encoder representations are high-dimensional, these terms must be approximated.
+
+We propose a far simpler and more efficient method. Instead of network representations, we choose $\hat { h }$ to simply be the output distribution $\hat { h } = f ( \mathbf { x } ) = p _ { \boldsymbol { \theta } } ( y | \mathbf { x } )$ of the classifier. Accordingly, our regularization objective is conditional mutual information (CMI) between the output distributions of the classifiers. For any pair of classifiers $f _ { j } , f _ { k }$ , we have:
+
+$$
+\mathcal { C M T } ( f _ { s } , f _ { t } ) = \mathbb { E } _ { y } \left[ \mathcal { D } _ { K L } \Big ( p ( f _ { s } , f _ { t } | y ) | | p ( f _ { s } | y ) p ( f _ { t } | y ) \Big ) \right]
+$$
+
+CMI is zero iff $f _ { s } \perp \perp f _ { t } | y$ for all values of $y$ . Enforcing conditional independence on the classifiers’ predicted output probabilities rather than underlying representations trades off granularity of the independence constraint for computational efficiency. We believe that this is a valuable trade-off. Since $\hat { y }$ has categorical support, (4) can be cheaply and exactly estimated from training data. As our experiments in Section 5 show, even on a noisier signal like output distributions, enforcing conditional independence is sufficient to learn a diverse ensemble.
+
+Even though a diverse ensemble implies pairwise conditionally independent classifiers, the converse is not necessarily true. Mutual information is also zero if one of the classifiers outputs random or constant class probabilities. In particular, optimizing a weighted sum of the cross-entropy term and the CMI term can be challenging — overly weak regularization produces an ensemble that is not diverse, whereas overly strong regularization tends towards solutions containing close-to-random classifiers. Instead, we propose adding another term to regularize for confident predictions:
+
+$$
+\mathcal { R } ( f ) = \sum _ { k = 1 } ^ { K } \| p ( \hat { y } | y = k ) - I _ { k } \|
+$$
+
+where $I _ { k }$ is the indicator function at $k$ . Put together, the overall loss objective is:
+
+$$
+\mathcal { L } ( \{ \theta _ { m } \} _ { m = 1 } ^ { M } ) = \sum _ { i = 1 } ^ { N } \sum _ { m = 1 } ^ { M } \log p _ { \theta _ { m } } ( y _ { i } | \mathbf { x } _ { i } ) + \lambda _ { 1 } \cdot \sum _ { s = 1 } ^ { M } \sum _ { t = 1 } ^ { s - 1 } \mathcal { C } \mathcal { M } \mathcal { Z } ( f _ { s } , f _ { t } ) + \lambda _ { 2 } \cdot \sum _ { m = 1 } ^ { M } \mathcal { R } ( f _ { m } )
+$$
+
+where $\lambda _ { 1 }$ and $\lambda _ { 2 }$ are hyperparameters controlling the strength of regularization. A solution that minimizes (6) contains an ensemble where: (i) each classifier is accurate (first term) and confident (third term), and (ii) different classifiers rely on different subspaces of $\mathbf { z }$ for prediction (second term). We name such an ensemble a Conditionally Independent Deep Ensemble (CoDE).
+
+# 3.3 CODE: COMPUTATIONAL DETAILS
+
+The hyperparameters of the method are $M , \lambda _ { 1 }$ , and $\lambda _ { 2 }$ . Unlike traditional ensembles, $M$ (ensemble size) will typically be small $M = 2$ for all our experiments) since $M$ cannot be larger than the number of conditionally independent predictive signals inherent in the dataset. As is typical for OOD problems, we assume access to validation data from the test environment for hyperparameter tuning.
+
+Objective (6) describes the situation where all $M$ classifiers are jointly optimized. Since $M$ is typically small, doing so is not difficult or computationally expensive (as might be with traditional ensembles). An alternative to joint optimization is to learn the classifiers in a sequential fashion. The analogue to (6) becomes:
+
+$$
+\mathcal { L } ( \theta _ { m } ) = \sum _ { i = 1 } ^ { N } \log p _ { \theta _ { m } } ( y _ { i } | \mathbf { x } _ { i } ) + \lambda _ { 1 } \cdot \sum _ { s = 1 } ^ { m - 1 } \mathcal { C } \mathcal { M } \mathcal { Z } ( \hat { y } _ { s } , \hat { y } _ { m } ) + \lambda _ { 2 } \cdot \mathcal { R } ( f _ { m } )
+$$
+
+Sequential optimization presents a natural way to determine $M$ , as we can terminate the training process when no more predictive classifiers can be learnt. However, it will fail if earlier classifiers in the sequence learn multiple predictive signals. We discuss this further in Section 5.
+
+# 4 RELATED WORK
+
+Ensemble Methods In statistics, ensembling traditionally refers to combining multiple predictors into a single model that outperforms the individual learners, typically by bagging (Breiman, 1996) or boosting (Schapire, 1990). Diversity in this context refers to minimizing correlation between individual learners, which reduces variance and improve generalization (Kuncheva & Whitaker, 2003). Deep ensembling (Lakshminarayanan et al., 2017) is an analogous approach in deep learning where multiple randomly-initialized networks are trained in parallel, however, they are generally used for the purpose of uncertainty estimation. Unlike these works, we consider diversity specifically in the context of datasets with multiple predictive signals, and learning a diverse ensemble as recovering all such signals for the purpose of OOD generalization.
+
+Various Approaches For Learning Diversity As an unsupervised task, diversity refers to learning disentangled representations where natural factors of variation in the dataset are encoded into distinct latent dimensions (Bengio et al., 2013; Higgins et al., 2018); however, recent work has proposed incorporating weak supervision in this process (Locatello et al., 2019; Shu et al., 2019; Brehmer et al., 2022). As a supervised problem without OOD shifts, diversity refers to learning functions that disagree outside training points. Methods in this space have generally made use of input gradients (Ross et al., 2017; 2018) and orthogonality (Mashhadi et al., 2021; Xu et al., 2021). Finally, diversity is considered in the context of distribution shifts — either to improve robustness against adversarial attacks (Pang et al., 2019), to disambiguate between perfectly correlated signals (Lee et al., 2022), or to evade the simplicity bias by learning more complex functions (Pagliardini et al., 2022; Teney et al., 2021). Our work is most closely aligned with this last category. Unlike the approaches above, we exploit information-theoretic measures as our objective.
+
+Shortcut Learning and Spurious Correlations Shortcut learning (Geirhos et al., 2020) involves distribution shifts arising from spurious correlations (Buolamwini & Gebru, 2018; Xiao et al., 2020; Moayeri et al., 2022) and neural network biases (architectural or simplicity biases) (Geirhos et al., 2018; Shah et al., 2020; Teney et al., 2021). Methods that tackle distribution shifts must use additional data and/or assumptions. Examples of additional data include having multiple training environments (Arjovsky et al., 2019), counterfactual examples (Teney et al., 2020), access to enough validation data to fine-tune the model (Kirichenko et al., 2022), or group labels (Sagawa et al., 2019; Puli et al., 2022). Examples of additional assumptions include exploiting the lottery ticket hypothesis (Zhang et al., 2021) or treating misclassified training examples by an initial model as a proxy for minority groups (Liu et al., 2021; Zhang et al., 2022). Unlike these methods, we aim to learn all predictive signals in the dataset, rather than performing well on a single test distribution. Furthermore, we use validation data for hyperparameter tuning only, without additional sources of data (e.g. group labels).
+
+Information Bottleneck and Conditional Independence The line of work most similar to ours also exploits the information bottleneck method to learn diversity. Sinha et al. (2020) minimizes the mutual information $\mathcal { T } ( \hat { z } _ { s } , \hat { z } _ { t } )$ between learnt representations $\hat { z } _ { m }$ , however, this term is unconditional and will simply learn weak (biased) predictors, as noted in Section 3. Rame & Cord (2021) introduce DICE, which minimizes the conditional term $\mathcal { C } \mathcal { M } \mathcal { I } ( \hat { z } _ { s } , \hat { z } _ { t } )$ . Pace et al. (2020) considers total correlation $\mathcal { T C } ( \hat { z } _ { 1 } , \dots , \hat { z } _ { M } )$ instead of pairwise terms. Unlike CoDE, both of these approaches compute mutual information terms on the high-dimensional representations $\hat { z } _ { m }$ . Their objectives are intractable and must be approximated. For example, DICE requires both variational approximations and a jointly trained adversarial discriminator that learns to distinguish pairwise classifiers. Compared to these approaches, CoDE is by far computationally advantageous as mutual information for categorical output distributions can be computed faster and exactly.
+
+# 5 EXPERIMENTS
+
+Section 5.1 presents experiments on ColoredMNIST, which is used both to demonstrate the viability of our approach and to highlight pivotal observations and ablations. Section 5.2 then evaluates CoDE on larger benchmark datasets for shortcut learning to show that it scales effectively.
+
+# 5.1 COLOREDMNIST
+
+Setup As described in Section 2, the original MNIST (LeCun et al., 1998) labels are binarized (0-4, 5-9) and used to generate true labels $y$ with noise $p _ { d }$ . $y$ then generates binary color labels with noise $p _ { c }$ , used to color the image (red or green). As per Arjovsky et al. (2019), we consider two test environments: the training distribution where $p _ { d } = 0 . 2 5$ and $p _ { c } = 0 . 1$ , and the adversarial distribution where $p _ { d } = 0 . 2 5$ but $p _ { c } = 0 . 9$ (hence the shortcut-label correlation is reversed).
+
+Evaluation Baselines and Metrics As is standard in existing work, we evaluate predictive accuracy on the training and adversarial distributions. In choosing baselines, we considered the following desiderata for fairness and comprehensiveness: (i) comparing to both ensembling and non-ensembling methods, (ii) amongst ensembling methods, comparing to both conditional independence-based methods and those that do not, and (iii) comparing only to methods that do not require additional sources of data besides validation data for hyperparameter tuning. We chose the following baselines:
+
+Table 1: Results on ColoredMNIST. A theoretically ideal classifier relying only on digit (denoted as “Invariant”) will be upper-bounded by the digit-label noise $p _ { d }$ $(7 5 \% )$ , hence any result above $7 5 \%$ is relying on the color shortcut. CoDE has the strongest performance on the adversarial distribution. \*We were unable to reproduce TC-Ensemble on ColoredMNIST, and are citing their results in lieu.   
+
+<table><tr><td colspan="6">Results on ColoredMNIST</td></tr><tr><td>(pd,Pc)</td><td>Training (0.25, 0.1)</td><td>(0.25, 0.9)</td><td>(0.25, 0.5)</td><td>AdversarialRandom-ColorRandom-Color + Perfect-Digit (0.0, 0.5)</td></tr><tr><td>Invariant</td><td>75</td><td>75</td><td>75</td><td>100</td></tr><tr><td>ERM</td><td>88.6</td><td>15.3</td><td>52.5</td><td>53.4</td></tr><tr><td>JTT</td><td>17.8</td><td>87.9</td><td>52.5</td><td>56.6</td></tr><tr><td>Ortho-Ensemble</td><td>89.8</td><td>11.1</td><td>50.3</td><td>49.2</td></tr><tr><td>TC-Ensemble</td><td>89.1</td><td>69.8*</td><td>-</td><td>-</td></tr><tr><td>CoDE</td><td>70.7</td><td>70.0</td><td>70.8</td><td>91.2</td></tr></table>
+
+1. ERM classifier (ERM): single, standard classifier trained with ERM 2. Just Train Twice (Liu et al., 2021) (JTT): an initial classifier is trained for a limited number of epochs; mis-classified examples are upweighted to train the final classifier 3. Ensembles using input gradient orthogonality (Teney et al., 2021) (Ortho-Ensemble): an ensemble where the regularizing term is the dot product of the two models’ input gradients 4. Ensembles using conditional total correlation (CTC) (Pace et al., 2020) (TC-Ensemble): an ensemble learnt by minimizing CTC over the encoder network’s representation
+
+Table 1 shows all results on ColoredMNIST. We discuss the most important findings below.
+
+# 1. Enforcing conditional independence on output distributions achieves diversity effectively.
+
+Since ColoredMNIST is an artificially-created dataset whose DGP we know satisfy latent conditional independence ${ \overset { \cdot } { p } } _ { c }$ and $p _ { d }$ are independently determined), it is the ideal dataset to evaluate our key claim. Indeed, the strong performance of CoDE shows that it is sufficient to enforce conditional independence on output distributions. The final predictor selected via fast adaptation achieves near-invariant results, suggesting that it has correctly learnt from digit rather than color.
+
+# 2. CoDE generalizes to multiple OOD test distributions, without overfitting on any one specific distribution.
+
+In Table 1, JTT achieved about $90 \%$ on the adversarial distribution, implying that it overfitted to the adversarial distribution — by learning the opposite shortcut (color) correlation rather than the true signal (digit). This is further confirmed with additional results on two other test environments (Random-Color and Random-Color $^ +$ Perfect-Digit) where $p _ { c } = 0 . 5$ . JTT is close to random on these two environments, suggesting that it is still relying on color as the predictive feature. In contrast, CoDE achieves $91 \%$ when $p _ { d } = 0 . 0$ , suggesting that it has learnt to predict using digit.
+
+While concerning, these results are not entirely surprising. A method like JTT did exactly what it was designed to do, which is to minimize classification errors on the adversarial test distribution. Since $p _ { c } = 0 . 1$ , the opposite color correlation is precisely this loss-minimizing function. In contrast, CoDE will not find such a solution because two classifiers that return opposite predictions using the same feature (color) are perfectly correlated, even when conditioned on $y$ .
+
+These results highlight the shortcomings of single classifier methods like JTT. Such methods are designed to generalize to a specific test distribution, in general, this does not imply that they have learnt the desired predictive signal — merely that they have learnt an arbitrary function that does well on the test distribution. In contrast, methods that enforce diversity, such as CoDE, explicitly recover meaningful predictive signals that can generalize to any test distribution where $p ( \mathbf { z } | y )$ changes.
+
+Table 2: Additional results on ColoredMNIST and CelebA.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">ColoredMNIST</td><td colspan="2">CelebA</td></tr><tr><td>Training</td><td>Adversarial</td><td>Ave</td><td>Worst</td></tr><tr><td>CoDE (sequential f1)</td><td>90.0</td><td>10.2</td><td>95.2</td><td>31.1</td></tr><tr><td>CoDE (sequential f2)</td><td>70.1</td><td>70.0</td><td>95.0</td><td>33.3</td></tr><tr><td>CoDE (sequential f3)</td><td>63.2</td><td>49.0</td><td></td><td></td></tr><tr><td>CoDE (sequential f5)</td><td>64.4</td><td>42.2</td><td></td><td></td></tr><tr><td>CoDE (joint M= 2)</td><td>73.4</td><td>60.2</td><td>89.2</td><td>83.3</td></tr><tr><td>CoDE (joint M = 3)</td><td>74.6</td><td>44.3</td><td></td><td></td></tr><tr><td>CoDE (joint M = 5)</td><td>71.9</td><td>43.1</td><td></td><td></td></tr></table>
+
+# 3. Joint and sequential optimization are suited to different datasets.
+
+From our experiments, we found that there is no clear preference between either choice in terms of generalization ability. Table 2 shows both joint and sequential results on the ColoredMNIST and CelebA datasets. For ColoredMNIST, we found that sequential training performed better than joint training. For CelebA, joint training yielded a stronger classifier.
+
+This might be explained by the biases of the ERM model. In ColoredMNIST, as both latent factors (color and digit) are noisy predictors and as color presents a particularly simple shortcut, the ERM model solely learns from color. As such, a second classifier model that is trained sequentially can learn to predict solely from the digit feature. In contrast, the ERM model in CelebA has likely picked up some combination of the spurious (gender) and true (hair color) features, possibly because gender gives rise to complex features that are not ncessarily simpler to learn. This corroborates previous findings indicating that ERM models can learn an arbitrary combination of all predictive signals (Zhang et al., 2021; Kirichenko et al., 2022). As such, when trained sequentially, the second model fails to learn from hair color alone.
+
+The advantages of sequential optimization are: (i) cheaper computational costs as $M$ increases, and (ii) providing a natural stopping point for training. The latter comes from the fact that we can select for $M$ by terminating the training process when the subsequent classifier is no longer predictive, which indicates that there are no further predictive factors to be learnt. In contrast, joint optimization is advantageous as it allows us to avoid the pathological sitation where earlier models learn combinations of predictive factors. As small values of $M$ work well for CoDEs, we note that the computational cost of CoDEs are not prohibitive.
+
+# 5.2 BENCHMARK DATASETS
+
+Setup We consider the following benchmark datasets:
+
+• CelebA Liu et al. (2018); Sagawa et al. (2019): A dataset of celebrity faces with various labelled attributes. We consider the benchmark task in (Sagawa et al., 2019) of predicting the binary hair color attribute (blond or not), with gender (female or male) as the spurious attribute. There are therefore four groups. • Waterbirds (Wah et al., 2011; Sagawa et al., 2019): Setup described in Section 2. There are also four groups as both latent factors (background and foreground) are binary. • MF-Dominoes (MNIST-FashionMNIST) (LeCun et al., 1998; Xiao et al., 2017; Shah et al., 2020; Pagliardini et al., 2022): Each input image concatenates an MNIST digit (0 or 1) with a FashionMNIST object (coat or dress). The true label is the FashionMNIST object; the simpler MNIST feature is the shortcut. The minority groups represent $5 \%$ of the data.
+
+Table 3 shows all results on the benchmark datasets.
+
+# 4. CoDE scales well to large datasets and retains effectiveness at preventing shortcut learning.
+
+<table><tr><td rowspan="2"></td><td colspan="2">CelebA</td><td colspan="2">Waterbirds</td><td colspan="2">MF-Dominoes</td></tr><tr><td>Method Ave</td><td>Worst</td><td>Ave</td><td>Worst</td><td>Ave</td><td>Worst</td></tr><tr><td>ERM</td><td>94.8</td><td>46.7</td><td>90.4</td><td>78.3</td><td>88.9</td><td>76.9</td></tr><tr><td>JTT</td><td>88.0*</td><td>81.1*</td><td>93.3*</td><td>86.7*</td><td>89.5</td><td>76.1</td></tr><tr><td>CoDE</td><td>89.2</td><td>83.3</td><td>91.5</td><td>79.4</td><td>92.1</td><td>91.4</td></tr></table>
+
+Table 3: Main results on all datasets. CoDE achieves better adversarial or wrost-group accuracy than the other methods on all datasets except Waterbirds. ∗ Results from the JTT paper. We share the same model and training environment as their paper.
+
+On CelebA and MF-Dominoes, CoDE achieves the best worst-group accuracy. Unlike the earlier ColoredMNIST dataset, we have no guarantees that the core assumption of latent conditional independence holds. However, the strong performance of CoDE on these datasets shows that such an assumption is generally valid and useful when scaled to more realistic datasets.
+
+We note that CoDE performs poorly on Waterbirds. In our experiments, we selected $M = 2$ as the ensemble size. Even though there are no guarantees what will be the two conditionally independent classifiers that CoDE learns, in the other datasets, the results show that they do each correspond to the shortcut and true signal. This implies that in these datasets: (a) there are no features conditionally independent to both the shortcut and true signals and yet also strongly predictive of the label, and (b) the shortcut or true signal cannot be decomposed themselves into conditionally independent signals. Our hypothesis is that (b) is not true for Waterbirds. As the dataset is varied and contains a range of land and water backgrounds, there could be multiple spurious signals in the background that are somehow conditionally independent, resulting in these signals being learnt. Another possibility is that the ensemble could have learnt an imperfect or partial foreground signal.
+
+# 5. Computational effectiveness is crucial to learn diverse ensembles at scale.
+
+Beyond ColoredMNIST, we found that it was computationally prohibitive to run Ortho-Ensemble, as the size of ensembles required to work well (48 or 96) was too high. We also noted that we could not implement TC-Ensembles successfully on larger datasets, noting that the original authors do not test on datasets besides ColoredMNIST either. We believe that this further highlights the importance of computational efficiency in diverse ensembling.
+
+# 6 DISCUSSION AND CONCLUSION
+
+Appendix B discusses potential failure modes of our method.
+
+We introduce CoDE, a method for learning an ensemble of diverse classifiers that rely on different predictive signals in the dataset. The key assumption made by CoDE conditional independence between predictive signals, which it enforces on classifiers’ output distributions. We find that CoDE works well in practice when applied to shortcut learning tasks. Future work includes: (a) evaluating CoDEs on other applications where multiple predictive signals exist, such as fairness-related tasks where we might want to learn classifiers that do not rely on sensitive attributes, and (b) considering other metrics for conditional independence that might provide more fine-grained signals than output distributions (e.g. minimizing mutual information between latent representations).
+
+# ETHICS STATEMENT
+
+Positive Impact Being robust to distribution shifts, CoDE will have a positive impact when deployed to high-stakes domains, where learning shortcut signals can have harmful social consequences. One such notable example is pneumonia prediction — models trained on pneumonia labels from chest $\mathrm { X }$ -ray scans have been shown to learn machine-specific artifacts in the background, which is a shortcut as hospitals have differing positivity rates and use different machines (Zech et al., 2018).
+
+Negative Impact There are no notable negative impacts of using CoDE specifically, besides the general potential for all machine learning models to be abused in the wrong hands.
+
+REPRODUCIBILITY STATEMENT
+
+We intend to release public code with a camera-ready version of the paper.
+
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+
+# A EXPERIMENTAL DETAILS
+
+Architecture and Training Details For ColoredMNIST, we use a CNN as the classifier, containing two convolutional layers and two fully-connected layers. Adam (Kingma & Ba, 2014) is used for optimization, with a learning rate of 0.001. For CelebA, Waterbirds, and MF-Dominoes, we use a ResNet-50 (He et al., 2016). SGD is used for optimization, with a learning rate of 0.001, momentum decay of 0.9, and weight decay of 0.001. Additionally, following previous work (e.g. Sagawa et al., 2019; Liu et al., 2021), the Waterbirds model is pre-trained on ImageNet (Deng et al., 2009) and includes data augmentation in the form of random horizontal flips and random resized cropping. For CelebA and Waterbirds, class reweighting is performed to ensure that there are roughly equal positive and negative labels. The random seed used for all experiments is 13.
+
+Hyperparameters for CoDE and Baselines CoDE. For all four datasets, we used $M = 2$ as the ensemble size, besides ablations for $M$ as detailed in Appendix B. The results in Table 3 were achieved with sequential training for ColoredMNIST and with joint training for the other three datasets. For ColoredMNIST, $\lambda _ { 1 } = 1 2 0 0$ and $\lambda _ { 2 } = 1 0$ . For CelebA, $\lambda _ { 1 } = 5 0 0$ and $\lambda _ { 2 } = 0 . 1$ . For Waterbirds, $\lambda _ { 1 } = 5 0 0$ and $\lambda _ { 2 } = 0 . 1$ . For MF-Dominoes, $\lambda _ { 1 } = 3 0 0$ and $\lambda _ { 2 } = 0 . 1$ . JTT. We performed a hyperparameter sweep with $T \in \{ 1 , 5 , 1 0 \}$ (number of epochs for initial model training) and $\alpha \in \{ 2 , 1 0 , 1 0 0 \}$ (upweighting factor for mis-classified examples). Orthogonal Ensembles. All classifiers share the same feature extractor (i.e. convolutional output for ColoredMNIST and ResNet-50 feature representation for the other three datasets). We experimented with different values of $M$ , however, values of $M$ above 16 (for ColoredMNIST) and above 4 (for the other three datasets) were prohitibively expensive. As such, we did not try $M = 4 8$ or $M = 9 6$ as used by Teney et al. (2021). For these smaller values of $M$ that we tried, we did not notice an improvement from the ERM model. Besides ColoredMNIST, we did not report these results.
+
+# B MODEL MIS-SPECIFICATION: POTENTIAL FAILURE MODES
+
+The success of any method tackling distribution shifts depends on how well the assumptions made have been upheld. We discuss the potential implications when the model is mis-specified and these assumptions are no longer valid.
+
+Conditional Dependence CoDE relies on the assumption that predictive signals are conditionally independent. We using the synthetic ColoredMNIST dataset to generate a DGP where such an assumption does not hold true. Instead of the standard setup where color labels are generated from the true labels, we generate color labels from the original (binarized) MNIST labels instead, at the same noise level $p _ { c } = 0 . 1$ . This means that the color and digit signals are now highly correlated. Both are still predictive since the true labels themselves were generated from MNIST labels.
+
+Table 4 shows the results of this experiment. As we expect, conditionally dependent features cannot be recovered by minimizing conditional mutual information. The ensemble either recovers one of the two features (when trained sequentially) or neither. This confirms our intuition that conditional independence must be correctly specified for CoDE to work. While these results demonstrate a failure mode of CoDE, conditional independence between predictive factors of interest does hold well in many natural image datasets, as shown in Table 3.
+
+Latent Mis-specification The size of the ensemble $M$ specifies how many predictive latent factors we believe generated the dataset. We can consider the mis-specification of $M$ in either direction: (i) if the true dimension of $\mathbf { z }$ is smaller than $M$ , and (ii) if the true dimension of $\mathbf { z }$ is larger than $M$ .
+
+In case (i), since the number of conditional independent components has been over-specified, whether the ensemble has been jointly or sequentially trained makes a difference. Consider the results on the ColoredMNIST dataset in Table 2 again. In the sequential regime, the first two classifiers $f _ { 1 }$ and $f _ { 2 }$ correspond to the color and digit classifiers respectively, however, the subsequent few classifiers $f _ { 3 }$ and $f _ { 5 } ^ { } ,$ ) do not learn anything meaningful and perform poorly on both training and adversarial distributions. However, as noted in Section 5, this does not pose a serious problem since we can use validation data to naturally determine the stopping point. On the other hand, over-specification of $M$ is more worrying in the joint regime, as there is no guarantee that any of the true latent factors are learnt at all. As Table 2 shows, for $M = 3$ or 5, the best-performing classifier does not generalize.
+
+Table 4: Results on ColoredMNIST with color-digit conditional dependence, on both joint and sequential training with $M = 2$ classifiers. When trained sequentially, the first classifier $f _ { 1 }$ learns the digit correlation since digit is most predictive in this setup. However, as color is no longer conditionally independent of digit, there is no predictive feature that can be learnt by the second classifier $f _ { 2 }$ , resulting in a close-to-random predictor. When trained jointly, neither of the classifiers correspond to the color or digit feature.   
+
+<table><tr><td>(pd,Pc)</td><td>Training (0.25, 0.1)</td><td>Adversarial (0.25, 0.9)</td><td>Random-Color (0.25, 0.5)</td><td>Perfect-Digit (0.0, 0.5)</td></tr><tr><td>CoDE (sequential f1)</td><td>77.1</td><td>63.2</td><td>70.6</td><td>90.5</td></tr><tr><td>CoDE (sequential f2)</td><td>53.6</td><td>50.1</td><td>51.5</td><td>54.5</td></tr><tr><td>CoDE (joint f1)</td><td>84.9</td><td>25.8</td><td>56.0</td><td>60.4</td></tr><tr><td>CoDE (joint f2)</td><td>54.3</td><td>73.0</td><td>64.4</td><td>77.5</td></tr></table>
+
+In case (ii), where the number of conditional independent components is under-specified, the learnt ensemble may correspond to any subset of the true latent factors and individual classifiers could also learn arbitrary combinations of the latent factors. For example, the trivial case where $M = 1$ is underspecified simply returns the ERM model. In general, since $M$ is a hyperparameter, latent mis-specification does not pose a serious problem as we can tune its value using the validation data.

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+        "text": "DOING FAST ADAPTATION FAST: CONDITIONALLY INDEPENDENT DEEP ENSEMBLES FOR DISTRIBUTION SHIFTS ",
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+        "text": "ABSTRACT ",
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+        "text": "Classifiers in a diverse ensemble capture distinct predictive signals, which is valuable for datasets containing multiple strongly predictive signals. Performing fast adaptation at test time allows us to generalize to distributions where certain signals are no longer predictive, or to avoid relying on sensitive or protected attributes. However, ensemble learning is often expensive, even more so when we need to enforce diversity constraints between the high-dimensional representations of the classifiers. Instead, we propose an efficient and fast method for learning ensemble diversity. We minimize conditional mutual information of the output distributions between classifiers, a quantity which can be cheaply and exactly computed from empirical data. The resulting ensemble contains individually strong predictors that are only dependent because they predict the label. We demonstrate the efficacy of our method on shortcut learning tasks. Performing fast adaptation on our ensemble selects shortcut-invariant models that generalize well to test distributions where the shortcuts are uncorrelated with the label. ",
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+        "text": "1 INTRODUCTION ",
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+        "text": "Some of the strongest scientific theories are supported by multiple sources of evidence, a principle described by 19th century philosopher William Whewell as “consilience”. Evolution is one such example, having been firmly corroborated by fields ranging from paleontology to genetics. In many real-world applications of machine learning, datasets can similarly contain multiple predictive signals that explain the label well. In these settings, a standard model typically learns from a combination of predictive features (Ross et al., 2018; Kirichenko et al., 2022). Such a model will fail to generalize to distribution shifts that break the correlation between certain signals and the label (Hovy & Søgaard, 2015; Hashimoto et al., 2018; Puli et al., 2022). ",
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+        "text": "This shortcoming can be addressed by learning a diverse set or ensemble of classifiers. Such methods typically exploit some notion of independence to learn multiple classifiers that rely on different predictive signals. We can then perform fast adaptation, using a small amount of out-of-distribution (OOD) validation data to select the model that generalizes best. Learning diversity is also beneficial in and of itself: these classifiers are empirically shown to be more human-interpretable than if we were to fit a single model (Ross et al., 2018), possibly because they learn disentangled representations that correspond to natural factors of variation (Shu et al., 2019). ",
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+        "text": "The key challenge is quantifying the right notion of diversity. Existing work has exploited concepts like input gradient or parameter orthogonality as a proxy for statistical independence (Teney et al., 2021; Xu et al., 2021). To tackle OOD generalization, which fundamentally requires additional assumptions or data beyond the observed training data (Bareinboim et al., 2022; Scholkopf et al., ¨ 2021), previous work have also assumed access to unlabelled test data and measured disagreement on those examples (Lee et al., 2022; Pagliardini et al., 2022). However, these objectives or assumptions are often prohibitive or unrealistic in real-world settings. For example, group-balanced test data is not always obtainable, e.g. when deploying a pneumonia model to multiple new hospitals whose patient profiles may change over time. Another costly example is enforcing input gradient orthogonality on high-dimensional covariates like images or text, where it can be challenging to avoid learning from orthogonal covariates of the same underlying feature, such as neighboring pixels. ",
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+    {
+        "type": "text",
+        "text": "To avoid the pitfalls of operating in high-dimensional input or parameter space, a promising line of work instead adopts the information-theoretic perspective and tackles the problem as representation learning. These approaches apply the information bottleneck method and minimize mutual information between the representations learnt by each classifier. Such an objective forces the classifiers to rely on distinctly meaningful features for prediction. Most notably, Pace et al. (2020) and Rame & Cord (2021) minimize mutual information between the classifier representations conditioned on the label. Since any pair of predictors cannot both be accurate while remaining unconditionally independent, the extra conditioning prevents learning weak classifiers. The resulting ensemble contains accurate classifiers that nevertheless rely on distinct predictive signals. The only core assumption is that the underlying predictive signals are themselves conditionally independent. ",
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+        "type": "text",
+        "text": "These approaches are conceptually appealing but practically challenging. Mutual information between high-dimensional representations is intractable and must be approximated, either via variational (e.g. Fischer, 2020) or contrastive (e.g. Oord et al., 2018) bounds. Furthermore, such approximations are computationally expensive, a problem that is compounded in the ensemble setting where we wish to train multiple classifiers speedily. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We seek to learn ensemble diversity fast and effectively. Our key insight is that it suffices to enforce conditional independence on the output distributions of the classifiers. Our first contribution is proposing conditional mutual information (CMI) between output distributions as the regularizing objective. Assuming conditionally independent predictive signals, enforcing CMI between output distributions also guarantees that the ensemble where separate predictive signals are learnt by separate classifiers is a minimizing solution. Since the output distribution is categorical, CMI can be cheaply and exactly computed from empirical data. In addition, our method avoids using additional sources of data that cannot be found in many real-world domains, such as unlabelled test data or “group” labels for each predictive signal in the dataset. We only permit a small amount of validation data from the test distribution for (1) hyperparameter tuning and (2) selecting the final predictor from our ensemble. We dub our approach as Conditionally Independent Deep Ensembles (CoDE). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our second contribution is evaluating CoDE on benchmark datasets for shortcut learning (Geirhos et al., 2020). Shortcuts are signals that are (i) highly but spuriously correlated to the label in the training distribution, possibly due to biases in data collection or other systematic pre-processing errors (Torralba & Efros, 2011), and (ii) preferentially learnt by a neural network, possibly due to simplicity biases (Shah et al., 2020) or architectural biases (e.g. convolutional neural networks (CNNs) relying on texture over shape (Baker et al., 2018)). An empirical risk minimizing (ERM) model will rely on shortcuts and fail to generalize to test distributions where they are no longer correlated to the label. This is a natural application for our method as the core assumption of conditional independence applies to many such datasets — for example, in natural images, the foreground is typically the label and is thus conditionally independent from the background (shortcut). We show that CoDE effectively recovers an ensemble where the shortcut features and the true signal are learnt by separate classifiers. ",
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+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES: SETUP AND NOTATION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In Section 3, we will fully motivate the assumptions behind our model of the data-generating process (DGP). However, we describe it here first to establish key terminology and concepts. ",
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+        "type": "text",
+        "text": "Data-Generating Process Let $\\mathbf { z }$ denote the set of latent factors that generate the set of observed features $\\mathbf { x } \\in \\mathbb { R } ^ { P }$ . Let $y \\in \\{ 0 , 1 , \\dotsc , K - 1 \\}$ denote the label. The data $p _ { e } ( \\mathbf { x } , y , \\mathbf { z } )$ is generated from a family of distributions indexed by $e$ , the environment. We only consider: (i) a single training environment $\\mathbf { \\boldsymbol { e } } _ { \\mathbf { \\lambda } } = t { \\boldsymbol { r } } _ { \\mathbf { \\lambda } } $ ), from which we have access to i.i.d. labelled training examples $D _ { t r } \\ = \\ \\{ { \\bf x } _ { i } , y _ { i } \\} _ { i = 1 } ^ { N }$ , and (ii) a test environment $\\mathit { \\Pi } _ { \\mathrm { ~ e ~ } } = \\mathit { \\Pi } _ { t e }$ ), from which we draw unlabelled test examples that our model should perform well on. We also allow access to a small set of labelled validation data $D _ { v a l } = \\{ \\mathbf { x } _ { i } , y _ { i } \\} _ { i = 1 } ^ { N ^ { \\prime } }$ from the test environment, which is used only for hyperparameter tuning and ensembling (i.e. constructing the final model from the set of learnt classifiers). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We make the following assumptions on the DGP: ",
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+    {
+        "type": "text",
+        "text": "(i) all label information is encoded by $\\mathbf { z }$ , i.e. $p _ { e } ( y | \\mathbf { x } , \\mathbf { z } ) = p _ { e } ( y | \\mathbf { z } )$ for all $e$ (ii) $p _ { e } ( \\mathbf { x } | \\mathbf { z } ) = p ( \\mathbf { x } | \\mathbf { z } )$ is invariant across all $e$ ",
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+    {
+        "type": "text",
+        "text": "(iii) $p _ { e } ( { \\bf z } ) > 0$ for all $e$ and $\\mathbf { z }$   \n(iv) $p _ { e } ( y ) > 0$ for all $e$ and $y$   \n(v) [Latent Conditional Independence] $z _ { i } \\perp \\perp z _ { j } \\mid y$ for all $e$ and $i , j$ ",
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+    {
+        "type": "text",
+        "text": "Based on these assumptions, we can factorize $p _ { e } ( \\mathbf { x } , y , \\mathbf { z } )$ as: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5fb6a370256322a0962a9947d8b9226219d54510301bc283217e37e847fc2d7a.jpg",
+        "text": "$$\np _ { e } ( \\mathbf { z } , \\mathbf { x } , y ) = p _ { e } ( y ) \\left( \\prod _ { i = 1 } ^ { L } p _ { e } ( z _ { i } | y ) \\right) p ( \\mathbf { x } | \\mathbf { z } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Example: ColoredMNIST As introduced in Arjovsky et al. (2019), $y$ is a binary label which determines color $( z _ { 1 } \\in \\{ \\mathrm { r e d } , \\mathrm { g r e e n } \\} )$ with probability $p _ { c }$ and digit $( z _ { 2 } \\in \\{ 0 \\ – 4 , 5 \\ – 9 \\} )$ ) with probability $p _ { d }$ . $p _ { c }$ and $p _ { d }$ are independently chosen. In the training distribution, $p _ { c } = 0 . 2 5$ and $p _ { d } = 0 . 1$ , as such, an ERM model will primarily learn from color. $p _ { c }$ and $p _ { d }$ can be arbitrary in the test distribution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Example: Waterbirds As introduced in Sagawa et al. (2019), $y$ is a binary label determining if the image represents a water or land bird. It perfectly determines the foreground $( z _ { 1 } \\in \\ \\left\\{ \\begin{array} { l l } { \\end{array} } \\right.$ water bird, land bird}) and is highly but spuriously correlated to the background $( z _ { 2 } \\in$ {water, land}) in the training distribution. An ERM model will learn from background features. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Group Robustness When $\\mathbf { z }$ is discrete, each possible value that $\\mathbf { z }$ can take is known as a group. Due to the spurious correlations created by $p _ { t r } ( z _ { i } | y )$ , groups that are highly represented in the training set are called “majority groups”, and poorly-represented groups are “minority groups”. Group robustness refers to the goal of generalizing well on all groups and is one natural way of evaluating if a model has been learning shortcuts. For example, both ColoredMNIST and Waterbirds admits four groups formed by the Cartesian product of $z _ { 1 }$ and $z _ { 2 }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Ensembles and Fast Adaptation A classifier $f ( \\mathbf { x } ) : = p _ { \\theta } ( y | \\mathbf { x } )$ is parametrized by $\\theta$ and outputs class probabilities. We will use $\\hat { y } : = p _ { \\theta } ( y )$ to denote the unconditional output distribution. We use the term “ensemble” loosely to refer to a set of or sequentially. (Section 4 clarifies the relations $M$ classifiers  to tradition $\\{ f _ { m } \\} _ { m = 1 } ^ { M }$ that can be learnt joinle methods.) After all $M$ classifiers are learnt, the final model $\\theta ^ { * }$ is selected using validation data $D _ { v a l }$ : ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/001c2f13b422603eecb84248118a943631e48b18f9a5ef9109c8ccaecd2fea37.jpg",
+        "text": "$$\n\\theta ^ { * } = \\arg \\operatorname* { m i n } _ { \\theta _ { m } , m \\in \\{ 1 , \\dots , M \\} } \\frac { 1 } { N ^ { \\prime } } \\sum _ { i = 1 } ^ { N ^ { \\prime } } \\log p _ { \\theta _ { m } } ( y _ { i } | \\mathbf { x } _ { i } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This process is referred to as fast adaptation. ",
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+    {
+        "type": "text",
+        "text": "3 CONDITIONALLY INDEPENDENT DEEP ENSEMBLES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To motivate our approach and the assumptions made in (1), we first define what it means to learn a diverse ensemble and explain why conditional independence is a sound measure of diversity. ",
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+    {
+        "type": "text",
+        "text": "3.1 DIVERSITY AS CONDITIONAL INDEPENDENCE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Diverse classifiers utilize separate predictive signals, intuitively, they predict the “same things for different reasons” (Rame & Cord, 2021). Our setup in Section 2 formalizes this notion of “different reasons” by explicitly defining the latent variable $\\mathbf { z }$ , which models the total underlying set of predictive signals that relate $\\mathbf { x }$ to $y$ . A classifier that learns a mapping from $\\mathbf { x }$ to $y$ can then be interpreted as implicitly inferring $\\mathbf { z }$ from $\\mathbf { x }$ and learning a mapping from $\\mathbf { z }$ to $y$ . We can thus define diverse classifiers that rely on separate predictive signals as learning from separate dimensions or subspaces of $\\mathbf { z }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To formalize the idea that a classifier $f$ learns using only a subspace of $\\mathbf { z }$ , one naive approach might be to define $f$ as relying only on the subspace ${ \\mathbf z } _ { [ a ] }$ if and only if (some distribution computed from) $f$ is independent of its complement ${ \\mathbf z } \\backslash { \\mathbf z } _ { [ a ] }$ . This definition is convenient as it suggests that the appropriate objective to learn a diverse ensemble is simply to enforce statistical independence between the classifiers. This follows because two classifiers that rely on overlapping subspaces of $\\mathbf { z }$ will necessarily be dependent. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "However, the definition above assumes that distinct predictive signals (i.e. subspaces of $\\mathbf { z }$ ) are themselves unconditionally independent. This is not always true when a dataset contains multiple strongly predictive signals. Dimensions of $\\mathbf { z }$ can be dependent by virtue of their correlation to $y$ Classifiers that learn from such signals will similarly be dependent. Shortcut learning is precisely a problem because meaningful and spurious features are highly correlated in the training environment. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This conundrum can be resolved by establishing independence of the latent factors with conditioning on $y$ . Doing so is equivalent to assuming that upon knowing the true label, observing one set of features yields no additional information about other features. This is usually a realistic assumption to make. As the Waterbirds example in Section 2 shows, backgrounds and foregrounds are often conditionally independent in the test distributions we care about. This motivates our assumption (v) of latent conditional independence in Section 2, where the individual factors $z _ { i }$ are conditionally independent given $y$ . We formalize this notion of “diversity as conditional independence” below. ",
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+    {
+        "type": "text",
+        "text": "Definition 3.1. Let $\\mathbf { z } _ { [ a ] } : = ( z _ { a _ { 1 } } , \\ldots , z _ { a _ { l } } )$ denote some subspace of $\\mathbf { z }$ . Let $\\hat { h } ( f )$ denote some distribution computed from $f$ . We say $f$ is invariant to ${ \\mathbf z } _ { [ a ] }$ if $\\hat { h } \\perp \\perp ( z _ { a _ { 1 } } , \\ldots , z _ { a _ { l } } ) | y$ . Let $\\mathbf { z } _ { [ i ] }$ be the maximal subset of $\\mathbf { z }$ that $f$ is invariant to. Then $f$ is said to rely on $\\mathbf { z } _ { - [ i ] } : = \\mathbf { z } \\backslash \\mathbf { z } _ { [ i ] }$ for prediction. ",
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+    {
+        "type": "text",
+        "text": "Definition 3.2. Let $f$ and $f ^ { \\prime }$ be a pair of classifiers that rely on $\\mathbf { z } _ { [ i ] }$ and $\\mathbf { z } _ { [ i ^ { \\prime } ] }$ respectively. $f$ and $f ^ { \\prime }$ are said to be diverse if $\\mathbf { z } _ { [ i ] } \\bigcap _ { . . } \\mathbf { z } _ { [ i ^ { \\prime } ] } = \\emptyset$ . An ensemble $\\{ f _ { m } \\} _ { m = 1 } ^ { M }$ is diverse if every pair of classifiers $f _ { j } , f _ { k }$ in the ensemble are diverse. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "It follows immediately from Definition 3.2 that diverse classifiers must themselves be conditionally independent, i.e. $\\hat { h } _ { i } \\perp \\perp \\hat { h } _ { j } | y$ . Our training objective for learning a diverse ensemble should therefore enforce conditional independence on all pairs of classifiers: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e4e8a0dbffae0fbcd6746d41ea494981c0525f1b325a81adf8c416c4bc03ced.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\arg \\underset { \\theta _ { 1 } , \\dots , \\theta _ { M } } { \\operatorname* { m a x } } \\sum _ { i = 1 } ^ { N } \\sum _ { m = 1 } ^ { M } \\log p _ { \\theta _ { m } } ( y _ { i } | \\mathbf { x } _ { i } ) } \\\\ { \\displaystyle \\mathrm { s u b j e c t ~ t o } \\hat { h } _ { s } \\perp \\hat { h } _ { t } \\vert y \\qquad \\forall s , t } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We can interpret (3) as follows: the main objective guarantees that the learnt ensemble contains individually strong predictors, whereas the constraint guarantees that each predictor is uninformative of the others when conditioned on the label. Put together, (3) learns classifiers that rely on conditionally independent subspaces of $\\mathbf { z }$ and thus provide no additional information about each other. As is typical in machine learning (Krogh & Hertz, 1991; Deb, 2014), we optimize an unconstrained analogue of (3) by expressing the constraint as a regularization term. ",
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+    {
+        "type": "text",
+        "text": "3.2 ENFORCING CONDITIONAL INDEPENDENCE VIA OUTPUT DISTRIBUTIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "It remains for us to decide on the distribution $\\hat { h }$ that we constrain, as well as the (unconstrained) regularization objective from (3). These choices are crucial in many ways. Since independence with respect to $\\hat { h }$ underpins the notions of invariance and diversity in Definitions 3.1 and 3.2, it must be informative about the underlying predictive signals that a classifier is relying on. Furthermore, $\\hat { h }$ and the regularization objective must be tractable. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Earlier work such as Pace et al. (2020) and Rame & Cord (2021) choose $\\hat { h }$ to be the representations learnt by the classifiers, e.g. by constructing $f = f _ { l } \\circ f _ { e }$ as a deep encoder network $f _ { e }$ that is attached to a linear classifier $f _ { l }$ and letting $\\hat { h } \\ = \\ f _ { e } ( \\mathbf { x } )$ . As the regularization objective for conditional independence, Rame & Cord (2021) compute pairwise conditional mutual information $\\mathcal { C M T } ( f _ { e , s } , f _ { e , t } )$ whereas Pace et al. (2020) compute total correlation $\\mathcal { T C } ( f _ { e , 1 } , \\dots , f _ { e , M } )$ . Since the encoder representations are high-dimensional, these terms must be approximated. ",
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+    {
+        "type": "text",
+        "text": "We propose a far simpler and more efficient method. Instead of network representations, we choose $\\hat { h }$ to simply be the output distribution $\\hat { h } = f ( \\mathbf { x } ) = p _ { \\boldsymbol { \\theta } } ( y | \\mathbf { x } )$ of the classifier. Accordingly, our regularization objective is conditional mutual information (CMI) between the output distributions of the classifiers. For any pair of classifiers $f _ { j } , f _ { k }$ , we have: ",
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+    {
+        "type": "equation",
+        "img_path": "images/663c12db3346b980918eac49e6b0f74452a9924ca7db2c6c7b339cb0aa43f621.jpg",
+        "text": "$$\n\\mathcal { C M T } ( f _ { s } , f _ { t } ) = \\mathbb { E } _ { y } \\left[ \\mathcal { D } _ { K L } \\Big ( p ( f _ { s } , f _ { t } | y ) | | p ( f _ { s } | y ) p ( f _ { t } | y ) \\Big ) \\right]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "CMI is zero iff $f _ { s } \\perp \\perp f _ { t } | y$ for all values of $y$ . Enforcing conditional independence on the classifiers’ predicted output probabilities rather than underlying representations trades off granularity of the independence constraint for computational efficiency. We believe that this is a valuable trade-off. Since $\\hat { y }$ has categorical support, (4) can be cheaply and exactly estimated from training data. As our experiments in Section 5 show, even on a noisier signal like output distributions, enforcing conditional independence is sufficient to learn a diverse ensemble. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Even though a diverse ensemble implies pairwise conditionally independent classifiers, the converse is not necessarily true. Mutual information is also zero if one of the classifiers outputs random or constant class probabilities. In particular, optimizing a weighted sum of the cross-entropy term and the CMI term can be challenging — overly weak regularization produces an ensemble that is not diverse, whereas overly strong regularization tends towards solutions containing close-to-random classifiers. Instead, we propose adding another term to regularize for confident predictions: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/30e979a8ac0e3b27a96e209e7b3d03a74ab1e7022bab12d04c9094409a5af617.jpg",
+        "text": "$$\n\\mathcal { R } ( f ) = \\sum _ { k = 1 } ^ { K } \\| p ( \\hat { y } | y = k ) - I _ { k } \\|\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $I _ { k }$ is the indicator function at $k$ . Put together, the overall loss objective is: ",
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+    {
+        "type": "equation",
+        "img_path": "images/ac82014a089a7337b07a043c5d569f12ebf1de2cb54a395ae9dbed7c17632475.jpg",
+        "text": "$$\n\\mathcal { L } ( \\{ \\theta _ { m } \\} _ { m = 1 } ^ { M } ) = \\sum _ { i = 1 } ^ { N } \\sum _ { m = 1 } ^ { M } \\log p _ { \\theta _ { m } } ( y _ { i } | \\mathbf { x } _ { i } ) + \\lambda _ { 1 } \\cdot \\sum _ { s = 1 } ^ { M } \\sum _ { t = 1 } ^ { s - 1 } \\mathcal { C } \\mathcal { M } \\mathcal { Z } ( f _ { s } , f _ { t } ) + \\lambda _ { 2 } \\cdot \\sum _ { m = 1 } ^ { M } \\mathcal { R } ( f _ { m } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\lambda _ { 1 }$ and $\\lambda _ { 2 }$ are hyperparameters controlling the strength of regularization. A solution that minimizes (6) contains an ensemble where: (i) each classifier is accurate (first term) and confident (third term), and (ii) different classifiers rely on different subspaces of $\\mathbf { z }$ for prediction (second term). We name such an ensemble a Conditionally Independent Deep Ensemble (CoDE). ",
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+    {
+        "type": "text",
+        "text": "3.3 CODE: COMPUTATIONAL DETAILS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The hyperparameters of the method are $M , \\lambda _ { 1 }$ , and $\\lambda _ { 2 }$ . Unlike traditional ensembles, $M$ (ensemble size) will typically be small $M = 2$ for all our experiments) since $M$ cannot be larger than the number of conditionally independent predictive signals inherent in the dataset. As is typical for OOD problems, we assume access to validation data from the test environment for hyperparameter tuning. ",
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+    {
+        "type": "text",
+        "text": "Objective (6) describes the situation where all $M$ classifiers are jointly optimized. Since $M$ is typically small, doing so is not difficult or computationally expensive (as might be with traditional ensembles). An alternative to joint optimization is to learn the classifiers in a sequential fashion. The analogue to (6) becomes: ",
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+    {
+        "type": "equation",
+        "img_path": "images/3635bb1f271b1688d0f076e2aa0021e2326da2684f767d91c174108efce5cc34.jpg",
+        "text": "$$\n\\mathcal { L } ( \\theta _ { m } ) = \\sum _ { i = 1 } ^ { N } \\log p _ { \\theta _ { m } } ( y _ { i } | \\mathbf { x } _ { i } ) + \\lambda _ { 1 } \\cdot \\sum _ { s = 1 } ^ { m - 1 } \\mathcal { C } \\mathcal { M } \\mathcal { Z } ( \\hat { y } _ { s } , \\hat { y } _ { m } ) + \\lambda _ { 2 } \\cdot \\mathcal { R } ( f _ { m } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Sequential optimization presents a natural way to determine $M$ , as we can terminate the training process when no more predictive classifiers can be learnt. However, it will fail if earlier classifiers in the sequence learn multiple predictive signals. We discuss this further in Section 5. ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Ensemble Methods In statistics, ensembling traditionally refers to combining multiple predictors into a single model that outperforms the individual learners, typically by bagging (Breiman, 1996) or boosting (Schapire, 1990). Diversity in this context refers to minimizing correlation between individual learners, which reduces variance and improve generalization (Kuncheva & Whitaker, 2003). Deep ensembling (Lakshminarayanan et al., 2017) is an analogous approach in deep learning where multiple randomly-initialized networks are trained in parallel, however, they are generally used for the purpose of uncertainty estimation. Unlike these works, we consider diversity specifically in the context of datasets with multiple predictive signals, and learning a diverse ensemble as recovering all such signals for the purpose of OOD generalization. ",
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+    {
+        "type": "text",
+        "text": "Various Approaches For Learning Diversity As an unsupervised task, diversity refers to learning disentangled representations where natural factors of variation in the dataset are encoded into distinct latent dimensions (Bengio et al., 2013; Higgins et al., 2018); however, recent work has proposed incorporating weak supervision in this process (Locatello et al., 2019; Shu et al., 2019; Brehmer et al., 2022). As a supervised problem without OOD shifts, diversity refers to learning functions that disagree outside training points. Methods in this space have generally made use of input gradients (Ross et al., 2017; 2018) and orthogonality (Mashhadi et al., 2021; Xu et al., 2021). Finally, diversity is considered in the context of distribution shifts — either to improve robustness against adversarial attacks (Pang et al., 2019), to disambiguate between perfectly correlated signals (Lee et al., 2022), or to evade the simplicity bias by learning more complex functions (Pagliardini et al., 2022; Teney et al., 2021). Our work is most closely aligned with this last category. Unlike the approaches above, we exploit information-theoretic measures as our objective. ",
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+    {
+        "type": "text",
+        "text": "Shortcut Learning and Spurious Correlations Shortcut learning (Geirhos et al., 2020) involves distribution shifts arising from spurious correlations (Buolamwini & Gebru, 2018; Xiao et al., 2020; Moayeri et al., 2022) and neural network biases (architectural or simplicity biases) (Geirhos et al., 2018; Shah et al., 2020; Teney et al., 2021). Methods that tackle distribution shifts must use additional data and/or assumptions. Examples of additional data include having multiple training environments (Arjovsky et al., 2019), counterfactual examples (Teney et al., 2020), access to enough validation data to fine-tune the model (Kirichenko et al., 2022), or group labels (Sagawa et al., 2019; Puli et al., 2022). Examples of additional assumptions include exploiting the lottery ticket hypothesis (Zhang et al., 2021) or treating misclassified training examples by an initial model as a proxy for minority groups (Liu et al., 2021; Zhang et al., 2022). Unlike these methods, we aim to learn all predictive signals in the dataset, rather than performing well on a single test distribution. Furthermore, we use validation data for hyperparameter tuning only, without additional sources of data (e.g. group labels). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Information Bottleneck and Conditional Independence The line of work most similar to ours also exploits the information bottleneck method to learn diversity. Sinha et al. (2020) minimizes the mutual information $\\mathcal { T } ( \\hat { z } _ { s } , \\hat { z } _ { t } )$ between learnt representations $\\hat { z } _ { m }$ , however, this term is unconditional and will simply learn weak (biased) predictors, as noted in Section 3. Rame & Cord (2021) introduce DICE, which minimizes the conditional term $\\mathcal { C } \\mathcal { M } \\mathcal { I } ( \\hat { z } _ { s } , \\hat { z } _ { t } )$ . Pace et al. (2020) considers total correlation $\\mathcal { T C } ( \\hat { z } _ { 1 } , \\dots , \\hat { z } _ { M } )$ instead of pairwise terms. Unlike CoDE, both of these approaches compute mutual information terms on the high-dimensional representations $\\hat { z } _ { m }$ . Their objectives are intractable and must be approximated. For example, DICE requires both variational approximations and a jointly trained adversarial discriminator that learns to distinguish pairwise classifiers. Compared to these approaches, CoDE is by far computationally advantageous as mutual information for categorical output distributions can be computed faster and exactly. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Section 5.1 presents experiments on ColoredMNIST, which is used both to demonstrate the viability of our approach and to highlight pivotal observations and ablations. Section 5.2 then evaluates CoDE on larger benchmark datasets for shortcut learning to show that it scales effectively. ",
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+    {
+        "type": "text",
+        "text": "5.1 COLOREDMNIST ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Setup As described in Section 2, the original MNIST (LeCun et al., 1998) labels are binarized (0-4, 5-9) and used to generate true labels $y$ with noise $p _ { d }$ . $y$ then generates binary color labels with noise $p _ { c }$ , used to color the image (red or green). As per Arjovsky et al. (2019), we consider two test environments: the training distribution where $p _ { d } = 0 . 2 5$ and $p _ { c } = 0 . 1$ , and the adversarial distribution where $p _ { d } = 0 . 2 5$ but $p _ { c } = 0 . 9$ (hence the shortcut-label correlation is reversed). ",
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+    {
+        "type": "text",
+        "text": "Evaluation Baselines and Metrics As is standard in existing work, we evaluate predictive accuracy on the training and adversarial distributions. In choosing baselines, we considered the following desiderata for fairness and comprehensiveness: (i) comparing to both ensembling and non-ensembling methods, (ii) amongst ensembling methods, comparing to both conditional independence-based methods and those that do not, and (iii) comparing only to methods that do not require additional sources of data besides validation data for hyperparameter tuning. We chose the following baselines: ",
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+    {
+        "type": "table",
+        "img_path": "images/42ba1a8bcc725fee74277fd64c4958a8b94e6b142b085c543d6dc70a3d2f8a44.jpg",
+        "table_caption": [
+            "Table 1: Results on ColoredMNIST. A theoretically ideal classifier relying only on digit (denoted as “Invariant”) will be upper-bounded by the digit-label noise $p _ { d }$ $(7 5 \\% )$ , hence any result above $7 5 \\%$ is relying on the color shortcut. CoDE has the strongest performance on the adversarial distribution. \\*We were unable to reproduce TC-Ensemble on ColoredMNIST, and are citing their results in lieu. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"6\">Results on ColoredMNIST</td></tr><tr><td>(pd,Pc)</td><td>Training (0.25, 0.1)</td><td>(0.25, 0.9)</td><td>(0.25, 0.5)</td><td>AdversarialRandom-ColorRandom-Color + Perfect-Digit (0.0, 0.5)</td></tr><tr><td>Invariant</td><td>75</td><td>75</td><td>75</td><td>100</td></tr><tr><td>ERM</td><td>88.6</td><td>15.3</td><td>52.5</td><td>53.4</td></tr><tr><td>JTT</td><td>17.8</td><td>87.9</td><td>52.5</td><td>56.6</td></tr><tr><td>Ortho-Ensemble</td><td>89.8</td><td>11.1</td><td>50.3</td><td>49.2</td></tr><tr><td>TC-Ensemble</td><td>89.1</td><td>69.8*</td><td>-</td><td>-</td></tr><tr><td>CoDE</td><td>70.7</td><td>70.0</td><td>70.8</td><td>91.2</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1. ERM classifier (ERM): single, standard classifier trained with ERM 2. Just Train Twice (Liu et al., 2021) (JTT): an initial classifier is trained for a limited number of epochs; mis-classified examples are upweighted to train the final classifier 3. Ensembles using input gradient orthogonality (Teney et al., 2021) (Ortho-Ensemble): an ensemble where the regularizing term is the dot product of the two models’ input gradients 4. Ensembles using conditional total correlation (CTC) (Pace et al., 2020) (TC-Ensemble): an ensemble learnt by minimizing CTC over the encoder network’s representation ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1 shows all results on ColoredMNIST. We discuss the most important findings below. ",
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+    {
+        "type": "text",
+        "text": "1. Enforcing conditional independence on output distributions achieves diversity effectively. ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Since ColoredMNIST is an artificially-created dataset whose DGP we know satisfy latent conditional independence ${ \\overset { \\cdot } { p } } _ { c }$ and $p _ { d }$ are independently determined), it is the ideal dataset to evaluate our key claim. Indeed, the strong performance of CoDE shows that it is sufficient to enforce conditional independence on output distributions. The final predictor selected via fast adaptation achieves near-invariant results, suggesting that it has correctly learnt from digit rather than color. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2. CoDE generalizes to multiple OOD test distributions, without overfitting on any one specific distribution. ",
+        "text_level": 1,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Table 1, JTT achieved about $90 \\%$ on the adversarial distribution, implying that it overfitted to the adversarial distribution — by learning the opposite shortcut (color) correlation rather than the true signal (digit). This is further confirmed with additional results on two other test environments (Random-Color and Random-Color $^ +$ Perfect-Digit) where $p _ { c } = 0 . 5$ . JTT is close to random on these two environments, suggesting that it is still relying on color as the predictive feature. In contrast, CoDE achieves $91 \\%$ when $p _ { d } = 0 . 0$ , suggesting that it has learnt to predict using digit. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "While concerning, these results are not entirely surprising. A method like JTT did exactly what it was designed to do, which is to minimize classification errors on the adversarial test distribution. Since $p _ { c } = 0 . 1$ , the opposite color correlation is precisely this loss-minimizing function. In contrast, CoDE will not find such a solution because two classifiers that return opposite predictions using the same feature (color) are perfectly correlated, even when conditioned on $y$ . ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "These results highlight the shortcomings of single classifier methods like JTT. Such methods are designed to generalize to a specific test distribution, in general, this does not imply that they have learnt the desired predictive signal — merely that they have learnt an arbitrary function that does well on the test distribution. In contrast, methods that enforce diversity, such as CoDE, explicitly recover meaningful predictive signals that can generalize to any test distribution where $p ( \\mathbf { z } | y )$ changes. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/a0527b5f8e897cd39dbbe6d9f094f8125afeb5bf189680630083f853f661b810.jpg",
+        "table_caption": [
+            "Table 2: Additional results on ColoredMNIST and CelebA. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">ColoredMNIST</td><td colspan=\"2\">CelebA</td></tr><tr><td>Training</td><td>Adversarial</td><td>Ave</td><td>Worst</td></tr><tr><td>CoDE (sequential f1)</td><td>90.0</td><td>10.2</td><td>95.2</td><td>31.1</td></tr><tr><td>CoDE (sequential f2)</td><td>70.1</td><td>70.0</td><td>95.0</td><td>33.3</td></tr><tr><td>CoDE (sequential f3)</td><td>63.2</td><td>49.0</td><td></td><td></td></tr><tr><td>CoDE (sequential f5)</td><td>64.4</td><td>42.2</td><td></td><td></td></tr><tr><td>CoDE (joint M= 2)</td><td>73.4</td><td>60.2</td><td>89.2</td><td>83.3</td></tr><tr><td>CoDE (joint M = 3)</td><td>74.6</td><td>44.3</td><td></td><td></td></tr><tr><td>CoDE (joint M = 5)</td><td>71.9</td><td>43.1</td><td></td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "3. Joint and sequential optimization are suited to different datasets. ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "From our experiments, we found that there is no clear preference between either choice in terms of generalization ability. Table 2 shows both joint and sequential results on the ColoredMNIST and CelebA datasets. For ColoredMNIST, we found that sequential training performed better than joint training. For CelebA, joint training yielded a stronger classifier. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This might be explained by the biases of the ERM model. In ColoredMNIST, as both latent factors (color and digit) are noisy predictors and as color presents a particularly simple shortcut, the ERM model solely learns from color. As such, a second classifier model that is trained sequentially can learn to predict solely from the digit feature. In contrast, the ERM model in CelebA has likely picked up some combination of the spurious (gender) and true (hair color) features, possibly because gender gives rise to complex features that are not ncessarily simpler to learn. This corroborates previous findings indicating that ERM models can learn an arbitrary combination of all predictive signals (Zhang et al., 2021; Kirichenko et al., 2022). As such, when trained sequentially, the second model fails to learn from hair color alone. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The advantages of sequential optimization are: (i) cheaper computational costs as $M$ increases, and (ii) providing a natural stopping point for training. The latter comes from the fact that we can select for $M$ by terminating the training process when the subsequent classifier is no longer predictive, which indicates that there are no further predictive factors to be learnt. In contrast, joint optimization is advantageous as it allows us to avoid the pathological sitation where earlier models learn combinations of predictive factors. As small values of $M$ work well for CoDEs, we note that the computational cost of CoDEs are not prohibitive. ",
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+    {
+        "type": "text",
+        "text": "5.2 BENCHMARK DATASETS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Setup We consider the following benchmark datasets: ",
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+    {
+        "type": "text",
+        "text": "• CelebA Liu et al. (2018); Sagawa et al. (2019): A dataset of celebrity faces with various labelled attributes. We consider the benchmark task in (Sagawa et al., 2019) of predicting the binary hair color attribute (blond or not), with gender (female or male) as the spurious attribute. There are therefore four groups. • Waterbirds (Wah et al., 2011; Sagawa et al., 2019): Setup described in Section 2. There are also four groups as both latent factors (background and foreground) are binary. • MF-Dominoes (MNIST-FashionMNIST) (LeCun et al., 1998; Xiao et al., 2017; Shah et al., 2020; Pagliardini et al., 2022): Each input image concatenates an MNIST digit (0 or 1) with a FashionMNIST object (coat or dress). The true label is the FashionMNIST object; the simpler MNIST feature is the shortcut. The minority groups represent $5 \\%$ of the data. ",
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+    {
+        "type": "text",
+        "text": "Table 3 shows all results on the benchmark datasets. ",
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+    {
+        "type": "text",
+        "text": "4. CoDE scales well to large datasets and retains effectiveness at preventing shortcut learning. ",
+        "text_level": 1,
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+    },
+    {
+        "type": "table",
+        "img_path": "images/df7e0a134cb3c4450a3948aeec2d7d2bae29f67283680aca42bc5dd0732341cb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">CelebA</td><td colspan=\"2\">Waterbirds</td><td colspan=\"2\">MF-Dominoes</td></tr><tr><td>Method Ave</td><td>Worst</td><td>Ave</td><td>Worst</td><td>Ave</td><td>Worst</td></tr><tr><td>ERM</td><td>94.8</td><td>46.7</td><td>90.4</td><td>78.3</td><td>88.9</td><td>76.9</td></tr><tr><td>JTT</td><td>88.0*</td><td>81.1*</td><td>93.3*</td><td>86.7*</td><td>89.5</td><td>76.1</td></tr><tr><td>CoDE</td><td>89.2</td><td>83.3</td><td>91.5</td><td>79.4</td><td>92.1</td><td>91.4</td></tr></table>",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Main results on all datasets. CoDE achieves better adversarial or wrost-group accuracy than the other methods on all datasets except Waterbirds. ∗ Results from the JTT paper. We share the same model and training environment as their paper. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "On CelebA and MF-Dominoes, CoDE achieves the best worst-group accuracy. Unlike the earlier ColoredMNIST dataset, we have no guarantees that the core assumption of latent conditional independence holds. However, the strong performance of CoDE on these datasets shows that such an assumption is generally valid and useful when scaled to more realistic datasets. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We note that CoDE performs poorly on Waterbirds. In our experiments, we selected $M = 2$ as the ensemble size. Even though there are no guarantees what will be the two conditionally independent classifiers that CoDE learns, in the other datasets, the results show that they do each correspond to the shortcut and true signal. This implies that in these datasets: (a) there are no features conditionally independent to both the shortcut and true signals and yet also strongly predictive of the label, and (b) the shortcut or true signal cannot be decomposed themselves into conditionally independent signals. Our hypothesis is that (b) is not true for Waterbirds. As the dataset is varied and contains a range of land and water backgrounds, there could be multiple spurious signals in the background that are somehow conditionally independent, resulting in these signals being learnt. Another possibility is that the ensemble could have learnt an imperfect or partial foreground signal. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "5. Computational effectiveness is crucial to learn diverse ensembles at scale. ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Beyond ColoredMNIST, we found that it was computationally prohibitive to run Ortho-Ensemble, as the size of ensembles required to work well (48 or 96) was too high. We also noted that we could not implement TC-Ensembles successfully on larger datasets, noting that the original authors do not test on datasets besides ColoredMNIST either. We believe that this further highlights the importance of computational efficiency in diverse ensembling. ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND CONCLUSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Appendix B discusses potential failure modes of our method. ",
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+    {
+        "type": "text",
+        "text": "We introduce CoDE, a method for learning an ensemble of diverse classifiers that rely on different predictive signals in the dataset. The key assumption made by CoDE conditional independence between predictive signals, which it enforces on classifiers’ output distributions. We find that CoDE works well in practice when applied to shortcut learning tasks. Future work includes: (a) evaluating CoDEs on other applications where multiple predictive signals exist, such as fairness-related tasks where we might want to learn classifiers that do not rely on sensitive attributes, and (b) considering other metrics for conditional independence that might provide more fine-grained signals than output distributions (e.g. minimizing mutual information between latent representations). ",
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+    {
+        "type": "text",
+        "text": "ETHICS STATEMENT ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Positive Impact Being robust to distribution shifts, CoDE will have a positive impact when deployed to high-stakes domains, where learning shortcut signals can have harmful social consequences. One such notable example is pneumonia prediction — models trained on pneumonia labels from chest $\\mathrm { X }$ -ray scans have been shown to learn machine-specific artifacts in the background, which is a shortcut as hospitals have differing positivity rates and use different machines (Zech et al., 2018). ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Negative Impact There are no notable negative impacts of using CoDE specifically, besides the general potential for all machine learning models to be abused in the wrong hands. ",
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "REPRODUCIBILITY STATEMENT ",
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+        "page_idx": 9
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+        "type": "text",
+        "text": "We intend to release public code with a camera-ready version of the paper. ",
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+        "text": "A EXPERIMENTAL DETAILS ",
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+        "text": "Architecture and Training Details For ColoredMNIST, we use a CNN as the classifier, containing two convolutional layers and two fully-connected layers. Adam (Kingma & Ba, 2014) is used for optimization, with a learning rate of 0.001. For CelebA, Waterbirds, and MF-Dominoes, we use a ResNet-50 (He et al., 2016). SGD is used for optimization, with a learning rate of 0.001, momentum decay of 0.9, and weight decay of 0.001. Additionally, following previous work (e.g. Sagawa et al., 2019; Liu et al., 2021), the Waterbirds model is pre-trained on ImageNet (Deng et al., 2009) and includes data augmentation in the form of random horizontal flips and random resized cropping. For CelebA and Waterbirds, class reweighting is performed to ensure that there are roughly equal positive and negative labels. The random seed used for all experiments is 13. ",
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+    {
+        "type": "text",
+        "text": "Hyperparameters for CoDE and Baselines CoDE. For all four datasets, we used $M = 2$ as the ensemble size, besides ablations for $M$ as detailed in Appendix B. The results in Table 3 were achieved with sequential training for ColoredMNIST and with joint training for the other three datasets. For ColoredMNIST, $\\lambda _ { 1 } = 1 2 0 0$ and $\\lambda _ { 2 } = 1 0$ . For CelebA, $\\lambda _ { 1 } = 5 0 0$ and $\\lambda _ { 2 } = 0 . 1$ . For Waterbirds, $\\lambda _ { 1 } = 5 0 0$ and $\\lambda _ { 2 } = 0 . 1$ . For MF-Dominoes, $\\lambda _ { 1 } = 3 0 0$ and $\\lambda _ { 2 } = 0 . 1$ . JTT. We performed a hyperparameter sweep with $T \\in \\{ 1 , 5 , 1 0 \\}$ (number of epochs for initial model training) and $\\alpha \\in \\{ 2 , 1 0 , 1 0 0 \\}$ (upweighting factor for mis-classified examples). Orthogonal Ensembles. All classifiers share the same feature extractor (i.e. convolutional output for ColoredMNIST and ResNet-50 feature representation for the other three datasets). We experimented with different values of $M$ , however, values of $M$ above 16 (for ColoredMNIST) and above 4 (for the other three datasets) were prohitibively expensive. As such, we did not try $M = 4 8$ or $M = 9 6$ as used by Teney et al. (2021). For these smaller values of $M$ that we tried, we did not notice an improvement from the ERM model. Besides ColoredMNIST, we did not report these results. ",
+        "bbox": [
+            174,
+            272,
+            825,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B MODEL MIS-SPECIFICATION: POTENTIAL FAILURE MODES",
+        "text_level": 1,
+        "bbox": [
+            176,
+            473,
+            699,
+            489
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The success of any method tackling distribution shifts depends on how well the assumptions made have been upheld. We discuss the potential implications when the model is mis-specified and these assumptions are no longer valid. ",
+        "bbox": [
+            178,
+            503,
+            823,
+            546
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Conditional Dependence CoDE relies on the assumption that predictive signals are conditionally independent. We using the synthetic ColoredMNIST dataset to generate a DGP where such an assumption does not hold true. Instead of the standard setup where color labels are generated from the true labels, we generate color labels from the original (binarized) MNIST labels instead, at the same noise level $p _ { c } = 0 . 1$ . This means that the color and digit signals are now highly correlated. Both are still predictive since the true labels themselves were generated from MNIST labels. ",
+        "bbox": [
+            174,
+            560,
+            825,
+            643
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 4 shows the results of this experiment. As we expect, conditionally dependent features cannot be recovered by minimizing conditional mutual information. The ensemble either recovers one of the two features (when trained sequentially) or neither. This confirms our intuition that conditional independence must be correctly specified for CoDE to work. While these results demonstrate a failure mode of CoDE, conditional independence between predictive factors of interest does hold well in many natural image datasets, as shown in Table 3. ",
+        "bbox": [
+            174,
+            651,
+            825,
+            736
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Latent Mis-specification The size of the ensemble $M$ specifies how many predictive latent factors we believe generated the dataset. We can consider the mis-specification of $M$ in either direction: (i) if the true dimension of $\\mathbf { z }$ is smaller than $M$ , and (ii) if the true dimension of $\\mathbf { z }$ is larger than $M$ . ",
+        "bbox": [
+            174,
+            750,
+            825,
+            791
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In case (i), since the number of conditional independent components has been over-specified, whether the ensemble has been jointly or sequentially trained makes a difference. Consider the results on the ColoredMNIST dataset in Table 2 again. In the sequential regime, the first two classifiers $f _ { 1 }$ and $f _ { 2 }$ correspond to the color and digit classifiers respectively, however, the subsequent few classifiers $f _ { 3 }$ and $f _ { 5 } ^ { } ,$ ) do not learn anything meaningful and perform poorly on both training and adversarial distributions. However, as noted in Section 5, this does not pose a serious problem since we can use validation data to naturally determine the stopping point. On the other hand, over-specification of $M$ is more worrying in the joint regime, as there is no guarantee that any of the true latent factors are learnt at all. As Table 2 shows, for $M = 3$ or 5, the best-performing classifier does not generalize. ",
+        "bbox": [
+            173,
+            797,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/fe195a4203196cb5e84fe532c6c5ade0e727c287130f1766eba073b330595b34.jpg",
+        "table_caption": [
+            "Table 4: Results on ColoredMNIST with color-digit conditional dependence, on both joint and sequential training with $M = 2$ classifiers. When trained sequentially, the first classifier $f _ { 1 }$ learns the digit correlation since digit is most predictive in this setup. However, as color is no longer conditionally independent of digit, there is no predictive feature that can be learnt by the second classifier $f _ { 2 }$ , resulting in a close-to-random predictor. When trained jointly, neither of the classifiers correspond to the color or digit feature. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>(pd,Pc)</td><td>Training (0.25, 0.1)</td><td>Adversarial (0.25, 0.9)</td><td>Random-Color (0.25, 0.5)</td><td>Perfect-Digit (0.0, 0.5)</td></tr><tr><td>CoDE (sequential f1)</td><td>77.1</td><td>63.2</td><td>70.6</td><td>90.5</td></tr><tr><td>CoDE (sequential f2)</td><td>53.6</td><td>50.1</td><td>51.5</td><td>54.5</td></tr><tr><td>CoDE (joint f1)</td><td>84.9</td><td>25.8</td><td>56.0</td><td>60.4</td></tr><tr><td>CoDE (joint f2)</td><td>54.3</td><td>73.0</td><td>64.4</td><td>77.5</td></tr></table>",
+        "bbox": [
+            200,
+            101,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In case (ii), where the number of conditional independent components is under-specified, the learnt ensemble may correspond to any subset of the true latent factors and individual classifiers could also learn arbitrary combinations of the latent factors. For example, the trivial case where $M = 1$ is underspecified simply returns the ERM model. In general, since $M$ is a hyperparameter, latent mis-specification does not pose a serious problem as we can tune its value using the validation data. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    }
+]

parse/dev/3IyL2XWDkG/3IyL2XWDkG.md

@@ -0,0 +1,392 @@
+# CAMEL: Communicative Agents for “Mind” Exploration of Large Language Model Society
+
+https://www.camel-ai.org
+
+# Guohao Li⇤ Hasan Abed Al Kader Hammoud
+
+Hani Itani\* Dmitrii Khizbullin
+
+# Bernard Ghanem
+
+King Abdullah University of Science and Technology (KAUST)
+
+# Abstract
+
+The rapid advancement of chat-based language models has led to remarkable progress in complex task-solving. However, their success heavily relies on human input to guide the conversation, which can be challenging and time-consuming. This paper explores the potential of building scalable techniques to facilitate autonomous cooperation among communicative agents, and provides insight into their “cognitive” processes. To address the challenges of achieving autonomous cooperation, we propose a novel communicative agent framework named roleplaying . Our approach involves using inception prompting to guide chat agents toward task completion while maintaining consistency with human intentions. We showcase how role-playing can be used to generate conversational data for studying the behaviors and capabilities of a society of agents, providing a valuable resource for investigating conversational language models. In particular, we conduct comprehensive studies on instruction-following cooperation in multi-agent settings. Our contributions include introducing a novel communicative agent framework, offering a scalable approach for studying the cooperative behaviors and capabilities of multi-agent systems, and open-sourcing our library to support research on communicative agents and beyond: https://github.com/camel-ai/camel.
+
+# 1 Introduction
+
+“What magical trick makes us intelligent? The trick is that there is no trick. The power of intelligence stems from our vast diversity, not from any single, perfect principle.”
+
+- Marvin Minsky, The Society of Mind, p. 308
+
+Confronted with the complexities of real-world tasks, solving them often requires multiple steps. The rapid progress of chat-based large-scale language models (LLMs) has yielded remarkable achievements in complex task-solving [82, 84, 116, 89, 5, 10, 122, 13]. Nevertheless, it is worth noting that their success is heavily reliant on human input to guide the conversation in the right direction. This reliance necessitates users to provide relevant and precise prompts based on their intentions and the chat agent’s feedback. This can be challenging, time-consuming, and sometimes impossible. Crafting effective prompts often demands a deep understanding and expertise of a particular domain of knowledge. Consider an individual who lacks trading expertise; they would find it difficult to create suitable prompts for directing a chat agent to develop a trading application. This predicament is raising a crucial question: can we replace human intervention with an autonomous communicative agent capable of steering the conversation toward task completion with minimal human supervision? To tackle this issue, it is crucial to conduct more research exploring the potential, capabilities, and limitations of communicative agents that operate entirely on their own to complete tasks. Understanding how multiple agents interact with each other is important for anticipating the future of artificial intelligence. The dynamics of collaborating or competing agents play a key role in determining the success of AI systems [6, 26, 27, 84, 99, 9, 10].
+
+This paper explores the potential of building scalable techniques to facilitate autonomous cooperation among communicative agents and provide insight into their “cognitive” processes. Several challenges arise when asking a society of agents to autonomously cooperate on completing tasks. Examples we encountered in our preliminary analysis include role flipping, assistant repeating instructions, flake replies, and infinite loop of messages. Therefore, it is critical to investigate ways to align these models with human intentions and to explore means enabling their effective cooperation. To address these issues, we propose a novel cooperative agent framework named role-playing to automate cooperation between communicative agents. Specifically, our proposed approach involves using role-playing with inception prompting to autonomously guide the communicative agents toward task completion. Only a preliminary idea is needed from human to guide the conversations toward complex task-solving.
+
+Our library, which we make publicly available, provides modular functionality, and includes implementations of different agents, examples of well-crafted prompts, and data explorers. We hope our library serves as a ground for future research in various areas such as multi-agent systems, cooperative AI, game theory simulations, social analysis, AI ethics, AI alignment, and beyond. In addition, our role-playing method provides a highly scalable way to generate conversational data for studying the behaviors and capabilities of chat agents. We showcase how role-playing can be used to let chat agents communicate with each other for task completion and record their conversations for behavior analysis and capability understanding. In particular, we consider two cooperative scenarios of role-playing and generate two large conversational, task-oriented, and instruction-following datasets: AI Society and Code. We also use our framework to collect two single-turn question-answer datasets, Math and Science, for LLM ability emergence study. Furthermore, we generate a Misalignment dataset that is a simulation of possible malicious applications which demonstrate the potential risks of an unaligned autonomous agent system. The datasets offer a valuable resource for investigating conversational language models, enabling them to comprehend and react to human language more effectively. Furthermore, our role-playing offers a scalable method of creating conversational instruction-following data, which can potentially enhance the development of more advanced language models. We show that solutions derived from our role-playing framework outperform those generated in a single shot by gpt-3.5-turbo [82] in both GPT4 and human evaluations. We also study knowledge emergence in LLMs by fine-tuning LLaMA [117] on progressively growing datasets generated through our framework. Additionally, we evaluate our code generation capabilities through benchmarking our final model on HumanEval [18] and HumanEval+ [69].
+
+Contributions. Our contributions are fourfold: (1) We introduce a novel cooperative agent framework, role-playing , that allows communicative agents to collaborate autonomously toward completing tasks while requiring minimal human intervention; (2) Our framework offers a scalable approach for studying the cooperative behaviors and capabilities of multi-agent systems. It illuminates the challenges of achieving autonomous cooperation, and provides strategies for addressing them. We showcase the potential power of multi-agent collaboration for complex-task solving; (3) We demonstrate the significant emergence of LLM training abilities by utilizing the datasets we have collected from simulating four distinct agent collaboration scenarios; (4) We have open-sourced our library, containing implementations of various agents, data generation pipelines, data analysis tools, and collected datasets, to support research on communicative agents and beyond.
+
+# 2 Related Work
+
+Communicative Agents. Communication between agents has been studied for a long time [76, 77]. There are many ways to facilitate communication between agents, and with agents [29, 90, 97]. Among these, natural language is considered the most natural form of communication [97]. By enabling agents to function as communicators themselves, they become capable of solving complex tasks [113, 85, 72, 3, 30, 111, 79, 41, 28, 102, 80, 106, 35, 49, 2, 51, 1, 55, 50, 65, 92]. Communication between AI agents can occur in a competitive setting [115, 108] or a cooperative setting [40, 27, 11, 137, 70]. Cooperative AI refers to artificial intelligence systems that are designed to work together with humans and other AI systems to achieve common goals [24, 125]. Cooperative AI systems take into account the needs and capabilities of other agents in the system and actively seek to collaborate and coordinate their actions with them, which has many potential benefits, including increased efficiency, improved decision-making, and the ability to tackle complex problems that are beyond the reach of any single agent. However, designing effective cooperative AI systems is still an active area of research, as it requires addressing a range of technical, ethical, and social challenges [27]. Our work enables communicative agents to engage in a conversation and cooperate with each other to solve assigned tasks. The agents, each assigned a distinct role, are expected to apply their expertise and knowledge to solve their common task.
+
+Instructional LLMs and Prompt Engineering. LLMs are trained on diverse text data and excel in text completion, with various downstream NLP applications [12, 22, 47, 131, 117]. However, InstructGPT suggests that LLMs may not align with user intent, proposing reinforcement learning from human feedback (RLHF) [23] and Instruction Fine-Tuning (IFT) [121] to improve LLMs’ relevance and appropriateness to user instructions. Special types of instruction or prompting methods , such as Chain-of-Thought (CoT) [123], zero-shot-CoT [61], and ReAct [126], have recently been developed to enhance the performance of LLMs on reasoning, arithmetic and decision making tasks [134, 118, 52, 73, 31, 103, 43, 64, 132, 46, 133, 105, 128, 25, 81, 109]. These techniques underpin the impressive capabilities of recent dialogue LLMs [106, 116, 36, 9, 82, 13], which aim to simulate human-like conversations and provide personalized and interactive experiences for users, exhibiting the behavior of conversational AI agents [33]. However, generating instruction datasets is a crucial challenge in building instruct-based LLMs, with existing datasets ranging from crowdsourced to generated. Hand-crafted instruction instances are available in [120], while leveraging previously crowdsourced NLP datasets is a less labor-intensive curation approach [121, 71, 78, 53]. LLMs have been explored for data generation in [101, 63, 68, 114], and Self-Instruct [119] proposes a semi-automated process for instruction instance generation. Unnatural-Instruction [48] collects instruction instances by prompting a language model with only three seed examples and paraphrasing the generated instances to expand the dataset. There is also a large chunk of work that has proposed methods for automatic dataset creation [67, 57, 19, 75, 20, 98, 59, 96, 129, 62, 130, 86, 8]. Another important challenge is prompt engineering. The quality of the prompt used to guide LLMs significantly affects its performance [91, 12, 66]. While LMs pre-trained on large data can implicitly learn tasks with few-shot prompting, hand-crafted prompts may not always suffice. Automated prompt generation methods have been proposed, such as gradient-guided search [104], mining-based and paraphrasing-based techniques [54], a meta-prompt [93], and automatic instruction selection and generation [136]. In this work, we introduce a conversational LLM auto-prompting method called Inception Prompting, which enables agents to prompt each other to solve tasks through Role-Playing. The AI user continuously provides instructions to the AI assistant for task-solving. This enables us to save the streaming instruction-solution pairs and create diverse, instructional, conversational, and task-oriented datasets. These datasets can be used to analyze the behavior and capabilities of LLMs and for future research for fine-tuning LLMs with conversational instructions.
+
+AI Alignment. AI alignment is a field that aims to ensure that AI systems adhere to their intended goals, interests, and values, as envisioned by their designers [4, 39, 110, 32, 38, 74, 10]. The first attempt at AI alignment was made through the "Three Laws of Robotics," which was introduced by Isaac Asimov in his science fiction stories [6]. Developing aligned AI systems is crucial for achieving desired objectives while avoiding unintended consequences. Research in AI alignment focuses on discouraging AI models from producing false, offensive, deceptive, or manipulative information that could result in various harms [56, 112, 42, 37]. Achieving a high level of alignment requires researchers to grapple with complex ethical, philosophical, and technical issues. We conduct extensive experiments to study different role-playing situations, which probe the alignment of LLMs.
+
+# 3 Methodology
+
+In this paper, we focus on studying communicative agents under cooperative settings where they share common interests. In particular, we study the assistant-user scenario, where a preliminary idea is given at the start. Agents will conceptualize the idea into a specific task and complete it autonomously through conversations.
+
+# 3.1 Role-playing Framework
+
+“What’s the most resilient parasite? An Idea. A single idea from the human mind can build cities. An idea can transform the world and rewrite all the rules. Which is why I have to steal it.”
+
+- Dom Cobb, Inception
+
+![](images/6abc6059b41ace03cab1cbc2501d41463903922476103d46fbfb309436a5b8fc.jpg)  
+Figure 1: CAMEL Role-Playing Framework. Our role-playing setup starts with the human user having an idea they want to implement, e.g. develop a trading bot for the stock market. The roles involved in this task would be an AI assistant agent who is a python programmer and an AI user agent who is a stock trader. The task is made more specific using our task specifier agent, leading to a well-defined task for the assistant to solve. Both AI user and AI assistant are provided with the specified task, after which they collaboratively communicate by chatting with each other in an instruction-following fashion to solve the specified task.
+
+Our proposed framework is a novel role-playing approach for studying multiple communicative agents. Specifically, we concentrate on task-oriented role-playing that involves one AI assistant and one AI user. After the multi-agent system receives a preliminary idea and the role assignment from human users, a task-specifier agent will provide a detailed description to make the idea specific. Afterwards, the AI assistant and AI user will cooperate on completing the specified task through multi-turn conversations until the AI user determines the task is done. The AI user is responsible for giving instructions to the AI assistant and directing the conversation toward task completion. On the other hand, the AI assistant is designed to follow the instructions from the AI user and respond with specific solutions. The whole role-playing framework is depicted in Figure 1.
+
+Human Input and Task Specifying. The role-playing session will be instantiated from an idea and selected roles by humans. As an example in Figure 1, a human has a preliminary idea to develop a trading bot for the stock market. Humans may or may not have the knowledge about how the idea can be realized. What is needed is only to designate the potential roles that can implement the idea. For instance, a Python Programmer could collaborate with a Stock Trader to realize the idea of developing a trading bot for the stock market. After the idea and roles are determined, the task specifier agent will brainstorm a specific task that the AI Assistant role can help with the AI user role to complete based on the input idea. An example of a specified task in this scenario could be: develop a trading bot with a sentiment analysis tool that can monitor social media platforms for positive or negative comments about a particular stock, and execute trades based on sentiment analysis results. The main motivation for introducing a task specifier is that conversational agents usually require a concrete task prompt for realizing the task which might be challenging or time-consuming for a non-domain expert. Therefore, the task specifier agent serves as an enhanced imagination module for the idea implementation. Please note that, when studying our framework at a large scale for AI society and Code scenarios, we generate roles and ideas automatically by prompting LLMs instead of relying on human inputs. For our generated Math and Science datasets we generated problem topics, subtopics, and problems automatically by prompting LLMs.
+
+AI Assistant-User Role Assignment. After the task specification, The AI assistant role and the AI user role will be assigned to the user agent and the assistant agent correspondingly to complete the specified task. In practice, a system message is passed to each agent declaring their role. We refer to the assistant system prompt/message by $\mathcal { P } _ { A }$ and that of the user by $\mathcal { P } _ { \mathcal { U } }$ . The system messages are passed to the agents before the conversations start. Let $\mathcal { F } _ { 1 }$ and $\mathcal { F } _ { 2 }$ denote two large-scale autoregressive language models [82]. When the system message is passed to those models respectively, we obtain $\mathcal { A }  \mathcal { F } _ { 1 } ^ { \mathcal { P } _ { A } }$ and $\mathcal { U }  \mathcal { F } _ { 2 } ^ { \mathcal { P } _ { \mathcal { U } } }$ which are referred to as the assistant and user agents respectively. In Figure 1, the AI assistant and the AI user are assigned the roles of a Python Programmer and a Stock Trader at the beginning of the role-playing session respectively. The AI user serves as a task planner, engaging in interactive planning to determine feasible steps for the AI assistant to execute. Meanwhile, the AI assistant acts as a task executor, offering solutions, executing planned steps, and providing responses to the AI user.
+
+Conversation Towards Task-Solving. After the role assignment is completed, the AI assistant $\mathcal { A }$ and AI user $\mathcal { U }$ will collaborate in an instruction-following manner to accomplish the task. In the AI assistant-user scenario, the AI user is responsible for providing instructions, and the assistant is expected to respond with a solution that fulfills the instructions. Formally, we denote the user instruction message obtained at time $t$ by $\mathcal { T } _ { t }$ and the assistant solution by $S _ { t }$ . The set of conversational messages obtained up until time $t$ is denoted by Equation (1) shown below:
+
+$$
+\mathcal { M } _ { t } = \{ ( \mathbb { Z } _ { 0 } , S _ { 0 } ) , . . . , ( \mathbb { Z } _ { t } , S _ { t } ) \} = \{ ( \mathbb { Z } _ { i } , S _ { i } ) \} | _ { i = 0 } ^ { t }
+$$
+
+At the next time step, $t + 1$ , the AI user $\mathcal { U }$ takes the historical conversation message set $\mathcal { M } _ { t }$ and provides a new instruction $\mathcal { T } _ { t + 1 }$ , as shown in Equation (2). The produced instruction message $\mathcal { T } _ { t + 1 }$ is then passed, along with message set $\mathcal { M } _ { t }$ , to the AI assistant $\mathcal { A }$ . The AI assistant will then respond with a solution, denoted by $\boldsymbol { S } _ { t + 1 }$ in Equation (3):
+
+$$
+\mathcal { T } _ { t + 1 } = \mathcal { U } ( \mathcal { M } t ) \qquad ( 2 ) \qquad \mathcal { S } t + 1 = \mathcal { A } ( \mathcal { M } t , \mathcal { I } t + 1 )
+$$
+
+After obtaining the solution $\boldsymbol { S } _ { t + 1 }$ to the instruction $\mathcal { T } _ { t + 1 }$ , the message set is updated using Equation (4) to obtain $\mathcal { M } _ { t + 1 }$ :
+
+$$
+\mathcal { M } _ { t + 1 }  \mathcal { M } _ { t } \cup ( \mathcal { T } _ { t + 1 } , S _ { t + 1 } )
+$$
+
+Note that the formulation above not only models AI-AI communicative scenarios, but it can also be easily extended to model human-AI communication or communication between more than two agents. Specifically, we can use message-passing graphs to model communication between an arbitrary number of agents. In Figure 1, we observe that the AI user initiates the installation and import of essential Python libraries for sentiment analysis and stock trading by instructing the AI assistant through conversations. This example is drawn from our experiments, and the entire conversation is available in the Appendix.
+
+Critic-In-The-Loop. To enhance the controllability of the role-playing framework, we introduce a critic agent capable of selecting proposals from or providing feedback to the role-playing agents. This enables tree-search-like decision-making for task-solving. In practice, the critic can be either an AI agent or a human. The detailed implementation and case studies can be found in the Appendix.
+
+# 3.2 Inception Prompting
+
+Since prompt engineering is crucial to our role-playing framework, this section delves deeply into our prompting techniques. Our prompt engineering occurs solely at the beginning of role-playing, for task specification and role assignment. Once the conversation phase commences, the AI assistant and AI user prompt each other automatically in a loop until termination. As such, we refer to our technique as Inception Prompting. Our Inception prompt consists of three prompts: the task specifier prompt $\mathcal { P } _ { T }$ , the assistant system prompt $\mathcal { P } _ { A }$ , and the user system prompt $\mathcal { P } _ { \mathcal { U } }$ . As an example, we consider the inception prompt of the $A I$ Society scenario. The templates for these prompts of $A I$ Society role-playing are shown in Figure 2. The task specifier prompt contains information about the roles of the AI assistant and AI user in the role-playing session. Therefore, the task specifier agent can take a preliminary task/idea as input and generate a specific task using imagination. The AI assistant system prompt $\mathcal { P } _ { A }$ and the AI user system prompt $\mathcal { P } _ { \mathcal { U } }$ are mostly symmetrical and include information about the assigned task and roles, communication protocols, termination conditions, and constraints or requirements to avoid unwanted behaviors. The prompt designs for both roles are crucial to achieve autonomous cooperation between agents. It is non-trivial to engineer prompts that ensure agents act in alignment with our intentions. We take the prompt templates from the AI Society in Figure 2 as an example to explain our key design choices. The prompts used for the Code scenario follow a similar sprint as the AI society scenario, but with some additional engineering related to programming languages. More details in the Appendix.
+
+# AI Society Inception Prompt
+
+# Task Specifier Prompt:
+
+Here is a task that <ASSISTANT_ROLE> will help <USER_ROLE> to complete: <TASK>.   
+Please make it more specific. Be creative and imaginative.   
+Please reply with the specified task in <WORD_LIMIT> words or less. Do not add anything else.
+
+# Assistant System Prompt:
+
+# User System Prompt:
+
+Never forget you are a <USER_ROLE> and I am a <ASSISTANT_ROLE>. Never flip roles! You will always instruct me.   
+We share a common interest in collaborating to successfully complete a task.   
+I must help you to complete the task.   
+Here is the task: <TASK>. Never forget our task!   
+You must instruct me based on my expertise and your needs to complete the task ONLY in the following two ways:
+
+Prompt Engineering. To delve deeper into the details in Figure 2, we start by chunking the various parts of the AI assistant system prompt $\mathcal { P } _ { A }$ shown below:
+
+• Never forget you are a <ASSISTANT_ROLE> and I am a <USER_ROLE>. This assigns the chosen role to the assistant agent and provides it with information about the user’s role. • Never flip roles! Never instruct me! This prevents agents from flipping roles. In some cases, we have observed the assistant and the user switching roles, where the assistant suddenly takes control and instructs the user, and the user follows those instructions. • You must decline my instruction honestly if you cannot perform the instruction due to physical, moral, legal reasons or your capability and explain the reasons. This prohibits the agent from producing harmful, false, illegal, and misleading information. • Unless I say the task is completed, you should always start with: Solution: <YOUR_SOLUTION>. <YOUR_SOLUTION> should be specific, and provide preferable implementations and examples for task-solving. This encourages the assistant always responds in a consistent format, avoiding any deviation from the structure of the conversation, and preventing vague or incomplete responses, which we refer to as flake responses, such as ${ } " \mathrm { I }$ will do something".
+
+• Always end your solution with: Next request. This ensures that the assistant keeps the conversation going by requesting a new instruction to solve.
+
+For the AI user system prompt $\mathcal { P } _ { \mathcal { U } }$ , we strive to maintain as much symmetry as possible with respect to the AI assistant system prompt. Apart from the opposite role assignment, the user system prompt differs from the assistant prompt in the following ways:
+
+• You must instruct me ... to complete the task ONLY in the following two ways: 1. Instruct with a necessary input: ...; 2. Instruct without any input: ... This follows the typical data structure of instruction-following, which allows the generated instruction-solution pairs to be easily used for fine-tuning LLMs. • Keep giving me instructions and necessary inputs until you think the task is completed. When the task is completed, you must only reply with a single word <CAMEL_TASK_DONE>. We introduce an end-of-task token, namely, <CAMEL_TASK_DONE>. This token is used once the user believes the task is done. This ensures that the chat is terminated when the user is satisfied. Without doing so, the agents might fall into a chatting loop where they keep on saying “thank you” to each other or “goodbye” indefinitely.
+
+# 4 Experiments
+
+In this section, we will discuss the various experiments that we conducted to arrive at our final design choices. Specifically, we will examine the interesting observations, challenging issues, and several examples we have encountered while enabling agents to communicate with each other under different prompt design choices to achieve autonomous cooperation. In our experiments, we employed two gpt-3.5-turbo agents, referred to as LLM agents for simplicity, with Inception Prompts, as described in Section 3.2, to simulate assistant-user cooperation. For our analysis, we set our attention on AI Society setting. We also gathered conversational data, named CAMEL AI Society and CAMEL Code datasets and problem-solution pairs data named CAMEL Math and CAMEL Science and analyzed and evaluated their quality. Moreover, we will discuss potential extensions of our framework and highlight both the risks and opportunities that future AI society might present.
+
+# Data Generation Prompts of AI Society
+
+# AI Society
+
+# Assistant Role Generation Prompt:
+
+You are a helpful assistant that can play many different roles. Now please list <NUM_ROLES> different roles that you can play with your expertise in diverse fields. Sort them by alphabetical order. No explanation required.
+
+# User Role Generation Prompt:
+
+Please list <NUM_ROLES> most common and diverse groups of internet users or occupations. Use singular form. No explanation. Sort them by alphabetical order. No explanation required.
+
+# Task Generation Prompt:
+
+List <NUM_TASKS> diverse tasks that <ASSISTANT_ROLE> can assist <USER_ROLE> cooperatively to achieve together. Be concise. Be creative.
+
+Figure 3: Data Generation Prompts. In order to maintain a scalable approach our data parameters are generated using an LLM model to reduce human involvement in the generation process. The generation prompts for both AI Society dataset are summarized in this figure.
+
+# 4.1 Role-Playing for AI Society
+
+To create our AI Society dataset, we have developed a scalable approach that follows a series of steps. Firstly, we prompt the LLM agent to generate possible roles for the assistant and the user. We achieve this by providing the LLM agent with specific prompts designed to elicit these roles. Next, we ask the LLM agent to generate a range of possible tasks that can be solved through collaboration between the assistant and user roles generated previously. After generating a range of possible tasks as described in the previous step, we then use the task specifier prompt passed to the LLM agent to make the task more specific. The prompts for assistant role generation, user role generation, and task generation are shown in Figure 5 (AI Society). For our AI society dataset, we generated 50 assistant roles, 50 user roles, and 10 tasks for each combination of roles yielding a total of 25,000 conversations. The generated assistant roles and user roles for AI Society as well as details about the generation of Code, Math and Science datasets can be found in the Appendix.
+
+Challenges and Observations. In this section, we explore the four main challenges that we identified during our analysis of the generated datasets. Our observations shed light on some interesting aspects of cooperative AI and the difficulties that arise in its development.
+
+• Role Flipping: One challenge we encountered was role flipping, where the assistant and user switch roles during the conversation. This issue typically arises when the assistant starts providing instructions or commands instead of following the user’s prompts, which can lead to confusion and a reversal of roles. To avoid role flipping, it is crucial for the assistant not to ask questions, as this can also contribute to the problem. • Assistant Repeats Instruction: Another challenge that we observed was the assistant simply repeating the user’s instructions without any role flipping occurring. • Flake Replies: We also observed instances where the assistant agent responds with a flake reply, often taking the form of "I will...". These messages do not contribute to the task at hand, as the assistant promises to take action but ultimately fails to follow through. • Infinite Loop of Messages: An interesting challenge that we encountered was when the assistant and user engage in an infinite loop of meaningless conversation, such as repeatedly thanking each other or saying goodbye without progressing the task. Interestingly, in some cases, the assistant and user are aware that they are stuck in a loop, but are unable to break out of it.
+
+The Appendix shows examples of each of the four challenges discussed above. Overall, our observations highlight the complexity of cooperative AI development and the need for continued exploration and innovation to overcome the challenges we face. By identifying these issues, we hope to contribute to the development of more effective and engaging cooperative AI systems.
+
+Termination Conditions. The conversation between the assistant and user agents is designed to follow a specific format to ensure consistent and accurate data generation. To ensure that both the user and assistant adhere to their respective roles and responsibilities, certain conditions have been set in place to terminate the chat if necessary. These conditions are outlined below:
+
+User No Instruct: If the user does not instruct the assistant for 3 rounds, conversation is ended.   
+• Assistant Instruct: If the assistant provides an instruction to the user, it indicates a role reversal, and the conversation is terminated.   
+• End of Task Token: If the user believes that the task has been solved, they are expected to say <CAMEL_TASK_DONE> to signify the completion of the task. Once this message is received, the conversation is terminated.   
+Assistant&User Token Limit: Given that gpt-3.5-turbo has a limitation on the number of tokens, the conversation is terminated if either the assistant or the user reach the token limit.   
+• Maximum Number of Messages: To keep the cost of generated chats in check, we have set a maximum limit of 40 messages. This limit guarantees a long enough conversation between the user and assistant while also ensuring that the data generated is not too costly to produce. The cost grows quadratically with the length of the conversation, making it essential to set a limit.
+
+# 5 Evaluation
+
+# 5.1 Agent Evaluation
+
+In order to assess the performance of CAMEL (Cooperative Role-playing Communication), we conduct two types of evaluations: (1) Human evaluation, and (2) GPT4 evaluation. We randomly select 100 tasks from our AI Society dataset for evaluation and 100 tasks from our Code dataset. Then, we employ the GPT4 model to summarize the content of the CAMEL conversation-based solution, presenting a consolidated final solution. Particularly, a GPT4 is used since it possesses a larger token limit which is suitable for summarization. Summarization also makes CAMEL agents’ solution undetectable by its format, allowing for a more fair comparison. Subsequently, this solution is compared with a single-shot solution generated by the gpt-3.5-turbo model for the same task. Sample tasks are provided in the Appendix.
+
+Human Evaluation. For this evaluation, we present both the CAMEL summarized agent solution and the gpt-3.5-turbo single-shot solution side-by-side to human participants. The identity behind each solution is not revealed. Participants are then asked to vote on whether one solution is superior to the other or if they are equally good. A total of 453 responses were collected during this evaluation. Note that, human evaluation is only done for AI Society, as assessing code is generally harder for humans (without running the code).
+
+GPT4 Evaluation. We engage a GPT4 agent to evaluate the effectiveness of Model 1 (CAMEL Agent solution) versus Model 2 (gpt-3.5-turbo single-shot solution) for each task. More specifically, we prompt GPT4 to score and decide which solution of the two solutions is better.
+
+Results. The summarized results of each evaluation are outlined in Table 1 which showcases that the CAMEL solution outperforms gpt-3.5-turbo single-shot solution in both the human evaluation and the GPT4 evaluation by a big margin. It is also worth noting that both human evaluation and GPT4 evaluation are highly aligned.
+
+Table 1: Agent Evaluation Results: Results of the evaluations of the CAMEL agent against gpt-3.5-turbo using both human evaluators and GPT4 consistently show that utilizing a multiagent cooperative approach is more effective than gpt $- 3 . 5$ -turbo’s single shot solution.   
+
+<table><tr><td>Dataset</td><td>Evaluation Type</td><td>Draw</td><td>gpt-3.5-turbo Wins</td><td>CAMEL Agents Win</td></tr><tr><td rowspan="2">AI Society</td><td>Human Evaluation</td><td>13.3%</td><td>10.4%</td><td>76.3%</td></tr><tr><td>GPT4 Evaluation</td><td>4.0%</td><td>23.0%</td><td>73.0%</td></tr><tr><td>Code</td><td>GPT4 Evaluation</td><td>0.0%</td><td>24.0%</td><td>76.0%</td></tr></table>
+
+# 5.2 GPT4 for ChatBot Evaluation
+
+In this section, we progressively fine-tune a LLaMA 7B model on our generated datasets. By progressively incorporating diverse datasets like AI society, code, math, and science, we expect fine-tuned model to demonstrate the ability to develop an increasingly sophisticated understanding of these domains.
+
+We initially start by training on AI society dataset, which aims to let the model learn about human interactions and societal dynamics. As additional datasets were introduced, such as code, the model gained knowledge of programming logic and syntax, enabling it to generate coherent and executable code snippets. The inclusion of the math dataset further expanded the model’s capabilities, allowing it to solve complex equations, reason about abstract concepts, and perform precise calculations. Finally, exposure to the science dataset broadened the model’s understanding of scientific theories, empirical observations, and experimental methods. The emergence of model capabilities is measured by evaluating the quality of the model responses, before and after training on the new domain, on a set of questions of varying difficulties from each domain. More precisely, the model is tested on 20 AI Society related tasks, 20 coding tasks, 20 math tasks and 60 science tasks.
+
+Those results are highlighted in Table 2 where we see that each time we add a dataset, the model performs better on the incorporated domain. Note that to measure the quality of the models’ responses, we follow the evaluation from Section T, which involves prompting a GPT4 agent to score and decide which solution is better. It is worth noting that an improvement on other domains is also observed in some cases such as when we train on Code we improve on Science. This is because our Code dataset contains problems that solve tasks in particular domains which include scientific domain. Similarly, training on AI Society improves code as AI Society contains the role of a "programmer" and hence coding related conversations. Finally, note that the draws observed in LLaMA-7B vs AI Society in Math reflects equally bad solutions compared to the draws observed in AI Society $\mathbf { + C o d e + M a t } ]$ h vs AI Society $+ \mathrm { C o d e } + \mathrm { M a t h } + \Omega$ cience where the draws are equally good solutions. This progression from AI society to code to math to science highlights the potential of AI models to acquire a versatile and adaptable knowledge base, paralleling the way humans gain expertise in diverse subjects. Sample tasks are provided in the Appendix.
+
+Table 2: Emergence of Knowledge. By progressively fine-tuning LLaMA on datasets from different domains, we observe the emergence of knowledge as the model transitions from AI society to code, math, and science. This finding is indicated by the fact that Model 2 almost always performs better than Model 1, especially on the added dataset.   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="3">Model 1</td><td colspan="3">Model 2</td><td rowspan="2">Draw</td><td rowspan="2">Model 1</td><td rowspan="2">Model 2</td></tr><tr><td>AI Society</td><td>CodeMath</td><td>Science</td><td>AI Society</td><td>Code1</td><td>Math Science</td><td></td></tr><tr><td>AI Society</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0</td><td>6</td><td>14</td></tr><tr><td>Code</td><td></td><td></td><td></td><td>交</td><td></td><td></td><td></td><td>0</td><td>0</td><td>20</td></tr><tr><td>Math</td><td></td><td></td><td></td><td>√</td><td></td><td></td><td></td><td>9</td><td>5</td><td>6</td></tr><tr><td>Science</td><td></td><td></td><td></td><td>√</td><td></td><td></td><td></td><td>0</td><td>13</td><td>47</td></tr><tr><td>AI Society</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>4</td><td>8</td><td>8</td></tr><tr><td>Code</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>9</td><td>10</td></tr><tr><td>Math</td><td>√</td><td></td><td></td><td></td><td>&lt;&lt;√</td><td></td><td></td><td>5</td><td>8</td><td>7</td></tr><tr><td>Science</td><td>√</td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>19</td><td>40</td></tr><tr><td>AI Society</td><td>√</td><td></td><td></td><td></td><td></td><td></td><td></td><td>5</td><td>6</td><td>9</td></tr><tr><td>Code</td><td></td><td>&gt;&lt;&gt;</td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>9</td><td>10</td></tr><tr><td>Math</td><td>√</td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>3</td><td>16</td></tr><tr><td>Science</td><td>√</td><td></td><td></td><td></td><td></td><td>&lt;&lt;√</td><td></td><td>3</td><td>8</td><td>49</td></tr><tr><td>AI Society</td><td>√</td><td></td><td></td><td></td><td></td><td></td><td></td><td>3</td><td>1</td><td>16</td></tr><tr><td>Code</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>1</td><td>8</td><td>11</td></tr><tr><td>Math</td><td>√</td><td></td><td></td><td></td><td></td><td></td><td></td><td>10</td><td>5</td><td>5</td></tr><tr><td>Science</td><td>√</td><td></td><td></td><td>√</td><td></td><td></td><td>&lt;&lt;&lt;</td><td>9</td><td>2</td><td>49</td></tr><tr><td>AI Society</td><td></td><td></td><td></td><td>√</td><td></td><td></td><td>√</td><td>0</td><td>0</td><td>20</td></tr><tr><td>Code</td><td></td><td></td><td></td><td></td><td></td><td>&lt;&lt;&lt;</td><td></td><td>0</td><td>0</td><td>20</td></tr><tr><td>Math</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0</td><td>0</td><td>20</td></tr><tr><td>Science</td><td></td><td></td><td></td><td></td><td></td><td></td><td>√</td><td>0</td><td>0</td><td>60</td></tr></table>
+
+# 5.3 HumanEval(+)
+
+Table 3: HumanEval(+) for Various Models. We test our CAMEL model, which is a LLaMa-7B fine-tuned on all our datasets (AI Society, Code, Math, Science) on HumanEval and HumanEval+ benchmarks, where we show competitive pass $@ k$ scores with LLaMa-7B and Vicuna-7B.   
+
+<table><tr><td></td><td colspan="2">HumanEval</td><td colspan="2">HumanEval+</td></tr><tr><td>pass@k [%]</td><td>k =1</td><td>k=100</td><td>k=1</td><td>k=100</td></tr><tr><td>gpt-3.5-turbo</td><td>69.4</td><td>94.0</td><td>61.7</td><td>89.8</td></tr><tr><td>LLaMA-7B</td><td>10.5</td><td>36.5</td><td>1</td><td>-</td></tr><tr><td>Vicuna-7B</td><td>11.0</td><td>42.9</td><td>9.9</td><td>34.7</td></tr><tr><td>CAMEL-7B</td><td>14.0</td><td>57.9</td><td>12.2</td><td>50.0</td></tr></table>
+
+To evaluate the coding task-solving capabilities of our CAMEL model, specifically the LLaMA7B fine-tuned on our comprehensive datasets, we rely on HumanEval [18] and HumanEval+ [69]. The results, as depicted in table 3, clearly demonstrate the remarkable performance of CAMEL. It surpasses not only the LLaMA-7B model but also Vicuna-7B [21] by a big margin. These findings underscore the critical role played by the generated datasets in enhancing LLaMA’s ability to tackle coding-related tasks.
+
+# 6 Conclusion
+
+In this paper, we explore the potential of autonomous cooperation among communicative agents and propose a novel cooperative agent framework named role-playing . Our approach enables communicative agents to collaborate autonomously toward completing tasks while requiring minimal human intervention, leading to better solutions are per our thorough evaluations. Through our analysis, we show that achieving autonomous cooperation is challenging due to issues like conversation deviation, role flipping, and termination conditions. Our framework offers a scalable approach for studying the cooperative behaviors and capabilities of multi-agent systems and provides strategies for addressing these challenges. Furthermore, our open-sourced library includes implementations of various agents, data generation pipelines, data analysis tools, and collected datasets, to support research on communicative agents and beyond. Our contributions offer valuable insights into the future of large language artificial intelligence models and cooperative AI systems.
+
+# 7 Acknowledgements
+
+This work was supported by SDAIA-KAUST Center of Excellence in Data Science and Artificial Intelligence (SDAIA-KAUST AI).
+
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parse/dev/3JvRnAzw_0/3JvRnAzw_0.md

@@ -0,0 +1,297 @@
+# ROBUST WEIGHT PERTURBATION FORADVERSARIAL TRAINING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Overfitting widely exists in adversarial robust training of deep networks. An effective and promising remedy is adversarial weight perturbation, which injects the worst-case weight perturbation during network training by maximizing the classification loss on adversarial examples. Adversarial weight perturbation helps reduce the robust generalization gap; however, it also undermines the robustness enhancement. A criterion that regulates the weight perturbation is therefore crucial for adversarial training. In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained perturbation. With LSC, we find that deep network first overfits the adversarial examples with small loss, and then gradually develops to overfit all adversarial examples in the later stage of training. Following this, we find that it is essential to conduct weight perturbation on adversarial data with small classification loss to eliminate overfitting in adversarial training. Weight perturbation on adversarial data with large classification loss is not necessary and may even lead to poor robustness. Based on these observations, we propose a robust perturbation strategy to constrain the extent of weight perturbation. The perturbation strategy prevents deep networks from overfitting while avoiding the side effect of excessive weight perturbation, significantly improving the robustness of adversarial training. Extensive experiments demonstrate the superiority of the proposed method over the state-of-the-art adversarial training methods.
+
+# 1 INTRODUCTION
+
+Although deep neural networks (DNNs) have led to impressive breakthroughs in a number of fields such as computer vision (He et al., 2016), speech recognition (Wang et al., 2017), and natural language processing (Devlin et al., 2018), they are extremely vulnerable to adversarial examples that are crafted by adding small and human-imperceptible perturbation to normal examples (Szegedy et al., 2013; Goodfellow et al., 2014).
+
+The vulnerability of DNNs has attracted extensive attention and led to a large number of defense techniques against adversarial examples. Across existing defenses, adversarial training (AT) is one of the strongest empirical defenses. AT directly incorporates adversarial examples into the training process to solve a min-max optimization problem (Madry et al., 2017), which can obtain models with moderate adversarial robustness and has not been comprehensively attacked (Athalye et al., 2018). However, different from the standard training scenario, overfitting is a dominant phenomenon in adversarial robust training of deep networks (Rice et al., 2020). After a certain point in AT, the robust performance on test data will continue to degrade with further training. This phenomenon, termed as robust overfitting, breaches the common practice in deep learning that using over-parameterized networks and training for as long as possible (Neyshabur et al., 2017; Belkin et al., 2019). Such anomaly in AT causes detrimental effects on the robust generalization performance and subsequent algorithm assessment (Rice et al., 2020; Chen et al., 2020b). Relief techniques that mitigate robust overfitting have thus become crucial for stable adversarial training.
+
+An effective and promising remedy for robust overfitting is Adversarial Weight Perturbation (AWP) (Wu et al., 2020), which forms a double-perturbation mechanism in the adversarial training framework that adversarially perturbs both inputs and weights:
+
+$$
+\operatorname* { m i n } _ { w } \operatorname* { m a x } _ { v \in \mathcal { V } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \operatorname* { m a x } _ { | | x _ { i } ^ { \prime } - x _ { i } | | _ { p } \leq \epsilon } \ell ( f _ { w + v } ( x _ { i } ^ { \prime } ) , y _ { i } ) ,
+$$
+
+where $n$ is the number of training examples, $\boldsymbol { x } _ { i } ^ { \prime }$ is the adversarial example of $x _ { i } , f _ { w }$ is the DNN with weight $w , \ell ( \cdot )$ is the loss function, $\epsilon$ is the maximum perturbation constraint for inputs (i.e., $| | x _ { i } ^ { \prime } - x _ { i } | | _ { p } \leq \epsilon )$ , and $\nu$ is the feasible perturbation region for weights (i.e., $\{ v \in \mathcal { V } : | | v | | _ { 2 } \leq$ $\gamma | | w | | _ { 2 } \bigr \}$ , where $\gamma$ is the constraint on weight perturbation size). The inner maximization is to find adversarial examples $x _ { i } ^ { \prime }$ within the $\epsilon$ -ball centered at normal examples $x _ { i }$ that maximizes the classification loss $\ell$ . On the other hand, the outer maximization is to find weight perturbation $\textbf {  { v } }$ that maximizes the loss $\ell$ on adversarial examples to flatten the weight loss landscape and reduce robust generalization gap. This is the problem of training a weight-perturbed robust classifier on adversarial examples. Therefore, how well the weight perturbation is found directly affects the performance of the outer minimization, i.e., the robustness of the classifier.
+
+Several attack methods have been used to solve the inner maximization problem in Eq.(1), such as Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014) and Projected Gradient Descent (PGD) (Madry et al., 2017). For the outer maximization problem, AWP (Wu et al., 2020) injects the worst-case weight perturbation to reduce robust generalization gap. However, the extent to which the weights should be perturbed has not been explored. Without an appropriate criterion to regulate the weight perturbation, the adversarial training procedure is difficult to unleash its full power. In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained perturbation, which sheds light on the nitty-gritty of robust overfitting in adversarial training, and this in turn motivates us to propose an improved weight perturbation strategy for better robustness. Our main contributions are follows:
+
+• We propose a principled criterion LSC to monitor the training status of different adversarial examples during network optimization. It provides a better understanding of robust overfitting in adversarial training, and it is also a good indicator for efficient weight perturbation. • With LSC, we find that deep network first overfits adversarial data with small classification loss and then gradually develops to overfit all adversarial data. Following this, we find that better perturbation of model weights is associated with perturbing on adversarial data with small classification loss. For adversarial data with large classification loss, weight perturbation is not necessary and can even be harmful. • We propose a robust perturbation strategy to constrain the extent of weight perturbation. Experiments show that the robust strategy significantly improves the robustness of adversarial training.
+
+# 2 RELATED WORK
+
+# 2.1 ADVERSARIAL ATTACKS
+
+Given a normal example $( x , y )$ , a DNN $f _ { w }$ , and maximum perturbation constraint $\epsilon$ . Let $\mathcal { X }$ denote the input feature space and $\mathcal { B } _ { \epsilon } ^ { p } ( x ) = \{ x ^ { \prime } \in \mathcal { X } : | | x ^ { \prime } - x | | _ { p } \leq \epsilon \}$ be the $\ell _ { p }$ -norm ball of radius $\epsilon$ centered at $x$ in $\mathcal { X }$ . The goal of adversarial attack is to find an adversarial example $x ^ { \prime } \in B _ { \epsilon } ^ { p } ( x )$ that can fool the DNN to produce an incorrect output ( $f _ { w } ( x ^ { \prime } ) \ne y$ ). Here we selectively introduce several commonly used adversarial attack methods.
+
+Fast Gradient Sign Method (FGSM). FGSM (Goodfellow et al., 2014) perturbs natural example $x$ for one step with step size $\epsilon$ along the gradient direction:
+
+$$
+\begin{array} { r } { x ^ { \prime } = x + \epsilon \cdot \mathrm { s i g n } \bigl ( \nabla _ { x } \ell ( f _ { w } ( x ) , y ) \bigr ) . } \end{array}
+$$
+
+Projected Gradient Descent (PGD). PGD (Madry et al., 2017) is a stronger iterative variant of FGSM, which perturbs normal example $x$ for multiple steps $K$ with a smaller step size $\alpha$ :
+
+$$
+x ^ { 0 } \sim \mathcal { U } ( B _ { \epsilon } ^ { p } ( x ) ) ,
+$$
+
+$$
+\boldsymbol { x } ^ { k } = \Pi _ { \mathcal { B } _ { \epsilon } ^ { p } ( x ) } ( \boldsymbol { x } ^ { k - 1 } + \alpha \cdot \mathrm { s i g n } ( \nabla _ { \boldsymbol { x } ^ { k - 1 } } \ell ( f _ { w } ( \boldsymbol { x } ^ { k - 1 } ) , \boldsymbol { y } ) ) ) ,
+$$
+
+where $\mathcal { U }$ denotes the uniform distribution, $x ^ { 0 }$ denotes the normal example disturbed by a small uniform random noise, $x ^ { k }$ denotes the adversarial example at step $k$ , and $\Pi _ { B _ { \epsilon } ^ { p } ( x ) }$ denotes the projection function that projects the adversarial example back into the set $B _ { \epsilon } ^ { p } ( x )$ if necessary.
+
+AutoAttack (AA). AA (Croce & Hein, 2020b) is an ensemble of complementary attacks, which consists of three white-box attacks (APGD-CE (Croce & Hein, 2020b), APGD-DLR (Croce & Hein, 2020b), and FAB (Croce & Hein, 2020a)) and a black-box attack (Square Attack (Andriushchenko et al., 2020)). AA regards models to be robust only if the models correctly classify all types of adversarial examples, which is among the most reliable evaluation of adversarial robustness to date.
+
+There are also other types of attacking methods, e.g., the CW attack (Carlini & Wagner, 2017), deformation attack (Engstrom et al., 2017; Xiao et al., 2018; Engstrom et al., 2019), Hamming distance based attack (Shamir et al., 2019), Frank-Wolfe based attack (Chen et al., 2020a) and adaptive attack (Tramer et al., 2020).
+
+# 2.2 ADVERSARIAL DEFENSE
+
+Since the discovery of adversarial examples, a large number of works have emerged for defending against adversarial attacks, such as input denoising (Guo et al., 2018; Liao et al., 2018; Wu et al., 2021), defensive distillation (Papernot et al., 2016; Carlini & Wagner, 2017), adversarial detection (Metzen et al., 2017; Tao et al., 2018), gradient regularization (Tramer et al., 2018; Ross & Doshi- \` Velez, 2018) and adversarial training (Goodfellow et al., 2014; Madry et al., 2017). Among them, adversarial training has been demonstrated to be the most effective method (Athalye et al., 2018). Based on adversarial training, a wide range of subsequent works are then proposed to further improve the model robustness (Xie et al., 2019; Mosbach et al., 2018; Kannan et al., 2018; Zhang et al., 2019; Cai et al., 2018; Wang et al., 2019a; Zhang et al., 2020a; Dong et al., 2018; Yang et al., 2019; Wang et al., 2019b; Song et al., 2020; Carmon et al., 2019; Zhai et al., 2019; Uesato et al., 2019; Hendrycks et al., 2019; Yan et al., 2021; Du et al., 2021). Here, we introduce two currently state-of-the-art adversarial training frameworks.
+
+TRADES. TRADES (Zhang et al., 2019) optimizes a regularized surrogate loss that is a trade-off between the natural accuracy and adversarial robustness:
+
+$$
+\ell ^ { { \mathrm { T R A D E S } } } ( \boldsymbol { w } ; \boldsymbol { x } , \boldsymbol { y } ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left\{ \operatorname { C E } ( f _ { w } ( x _ { i } ) , y _ { i } ) + \beta \cdot \operatorname* { m a x } _ { \boldsymbol { x } ^ { \prime } \in B _ { \epsilon } ^ { p } ( \boldsymbol { x } ) } \operatorname { K L } ( f _ { w } ( x _ { i } ) | | f _ { w } ( x _ { i } ^ { \prime } ) ) \right\} ,
+$$
+
+where CE is the cross-entropy loss that encourages the network to maximize the natural accuracy, KL is the Kullback-Leibler divergence that encourages to improve the robust accuracy, and $\beta$ is the hyperparameter to control the trade-off between natural accuracy and adversarial robustness.
+
+Robust Self-Training (RST). RST (Carmon et al., 2019) utilize additional 500K unlabeled data extracted from the 80 Million Tiny Images dataset (Torralba et al., 2008). RST first leverages the surrogate natural model to generate pseudo-labels for these unlabeled data, and then adversarially trains the network with both additional pseudo-labeled unlabeled data $( \tilde { x } , \tilde { y } )$ and original labeled data $( x , y )$ in a supervised setting:
+
+$$
+\ell ^ { \mathrm { R S T } } ( \boldsymbol { w } ; x , y , \tilde { x } , \tilde { y } ) = \ell ^ { \mathrm { T R A D E S } } ( \boldsymbol { w } ; x , y ) + \lambda \cdot \ell ^ { \mathrm { T R A D E S } } ( \boldsymbol { w } ; \tilde { x } , \tilde { y } ) ,
+$$
+
+where $\lambda$ is the weight on unlabeled data.
+
+# 2.3 ROBUST OVERFITTING
+
+Nowadays, there are effective countermeasures to alleviate the overfitting in standard training. But in adversarial training, robust overfitting widely exists and those common countermeasures used in standard training help little (Rice et al., 2020). Schmidt et al. (2018) explains robust overfitting partially from the perspective of sample complexity, and is supported by empirical results in derivative works, such as adversarial training with semi-supervised learning (Carmon et al., 2019; Uesato et al., 2019; Zhai et al., 2019), robust local feature (Song et al., 2020) and data interpolation (Zhang & Xu, 2019; Lee et al., 2020; Chen et al., 2021). Separate works have also attempt to mitigate robust overfitting by the unequal treatment of data (Zhang et al., 2020b) and weight smoothing (Chen et al., 2020b). Recent study (Wu et al., 2020) reveals the connection between the flatness of weight loss landscape and robust generalization gap, and proposes to incorporate adversarial weight perturbation mechanism in the adversarial training framework. Despite the efficacy of adversarial weight perturbation in suppressing the robust overfitting in adversarial training, a deeper understanding of the cause of robust overfitting and a clear direction for valid weight perturbation is largely missing. The outer maximization in Eq.(1) lacks an effective criterion to regulate and constrain the extent of weight perturbation, which in turn influences the optimization of the outer minimization problem. In this paper, we propose such a criterion and provide new understanding of the robust overfitting in adversarial training. Following this, we design a robust weight perturbation strategy that significantly improves the robustness of adversarial training.
+
+![](images/cf8c9c1611ba4804d2a3319c5b7ff65a8e692709372051083b7ff11770673d20.jpg)  
+Figure 1: (a): Test robustness of AWP with varying weight perturbation size; (b): The learning curve of vanilla AT; (c): Test robustness of AWP with varying LSC range.
+
+# 3 LOSS STATIONARY CONDITION
+
+In this section, we first empirically investigate the relationship between weight perturbation robustness and adversarial robustness, and then propose a new criterion to monitor the training status of different adversarial examples in the learning process of adversarial training, which leads to a new perspective of robust overfitting. To this end, some discussions about robust overfitting and adversarial weight perturbation are provided.
+
+Does Weight Perturbation Robustness Lead to Better Adversarial Robustness? First, we investigate whether the robustness against weight perturbation is beneficial to the adversarial robustness. In particular, we train PreAct ResNet-18 with AWP on CIFAR-10 using varying weight perturbation size from $0 \gamma , \gamma / 8 , \gamma / 4 , \gamma / 2 , \gamma , 2 \gamma , 4 \gamma$ to $8 \gamma$ . In each setting, we evaluate the robustness of the model against 20-step PGD (PGD-20) attacks on CIFAR-10 test images. As shown in Figure 1(a), when varying weight perturbation size, the best adversarial robustness has a certain improvement in the early stage. When weight perturbation size is large, the best adversarial robustness begins to decrease significantly as the size of the perturbation increases. It might be explained by the fact that the network has to sacrifice adversarial robustness to allocate more capacity to defend against weight perturbation, which implies that weight perturbation robustness and adversarial robustness are not actually mutually beneficial. The performance gain of AWP is mainly due to suppressing robust overfitting.
+
+Loss Stationary Condition. In order to further understand the robust overfitting, we propose a criterion that divides the training adversarial examples into different groups according to their classification loss:
+
+$$
+\mathrm { L S C } [ p , q ] = \{ x ^ { \prime } \in \mathcal { X } \mid p \leq \ell ( f _ { w } ( x ^ { \prime } ) , y ) \leq q \} ,
+$$
+
+![](images/029122678a72cbfbbe80142edc40947c335d750d1a999a146521906d2088d916.jpg)  
+Figure 2: The weight loss landscape of different LSC groups on different checkpoints.
+
+where $p \leq q$ . The adversarial examples in the group all satisfy their classification loss within a certain range, which is termed Loss Stationary Condition (LSC). The proposed criterion LSC allows the analysis of training status of different adversarial examples independently, and provides more insights into the robust overfitting.
+
+LSC View of Robust Overfitting. To provide details of the robust overfitting in adversarial training, we train a PreAct ResNet-18 for 200 epochs on CIFAR-10 using PGD-10 with step size $\epsilon / 4$ , maximum perturbation $\epsilon = 8 / 2 5 5$ , following the standard setting in Madry et al. (2017). The learning curve is shown in Figure 1(b). For each intermediate model, we then apply the same PGD-10 attack on CIFAR-10 training images to craft adversarial examples, and divide the crafted adversarial examples into 6 consecutive LSC groups ranging from 0.0 to 3.0. Then, we use the weight loss landscape to characterize the training status of the adversarial examples in each LSC group, which plots the classification loss change when perturbing the model weight $\pmb { w }$ by a random noise $^ d$ with magnitude $\mu$ :
+
+$$
+g _ { j } ( \mu ) = \frac { 1 } { n _ { j } } \sum _ { i = 1 } ^ { n _ { j } } \ell ( f _ { w + \mu d } ( x _ { i j } ^ { \prime } ) , y _ { i j } ) , \ x _ { i j } ^ { \prime } \in \mathrm { L S C } [ p _ { j } , q _ { j } ] ,
+$$
+
+where $j$ is the number of groups, $n _ { j }$ is the number of adversarial examples in $j$ -th LSC group, and $^ d$ is filter normalized by $\begin{array} { r } { d \gets \frac { d } { | | d | | } | | \boldsymbol { w } | | } \end{array}$ following Li et al. (2017). It is worth noting that weight loss landscape has been widely used to characterize the generalization gap (Neyshabur et al., 2017; Foret et al., 2020; Wu et al., 2020). Here, we use it to characterize the training status of different adversarial examples. For training adversarial examples, the higher the degree of overfitting by the model, the more sensitive its loss is to model weight perturbations, thus making the weight loss landscape sharper. Here the weight loss curve sharpness is served as a comparable measurement of overfitting strength. Besides, another key difference to previous works lies on the LSC criterion used for visualization, which provides more insights into the robust overfitting.
+
+We show the weight loss curve of each LSC group on different checkpoints in Figure 2. In the early stage of training (between 100 and 120 epoch), it can be seen that the weight loss curve of the LSC group with small loss is obviously sharper than that of the LSC group with large loss, which indicates that the adversarial examples with small classification loss were first overfitted. As the training progresses, the weight loss curves of all LSC groups become very sharp, which shows that the network overfits all adversarial examples. These observations suggest that robust overfitting exists a diffusion process: the model will first memorize some easy-to-learn adversarial examples, and then spread to the entire training dataset.
+
+Algorithm 1 Robust Weight Perturbation   
+
+<table><tr><td>igorlhhTKobustWergntFerturbaton Input: Network fw, training data S,mini-batch B,batch size n,learning rate n,PGD step size α,</td></tr><tr><td>PGD steps K1,PGD constraint ε, RWP steps K2, RWP constraint γ, minimum LSC value Cmin· Output: Adversarially robust model fw · repeat</td></tr><tr><td>Read mini-batch xp from training set S. xB ← xB +δ,where δ ~ Uniform(-∈,∈) for k=1 to K1 do</td></tr><tr><td>x&#x27; ←II(xB+α·sign(Vxsl(fw(xB),y)))</td></tr><tr><td>end for</td></tr><tr><td>Initialize v = 0</td></tr><tr><td>for k=1 to K2 do</td></tr><tr><td>V = 1B(l(fw+u(xs),y)≤ Cmin)</td></tr><tr><td>if∑V=0 then</td></tr><tr><td>break</td></tr><tr><td></td></tr><tr><td>else</td></tr><tr><td>v ← v+ Vu(V ·l(fw+v(xs),y))</td></tr><tr><td>U←Yw</td></tr><tr><td>end if</td></tr><tr><td>w ←(w+v)-nVw+vn∑i=1l(fw+u(x&#x27;g),y(i))-v</td></tr><tr><td>end for</td></tr></table>
+
+LSC view of Adversarial Weight Perturbation. To provide more insight into how AWP suppresses robust overfitting, we train PreAct ResNet-18 on CIFAR-10 by varying the LSC group that performs adversarial weight perturbation. In each setting, we evaluate the robustness of the model against PGD-20 attacks on CIFAR-10 test images. As shown in Figure 1(c), when varying the LSC range, we can observe that conducting adversarial weight perturbation on adversarial examples with small classification loss is sufficient to suppress robust overfitting. Recalling the diffusion process in robust overfitting, we can infer that to eliminate robust overfitting, it is essential to prevent the model from memorizing the easy-to-learn adversarial examples. Besides, it is observed that conducting adversarial weight perturbation on adversarial examples with large classification loss leads to worse adversarial robustness, which again verifies that the robustness against weight perturbation will not bring adversarial robustness gain, or even on the contrary, it undermines the adversarial robustness enhancement.
+
+Do We Really Need the Worst-case Weight Perturbation? As aforementioned, the robustness against weight perturbation is detrimental to the adversarial robustness enhancement. Therefore, to purely prevent the network from memorizing the adversarial examples with small classification loss, conducting worst-case weight perturbation on these adversarial examples is not necessary, since it will also deteriorate the adversarial robustness. In the next section, we will propose a robust perturbation strategy to address this issue.
+
+# 4 ROBUST WEIGHT PERTURBATION
+
+In this section, we introduce the proposed robust weight perturbation strategy and its algorithmic realization.
+
+As mentioned in Section 3, conducting adversarial weight perturbation on adversarial examples with small classification loss is enough to prevent robust overfitting and leads to higher robustness. However, conducting adversarial weight perturbation on adversarial examples with large classification loss may not be helpful. Recalling the criterion LSC proposed in Section 3, we have seen that it is closely correlated with the tendency of adversarial example to be overfitted. Thus, it can be used to regulate the extent of weight perturbation at a fine-grained level. Therefore, we propose to train network with adversarial examples that are all above a minimum LSC value, so as to ensure that no robust overfitting occurs while avoiding the side effect of excessive weight perturbation. Let $c _ { m i n }$ be the minimum LSC value. Instead of generating weight perturbation $\textbf {  { v } }$ via outer maximization in Eq.(1), we generate $\pmb { v }$ as follows:
+
+$$
+\begin{array} { r l } { \pmb { v } ^ { k + 1 } = \pmb { v } ^ { k } + \nabla _ { \pmb { v } ^ { k } } \cfrac { 1 } { n } \displaystyle \sum _ { i = 1 } ^ { n } \mathbb { 1 } ( x _ { i } ^ { \prime } , y _ { i } ) \ell ( f _ { \pmb { w + v } ^ { k } } ( x _ { i } ^ { \prime } ) , y _ { i } ) , } & { } \\ { \mathrm { w h e r e } \quad \mathbb { 1 } ( x _ { i } ^ { \prime } , y _ { i } ) = \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f ~ } \ell ( f _ { \pmb { w + v } ^ { k } } ( x _ { i } ^ { \prime } ) , y _ { i } ) > c _ { m i n } } \\ { 1 } & { \mathrm { i f ~ } \ell ( f _ { \pmb { w + v } ^ { k } } ( x _ { i } ^ { \prime } ) , y _ { i } ) \leq c _ { m i n } } \end{array} \right. } \end{array}
+$$
+
+The proposed Robust Weight Perturbation (RWP) algorithm is shown in Algorithm 1. We use PGD attack (Madry et al., 2017) to generate the training adversarial examples, which can be also extended to other variants such as TRADES (Zhang et al., 2019) and RST (Carmon et al., 2019). The mimimum LSC value $c _ { m i n }$ controls the minimum classification loss (minimum weight perturbation strength) of the adversarial examples during network training. In the early stages of training, the classification loss of adversarial example is generally larger than $c _ { m i n }$ corresponding to no weight perturbation process. The classification loss of adversarial examples then decreases as training progresses. At each optimization step, we monitor the classification loss of the adversarial example and conduct the weight perturbation process for adversarial examples whose classification loss is already smaller than $c _ { m i n }$ , enabled by an indicator control vector $V$ . At each perturbation step, the weight perturbation $\pmb { v }$ will be updated to increase the classification loss of the corresponding adversarial example. When the classification loss of training adversarial examples is all higher than $c _ { m i n }$ or the number of perturbation step reaches the defined value, we stop the weight perturbation process and inject the generated weight perturbation $\textbf {  { v } }$ for adversarial training.
+
+# 5 EXPERIMENTS
+
+In this section, we conduct comprehensive experiments to evaluate the effectiveness of RWP including its experimental settings, robustness evaluation and ablation studies.
+
+# 5.1 EXPERIMENTAL SETUP
+
+Baselines and Implementation Details. Our implementation is based on PyTorch and the code as well as other related resources will be released for public use and verification. We conduct extensive experiments across three benchmark datasets (CIFAR-10 (Krizhevsky et al., 2009), CIFAR100 (Krizhevsky et al., 2009) and SVHN (Netzer et al., 2011)) and two threat models $L _ { \infty }$ and $L _ { 2 }$ ). We use PreAct ResNet-18 (He et al., 2016) and Wide ResNet (WRN-28-10 and WRN-34-10) (Zagoruyko & Komodakis, 2016) as the network structure following Wu et al. (2020). We compare the performance of the proposed method on a number of baseline methods: 1) standard adversarial training without weight perturbation, including vanilla AT (Madry et al., 2017), TRADES (Zhang et al., 2019) and RST (Carmon et al., 2019); 2) adversarial training with adversarial weight perturbation (AWP) (Wu et al., 2020). For training, the network is trained for 200 epochs using SGD with momentum 0.9, weight decay $5 \times 1 0 ^ { - 4 }$ , and an initial learning rate of 0.1. The learning rate is divided by 10 at the 100-th and 150-th epoch. Standard data augmentation including random crops with 4 pixels of padding and random horizontal flips are applied. For testing, model robustness is evaluated by measuring the accuracy of the model under different adversarial attacks. For hyperparameters in RWP, we set perturbation step $K _ { 2 } = 1 0$ for all datasets. The minimum LSC value $c _ { m i n } = 1 . 7$ for CIFAR-10, $c _ { m i n } = 2 . 2$ for SVHN and $c _ { m i n } = 4 . 0$ for CIFAR-100. The weight perturbation budget of $\gamma = 0 . 0 1$ for AT-RWP, $\gamma = 0 . 0 0 5$ for TRADES-RWP and RST-RWP following literature (Wu et al., 2020). Other hyper-parameters of the baselines are configured as per their original papers.
+
+Adversarial Setting. The training attack is 10-step PGD attack with random start. We follow the same settings in Rice et al. (2020) : for $L _ { \infty }$ threat model, $\epsilon = 8 / 2 5 5$ , step size $\alpha = 1 / 2 5 5$ for SVHN, and $\alpha = 2 / 2 5 5$ for both CIFAR10 and CIFAR100; for $L _ { 2 }$ threat model, $\epsilon = 1 2 8 / 2 5 5$ , step size $\alpha = 1 5 / 2 5 5$ for all datasets. The test attacks used for robustness evaluation are generated from the original test set images by attacking the defense models using different attacking methods, including: FGSM, PGD-20, PGD-100, $\mathrm { C } \& \mathbf { W } _ { \infty }$ $L _ { \infty }$ version of C&W optimized by PGD for 100 steps) and Auto Attack (AA).
+
+# 5.2 ROBUSTNESS EVALUATION
+
+Performance Evaluations. To validate the effectiveness of the proposed RWP, we conduct performance evaluation on vanilla AT, AT-AWP and AT-RWP across different benchmark datasets and threat models using PreAct ResNet-18. We report the accuracy on the test images under PGD-20 attack. The evaluation results are summarized in Table 1. “Best” denotes the highest robustness that ever achieved at different checkpoints and ”last” denotes the robustness at the last epoch checkpoint. It is observed vanilla AT suffers from severe robust overfitting (the performance gap between ”best” and ”last” is very large). AT-AWP and AT-RWP method narrow the performance gap significantly over the vanilla AT model due to suppression of robust overfitting. Moreover, on CIFAR-10 dataset under the $L _ { \infty }$ attack, vanilla AT achieves $5 2 . 7 9 \%$ ”best” test robustness. The AT-AWP approach boosts the performance to $5 5 . 3 9 \%$ . The proposed approach further outperforms both methods by a large margin, improving over vanilla AT by $5 . 7 6 \%$ , and is $3 . 1 6 \%$ better than AT-AWP, achieving $5 8 . 5 5 \%$ accuracy under the standard 20 steps PGD attack. Similar patten has been observed on other datasets and threat model. AT-RWP consistently improves the test robustness across a wide range of datasets and threat models, demonstrating the effectiveness of the proposed approach.
+
+Table 1: Test robustness $( \% )$ of AT, AT-AWP and AT-RWP using PreAct ResNet-18.   
+
+<table><tr><td rowspan="2">Threat Model</td><td rowspan="2">Method</td><td colspan="2">SVHN</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td></tr><tr><td>Best</td><td>Last</td><td>Best</td><td>Last</td><td>Best</td><td>Last</td></tr><tr><td rowspan="3">L8</td><td>AT</td><td>53.36</td><td>44.49</td><td>52.79</td><td>44.44</td><td>27.22</td><td>20.82</td></tr><tr><td>AT-AWP</td><td>59.12</td><td>55.87</td><td>55.39</td><td>54.73</td><td>30.71</td><td>30.28</td></tr><tr><td>AT-RWP</td><td>61.15</td><td>57.45</td><td>58.55</td><td>58.01</td><td>31.17</td><td>30.64</td></tr><tr><td rowspan="3">L2</td><td>AT</td><td>66.87</td><td>65.03</td><td>69.15</td><td>65.93</td><td>41.33</td><td>35.27</td></tr><tr><td>AT-AWP</td><td>72.57</td><td>67.73</td><td>72.69</td><td>72.08</td><td>45.60</td><td>44.66</td></tr><tr><td>AT-RWP</td><td>73.35</td><td>69.48</td><td>74.47</td><td>73.84</td><td>45.71</td><td>45.05</td></tr></table>
+
+Benchmarking the state-of-the-art Robustness. To manifest the full power of our proposed perturbation strategy and also benchmark the state-of-the-art robustness on CIFAR-10 under $\mathrm { L } _ { \infty }$ threat model, we conduct experiments on the large capacity network with different baseline methods. We train Wide ResNet-34-10 for AT and TRADES, and Wide ResNet-28-10 for RST following their original papers. We evaluate the adversarial robustness of trained model with various test attack and report the “best” test robustness, with the results shown in Table 2. “Natural” denotes the accuracy on natural test data. First, it is observed that the natural accuracy of RWP model consistently outperforms AWP by a large margin. It is due to the benefits that our RWP avoids the excessive weight perturbation. Moreover, RWP achieves the best adversarial robustness against all types of attack across a wide range of baseline methods, which verifies that RWP is effective in general and improves adversarial robustness reliably rather than improper tuning of hyper-parameters of attacks, gradient obfuscation or masking.
+
+Table 2: Test robustness $( \% )$ on CIFAR-10 using Wide ResNet under $L _ { \infty }$ threat model.   
+
+<table><tr><td>Defense</td><td>Natural</td><td>FGSM</td><td>PGD-20</td><td>PGD-100</td><td>C&amp;W</td><td>AA</td></tr><tr><td>AT</td><td>86.07</td><td>61.76</td><td>56.10</td><td>55.79</td><td>54.19</td><td>52.60</td></tr><tr><td>AT-AWP</td><td>85.57</td><td>62.90</td><td>58.14</td><td>57.94</td><td>55.96</td><td>54.04</td></tr><tr><td>AT-RWP</td><td>86.86</td><td>66.22</td><td>62.87</td><td>62.87</td><td>56.62</td><td>54.61</td></tr><tr><td>TRADES</td><td>84.65</td><td>61.32</td><td>56.33</td><td>56.07</td><td>54.20</td><td>53.08</td></tr><tr><td>TRADES-AWP</td><td>85.36</td><td>63.49</td><td>59.27</td><td>59.12</td><td>57.07</td><td>56.17</td></tr><tr><td>TRADES-RWP</td><td>86.14</td><td>64.70</td><td>60.45</td><td>60.30</td><td>58.07</td><td>57.20</td></tr><tr><td>RST</td><td>89.69</td><td>69.60</td><td>62.60</td><td>62.22</td><td>60.47</td><td>59.53</td></tr><tr><td>RST-AWP</td><td>88.25</td><td>67.94</td><td>63.73</td><td>63.58</td><td>61.62</td><td>60.05</td></tr><tr><td>RST-RWP</td><td>88.87</td><td>69.71</td><td>64.11</td><td>63.92</td><td>62.03</td><td>60.36</td></tr></table>
+
+![](images/05cc9a011b1b361f1fc69f9f40b1f1f4a73327c2b02695b658d3ce690a694601.jpg)  
+Figure 3: The ablation study experiments on CIFAR-10.
+
+# 5.3 ABLATION STUDIES
+
+In this part, we investigate the impacts of algorithmic components using AT-RWP on PreAct ResNet18 under $L _ { \infty }$ threat model with $\epsilon = 8 / 2 5 5$ and $\alpha = 2 / 2 5 5$ following the same setting in section 5.1. The training/test attacks are PGD-10/PGD-20 respectively.
+
+The Importance of Minimum LSC Value. We empirically verify the effectiveness of minimum LSC value $c _ { m i n }$ , by comparing the performance of models trained using different weight perturbation schemes: 1) AT: standard adversarial training without weight perturbation (equivalent to $c _ { m i n } = 0 \mathrm { { . } }$ ); 2) AWP: weight perturbation generated via outer maximization in Eq.(1) (equivalent to $c _ { m i n } = \infty$ ); 3) RWP: weight perturbation generated using the proposed robust strategy with different $c _ { m i n }$ values. All other hyper-parameters are kept exactly the same other than the perturbation scheme used. The results are summarized in Table 3(a). It is observed that the test robustness of RWP model first increases and then decreases as the minimum LSC value increases, and the best test robustness is obtained at $c _ { m i n } = 1 . 7$ . It is evident that RWP with a wide range of $c _ { m i n }$ outperforms both AT and AWP model, demonstrating its effectiveness. Furthermore, as it is the major component that is different from the AWP pipeline, this result suggests that LSC criterion constraints is the main contributor to the improved adversarial robustness.
+
+The Impact of Step Number. We further investigate the effect of step number $K _ { 2 }$ , by comparing the performances of model trained using different perturbation steps. The step number $K _ { 2 }$ for RWP varies from 1 to 10. The results are shown in Figure 3(b). As expected, when $K _ { 2 }$ is small, increasing $K _ { 2 }$ leads higher test robustness. When $K _ { 2 }$ increases from 7 to 10, the performance is flat, which suggests that the generating weight perturbation is sufficient to comprehensively avoid robust overfitting. Note that extra iterations will not bring computational overhead when classification loss of adversarial examples in the batch exceeds minimum LSC value $c _ { m i n }$ , as shown in Algorithm 1. Therefore, we uniformly use $K _ { 2 } = 1 0$ in our implementation.
+
+Effect on Adversarial Robustness and Robust Overfitting. We then visualize the learning curve of AT, AWP and RWP in Figure 3(c). We observe that the test robustness of RWP model continues to increase as the training progresses. In addition, RWP outperforms AWP with a clear margin in the later stage of training. Such observations exactly reflect the nature of our approach which aims to prevent robust overfitting as well as enhance adversarial robustness.
+
+# 6 CONCLUSION
+
+In this paper, we proposed a criterion, Loss Stationary Condition (LSC) for constrained perturbation, to monitor the training status of different adversarial examples during network optimization. The proposed criterion provides a new understanding of robust overfitting in adversarial training. Based on LSC, we found that elimination of robust overfitting and higher robustness of adversarial training can be achieved by weight perturbation on adversarial examples with small classification loss, rather than adversarial examples with large classification loss. Following this, we proposed a Robust Weight Perturbation (RWP) strategy to monitor and regulate the extent of weight perturbation. Comprehensive experiments show that RWP is generic and can improve the state-of-the-art adversarial robustness across different adversarial training approaches, network architectures, threat models and benchmark datasets.
+
+# REPRODUCIBILITY STATEMENT
+
+For sake of reproducibility of our algorithm, we make the following efforts: (i) In Section 5.1, we clearly state the implementation details, including benchmark datasets, network structure, baselines, training and test parameter setting as well as training and test attack setting. (ii) In Section 5.3, we evaluate the sensitivity of the algorithm to hyperparameters and show the detailed hyperparameter tuning process. (iii) At last, we open-source the source code of RWP algorithm, available at supplementary material.
+
+# REFERENCES
+
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+
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+
+# A APPENDIX
+
+In this part, we verify the generalities of diffusion process in robust overfitting (the model will first memorize some easy-to-learn adversarial examples, and then spread to the entire training dataset) across different threat models, datasets and network architectures. Specifically, we remove the training examples whose loss value is lower than the LSC value during adversarial training. The learning curve and the rate of removal are shown in Figure 4. We can observe that if these easy-to-learn adversarial examples are not included in the training data, robust overfitting will not occur during adversarial training, which verified our conclusion.
+
+![](images/63a545c53b23ec9d26c6ce6938c8f965e1898870c7e3b47b57c0bcfb4fd6b0cf.jpg)  
+Figure 4: The learning curve and removing rate of adversarial training under (a) Cifar10 - $L _ { \infty }$ - PreAct ResNet18; (b) Cifar10 - $L _ { 2 }$ - PreAct ResNet18; (c) Cifar $1 0 0 - L _ { \infty }$ - PreAct ResNet18; (d) Cifar $1 0 0 - L _ { \infty }$ - Wide ResNet.

parse/dev/3JvRnAzw_0/3JvRnAzw_0_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "ROBUST WEIGHT PERTURBATION FORADVERSARIAL TRAINING",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            633,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Overfitting widely exists in adversarial robust training of deep networks. An effective and promising remedy is adversarial weight perturbation, which injects the worst-case weight perturbation during network training by maximizing the classification loss on adversarial examples. Adversarial weight perturbation helps reduce the robust generalization gap; however, it also undermines the robustness enhancement. A criterion that regulates the weight perturbation is therefore crucial for adversarial training. In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained perturbation. With LSC, we find that deep network first overfits the adversarial examples with small loss, and then gradually develops to overfit all adversarial examples in the later stage of training. Following this, we find that it is essential to conduct weight perturbation on adversarial data with small classification loss to eliminate overfitting in adversarial training. Weight perturbation on adversarial data with large classification loss is not necessary and may even lead to poor robustness. Based on these observations, we propose a robust perturbation strategy to constrain the extent of weight perturbation. The perturbation strategy prevents deep networks from overfitting while avoiding the side effect of excessive weight perturbation, significantly improving the robustness of adversarial training. Extensive experiments demonstrate the superiority of the proposed method over the state-of-the-art adversarial training methods. ",
+        "bbox": [
+            233,
+            273,
+            764,
+            550
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            593,
+            334,
+            609
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although deep neural networks (DNNs) have led to impressive breakthroughs in a number of fields such as computer vision (He et al., 2016), speech recognition (Wang et al., 2017), and natural language processing (Devlin et al., 2018), they are extremely vulnerable to adversarial examples that are crafted by adding small and human-imperceptible perturbation to normal examples (Szegedy et al., 2013; Goodfellow et al., 2014). ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The vulnerability of DNNs has attracted extensive attention and led to a large number of defense techniques against adversarial examples. Across existing defenses, adversarial training (AT) is one of the strongest empirical defenses. AT directly incorporates adversarial examples into the training process to solve a min-max optimization problem (Madry et al., 2017), which can obtain models with moderate adversarial robustness and has not been comprehensively attacked (Athalye et al., 2018). However, different from the standard training scenario, overfitting is a dominant phenomenon in adversarial robust training of deep networks (Rice et al., 2020). After a certain point in AT, the robust performance on test data will continue to degrade with further training. This phenomenon, termed as robust overfitting, breaches the common practice in deep learning that using over-parameterized networks and training for as long as possible (Neyshabur et al., 2017; Belkin et al., 2019). Such anomaly in AT causes detrimental effects on the robust generalization performance and subsequent algorithm assessment (Rice et al., 2020; Chen et al., 2020b). Relief techniques that mitigate robust overfitting have thus become crucial for stable adversarial training. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "An effective and promising remedy for robust overfitting is Adversarial Weight Perturbation (AWP) (Wu et al., 2020), which forms a double-perturbation mechanism in the adversarial training framework that adversarially perturbs both inputs and weights: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7291a5ecc48ec28aa86438dd16259ae630db3e03eae87e519da7d54998a58aef.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { w } \\operatorname* { m a x } _ { v \\in \\mathcal { V } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\operatorname* { m a x } _ { | | x _ { i } ^ { \\prime } - x _ { i } | | _ { p } \\leq \\epsilon } \\ell ( f _ { w + v } ( x _ { i } ^ { \\prime } ) , y _ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            126,
+            647,
+            167
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $n$ is the number of training examples, $\\boldsymbol { x } _ { i } ^ { \\prime }$ is the adversarial example of $x _ { i } , f _ { w }$ is the DNN with weight $w , \\ell ( \\cdot )$ is the loss function, $\\epsilon$ is the maximum perturbation constraint for inputs (i.e., $| | x _ { i } ^ { \\prime } - x _ { i } | | _ { p } \\leq \\epsilon )$ , and $\\nu$ is the feasible perturbation region for weights (i.e., $\\{ v \\in \\mathcal { V } : | | v | | _ { 2 } \\leq$ $\\gamma | | w | | _ { 2 } \\bigr \\}$ , where $\\gamma$ is the constraint on weight perturbation size). The inner maximization is to find adversarial examples $x _ { i } ^ { \\prime }$ within the $\\epsilon$ -ball centered at normal examples $x _ { i }$ that maximizes the classification loss $\\ell$ . On the other hand, the outer maximization is to find weight perturbation $\\textbf {  { v } }$ that maximizes the loss $\\ell$ on adversarial examples to flatten the weight loss landscape and reduce robust generalization gap. This is the problem of training a weight-perturbed robust classifier on adversarial examples. Therefore, how well the weight perturbation is found directly affects the performance of the outer minimization, i.e., the robustness of the classifier. ",
+        "bbox": [
+            173,
+            174,
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+            314
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Several attack methods have been used to solve the inner maximization problem in Eq.(1), such as Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014) and Projected Gradient Descent (PGD) (Madry et al., 2017). For the outer maximization problem, AWP (Wu et al., 2020) injects the worst-case weight perturbation to reduce robust generalization gap. However, the extent to which the weights should be perturbed has not been explored. Without an appropriate criterion to regulate the weight perturbation, the adversarial training procedure is difficult to unleash its full power. In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained perturbation, which sheds light on the nitty-gritty of robust overfitting in adversarial training, and this in turn motivates us to propose an improved weight perturbation strategy for better robustness. Our main contributions are follows: ",
+        "bbox": [
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+            320,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose a principled criterion LSC to monitor the training status of different adversarial examples during network optimization. It provides a better understanding of robust overfitting in adversarial training, and it is also a good indicator for efficient weight perturbation. • With LSC, we find that deep network first overfits adversarial data with small classification loss and then gradually develops to overfit all adversarial data. Following this, we find that better perturbation of model weights is associated with perturbing on adversarial data with small classification loss. For adversarial data with large classification loss, weight perturbation is not necessary and can even be harmful. • We propose a robust perturbation strategy to constrain the extent of weight perturbation. Experiments show that the robust strategy significantly improves the robustness of adversarial training. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            646,
+            344,
+            662
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 ADVERSARIAL ATTACKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            679,
+            387,
+            693
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Given a normal example $( x , y )$ , a DNN $f _ { w }$ , and maximum perturbation constraint $\\epsilon$ . Let $\\mathcal { X }$ denote the input feature space and $\\mathcal { B } _ { \\epsilon } ^ { p } ( x ) = \\{ x ^ { \\prime } \\in \\mathcal { X } : | | x ^ { \\prime } - x | | _ { p } \\leq \\epsilon \\}$ be the $\\ell _ { p }$ -norm ball of radius $\\epsilon$ centered at $x$ in $\\mathcal { X }$ . The goal of adversarial attack is to find an adversarial example $x ^ { \\prime } \\in B _ { \\epsilon } ^ { p } ( x )$ that can fool the DNN to produce an incorrect output ( $f _ { w } ( x ^ { \\prime } ) \\ne y$ ). Here we selectively introduce several commonly used adversarial attack methods. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Fast Gradient Sign Method (FGSM). FGSM (Goodfellow et al., 2014) perturbs natural example $x$ for one step with step size $\\epsilon$ along the gradient direction: ",
+        "bbox": [
+            173,
+            781,
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/952e1c47e1823c3227cdfaac7c6095e657112ff88a6291e8c8541528e5f252c1.jpg",
+        "text": "$$\n\\begin{array} { r } { x ^ { \\prime } = x + \\epsilon \\cdot \\mathrm { s i g n } \\bigl ( \\nabla _ { x } \\ell ( f _ { w } ( x ) , y ) \\bigr ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            818,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Projected Gradient Descent (PGD). PGD (Madry et al., 2017) is a stronger iterative variant of FGSM, which perturbs normal example $x$ for multiple steps $K$ with a smaller step size $\\alpha$ : ",
+        "bbox": [
+            171,
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/79264faa6ff45f1aef7f22fa537f2dd6fffda973b3d97ded9f3ec049a5982a2f.jpg",
+        "text": "$$\nx ^ { 0 } \\sim \\mathcal { U } ( B _ { \\epsilon } ^ { p } ( x ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            442,
+            878,
+            553,
+            897
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a73b5ef623b75557caaff3282b72e798941ef060aaa0e6629ef5f7416d1598c.jpg",
+        "text": "$$\n\\boldsymbol { x } ^ { k } = \\Pi _ { \\mathcal { B } _ { \\epsilon } ^ { p } ( x ) } ( \\boldsymbol { x } ^ { k - 1 } + \\alpha \\cdot \\mathrm { s i g n } ( \\nabla _ { \\boldsymbol { x } ^ { k - 1 } } \\ell ( f _ { w } ( \\boldsymbol { x } ^ { k - 1 } ) , \\boldsymbol { y } ) ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            906,
+            681,
+            926
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { U }$ denotes the uniform distribution, $x ^ { 0 }$ denotes the normal example disturbed by a small uniform random noise, $x ^ { k }$ denotes the adversarial example at step $k$ , and $\\Pi _ { B _ { \\epsilon } ^ { p } ( x ) }$ denotes the projection function that projects the adversarial example back into the set $B _ { \\epsilon } ^ { p } ( x )$ if necessary. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "AutoAttack (AA). AA (Croce & Hein, 2020b) is an ensemble of complementary attacks, which consists of three white-box attacks (APGD-CE (Croce & Hein, 2020b), APGD-DLR (Croce & Hein, 2020b), and FAB (Croce & Hein, 2020a)) and a black-box attack (Square Attack (Andriushchenko et al., 2020)). AA regards models to be robust only if the models correctly classify all types of adversarial examples, which is among the most reliable evaluation of adversarial robustness to date. ",
+        "bbox": [
+            174,
+            154,
+            825,
+            224
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are also other types of attacking methods, e.g., the CW attack (Carlini & Wagner, 2017), deformation attack (Engstrom et al., 2017; Xiao et al., 2018; Engstrom et al., 2019), Hamming distance based attack (Shamir et al., 2019), Frank-Wolfe based attack (Chen et al., 2020a) and adaptive attack (Tramer et al., 2020). ",
+        "bbox": [
+            174,
+            231,
+            825,
+            286
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 ADVERSARIAL DEFENSE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            304,
+            387,
+            319
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Since the discovery of adversarial examples, a large number of works have emerged for defending against adversarial attacks, such as input denoising (Guo et al., 2018; Liao et al., 2018; Wu et al., 2021), defensive distillation (Papernot et al., 2016; Carlini & Wagner, 2017), adversarial detection (Metzen et al., 2017; Tao et al., 2018), gradient regularization (Tramer et al., 2018; Ross & Doshi- \\` Velez, 2018) and adversarial training (Goodfellow et al., 2014; Madry et al., 2017). Among them, adversarial training has been demonstrated to be the most effective method (Athalye et al., 2018). Based on adversarial training, a wide range of subsequent works are then proposed to further improve the model robustness (Xie et al., 2019; Mosbach et al., 2018; Kannan et al., 2018; Zhang et al., 2019; Cai et al., 2018; Wang et al., 2019a; Zhang et al., 2020a; Dong et al., 2018; Yang et al., 2019; Wang et al., 2019b; Song et al., 2020; Carmon et al., 2019; Zhai et al., 2019; Uesato et al., 2019; Hendrycks et al., 2019; Yan et al., 2021; Du et al., 2021). Here, we introduce two currently state-of-the-art adversarial training frameworks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "TRADES. TRADES (Zhang et al., 2019) optimizes a regularized surrogate loss that is a trade-off between the natural accuracy and adversarial robustness: ",
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+    {
+        "type": "equation",
+        "img_path": "images/5038702dda60b9f60e6298c190f52bf80e5d5a1bf4f001e081811522069bab56.jpg",
+        "text": "$$\n\\ell ^ { { \\mathrm { T R A D E S } } } ( \\boldsymbol { w } ; \\boldsymbol { x } , \\boldsymbol { y } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\left\\{ \\operatorname { C E } ( f _ { w } ( x _ { i } ) , y _ { i } ) + \\beta \\cdot \\operatorname* { m a x } _ { \\boldsymbol { x } ^ { \\prime } \\in B _ { \\epsilon } ^ { p } ( \\boldsymbol { x } ) } \\operatorname { K L } ( f _ { w } ( x _ { i } ) | | f _ { w } ( x _ { i } ^ { \\prime } ) ) \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where CE is the cross-entropy loss that encourages the network to maximize the natural accuracy, KL is the Kullback-Leibler divergence that encourages to improve the robust accuracy, and $\\beta$ is the hyperparameter to control the trade-off between natural accuracy and adversarial robustness. ",
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+    {
+        "type": "text",
+        "text": "Robust Self-Training (RST). RST (Carmon et al., 2019) utilize additional 500K unlabeled data extracted from the 80 Million Tiny Images dataset (Torralba et al., 2008). RST first leverages the surrogate natural model to generate pseudo-labels for these unlabeled data, and then adversarially trains the network with both additional pseudo-labeled unlabeled data $( \\tilde { x } , \\tilde { y } )$ and original labeled data $( x , y )$ in a supervised setting: ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/51daceb8a8209ae90ce8723bce03d88fd94b9ff19aa9e7b78ad84ff7d7640980.jpg",
+        "text": "$$\n\\ell ^ { \\mathrm { R S T } } ( \\boldsymbol { w } ; x , y , \\tilde { x } , \\tilde { y } ) = \\ell ^ { \\mathrm { T R A D E S } } ( \\boldsymbol { w } ; x , y ) + \\lambda \\cdot \\ell ^ { \\mathrm { T R A D E S } } ( \\boldsymbol { w } ; \\tilde { x } , \\tilde { y } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\lambda$ is the weight on unlabeled data. ",
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+    {
+        "type": "text",
+        "text": "2.3 ROBUST OVERFITTING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Nowadays, there are effective countermeasures to alleviate the overfitting in standard training. But in adversarial training, robust overfitting widely exists and those common countermeasures used in standard training help little (Rice et al., 2020). Schmidt et al. (2018) explains robust overfitting partially from the perspective of sample complexity, and is supported by empirical results in derivative works, such as adversarial training with semi-supervised learning (Carmon et al., 2019; Uesato et al., 2019; Zhai et al., 2019), robust local feature (Song et al., 2020) and data interpolation (Zhang & Xu, 2019; Lee et al., 2020; Chen et al., 2021). Separate works have also attempt to mitigate robust overfitting by the unequal treatment of data (Zhang et al., 2020b) and weight smoothing (Chen et al., 2020b). Recent study (Wu et al., 2020) reveals the connection between the flatness of weight loss landscape and robust generalization gap, and proposes to incorporate adversarial weight perturbation mechanism in the adversarial training framework. Despite the efficacy of adversarial weight perturbation in suppressing the robust overfitting in adversarial training, a deeper understanding of the cause of robust overfitting and a clear direction for valid weight perturbation is largely missing. The outer maximization in Eq.(1) lacks an effective criterion to regulate and constrain the extent of weight perturbation, which in turn influences the optimization of the outer minimization problem. In this paper, we propose such a criterion and provide new understanding of the robust overfitting in adversarial training. Following this, we design a robust weight perturbation strategy that significantly improves the robustness of adversarial training. ",
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+        "image_caption": [
+            "Figure 1: (a): Test robustness of AWP with varying weight perturbation size; (b): The learning curve of vanilla AT; (c): Test robustness of AWP with varying LSC range. "
+        ],
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+        "type": "text",
+        "text": "3 LOSS STATIONARY CONDITION ",
+        "text_level": 1,
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+        "text": "In this section, we first empirically investigate the relationship between weight perturbation robustness and adversarial robustness, and then propose a new criterion to monitor the training status of different adversarial examples in the learning process of adversarial training, which leads to a new perspective of robust overfitting. To this end, some discussions about robust overfitting and adversarial weight perturbation are provided. ",
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+        "type": "text",
+        "text": "Does Weight Perturbation Robustness Lead to Better Adversarial Robustness? First, we investigate whether the robustness against weight perturbation is beneficial to the adversarial robustness. In particular, we train PreAct ResNet-18 with AWP on CIFAR-10 using varying weight perturbation size from $0 \\gamma , \\gamma / 8 , \\gamma / 4 , \\gamma / 2 , \\gamma , 2 \\gamma , 4 \\gamma$ to $8 \\gamma$ . In each setting, we evaluate the robustness of the model against 20-step PGD (PGD-20) attacks on CIFAR-10 test images. As shown in Figure 1(a), when varying weight perturbation size, the best adversarial robustness has a certain improvement in the early stage. When weight perturbation size is large, the best adversarial robustness begins to decrease significantly as the size of the perturbation increases. It might be explained by the fact that the network has to sacrifice adversarial robustness to allocate more capacity to defend against weight perturbation, which implies that weight perturbation robustness and adversarial robustness are not actually mutually beneficial. The performance gain of AWP is mainly due to suppressing robust overfitting. ",
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+    {
+        "type": "text",
+        "text": "Loss Stationary Condition. In order to further understand the robust overfitting, we propose a criterion that divides the training adversarial examples into different groups according to their classification loss: ",
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+        "type": "equation",
+        "img_path": "images/bb5610cf813f89d661a182e85eec8c7f431ffac34cd888580cdada2d5705a4f1.jpg",
+        "text": "$$\n\\mathrm { L S C } [ p , q ] = \\{ x ^ { \\prime } \\in \\mathcal { X } \\mid p \\leq \\ell ( f _ { w } ( x ^ { \\prime } ) , y ) \\leq q \\} ,\n$$",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/029122678a72cbfbbe80142edc40947c335d750d1a999a146521906d2088d916.jpg",
+        "image_caption": [
+            "Figure 2: The weight loss landscape of different LSC groups on different checkpoints. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $p \\leq q$ . The adversarial examples in the group all satisfy their classification loss within a certain range, which is termed Loss Stationary Condition (LSC). The proposed criterion LSC allows the analysis of training status of different adversarial examples independently, and provides more insights into the robust overfitting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "LSC View of Robust Overfitting. To provide details of the robust overfitting in adversarial training, we train a PreAct ResNet-18 for 200 epochs on CIFAR-10 using PGD-10 with step size $\\epsilon / 4$ , maximum perturbation $\\epsilon = 8 / 2 5 5$ , following the standard setting in Madry et al. (2017). The learning curve is shown in Figure 1(b). For each intermediate model, we then apply the same PGD-10 attack on CIFAR-10 training images to craft adversarial examples, and divide the crafted adversarial examples into 6 consecutive LSC groups ranging from 0.0 to 3.0. Then, we use the weight loss landscape to characterize the training status of the adversarial examples in each LSC group, which plots the classification loss change when perturbing the model weight $\\pmb { w }$ by a random noise $^ d$ with magnitude $\\mu$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/75abf559daaca9833f7b426bdcb43c93e5379ab29bce5e90e5fdaed3dd53c7a8.jpg",
+        "text": "$$\ng _ { j } ( \\mu ) = \\frac { 1 } { n _ { j } } \\sum _ { i = 1 } ^ { n _ { j } } \\ell ( f _ { w + \\mu d } ( x _ { i j } ^ { \\prime } ) , y _ { i j } ) , \\ x _ { i j } ^ { \\prime } \\in \\mathrm { L S C } [ p _ { j } , q _ { j } ] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $j$ is the number of groups, $n _ { j }$ is the number of adversarial examples in $j$ -th LSC group, and $^ d$ is filter normalized by $\\begin{array} { r } { d \\gets \\frac { d } { | | d | | } | | \\boldsymbol { w } | | } \\end{array}$ following Li et al. (2017). It is worth noting that weight loss landscape has been widely used to characterize the generalization gap (Neyshabur et al., 2017; Foret et al., 2020; Wu et al., 2020). Here, we use it to characterize the training status of different adversarial examples. For training adversarial examples, the higher the degree of overfitting by the model, the more sensitive its loss is to model weight perturbations, thus making the weight loss landscape sharper. Here the weight loss curve sharpness is served as a comparable measurement of overfitting strength. Besides, another key difference to previous works lies on the LSC criterion used for visualization, which provides more insights into the robust overfitting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We show the weight loss curve of each LSC group on different checkpoints in Figure 2. In the early stage of training (between 100 and 120 epoch), it can be seen that the weight loss curve of the LSC group with small loss is obviously sharper than that of the LSC group with large loss, which indicates that the adversarial examples with small classification loss were first overfitted. As the training progresses, the weight loss curves of all LSC groups become very sharp, which shows that the network overfits all adversarial examples. These observations suggest that robust overfitting exists a diffusion process: the model will first memorize some easy-to-learn adversarial examples, and then spread to the entire training dataset. ",
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+    {
+        "type": "table",
+        "img_path": "images/169fb02ee0a43a98b3db59ea741d4fd545c4822786f0625739702233feece071.jpg",
+        "table_caption": [
+            "Algorithm 1 Robust Weight Perturbation "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>igorlhhTKobustWergntFerturbaton Input: Network fw, training data S,mini-batch B,batch size n,learning rate n,PGD step size α,</td></tr><tr><td>PGD steps K1,PGD constraint ε, RWP steps K2, RWP constraint γ, minimum LSC value Cmin· Output: Adversarially robust model fw · repeat</td></tr><tr><td>Read mini-batch xp from training set S. xB ← xB +δ,where δ ~ Uniform(-∈,∈) for k=1 to K1 do</td></tr><tr><td>x&#x27; ←II(xB+α·sign(Vxsl(fw(xB),y)))</td></tr><tr><td>end for</td></tr><tr><td>Initialize v = 0</td></tr><tr><td>for k=1 to K2 do</td></tr><tr><td>V = 1B(l(fw+u(xs),y)≤ Cmin)</td></tr><tr><td>if∑V=0 then</td></tr><tr><td>break</td></tr><tr><td></td></tr><tr><td>else</td></tr><tr><td>v ← v+ Vu(V ·l(fw+v(xs),y))</td></tr><tr><td>U←Yw</td></tr><tr><td>end if</td></tr><tr><td>w ←(w+v)-nVw+vn∑i=1l(fw+u(x&#x27;g),y(i))-v</td></tr><tr><td>end for</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "LSC view of Adversarial Weight Perturbation. To provide more insight into how AWP suppresses robust overfitting, we train PreAct ResNet-18 on CIFAR-10 by varying the LSC group that performs adversarial weight perturbation. In each setting, we evaluate the robustness of the model against PGD-20 attacks on CIFAR-10 test images. As shown in Figure 1(c), when varying the LSC range, we can observe that conducting adversarial weight perturbation on adversarial examples with small classification loss is sufficient to suppress robust overfitting. Recalling the diffusion process in robust overfitting, we can infer that to eliminate robust overfitting, it is essential to prevent the model from memorizing the easy-to-learn adversarial examples. Besides, it is observed that conducting adversarial weight perturbation on adversarial examples with large classification loss leads to worse adversarial robustness, which again verifies that the robustness against weight perturbation will not bring adversarial robustness gain, or even on the contrary, it undermines the adversarial robustness enhancement. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Do We Really Need the Worst-case Weight Perturbation? As aforementioned, the robustness against weight perturbation is detrimental to the adversarial robustness enhancement. Therefore, to purely prevent the network from memorizing the adversarial examples with small classification loss, conducting worst-case weight perturbation on these adversarial examples is not necessary, since it will also deteriorate the adversarial robustness. In the next section, we will propose a robust perturbation strategy to address this issue. ",
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+    {
+        "type": "text",
+        "text": "4 ROBUST WEIGHT PERTURBATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we introduce the proposed robust weight perturbation strategy and its algorithmic realization. ",
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+    {
+        "type": "text",
+        "text": "As mentioned in Section 3, conducting adversarial weight perturbation on adversarial examples with small classification loss is enough to prevent robust overfitting and leads to higher robustness. However, conducting adversarial weight perturbation on adversarial examples with large classification loss may not be helpful. Recalling the criterion LSC proposed in Section 3, we have seen that it is closely correlated with the tendency of adversarial example to be overfitted. Thus, it can be used to regulate the extent of weight perturbation at a fine-grained level. Therefore, we propose to train network with adversarial examples that are all above a minimum LSC value, so as to ensure that no robust overfitting occurs while avoiding the side effect of excessive weight perturbation. Let $c _ { m i n }$ be the minimum LSC value. Instead of generating weight perturbation $\\textbf {  { v } }$ via outer maximization in Eq.(1), we generate $\\pmb { v }$ as follows: ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "equation",
+        "img_path": "images/7d0912a7f1e94c34b52730fda58bc81a07a4b7a8a81559896aed4b3e0f410fc9.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\pmb { v } ^ { k + 1 } = \\pmb { v } ^ { k } + \\nabla _ { \\pmb { v } ^ { k } } \\cfrac { 1 } { n } \\displaystyle \\sum _ { i = 1 } ^ { n } \\mathbb { 1 } ( x _ { i } ^ { \\prime } , y _ { i } ) \\ell ( f _ { \\pmb { w + v } ^ { k } } ( x _ { i } ^ { \\prime } ) , y _ { i } ) , } & { } \\\\ { \\mathrm { w h e r e } \\quad \\mathbb { 1 } ( x _ { i } ^ { \\prime } , y _ { i } ) = \\left\\{ \\begin{array} { l l } { 0 } & { \\mathrm { i f ~ } \\ell ( f _ { \\pmb { w + v } ^ { k } } ( x _ { i } ^ { \\prime } ) , y _ { i } ) > c _ { m i n } } \\\\ { 1 } & { \\mathrm { i f ~ } \\ell ( f _ { \\pmb { w + v } ^ { k } } ( x _ { i } ^ { \\prime } ) , y _ { i } ) \\leq c _ { m i n } } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The proposed Robust Weight Perturbation (RWP) algorithm is shown in Algorithm 1. We use PGD attack (Madry et al., 2017) to generate the training adversarial examples, which can be also extended to other variants such as TRADES (Zhang et al., 2019) and RST (Carmon et al., 2019). The mimimum LSC value $c _ { m i n }$ controls the minimum classification loss (minimum weight perturbation strength) of the adversarial examples during network training. In the early stages of training, the classification loss of adversarial example is generally larger than $c _ { m i n }$ corresponding to no weight perturbation process. The classification loss of adversarial examples then decreases as training progresses. At each optimization step, we monitor the classification loss of the adversarial example and conduct the weight perturbation process for adversarial examples whose classification loss is already smaller than $c _ { m i n }$ , enabled by an indicator control vector $V$ . At each perturbation step, the weight perturbation $\\pmb { v }$ will be updated to increase the classification loss of the corresponding adversarial example. When the classification loss of training adversarial examples is all higher than $c _ { m i n }$ or the number of perturbation step reaches the defined value, we stop the weight perturbation process and inject the generated weight perturbation $\\textbf {  { v } }$ for adversarial training. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we conduct comprehensive experiments to evaluate the effectiveness of RWP including its experimental settings, robustness evaluation and ablation studies. ",
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+    {
+        "type": "text",
+        "text": "5.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Baselines and Implementation Details. Our implementation is based on PyTorch and the code as well as other related resources will be released for public use and verification. We conduct extensive experiments across three benchmark datasets (CIFAR-10 (Krizhevsky et al., 2009), CIFAR100 (Krizhevsky et al., 2009) and SVHN (Netzer et al., 2011)) and two threat models $L _ { \\infty }$ and $L _ { 2 }$ ). We use PreAct ResNet-18 (He et al., 2016) and Wide ResNet (WRN-28-10 and WRN-34-10) (Zagoruyko & Komodakis, 2016) as the network structure following Wu et al. (2020). We compare the performance of the proposed method on a number of baseline methods: 1) standard adversarial training without weight perturbation, including vanilla AT (Madry et al., 2017), TRADES (Zhang et al., 2019) and RST (Carmon et al., 2019); 2) adversarial training with adversarial weight perturbation (AWP) (Wu et al., 2020). For training, the network is trained for 200 epochs using SGD with momentum 0.9, weight decay $5 \\times 1 0 ^ { - 4 }$ , and an initial learning rate of 0.1. The learning rate is divided by 10 at the 100-th and 150-th epoch. Standard data augmentation including random crops with 4 pixels of padding and random horizontal flips are applied. For testing, model robustness is evaluated by measuring the accuracy of the model under different adversarial attacks. For hyperparameters in RWP, we set perturbation step $K _ { 2 } = 1 0$ for all datasets. The minimum LSC value $c _ { m i n } = 1 . 7$ for CIFAR-10, $c _ { m i n } = 2 . 2$ for SVHN and $c _ { m i n } = 4 . 0$ for CIFAR-100. The weight perturbation budget of $\\gamma = 0 . 0 1$ for AT-RWP, $\\gamma = 0 . 0 0 5$ for TRADES-RWP and RST-RWP following literature (Wu et al., 2020). Other hyper-parameters of the baselines are configured as per their original papers. ",
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+    {
+        "type": "text",
+        "text": "Adversarial Setting. The training attack is 10-step PGD attack with random start. We follow the same settings in Rice et al. (2020) : for $L _ { \\infty }$ threat model, $\\epsilon = 8 / 2 5 5$ , step size $\\alpha = 1 / 2 5 5$ for SVHN, and $\\alpha = 2 / 2 5 5$ for both CIFAR10 and CIFAR100; for $L _ { 2 }$ threat model, $\\epsilon = 1 2 8 / 2 5 5$ , step size $\\alpha = 1 5 / 2 5 5$ for all datasets. The test attacks used for robustness evaluation are generated from the original test set images by attacking the defense models using different attacking methods, including: FGSM, PGD-20, PGD-100, $\\mathrm { C } \\& \\mathbf { W } _ { \\infty }$ $L _ { \\infty }$ version of C&W optimized by PGD for 100 steps) and Auto Attack (AA). ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "5.2 ROBUSTNESS EVALUATION ",
+        "text_level": 1,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Performance Evaluations. To validate the effectiveness of the proposed RWP, we conduct performance evaluation on vanilla AT, AT-AWP and AT-RWP across different benchmark datasets and threat models using PreAct ResNet-18. We report the accuracy on the test images under PGD-20 attack. The evaluation results are summarized in Table 1. “Best” denotes the highest robustness that ever achieved at different checkpoints and ”last” denotes the robustness at the last epoch checkpoint. It is observed vanilla AT suffers from severe robust overfitting (the performance gap between ”best” and ”last” is very large). AT-AWP and AT-RWP method narrow the performance gap significantly over the vanilla AT model due to suppression of robust overfitting. Moreover, on CIFAR-10 dataset under the $L _ { \\infty }$ attack, vanilla AT achieves $5 2 . 7 9 \\%$ ”best” test robustness. The AT-AWP approach boosts the performance to $5 5 . 3 9 \\%$ . The proposed approach further outperforms both methods by a large margin, improving over vanilla AT by $5 . 7 6 \\%$ , and is $3 . 1 6 \\%$ better than AT-AWP, achieving $5 8 . 5 5 \\%$ accuracy under the standard 20 steps PGD attack. Similar patten has been observed on other datasets and threat model. AT-RWP consistently improves the test robustness across a wide range of datasets and threat models, demonstrating the effectiveness of the proposed approach. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/45802b10bc18e868a421b53cff82ff389c7a141952169fe23eae3703c4e7ad74.jpg",
+        "table_caption": [
+            "Table 1: Test robustness $( \\% )$ of AT, AT-AWP and AT-RWP using PreAct ResNet-18. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Threat Model</td><td rowspan=\"2\">Method</td><td colspan=\"2\">SVHN</td><td colspan=\"2\">CIFAR-10</td><td colspan=\"2\">CIFAR-100</td></tr><tr><td>Best</td><td>Last</td><td>Best</td><td>Last</td><td>Best</td><td>Last</td></tr><tr><td rowspan=\"3\">L8</td><td>AT</td><td>53.36</td><td>44.49</td><td>52.79</td><td>44.44</td><td>27.22</td><td>20.82</td></tr><tr><td>AT-AWP</td><td>59.12</td><td>55.87</td><td>55.39</td><td>54.73</td><td>30.71</td><td>30.28</td></tr><tr><td>AT-RWP</td><td>61.15</td><td>57.45</td><td>58.55</td><td>58.01</td><td>31.17</td><td>30.64</td></tr><tr><td rowspan=\"3\">L2</td><td>AT</td><td>66.87</td><td>65.03</td><td>69.15</td><td>65.93</td><td>41.33</td><td>35.27</td></tr><tr><td>AT-AWP</td><td>72.57</td><td>67.73</td><td>72.69</td><td>72.08</td><td>45.60</td><td>44.66</td></tr><tr><td>AT-RWP</td><td>73.35</td><td>69.48</td><td>74.47</td><td>73.84</td><td>45.71</td><td>45.05</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Benchmarking the state-of-the-art Robustness. To manifest the full power of our proposed perturbation strategy and also benchmark the state-of-the-art robustness on CIFAR-10 under $\\mathrm { L } _ { \\infty }$ threat model, we conduct experiments on the large capacity network with different baseline methods. We train Wide ResNet-34-10 for AT and TRADES, and Wide ResNet-28-10 for RST following their original papers. We evaluate the adversarial robustness of trained model with various test attack and report the “best” test robustness, with the results shown in Table 2. “Natural” denotes the accuracy on natural test data. First, it is observed that the natural accuracy of RWP model consistently outperforms AWP by a large margin. It is due to the benefits that our RWP avoids the excessive weight perturbation. Moreover, RWP achieves the best adversarial robustness against all types of attack across a wide range of baseline methods, which verifies that RWP is effective in general and improves adversarial robustness reliably rather than improper tuning of hyper-parameters of attacks, gradient obfuscation or masking. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c358e97c8b763776c5cf4fa2be767217492a15a71f8f43feb9865a88292bf574.jpg",
+        "table_caption": [
+            "Table 2: Test robustness $( \\% )$ on CIFAR-10 using Wide ResNet under $L _ { \\infty }$ threat model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Defense</td><td>Natural</td><td>FGSM</td><td>PGD-20</td><td>PGD-100</td><td>C&amp;W</td><td>AA</td></tr><tr><td>AT</td><td>86.07</td><td>61.76</td><td>56.10</td><td>55.79</td><td>54.19</td><td>52.60</td></tr><tr><td>AT-AWP</td><td>85.57</td><td>62.90</td><td>58.14</td><td>57.94</td><td>55.96</td><td>54.04</td></tr><tr><td>AT-RWP</td><td>86.86</td><td>66.22</td><td>62.87</td><td>62.87</td><td>56.62</td><td>54.61</td></tr><tr><td>TRADES</td><td>84.65</td><td>61.32</td><td>56.33</td><td>56.07</td><td>54.20</td><td>53.08</td></tr><tr><td>TRADES-AWP</td><td>85.36</td><td>63.49</td><td>59.27</td><td>59.12</td><td>57.07</td><td>56.17</td></tr><tr><td>TRADES-RWP</td><td>86.14</td><td>64.70</td><td>60.45</td><td>60.30</td><td>58.07</td><td>57.20</td></tr><tr><td>RST</td><td>89.69</td><td>69.60</td><td>62.60</td><td>62.22</td><td>60.47</td><td>59.53</td></tr><tr><td>RST-AWP</td><td>88.25</td><td>67.94</td><td>63.73</td><td>63.58</td><td>61.62</td><td>60.05</td></tr><tr><td>RST-RWP</td><td>88.87</td><td>69.71</td><td>64.11</td><td>63.92</td><td>62.03</td><td>60.36</td></tr></table>",
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+    {
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+        "img_path": "images/05cc9a011b1b361f1fc69f9f40b1f1f4a73327c2b02695b658d3ce690a694601.jpg",
+        "image_caption": [
+            "Figure 3: The ablation study experiments on CIFAR-10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.3 ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this part, we investigate the impacts of algorithmic components using AT-RWP on PreAct ResNet18 under $L _ { \\infty }$ threat model with $\\epsilon = 8 / 2 5 5$ and $\\alpha = 2 / 2 5 5$ following the same setting in section 5.1. The training/test attacks are PGD-10/PGD-20 respectively. ",
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+    },
+    {
+        "type": "text",
+        "text": "The Importance of Minimum LSC Value. We empirically verify the effectiveness of minimum LSC value $c _ { m i n }$ , by comparing the performance of models trained using different weight perturbation schemes: 1) AT: standard adversarial training without weight perturbation (equivalent to $c _ { m i n } = 0 \\mathrm { { . } }$ ); 2) AWP: weight perturbation generated via outer maximization in Eq.(1) (equivalent to $c _ { m i n } = \\infty$ ); 3) RWP: weight perturbation generated using the proposed robust strategy with different $c _ { m i n }$ values. All other hyper-parameters are kept exactly the same other than the perturbation scheme used. The results are summarized in Table 3(a). It is observed that the test robustness of RWP model first increases and then decreases as the minimum LSC value increases, and the best test robustness is obtained at $c _ { m i n } = 1 . 7$ . It is evident that RWP with a wide range of $c _ { m i n }$ outperforms both AT and AWP model, demonstrating its effectiveness. Furthermore, as it is the major component that is different from the AWP pipeline, this result suggests that LSC criterion constraints is the main contributor to the improved adversarial robustness. ",
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+    {
+        "type": "text",
+        "text": "The Impact of Step Number. We further investigate the effect of step number $K _ { 2 }$ , by comparing the performances of model trained using different perturbation steps. The step number $K _ { 2 }$ for RWP varies from 1 to 10. The results are shown in Figure 3(b). As expected, when $K _ { 2 }$ is small, increasing $K _ { 2 }$ leads higher test robustness. When $K _ { 2 }$ increases from 7 to 10, the performance is flat, which suggests that the generating weight perturbation is sufficient to comprehensively avoid robust overfitting. Note that extra iterations will not bring computational overhead when classification loss of adversarial examples in the batch exceeds minimum LSC value $c _ { m i n }$ , as shown in Algorithm 1. Therefore, we uniformly use $K _ { 2 } = 1 0$ in our implementation. ",
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+    {
+        "type": "text",
+        "text": "Effect on Adversarial Robustness and Robust Overfitting. We then visualize the learning curve of AT, AWP and RWP in Figure 3(c). We observe that the test robustness of RWP model continues to increase as the training progresses. In addition, RWP outperforms AWP with a clear margin in the later stage of training. Such observations exactly reflect the nature of our approach which aims to prevent robust overfitting as well as enhance adversarial robustness. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we proposed a criterion, Loss Stationary Condition (LSC) for constrained perturbation, to monitor the training status of different adversarial examples during network optimization. The proposed criterion provides a new understanding of robust overfitting in adversarial training. Based on LSC, we found that elimination of robust overfitting and higher robustness of adversarial training can be achieved by weight perturbation on adversarial examples with small classification loss, rather than adversarial examples with large classification loss. Following this, we proposed a Robust Weight Perturbation (RWP) strategy to monitor and regulate the extent of weight perturbation. Comprehensive experiments show that RWP is generic and can improve the state-of-the-art adversarial robustness across different adversarial training approaches, network architectures, threat models and benchmark datasets. ",
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+    {
+        "type": "text",
+        "text": "REPRODUCIBILITY STATEMENT ",
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+        "type": "text",
+        "text": "For sake of reproducibility of our algorithm, we make the following efforts: (i) In Section 5.1, we clearly state the implementation details, including benchmark datasets, network structure, baselines, training and test parameter setting as well as training and test attack setting. (ii) In Section 5.3, we evaluate the sensitivity of the algorithm to hyperparameters and show the detailed hyperparameter tuning process. (iii) At last, we open-source the source code of RWP algorithm, available at supplementary material. ",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            440,
+            297,
+            455
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In this part, we verify the generalities of diffusion process in robust overfitting (the model will first memorize some easy-to-learn adversarial examples, and then spread to the entire training dataset) across different threat models, datasets and network architectures. Specifically, we remove the training examples whose loss value is lower than the LSC value during adversarial training. The learning curve and the rate of removal are shown in Figure 4. We can observe that if these easy-to-learn adversarial examples are not included in the training data, robust overfitting will not occur during adversarial training, which verified our conclusion. ",
+        "bbox": [
+            173,
+            472,
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/63a545c53b23ec9d26c6ce6938c8f965e1898870c7e3b47b57c0bcfb4fd6b0cf.jpg",
+        "image_caption": [
+            "Figure 4: The learning curve and removing rate of adversarial training under (a) Cifar10 - $L _ { \\infty }$ - PreAct ResNet18; (b) Cifar10 - $L _ { 2 }$ - PreAct ResNet18; (c) Cifar $1 0 0 - L _ { \\infty }$ - PreAct ResNet18; (d) Cifar $1 0 0 - L _ { \\infty }$ - Wide ResNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            588,
+            815,
+            867
+        ],
+        "page_idx": 12
+    }
+]

parse/dev/3Pbra-_u76D/3Pbra-_u76D_content_list.json

@@ -0,0 +1,1857 @@
+[
+    {
+        "type": "text",
+        "text": "RETHINKING NETWORK DESIGN AND LOCAL GEOMETRY IN POINT CLOUD: A SIMPLE RESIDUAL MLP FRAMEWORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Xu $\\mathbf { M a } ^ { 1 }$ , Can ${ \\bf { Q } i n } ^ { 1 }$ , Haoxuan $\\mathbf { Y o u } ^ { 2 }$ , Haoxi Ran1, Yun Fu1 1Northeastern University, Boston, MA, USA 2Columbia University, New York, NY, USA {ma.xu1,qin.ca,ran.h}@northeastern.edu {haoxuanyou,ranhaoxi}@gmail.com yunfu@ece.neu.edu ",
+        "bbox": [
+            184,
+            194,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            318,
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+            333
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Point cloud analysis is challenging due to irregularity and unordered data structure. To capture the 3D geometries, prior works mainly rely on exploring sophisticated local geometric extractors using convolution, graph, or attention mechanisms. These methods, however, incur unfavorable latency during inference, and the performance saturates over the past few years. In this paper, we present a novel perspective on this task. We notice that detailed local geometrical information probably is not the key to point cloud analysis – we introduce a pure residual MLP network, called PointMLP, which integrates no “sophisticated” local geometrical extractors but still performs very competitively. Equipped with a proposed lightweight geometric affine module, PointMLP delivers the new stateof-the-art on multiple datasets. On the real-world ScanObjectNN dataset, our method even surpasses the prior best method by $3 . 3 \\%$ accuracy. We emphasize that PointMLP achieves this strong performance without any sophisticated operations, hence leading to a superior inference speed. Compared to most recent CurveNet, PointMLP trains $2 \\times$ faster, tests $7 \\times$ faster, and is more accurate on ModelNet40 benchmark. We hope our PointMLP may help the community towards a better understanding of point cloud analysis. The code is available at https://github.com/ma-xu/pointMLP-pytorch. ",
+        "bbox": [
+            233,
+            347,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            621,
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+            637
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Lately, point cloud analysis has emerged as a popular topic in 3D understanding, attracting attention from academia and industry (Qi et al., 2017a; Shi et al., 2019; Xu et al., 2020). Different from 2D images represented by regular dense pixels, point clouds are composed of unordered and irregular sets of points $\\mathcal { P } \\in \\bar { \\mathbb { R } ^ { N \\times \\tilde { 3 } } }$ , making it infeasible to apply image processing methods to point cloud analysis directly. Meanwhile, the nature of sparseness and the presence of noises further restrict the performance. In the past few years, endowing with neural networks, point cloud analysis has seen a great improvement in various applications, including 3D shape classification (Qi et al., 2017a), semantic segmentation (Hu et al., 2020) and object detection (Shi & Rajkumar, 2020), etc. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent efforts have shown promising results for point cloud analysis by exploring local geometric information, using convolution (Li et al., 2021a), graph (Li et al., 2021a), or attention mechanism (Guo et al., 2021) (see Section 2 for details). These methods, despite their gratifying results, have mainly relied on the premise that an elaborate local extractor is essential for point cloud analysis, leading to the competition for careful designs that explore fine local geometric properties. Nevertheless, sophisticated extractors are not without drawbacks. On the one hand, due to prohibitive computations and the overhead of memory access, these sophisticated extractors hamper the efficiency of applications in natural scenes. As an example, until now, most 3D point cloud applications are still based on the simple PointNet ( and PointNet++) or the voxel-based methods (Liu et al., 2021; Li et al., 2021b; Zhang et al., 2021). However, applications that employ the aforementioned advanced methods are rare in literature. On the other hand, the booming sophisticated extractors saturate the performance since they already describe the local geometric properties well. A more complicated design is no longer to improve the performance further. These phenomena suggest that we may need to stop the race of local feature extraction designing, rethinking the necessity of elaborate local feature extractors and further revisiting the succinct design philosophy in point cloud analysis. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we aim at the ambitious goal of building a deep network for point cloud analysis using only residual feed-forward MLPs, without any delicate local feature explorations. By doing so, we eschew the prohibitive computations and continued memory access caused by the sophisticated local geometric extractors and enjoy the advantage of efficiency from the highly-optimized MLPs. To further improve the performance and generalization ability, We introduce a lightweight local geometric affine module that adaptively transforms the point feature in a local region. We term our new network architecture as PointMLP. In the sense of MLPbased design philosophy, our PointMLP is similar to PointNet and PointNet+ $^ +$ (Qi et al., 2017a;b). However, our model is more generic and exhibits promising performance. Different from the models with sophisticated local geometric extractors (e.g., DeepGCNs (Li et al., 2019), RSCNN (Liu et al., ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/635ed58352f670fed8bd190e81fdebc6d88479acd157fa0f7e28451bbf4cf84e.jpg",
+        "image_caption": [
+            "Figure 1: Accuracy-speed tradeoff on ModelNet40. Our PointMLP performs best. Please refer to Section 4 for details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            521,
+            183,
+            818,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2019b), RPNet (Ran et al., 2021).), our PointMLP is conceptually simpler and achieves results on par or even better than these state-of-the-art methods (see Figure 1). Keep in mind that we did not challenge the advantages of these local geometric extractors and we acknowledge their contributions; however, a more succinct framework should be studied considering both the efficiency and accuracy. In Table 1, we systemically compare our PointMLP with some representative methods. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Even though the design philosophy is simple, PointMLP (as well as the elite version) exhibits superior performance on 3D point cloud analysis. Specifically, we achieve the state-of-the-art classification performance, $9 4 . 5 \\%$ , on the ModelNet40 benchmark, and we outperform related works by $3 . 3 \\%$ accuracy on the real-world ScanObjectNN dataset, with a significantly higher inference speed. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Point cloud analysis. There are mainly two streams to process point cloud. Since the point cloud data structure is irregular and unordered, some works consider projecting the original point clouds to intermediate voxels (Maturana & Scherer, 2015; Shi et al., 2020) or images (You et al., 2018; Li et al., 2020), translating the challenging 3D task into a well-explored 2D image problem. In this regime, point clouds understanding is largely boosted and enjoys the fast processing speed from 2D images or voxels. Albeit efficient, information loss caused by projection degrades the representational quality of details for point clouds (Yang et al., 2019). To this end, some methods are proposed to process the original point cloud sets directly. PointNet (Qi et al., 2017a) is a pioneering work that directly consumes unordered point sets as inputs using shared MLPs. Based on PointNet, PointNet+ $^ { - + }$ (Qi et al., 2017b) further introduced a hierarchical feature learning paradigm to capture the local geometric structures recursively. Owing to the local point representation (and multi-scale information), PointNet $^ { + + }$ exhibits promising results and has been the cornerstone of modern point cloud methods (Wang et al., 2019; Fan et al., 2021; Xu et al., 2021a). Our PointMLP also follows the design philosophy of PointNet++ but explores a simpler yet much deeper network architecture. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Local geometry exploration. As PointNet+ $^ +$ built the generic point cloud analysis network framework, the recent research focus is shifted to how to generate better regional points representation. Predominantly, the explorations of local points representation can be divided into three categories: convolution-, graph-, and attention-based methods. One of the most distinguished convolution-based methods is PointConv (Wu et al., 2019). By approximating continuous weight and density functions in convolutional filters using an MLP, PointConv is able to extend the dynamic filter to a new convolution operation. Also, PAConv (Xu et al., 2021a) constructs the convolution kernel by dynamically assembling basic weight matrices stored in a weight bank. Without modifying network configurations, PAConv can be seamlessly integrated into classical MLP-based pipelines. Unlike convolution-based methods, Graph-based methods investigate mutually correlated relationships among points with a graph. In Wang et al. (2019), an EdgeConv is proposed to generate edge features that describe the relationships between a point and its neighbors. By doing so, a local graph is built, and the point relationships are well preserved. In 3D-GCN (Lin et al., 2021), authors aim at deriving deformable 3D kernels using a 3D Graph Convolution Network. Closely related to graphbased methods, the attention-based methods exhibit excellent ability on relationship exploration as well, like PCT (Guo et al., 2021) and Point Transformer (Zhao et al., 2021; Engel et al., 2020). With the development of local geometry exploration, the performances on various tasks appear to be saturated. Continuing on this track would bring minimal improvements. In this paper, we showcase that even without the carefully designed operations for local geometry exploration, a pure deep hierarchical MLP architecture is able to exhibit gratifying performances and even better results. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Deep network architecture for point cloud. Interestingly, the development of point cloud analysis is closely related to the evolution of the image processing network. In the early era, works in the image processing field simply stack several learning layers to probe the performance limitations (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; Dong et al., 2014). Then, the great success of deep learning was significantly promoted by deep neural architectures like ResNet (He et al., 2016), which brings a profound impact to various research fields. Recently, attention-based models, including atten",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/dfe41ca7f89998d8aa2e8f7f73f2de3943bc56780ab345030028df2f62d3ea97.jpg",
+        "table_caption": [
+            "Table 1: Systematic comparison among some representative methods. “Deep” indicates that a model is expandable along depth. “Opt.” stands for the principal operator. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>hierarchy</td><td>locality</td><td>deep</td><td>opt.</td></tr><tr><td>PointNet</td><td>x／</td><td>X</td><td></td><td>MLP</td></tr><tr><td>PointNet++</td><td></td><td></td><td></td><td>MLP</td></tr><tr><td>DGCNN</td><td>X</td><td></td><td></td><td>GCN</td></tr><tr><td>DeepGCNs</td><td></td><td></td><td>xxx/</td><td>GCN</td></tr><tr><td>PointConv</td><td></td><td></td><td>X</td><td>Conv.</td></tr><tr><td>Point Trans.</td><td></td><td></td><td></td><td>Atten.</td></tr><tr><td>PointMLP</td><td></td><td></td><td></td><td>MLP</td></tr></table>",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "tion blocks (Wang et al., 2018) and Transformer architectures (Dosovitskiy et al., 2021), further flesh out the community. Most recently, the succinct deep MLP architectures have attracted a lot of attention due to their efficiency and generality. Point cloud analysis follows the same develop history as well, from MLP-based PointNet (Qi et al., 2017a), deep hierarchical PointNet+ $^ +$ (Qi et al., 2017b), convolution-/graph-/relation- based methods (Wu et al., 2019; Wang et al., 2019; Ran et al., 2021), to state-of-the-art Transformer-based models (Guo et al., 2021; Zhao et al., 2021). In this paper, we abandon sophisticated details and present a simple yet effective deep residual MLP network for point cloud analysis. Instead of following the tendency in the vision community deliberately, we are in pursuit of an inherently simple and empirically powerful architecture for point cloud analysis. ",
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+        "type": "text",
+        "text": "3 DEEP RESIDUAL MLP FOR POINT CLOUD ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We propose to learn the point cloud representation by a simple feed-forward residual MLP network (named PointMLP), which hierarchically aggregates the local features extracted by MLPs, and abandons the use of delicate local geometric extractors. To further improve the robustness and improve the performance, we also introduce a lightweight geometric affine module to transform the local points to a normal distribution. The detailed framework of our method is illustrated in Figure 2. ",
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+        "type": "text",
+        "text": "3.1 REVISITING POINT-BASED METHODS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The design of point-based methods for point cloud analysis dates back to the PointNet and Point$\\mathrm { N e t } { + + }$ papers (Qi et al., 2017a;b), if not earlier. The motivation behind this direction is to directly consume point clouds from the beginning and avoid unnecessary rendering processes. ",
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+        "type": "text",
+        "text": "Given a set of points $\\mathcal P = \\{ p _ { i } | i = 1 , \\cdots , N \\} \\in \\mathbb { R } ^ { N \\times 3 }$ , where $N$ indicates the number of points in a $( x , y , z )$ Cartesian space, point-based methods aims to directly learn the underlying representation $f$ of $\\mathcal { P }$ using neural networks. One of the most pioneering works is PointNet $^ { + + }$ , which learns hierarchical features by stacking multiple learning stages. In each stage, $N _ { s }$ points are re-sampled by the farthest point sampling (FPS) algorithm where $s$ indexes the stage and $K$ neighbors are employed for each sampled point and aggregated by max-pooling to capture local structures. Conceptually, the kernel operation of PointNet++ can be formulated as: ",
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+        "img_path": "images/295efa483234887acc5f296fe8f2504323ab06afe3fc5017c4d57094223bdd15.jpg",
+        "image_caption": [
+            "Figure 2: Overview of one stage in PointMLP. Given an input point cloud, PointMLP progressively extracts local features using residual point MLP blocks. In each stage, we first transform the local points using a geometric affine module, then they are extracted before and after the aggregation operation, respectively. PointMLP progressively enlarges the receptive field and models complete point cloud geometric information by repeating multiple stages. "
+        ],
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/37c5053951f9d79355dc87606ae0d783c54bbe0221d904711eb57540e637ba31.jpg",
+        "text": "$$\ng _ { i } = \\mathcal { A } \\left( \\Phi \\left( f _ { i , j } \\right) \\middle | j = 1 , \\cdot \\cdot \\ , K \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { \\mathcal { A } } \\left( \\cdot \\right)$ means aggregation function (max-pooling in PointNet++), $\\Phi \\left( \\cdot \\right)$ denotes the local feature extraction function (MLP in PointNet $^ { + + }$ ), and $f _ { i , j }$ is the $j$ -th neighbor point feature of $i$ -th sampled point. By doing so, PointNet+ $^ +$ is able to effectively capture local geometric information and progressively enlarge the receptive fields by repeating the operation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the sense of network architecture design, PointNet $^ { + + }$ exhibits a universal pipeline for point cloud analysis. Following this pipeline, some plug-and-play methods have been proposed, mainly focusing on the local feature extractor $\\Phi \\left( \\cdot \\right)$ (Xu et al., 2021a; Liu et al., 2019b; Thomas et al., 2019; Zhao et al., 2021). Generally, these local feature extractors thoroughly explore the local geometric information using convolution, graph, or self-attention mechanisms. In RSCNN (Liu et al., 2019b), the extractor is mainly achieved by exploring point relations as follow: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/3e10cd6d43b505d311b761b00a349e72b4d5e500ae760188f2b68431b15aba4c.jpg",
+        "text": "$$\n\\begin{array} { r } { \\Phi \\left( f _ { i , j } \\right) = \\mathrm { M L P } \\left( \\left[ \\left. x _ { i , j } - x _ { i } \\right. _ { 2 } , x _ { i , j } - x _ { i } , x _ { i , j } , x _ { i } \\right] \\right) * f _ { i , j } , \\forall j \\in \\left\\{ 1 , \\cdots , K \\right\\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $[ \\cdot ]$ is the concatenation operation and MLP is a small network composed of a Fully-connected (FC) layer, Batch Normalization layer, and activation function. Unlike RSCNN, Point Transformer introduces the self-attention mechanism into point cloud analysis and considers the similarities between pair-wise points in a local region. To this end, it re-formulates the extractor as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/27a19c126266e371609ed0175f575a9473d7818fb167bb195778a3421eabdcd9.jpg",
+        "text": "$$\n\\Phi \\left( f _ { i } \\right) = \\sum _ { j = 1 } ^ { k } { { \\rho \\left( { \\gamma \\left( { \\varphi \\left( { { f _ { i } } } \\right) - \\psi \\left( { { f _ { i , j } } } \\right) + \\delta } \\right) } \\right) } \\odot \\left( { \\alpha \\left( { { f _ { i , j } } + \\delta } \\right) } \\right) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\gamma , \\varphi , \\psi$ and $\\alpha$ are linear mapping function, “ $\\odot$ ” is a Hadamard product, and $\\rho$ is a softmax normalization. In particular, Point Transformer introduces a relative position encoding, $\\delta \\ = \\ \\theta \\left( x _ { i } - x _ { i , j } \\right)$ , where the relative position is encoded by two FC layers with a ReLU nonlinearity layer, into both attention weights and features. The lightweight positional encoder largely improves the performance of Point Transformer. ",
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+        "type": "text",
+        "text": "While these methods can easily take the advantage of detailed local geometric information and usually exhibit promising results, two issues limit their development. First, with the introduction of delicate extractors, the computational complexity is largely increased, leading to prohibitive inference latency 1. For example, the FLOPs of Equation 3 in Point Transformer would be $1 4 K d ^ { 2 }$ , ignoring the summation and subtraction operations. Compared with the conventional FC layer that enjoys $2 K d ^ { 2 }$ FLOPs, it increases the computations by times. Notice that the memory access cost is not considered yet. Second, with the development of local feature extractors, the performance gain has started to saturate on popular benchmarks. Moreover, empirical analysis in Liu et al. (2020) reveals that most sophisticated local extractors make surprisingly similar contributions to the network performance under the same network input. Both limitations encourage us to develop a new method that circumvents the employment of sophisticated local extractors, and provides gratifying results. ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "3.2 FRAMEWORK OF POINTMLP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to get rid of the restrictions mentioned above, we present a simple yet effective MLP-based network for point cloud analysis that no sophisticated or heavy operations are introduced. The key operation of our PointMLP can be formulated as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/bdde877833ad9bb03b926c34d879a3aaf3cfdcf1a99e4d43024f7620131817e0.jpg",
+        "text": "$$\ng _ { i } = \\Phi _ { p o s } ( A ( \\Phi _ { p r e } ( f _ { i , j } ) , | j = 1 , \\cdot \\cdot \\cdot , K ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\Phi _ { p r e }$ (·) and $\\Phi _ { p o s } \\left( \\cdot \\right)$ are residual point MLP blocks: the shared $\\Phi _ { p r e } \\left( \\cdot \\right)$ is designed to learn shared weights from a local region while the $\\Phi _ { p o s } \\left( \\cdot \\right)$ is leveraged to extract deep aggregated features. In detail, the mapping function can be written as a series of homogeneous residual MLP blocks, MLP $( x ) + x$ , in which MLP is combined by FC, normalization and activation layers (repeated two times). Following Qi et al. (2017a), we consider the aggregation function $\\boldsymbol { \\mathcal { A } } \\left( \\cdot \\right)$ as max-pooling operation. Equation 4 describes one stage of of PointMLP. For a hierarchical and deep network, we recursively repeat the operation by $s$ stages. Albeit the framework of PointMLP is succinct, it exhibits some prominent merits. 1) Since PointMLP only leverages MLPs, it is naturally invariant to permutation, which perfectly fits the characteristic of point cloud. 2) By incorporating residual connections, PointMLP can be easily extended to dozens layers, resulting deep feature representations. 3) In addition, since there is no sophisticated extractors included and the main operation is only highly optimized feed-forward MLPs, even we introduce more layers, our PointMLP still performs efficiently. Unless explicitly stated, the networks in our experiments use four stages, and two residual blocks in both $\\Phi _ { p r e }$ (·) and $\\Phi _ { p o s }$ (·). We employ $\\mathbf { k }$ -nearest neighbors algorithm (kNN) to select the neighbors and set the number $K$ to 24. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 GEOMETRIC AFFINE MODULE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While it may be easy to simply increase the depth by considering more stages or stacking more blocks in $\\Phi _ { p r e }$ and $\\Phi _ { p o s }$ , we notice that a simple deep MLP structure will decrease the accuracy and stability, making the model less robust. This is perhaps caused by the sparse and irregular geometric structures in local regions. Diverse geometric structures among different local regions may require different extractors but shared residual MLPs struggle at achieving this. We flesh out this intuition and develop a lightweight geometric affine module to tackle this problem. Let $\\{ f _ { i , j } \\} _ { j = 1 , \\cdots , k } \\in$ $\\mathbb { R } ^ { k \\times d }$ be the grouped local neighbors of $f _ { i } \\in \\mathbb { R } ^ { d }$ containing $k$ points, and each neighbor point $f _ { i , j }$ is a $d$ -dimensional vector. We transform the local neighbor points by the following formulation: ",
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+    {
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+        "img_path": "images/6c894d8ad679ca570a82f2f24d39f496d18afa38f752f1d31e55d7acbe76a192.jpg",
+        "text": "$$\n\\{ f _ { i , j } \\} = \\alpha \\odot \\frac { \\{ f _ { i , j } \\} - f _ { i } } { \\sigma + \\epsilon } + \\beta , ~ \\sigma = \\sqrt { \\frac { 1 } { k \\times n \\times d } \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { k } { ( f _ { i , j } - f _ { i } ) ^ { 2 } } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "where $\\alpha \\in \\mathbb { R } ^ { d }$ and $\\beta \\in \\mathbb { R } ^ { d }$ are learnable parameters, $\\odot$ indicates Hadamard production, and $\\epsilon =$ $1 e ^ { - 5 }$ is a small number for numerical stability (Ioffe & Szegedy, 2015; Wu & He, 2018; Dixon & Massey Jr, 1951). Note that $\\sigma$ is a scalar describes the feature deviation across all local groups and channels. By doing so, we transform the local points to a normal distribution while maintaining original geometric properties. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.4 COMPUTATIONAL COMPLEXITY AND ELITE VERSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Although the FC layer is highly optimized by mainstream deep learning framework, the theoretical number of parameters and computational complexity are still high. To further improve the efficiency, we introduce a lightweight version of PointMLP named as pointMLP-elite, with less than 0.7M parameters and prominent inference speed (176 samples/second on ModelNet40 benchmark). ",
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+    {
+        "type": "table",
+        "img_path": "images/103b1fdb6628cc296abe486b55f88d5eb59eb1298b6caf4f1b859b6afc86c21d.jpg",
+        "table_caption": [
+            "Table 2: Classification results on ModelNet40 dataset. With only 1k points, our method achieves state-of-the-art results on both class mean accuracy (mAcc) and overall accuracy (OA) metrics. We also report the speed of some open-sourced methods by samples/second tested on one Tesla V100- pcie GPU and four cores AMD EPYC $7 3 5 1 @ 2 . 6 0 \\mathrm { G H z }$ CPU. \\* For KPConv, we take the results from the original paper. The best is marked in bold and second best is in blue. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=3>Method</td><td rowspan=1 colspan=1>Inputs  mAcc(%)  OA(%)</td><td rowspan=1 colspan=1>Train   TestParam.speed speed</td></tr><tr><td rowspan=1 colspan=3>PointNet (Qi et al., 2017a)PointNet++ (Qi et al., 2017b)PointNet++ (Qi et al., 2017b)</td><td rowspan=1 colspan=1>1k P      86.0      89.21kP        -        90.75k P+N      -        91.9</td><td rowspan=1 colspan=1>1.41M  223.8  308.51.41M</td></tr><tr><td rowspan=10 colspan=3>PointCNN (Li et al., 2018b)PointConv (Wu et al., 2019)KPConv (Thomas et al., 2019)DGCNN (Wang et al., 2019)RS-CNN (Liu et al., 2019b)DensePoint (Liu et al., 2019a)PointASNL (Yan et al., 2020)PosPool (Liu et al., 2020)Point Trans. (Engel et al., 2020)</td><td rowspan=2 colspan=1>1k P      88.1      92.5</td><td rowspan=4 colspan=1>18.6M  17.9   10.215.2M  31.0*  80.0*</td></tr><tr><td rowspan=2 colspan=1>18.6M  17.915.2M  31.0*</td></tr><tr><td rowspan=1 colspan=1>1k P+N       -        92.5</td></tr><tr><td rowspan=1 colspan=1>7kP        -        92.9</td><td rowspan=1 colspan=1>15.2M  31.0*</td></tr><tr><td rowspan=1 colspan=1>1k P      90.2      92.9</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>1k P        -        92.9</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>1k P        1        93.2</td><td rowspan=5 colspan=1>8.39M   16.3   112</td></tr><tr><td rowspan=1 colspan=1>1k P        =        92.9</td></tr><tr><td rowspan=1 colspan=1>5k P        =        93.2</td></tr><tr><td rowspan=1 colspan=2>020）</td><td rowspan=1 colspan=1>1k P        =        92.8</td></tr><tr><td rowspan=7 colspan=3>GBNet (Qiu et al., 2021b)GDANet (Xu et al., 2021b)PA-DGC (Xu et al., 2021a)MLMSPT (Han et al., 2021)PCT (Guo et al., 2021)Point Trans. (Zhao et al.,2021)CurveNet (Xiang et al., 2021)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>1k P      91.0      93.8</td></tr><tr><td rowspan=1 colspan=1>1k P        =        93.8</td><td rowspan=6 colspan=1>0.93M   26.3   14.02.04M  20.8   15.0</td></tr><tr><td rowspan=1 colspan=1>1k P                  93.9</td></tr><tr><td rowspan=1 colspan=1>1k P                  92.9</td></tr><tr><td rowspan=1 colspan=1>1k P        =        93.2</td></tr><tr><td rowspan=1 colspan=1>1k P      90.6      93.7</td></tr><tr><td rowspan=1 colspan=1>1k P        -       94.2</td></tr><tr><td rowspan=4 colspan=3>PointMLP w/o vot.PointMLP w/ vot.PointMLP-elite w/o vot.PointMLP-elite w/ vot.</td><td rowspan=1 colspan=1>1k P      91.3      94.1</td><td rowspan=1 colspan=1>12.6M  47.1    112</td></tr><tr><td rowspan=1 colspan=1>1k P      91.4      94.5</td><td rowspan=1 colspan=1>12.6M  47.1    112</td></tr><tr><td rowspan=1 colspan=1>1k P      90.9      93.6</td><td rowspan=1 colspan=1>0.68M   116    176</td></tr><tr><td rowspan=1 colspan=1>1k P      90.7      94.0</td><td rowspan=1 colspan=1>0.68M   116    176</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Inspired by He et al. (2016); Hu et al. (2018), we present a bottleneck structure for the mapping function $\\Phi _ { p r e }$ and $\\Phi _ { p o s }$ . We opt to reduce the channel number of the intermediate FC layer by a factor of $r$ and increase the channel number as the original feature map. This strategy is opposite to the design in Vaswani et al. (2017); Touvron et al. (2021) which increases the intermediate feature dimensions. Empirically, we do not observe a significant performance drop. This method reduce the parameters of residual MLP blocks from $2 d ^ { 2 }$ to $\\scriptstyle { \\frac { 2 } { r } } d ^ { 2 }$ . By default, we set $r$ to 4 in PointMLPelite. Besides, we also slightly adjust the network architecture, reducing both the MLP blocks and embedding dimension number (see appendix for details). Inspired by Xie et al. (2017), we also investigated a grouped FC operation in the network that divides one FC layer into $g$ groups of subFC layers, like group convolution layer. However, we empirically found that this strategy would largely hamper the performance. As a result, we did not consider it in our implementation. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we comprehensively evaluate PointMLP on several benchmarks. Detailed ablation studies demonstrate the effectiveness of PointMLP with both quantitative and qualitative analysis. ",
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+    {
+        "type": "text",
+        "text": "4.1 SHAPE CLASSIFICATION ON MODELNET40 ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first evaluate PointMLP on the ModelNet40 (Wu et al., 2015) benchmark, which contains 9,843 training and 2,468 testing meshed CAD models belonging to 40 categories. Following the standard practice in the community, we report the class-average accuracy (mAcc) and overall accuracy (OA) on the testing set. We train all models for 300 epochs using SGD optimizer. ",
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+        "img_path": "images/071941c75a0797a644a63c156f0143a299fb274fe0ca60d40c36dc15a5458c0d.jpg",
+        "image_caption": [
+            "Figure 3: Four run results (mean $\\pm$ std) of PointMLP with/without our geometric affine module on ScanObjectNN test set. We zoom in on the details of PointMLP40 to show the stability difference. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Experimental results are presented in Table 2. Among these methods, our PointMLP clearly outperforms state-of-the-art method CurveNet by $0 . 3 \\%$ $9 4 . 5 \\%$ vs. $9 4 . 2 \\%$ ) overall accuracy with only 1k points. Note that this improvement could be considered as a promising achievement since the results on ModelNet40 recent methods have been saturated around $94 \\%$ for a long time. Even without the voting strategy (Liu et al., 2019b), our PointMLP still performs on par or even better than other methods that are tested with voting strategy. ",
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+    {
+        "type": "text",
+        "text": "Despite having better accuracy, our method is much faster than the methods with sophisticated local geometric extractors. We compare PointMLP to several open-sourced methods and report the parameters, classification accuracy, training, and testing speed. As we stated previously, a key intuition behind this experiment is that model complexity can not directly reflect efficiency. For example, CurveNet is lightweight and delivers a strong result, whereas the inference cost is prohibitive (15 samples/second). On the contrary, our PointMLP presents a high inference speed $( \\mathbf { 1 1 2 \\ s a m } .$ - ples/second). To further reduce the model size and speed up the inference, we present a lightweight PointMLP-elite, which significantly reduces the number of parameters to 0.68M, while maintaining high-performance $9 0 . 9 \\%$ mAcc and $9 4 . 0 \\%$ OA on ModelNet40. With PointMLP-elite, we further speed up the inference to 176 samples/second. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 SHAPE CLASSIFICATION ON SCANOBJECTNN ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "While ModelNet40 is the de-facto canonical benchmark for point cloud analysis, it may not meet the requirement of modern methods due to its synthetic nature and the fast development of point cloud analysis. To this end, we also conduct experiments on the ScanObjectNN benchmark (Uy et al., 2019). ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ScanObjectNN is a recently released point cloud benchmark that contains 15,000 objects that are categorized into 15 classes with 2,902 unique object instances in the real world. Due to the existence of background, noise, and occlusions, this benchmark poses significant challenges to existing point cloud analysis methods. We consider the hard",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c1d2456c590a0b43cd91f10a9cd9fe4e7dfe64953c9aad84000bf8617ead6a5c.jpg",
+        "table_caption": [
+            "Table 3: Classification results on ScanObjectNN dataset. We examine all methods on the most challenging variant $( \\mathrm { P B } _ { - } \\mathrm { T } 5 0 \\mathrm { \\_ R S } )$ . For our pointMLP and PointMLP-elite, we train and test for four runs and report mean $\\pm$ std results. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>mAcc(%)</td><td>OA(%)</td></tr><tr><td>3DmFV PointNet (Qi et al., 2017a) SpiderCNN (Xu et al., 2018) PointNet++ (Qi et al., 2017b)</td><td>58.1 63.4 69.8</td><td>63 68.2 73.7</td></tr><tr><td>DGCNN (Wang et al., 2019) PointCNN (Li et al., 2018b) BGA-DGCNN (Uy et al., 2019) BGA-PN++ (Uy et al., 2019)</td><td>75.4 73.6 75.1 75.7 77.5</td><td>77.9 78.1 78.5 79.7</td></tr><tr><td>DRNet (Qiu et al.,2021a) GBNet (Qiu et al., 2021b) SimpleView (Goyal et al., 2021) PRANet (Cheng et al., 2021)</td><td>78.0 77.8 =</td><td>80.2 80.3 80.5 80.5±0.3</td></tr><tr><td>MVTN (Hamdi et al., 2021) PointMLP (ours) PointMLP-elite (ours)</td><td>79.1 - 83.9±0.5</td><td>82.1 82.8 85.4±0.3</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "est perturbed variant (PB T50 RS) in our experiments. We train our model using an SGD optimizer for 200 epochs with a batch size of 32. For a better illustration, we train and test our method for four runs and report the mean $\\pm$ standard deviation in Table 3. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/4714faffbdb818a3d25aa4d667dfd5a946aa4697fe16c249bde466faf50d5a8a.jpg",
+        "table_caption": [
+            "Table 4: Classification accuracy of pointMLP on ScanObjectNN test set using 24, 40, and 56 layers, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Depth</td><td>mAcc(%)</td><td>0A(%)</td></tr><tr><td>24 layers</td><td>83.4±0.4</td><td>84.8±0.5</td></tr><tr><td>40 layers</td><td>83.9±0.5</td><td>85.4±0.3</td></tr><tr><td>56 layers</td><td>83.2±0.2</td><td>85.0±0.1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/fffd4a3c4c065d15caaaf03ea8e8f208a399f656de0a5aab528d83266347fbbc.jpg",
+        "table_caption": [
+            "Table 5: Component ablation studies on ScanObjectNN test set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Tpre</td><td>Tpos</td><td>Affine</td><td>mAcc(%)</td><td>0A(%)</td></tr><tr><td></td><td></td><td></td><td>80.8±0.4</td><td>82.8±0.0</td></tr><tr><td>x/</td><td>×</td><td>√</td><td>83.3±0.3</td><td>84.7±0.2</td></tr><tr><td></td><td>√</td><td>×</td><td>79.1±1.7</td><td>81.5±1.4</td></tr><tr><td></td><td></td><td>「</td><td>83.9±0.5</td><td>85.4±0.3</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Empirically, our PointMLP surpasses all methods by a significant improvement on both class mean accuracy (mAcc) and the overall accuracy (OA). For example, we outperform PRANet by $4 . 8 \\%$ mAcc and $3 . 3 \\%$ OA. Even compared with the heavy multi-view projection method MVTN (12 views), our PointMLP still performs much better $( 8 5 . 3 9 \\% 8 2 . 8 \\% )$ . Notice that we achieve this by fewer training epochs and did not consider the voting strategy. Moreover, we notice that our method achieves the smallest gap between class mean accuracy and overall accuracy. This phenomenon indicates that PointMLP did not bias to a particular category, showing decent robustness. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Network Depth. Network depth has been exploited in many tasks but is rare in point cloud analysis. We first investigate the performance of PointMLP with different depths in Table 4. We vary the network depth by setting the number of homogeneous residual MLP blocks to 1, 2, and 3, respectively, resulting in 24, 40, and 56-layers PointMLP variants. Detailed depth formulation can be found in Appendix D. At first glance, we notice that simply increasing the depth would not always bring better performance; an appropriate depth would be a good solution. Additionally, the model gets stable with more layers introduced, as demonstrated by the decreasing standard deviation. When the depth is set to 40, we achieve the best tradeoff between accuracy and stability $( 8 5 . 4 \\%$ mean accuracy and 0.3 standard deviations). Remarkably, PointMLP consistently achieves gratifying results that outperform recent methods, regardless of the depth. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Geometric Affine Module. Other work provides sophisticated local geometric extractors to explore geometric structures. Instead, our PointMLP discards these burdensome modules and introduces a lightweight geometric affine module. Figure 3 presents the results of PointMLP with/without the geometric affine module. By integrating the module, we systematically improve the performance of PointMLP by about $3 \\%$ for all variants. The reasons for this large improvement are two-fold. First, the geometric affine module maps local input features to a normal distribution, which eases the training of PointMLP. Second, the geometric affine module implicitly encodes the local geometrical information by the channel-wise distance to local centroid and variance, remedying the deficiency of geometric information. Besides the gratifying improvements, the geometric affine module also largely boosts the stability of PointMLP, suggesting better robustness. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Component ablation study. Table 5 reports the results on ScanObjectNN of removing each individual component in PointMLP. Consistent with Figure 3, geometric affine module plays an important role in PointMLP, improving the base architecture by $3 . 9 \\%$ . Remarkably, even without this module, which is an unfair setting for PointMLP, our base network stills achieves ",
+        "bbox": [
+            173,
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/64f9834890660778d301ec0a08af23475a0ca068c31e381a3a3c9289448af64f.jpg",
+        "image_caption": [
+            "Figure 4: Loss landscape along two rand directions. By introducing residual connection, we ease the optimization of PointMLP and achieve a flat landscape like a simple shallow network intuitively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ceda5cee5916d197091d2037ad09ed41cac5d45162aec97b1b36400de1470430.jpg",
+        "table_caption": [
+            "Table 6: Part segmentation results on the ShapeNetPart dataset. Empirically, our method is much faster than the best method KPConv, and presents a competitive performance. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Cls. mIoU</td><td rowspan=\"2\">Inst. mIoU</td><td rowspan=\"2\">aero</td><td rowspan=\"2\">bag</td><td rowspan=\"2\">cap</td><td rowspan=\"2\">car</td><td rowspan=\"2\">chair</td><td rowspan=\"2\">aerp- hone</td><td rowspan=\"2\">guitar</td><td rowspan=\"2\">knife</td><td rowspan=\"2\">lamp</td><td rowspan=\"2\">laptop</td><td rowspan=\"2\">motor- bike</td><td rowspan=\"2\">mug pistol</td><td rowspan=\"2\"></td><td rowspan=\"2\">rocket skate- board</td><td rowspan=\"2\"></td><td rowspan=\"2\">table</td></tr><tr><td></td></tr><tr><td>PointNet</td><td>80.4</td><td>83.7</td><td>83.4</td><td>78.7</td><td>82.5</td><td>74.9</td><td>89.6</td><td>73.0</td><td>91.5</td><td>85.9</td><td>80.8</td><td>95.3</td><td>65.2</td><td>93.0</td><td>81.2</td><td>57.9</td><td>72.8</td><td>80.6</td></tr><tr><td>PointNet++</td><td>81.9</td><td>85.1</td><td>82.4</td><td>79.0</td><td>87.7</td><td>77.3</td><td>90.8</td><td>71.8</td><td>91.0</td><td>85.9</td><td>83.7</td><td>95.3</td><td>71.6</td><td>94.1</td><td>81.3</td><td>58.7</td><td>76.4</td><td>82.6</td></tr><tr><td>Kd-Net</td><td>-</td><td>82.3</td><td>80.1</td><td>74.6</td><td>74.3</td><td>70.3</td><td>88.6</td><td>73.5</td><td>90.2</td><td>87.2</td><td>81.0</td><td>94.9</td><td>57.4</td><td>86.7</td><td>78.1</td><td>51.8</td><td>69.9</td><td>80.3</td></tr><tr><td>SO-Net</td><td>-</td><td>84.9</td><td>82.8</td><td>77.8</td><td>88.0</td><td>77.3</td><td>90.6</td><td>73.5</td><td>90.7</td><td>83.9</td><td>82.8</td><td>94.8</td><td>69.1</td><td>94.2</td><td>80.9</td><td>53.1</td><td>72.9</td><td>83.0</td></tr><tr><td>PCNN</td><td>81.8</td><td>85.1</td><td>82.4</td><td>80.1</td><td>85.5</td><td>79.5</td><td>90.8</td><td>73.2</td><td>91.3</td><td>86.0</td><td>85.0</td><td>95.7</td><td>73.2</td><td>94.8</td><td>83.3</td><td>51.0</td><td>75.0</td><td>81.8</td></tr><tr><td>DGCNN</td><td>82.3</td><td>85.2</td><td>84.0</td><td>83.4</td><td>86.7</td><td>77.8</td><td>90.6</td><td>74.7</td><td>91.2</td><td>87.5</td><td>82.8</td><td>95.7</td><td>66.3</td><td>94.9</td><td>81.1</td><td>63.5</td><td>74.5</td><td>82.6</td></tr><tr><td>P2Sequence</td><td>-</td><td>85.2</td><td>82.6</td><td>81.8</td><td>87.5</td><td>77.3</td><td>90.8</td><td>77.1</td><td>91.1</td><td>86.9</td><td>83.9</td><td>95.7</td><td>70.8</td><td>94.6</td><td>79.3</td><td>58.1</td><td>75.2</td><td>82.8</td></tr><tr><td>PointCNN</td><td>84.6</td><td>86.1</td><td>84.1</td><td>86.5</td><td>86.0</td><td>80.8</td><td>90.6</td><td>79.7</td><td>92.3</td><td>88.4</td><td>85.3</td><td>96.1</td><td>77.2</td><td>95.2</td><td>84.2</td><td>64.2</td><td>80.0</td><td>83.0</td></tr><tr><td>PointASNL</td><td>-</td><td>86.1</td><td>84.1</td><td>84.7</td><td>87.9</td><td>79.7</td><td>92.2</td><td>73.7</td><td>91.0</td><td>87.2</td><td>84.2</td><td>95.8</td><td>74.4</td><td>95.2</td><td>81.0</td><td>63.0</td><td>76.3</td><td>83.2</td></tr><tr><td>RS-CNN</td><td>84.0</td><td>86.2</td><td>83.5</td><td>84.8</td><td>88.8</td><td>79.6</td><td>91.2</td><td>81.1</td><td>91.6</td><td>88.4</td><td>86.0</td><td>96.0</td><td>73.7</td><td>94.1</td><td>83.4</td><td>60.5</td><td>77.7</td><td>83.6</td></tr><tr><td>SynSpec</td><td>82.0</td><td>84.7</td><td>81.6</td><td>81.7</td><td>81.9</td><td>75.2</td><td>90.2</td><td>74.9</td><td>93.0</td><td>86.1</td><td>84.7</td><td>95.6</td><td>66.7</td><td>92.7</td><td>81.6</td><td>60.6</td><td>82.9</td><td>82.1</td></tr><tr><td>SPLATNet</td><td>83.7</td><td>85.4</td><td>83.2</td><td>84.3</td><td>89.1</td><td>80.3</td><td>90.7</td><td>75.5</td><td>92.1</td><td>87.1</td><td>83.9</td><td>96.3</td><td>75.6</td><td>95.8</td><td>83.8</td><td>64.0</td><td>75.5</td><td>81.8</td></tr><tr><td>SpiderCNN</td><td>82.4 85.1</td><td>85.3 86.4</td><td>83.5</td><td>81.0</td><td>87.2</td><td>77.5</td><td>90.7</td><td>76.8</td><td>91.1 92.6</td><td>87.3 88.4</td><td>83.3</td><td>95.8</td><td>70.2</td><td>93.5</td><td>82.7</td><td>59.7</td><td>75.8</td><td>82.8</td></tr><tr><td>KPConv PA-DGC</td><td>84.6</td><td>86.1</td><td>84.6</td><td>86.3</td><td>87.2</td><td>81.1</td><td>91.1 90.6</td><td>77.8 80.8</td><td>92.0</td><td>88.7</td><td>82.7</td><td>96.2</td><td>78.1 73.9</td><td>95.8 94.7</td><td>85.4 84.7</td><td>69.0</td><td>82.0</td><td>83.6 84.0</td></tr><tr><td></td><td></td><td></td><td>84.3</td><td>85.0</td><td>90.4</td><td>79.7</td><td></td><td></td><td></td><td></td><td>82.2</td><td>95.9</td><td></td><td></td><td></td><td>65.9</td><td>81.4</td><td></td></tr><tr><td>PointMLP</td><td>84.6</td><td>86.1</td><td>83.5</td><td>83.4</td><td>87.5</td><td>80.54</td><td>90.3</td><td>78.2</td><td>92.2</td><td>88.1</td><td>82.6</td><td>96.2</td><td>77.5</td><td>95.8</td><td>85.4</td><td>64.6</td><td>83.3</td><td>84.3</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "$8 1 . 5 \\pm 1 . 4 \\%$ OA, outperforming most related methods (see Table 3). Removing $\\Phi _ { p r e }$ function (MLPs before aggregator $\\mathcal { A }$ ), the performance drops $2 . 6 \\%$ overall accuracy. Combining all these components together, we achieve the best result $8 5 . 4 \\%$ OA. See Appendix C for more ablations. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Loss landscape. We depict the 3D loss landscape (Li et al., 2018a) in Figure 4. Simply increasing the network depth may not achieve a better representation and even hamper the results. When removing the residual connection in PointMLP, the loss landscape turns sharp, and the performance plummets to $8 8 . 1 \\%$ $6 \\%$ drop) on ModelNet40. With residual connection, we greatly ease the optimization course of PointMLP and make it possible to train a deep network. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.4 PART SEGMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            478,
+            362,
+            491
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our PointMLP can also be generalized to other 3D point cloud tasks. We next test PointMLP for 3D shape part segmentation task on the ShapeNetPart benchmark (Yi et al., 2016). The shapeNetPart dataset consists of 16,881 shapes with 16 classes belonging to 50 parts labels in total. In each class, the number of parts is between 2 and 6. We follow the settings from Qi et al. (2017b) that randomly select 2048 points as input for a fair comparison. We compare our methods with several recent works, ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/937de1c2245f3fbccbf26ca1813370b71e9c2b3909037a38cd4c4f9c3749559f.jpg",
+        "image_caption": [
+            "Figure 5: Part segmentation results on ShapeNetPart. Top line is ground truth and bottom line is our prediction. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            437,
+            500,
+            820,
+            650
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "including SyncSpecCNN (Yi et al., 2017), SPLATNet (Su et al., 2018), etc. We also visualize the segmentation ground truths and predictions in Figure 5. Intuitively, the predictions of our PointMLP are close to the ground truth. Best viewed in color. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            758,
+            318,
+            775
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a simple yet powerful architecture named PointMLP for point cloud analysis. The key insight behind PointMLP is that a sophisticated local geometric extractor may not be crucial for performance. We begin with representing local points with simple residual MLPs as they are permutation-invariant and straightforward. Then we introduce a lightweight geometric affine module to boost the performance. To improve efficiency further, we also introduce a lightweight counterpart, dubbed as PointMLP-elite. Experimental results have shown that PointMLP outperforms related work on different benchmarks beyond simplicity and efficiency. We hope this novel idea will inspire the community to rethink the network design and local geometry in point cloud. ",
+        "bbox": [
+            173,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+        "text": "A POINTMLP DETAIL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We detail the architecture of PointMLP in Figure 6 (as well as PointMLP-elite in Figure 7) for a better understanding. Compared with PointMLP, the elite version mainly adjusts three configurations: 1) it reduces the number of residual point (Resp) MLP blocks; 2) it reduces the embedding dimension from 64 to 32, hence the overall model overhead is significantly alleviated; 3) by introducing a bottleneck structure, PointMLP further reduces the parameters by four times. ",
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+        "text": "For part segmentation task, we use the framework presented in PointNet (Qi et al., 2017a) and replace the backbone to our PointMLP. With the only modification, we improve the performance from 85.1 to 86.1 Instance mIoU. ",
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+        "type": "image",
+        "img_path": "images/61378541d7d7ab1bfa651ab48ae34b0ce8784c4821b4f81db8424ae8c2f88c60.jpg",
+        "image_caption": [
+            "Figure 6: Detail architecture of PointMLP for classification. "
+        ],
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+    {
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+        "img_path": "images/29e776509eac6ea6a31083e3cafa6b828bf28a0aa1fe7f9f5e9a64cef2650dab.jpg",
+        "image_caption": [
+            "Figure 7: Detail architecture of PointMLP-elite for classification. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "B DETAIL EXPERIMENTAL SETTING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            611,
+            490,
+            627
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.1 MODELNET40 AND SCANOBJECTNN ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            645,
+            477,
+            660
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Our implementations are based on PyTorch. For ModelNet40, we train models for 300 epochs on one Tesla V100 GPU with a batch size of 32. All our models are trained using synchronous SGD with a Nesterov momentum of 0.9 and a weight decay of 0.0002. The learning rate is set to 0.1 initially. We use the cosine annealing scheduler (Loshchilov & Hutter, 2017) to adjust the learning rate. For each sample, we randomly select 1024 points and consider the same augmentation strategy as Qi et al. (2017b). The setting for ScanObjectNN is similar to ModelNet40, except we train all models for only 200 epochs. ",
+        "bbox": [
+            173,
+            672,
+            825,
+            771
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For the reported speed in Table 2, we test the open-source code on a Tesla V100-pcie GPU. All the source codes we used are listed2 in the footnote. ",
+        "bbox": [
+            174,
+            779,
+            823,
+            806
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.2 SHAPENETPART ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            331,
+            117
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Our setting for part segmentation task is following PointNet (Qi et al., 2017a). We randomly sample 2048 points for each sample and re-scale the input in a range of [0.67, 1.5]. Note that we did not test the result using a multi-scale testing strategy, which could further improve the performance, but is not realizable in real-world applications. Hence, we only report the single-scale results. Even the comparison is unfair, we still achieve competitive performance. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            199
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C MORE DETAILED ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            219,
+            526,
+            236
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Skip connection. Figure 4 shows the loss landscapes of our PointMLP with and without skip connections. We also consider adding skip connections to PointNet+ $^ +$ to validate the effectiveness of skip connections. Due to the structure of PointNe $^ { + + }$ , only two skip connections could be added without modifying the original architecture of PointNet++. By adding the skip connections, we achieve a classification accuracy of $9 2 . 7 \\%$ on ModelNet40 in our re-implementation. ",
+        "bbox": [
+            174,
+            251,
+            825,
+            321
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Pre-MLP block vs. Pos-MLP block. we also modified the configuration of our PointMLP and retrained the model to investigate the importance of Pre-MLP and Pos-MLP blocks. In our original implementation, we set the pre-MLP block list to [2, 2, 2, 2] and the pos-MLP blocks list to [2, 2, 2, 2]. Here, we remove the pos-MLP blocks and change the pre-MLP blocks to [4, 4, 4, 4] to match the block number. The 3-layer classifier can be considered as the MLP at the end of the last stage. We trained the models two times and got an average OA of $8 4 . 1 3 \\%$ $8 3 . 8 7 \\%$ and $8 4 . 3 9 \\%$ ), which is lower than vanilla PointMLP $8 5 . 4 \\%$ , and even the result in Table 5 second-row $8 4 . 7 \\%$ . This result indicates that pos-MLP does benefit our PointMLP, and simply adding more pre-MLP blocks does not help. We acknowledge that the effect of pos-MLP is not as strong as other components and believe that a detailed fine-tuning of the configurations would deliver an even better performance-efficiency balance. ",
+        "bbox": [
+            173,
+            328,
+            825,
+            481
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Geometric Affine Module Applications. Geometric affine module plays an essential role in our PointMLP, exhibiting promising performance improvements. While this module can be considered as a plug-and-play method, the overlap with some local geometric extractors in other methods may limit its application. Here we integrate the module to two popular methods, PointNet $^ { - + }$ and DGCNN, for illustration and experiment on the ModelNet40 benchmark. By integrating the geometric affine module, we improve the performance of PointNet $^ { + + }$ to $9 3 . 3 \\%$ , achieving an improvement of $1 . 4 \\%$ . However, when integrating the module to DGCNN, we get a performance of $9 2 . 8 \\%$ , which is slightly lower than the original results $( 9 2 . 9 \\% )$ . Note that both results are tested without voting. ",
+        "bbox": [
+            173,
+            487,
+            825,
+            599
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D POINTMLP DEPTH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            618,
+            369,
+            636
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Here we format the detailed formulation of layer number in our PointMLP. For the sake of clarity, we ignore Batch Normalization layers and activation functions. Let $\\mathrm { P r e } _ { i }$ and $\\mathrm { P o s } _ { i }$ indicate the repeating number of the $\\Phi _ { p r e }$ block (which includes 3 layers) and $\\Phi _ { p o s }$ block (which includes 2 layers) in $i$ -th stage, respectively. Note that we have one layer in feature embedding in the beginning, one layer for channel number matching in each stage, and three layers in the classifier. Hence, the total number of learnable layers $L$ would be ",
+        "bbox": [
+            173,
+            650,
+            825,
+            734
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/544c088adc6a9d48a769bf2d202a315813c99ecab6063d0edbef759009e7c0e7.jpg",
+        "text": "$$\nL = 1 + \\sum _ { i = 1 } ^ { 4 } \\left( 1 + 2 \\times \\mathrm { P r e } _ { i } + 2 \\times \\mathrm { P o s } _ { i } \\right) + 3 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            741,
+            650,
+            785
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "As a result, the depth configuration of our network (24, 40, and 56) can be summarized as: ",
+        "bbox": [
+            168,
+            790,
+            764,
+            806
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/6e8b0d057b0bcac5444ea4b490a91e8d57b8eda1cc8fe2948e1d3310c33751e2.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td>Depth|[Pre1,Pre2,Pre3,Pre4]|[Pos1,Pos2,Pos3,Pos4]</td></tr><tr><td>24</td><td>[1,1,1,1]</td><td>[1,1,1,1]</td></tr><tr><td>40</td><td>[2,2,2,2]</td><td>[2,2,2,2]</td></tr><tr><td>56</td><td>[3,3,3,3]</td><td>[3,3,3,3]</td></tr></table>",
+        "bbox": [
+            287,
+            818,
+            710,
+            890
+        ],
+        "page_idx": 14
+    }
+]

parse/dev/5Xc1ecxO1h/5Xc1ecxO1h.md

@@ -0,0 +1,284 @@
+# Tree of Thoughts: Deliberate Problem Solving with Large Language Models
+
+Shunyu Yao Princeton University
+
+Dian Yu Google DeepMind
+
+Jeffrey Zhao Google DeepMind
+
+Izhak Shafran Google DeepMind
+
+Thomas L. Griffiths Princeton University
+
+Yuan Cao Google DeepMind
+
+Karthik Narasimhan Princeton University
+
+# Abstract
+
+Language models are increasingly being deployed for general problem solving across a wide range of tasks,but are still confined to token-level, left-to-right decision-making processes during inference. This means they can fall short in tasks that require exploration, strategic lookahead, or where initial decisions play a pivotal role. To surmount these challenges,we introduce a new framework for language model inference,“Tree of Thoughts”(ToT),which generalizes over the popular“Chain of Thought” approach to prompting language models,and enables exploration over coherent units of text ("thoughts") that serve as intermediate steps toward problem solving. ToT allows LMs to perform deliberate decision making by considering multiple different reasoning paths and self-evaluating choices to decide the next course of action, as well as looking ahead or backtracking when necessary to make global choices. Our experiments show that ToT significantly enhances language models’ problem-solving abilities on three novel tasks requiring non-trivial planning or search: Game of 24, Creative Writing,and Mini Crosswords. For instance, in Game of 24, while GPT-4 with chain-of-thought prompting only solved $4 \%$ of tasks,our method achieved a success rate of $74 \%$ . Code repo with all prompts: https://github.com/princeton-nlp/tree-of-thought-llm.
+
+# 1Introduction
+
+Originally designed to generate text, scaled-up versions of language models (LMs) such as GPT [25, 26,1,23] and PaLM [5] have been shown to be increasingly capable of performing an ever wider range of tasks requiring mathematical, symbolic, commonsense,and knowledge reasoning. It is perhaps surprising that underlying all this progress is still the original autoregressive mechanism for generating text, which makes token-level decisions one by one and in a left-to-right fashion. Is such a simple mechanism sufficient for a LM to be built toward a general problem solver? If not, what problems would challenge the current paradigm,and what should be alternative mechanisms?
+
+The literature on human cognition provides some clues to answer these questions. Research on “dual process” models suggests that people have two modes in which they engage with decisions - a fast, automatic,unconscious mode ("System 1") and a slow, deliberate, conscious mode ("System 2") [30,31,16,15]. These two modes have previously been connected to a variety of mathematical models used in machine learning. For example, research on reinforcement learning in humans and other animals has explored the circumstances under which they engage in associative “model free" learning or more deliberative“model based” planning [7]. The simple associative token-level choices of LMs are also reminiscent of “System 1",and thus might benefit from augmentation by a more deliberate “System $2 ^ { \circ }$ planning process that (1) maintains and explores diverse alternatives for current choices instead of just picking one,and (2) evaluates its current status and actively looks ahead or backtracks to make more global decisions.
+
+![](images/b7fb3f215b6ad763cd26c1185478d0e2a29d7cb659f6ab760a80492664d522bb.jpg)  
+Figure 1: Schematic ilustrating Various approaches to problem solving with LLMs. Each rectangle box represents a thought, which is a coherent language sequence that serves as an intermediate step toward problem solving. See concrete examples of how thoughts are generated, evaluated, and searched in Figures 2,4,6.
+
+To design such a planning process, we return to the origins of artificial intelligence (and cognitive science), drawing inspiration from the planning processes explored by Newell, Shaw,and Simon starting in the 195Os [21,22]. Newell and colleagues characterized problem solving [21] as search through a combinatorial problem space,represented as a tree.We thus propose the Tree of Thoughts (ToT) framework for general problem solving with language models. As Figure 1 illustrates, while existing methods (detailed below) sample continuous language sequences for problem solving, ToT actively maintains a tree of thoughts,where each thought is a coherent language sequence that serves as an intermediate step toward problem solving (Table 1). Such a high-level semantic unit allows the LM to self-evaluate the progress different intermediate thoughts make towards solving the problem through a deliberate reasoning process that is also instantiated in language (Figures 2,4,6). This implementation of search heuristics via LM self-evaluation and deliberation is novel,as previous search heuristics are either programmed or learned. Finally，we combine this language-based capability to generate and evaluate diverse thoughts with search algorithms,such as breadth-first search (BFS) or depth-first search (DFS), which allow systematic exploration of the tree of thoughts with lookahead and backtracking.
+
+Empirically, we propose three new problems that challenge existing LM inference methods even with the state-of-the-art language model, GPT-4 [23]: Game of 24, Creative Writing,and Crosswords (Table 1). These tasks require deductive,mathematical, commonsense,lexical reasoning abilities, and a way to incorporate systematic planning or search. We show ToT obtains superior results on allthree tasks by being general and flexible enough to support different levels of thoughts,different ways to generate and evaluate thoughts,and diferent search algorithms that adapt to the nature of different problems. We also analyze how such choices affect model performances via systematic ablations and discuss future directions to better train and use LMs.
+
+# 2Background
+
+We first formalize some existing methods that use large language models for problem-solving, which our approach is inspired by and later compared with. We use $p _ { \theta }$ to denote a pre-trained LM with parameters $\theta$ ，and lowercase letters $x , y , z , s , \cdots$ to denote a language sequence, i.e. $x =$ $( x [ 1 ] , \hat { \cdot } \cdot \cdot , x [ n ] )$ where each $x [ i ]$ is a token, so that $\begin{array} { r } { p _ { \theta } ( x ) = \prod _ { i = 1 } ^ { n } p _ { \theta } ( x [ i ] | x [ 1 . . . i ] ) } \end{array}$ . We use uppercase letters $S , \cdots$ to denote a collection of language sequences.
+
+Input-output (IO) prompting is the most common way to turn a problem input $x$ into output $y$ with LM: $y \sim p _ { \boldsymbol \theta } ( y | \mathrm { p r o m p t } _ { I O } ( x ) )$ ，where $\mathsf { p r o m p t } _ { I O } ( x )$ wraps input $x$ with task instructions and/or few-shotinputoutputexamples.Forsimplicityetsde $p _ { \theta } ^ { \mathrm { p r o m p t } } ( \mathsf { o u t p u t } \mid \mathsf { i n p u t } ) =$ $p _ { \theta } ( \mathrm { { o u t p u t } \mid p r o m p t ( i n p u t ) } )$ , so that IO prompting can be formulated as $y \sim p _ { \theta } ^ { I O } ( y | x )$ ：
+
+Chain-of-thought $\mathbf { ( C o T ) }$ prompting [38] was proposed to address cases where the mapping of input $x$ to output $y$ is non-trivial (e.g. when $x$ is a math question and $y$ is the final numerical answer). The key idea is to introduce a chain of thoughts $z _ { 1 } , \cdots , z _ { n }$ to bridge $x$ and $y$ ，where each $z _ { i }$ is a coherent language sequence that serves as a meaningful intermediate step toward problem solving (e.g. $z _ { i }$ could be an intermediate equation for math QA). To solve problems with CoT, each thought $\boldsymbol { z } _ { i } \sim p _ { \theta } ^ { C o T } ( \boldsymbol { z } _ { i } \mid x , \boldsymbol { z } _ { 1 \cdots i - 1 } )$ is sampledsequentially,thentheoutput $y \sim p _ { \theta } ^ { C o T } ( y | x , z _ { 1 } . . . n )$ In practice, $[ z _ { 1 \cdots n } , y ] \sim p _ { \theta } ^ { C o T } ( z _ { 1 \cdots n } , y | x )$ is sampled asacontiuouslanguagesequenceadthe decomposition of thoughts (e.g.is each $z _ { i }$ a phrase, a sentence, or a paragraph) is left ambiguous.
+
+Self-consistency with CoT (CoT-SC) [36] is an ensemble approach that samples $k$ i.i.d. chains of thought: $[ z _ { 1 \cdots n } ^ { ( i ) } , y ^ { ( i ) } ] \sim p _ { \theta } ^ { C o T } ( z _ { 1 \cdots n } , y | x )$ $( i = 1 \cdots k )$ thenretusteostfrequentoutput arg $\operatorname* { m a x } _ { y }$ # $\{ i \mid y ^ { ( i ) } = y \}$ .CoT-SC improves upon CoT, because there are generally different thought processes for the same problem (e.g. different ways to prove the same theorem), and the output decision can be more faithful by exploring a richer set of thoughts. However, within each chain there is no local exploration of different thought steps,and the “most frequent” heuristic only applies when the output space is limited (e.g. multi-choice QA).
+
+# 3Tree of Thoughts: Deliberate Problem Solving with LM
+
+A genuine problem-solving process involves the repeated use of available information to initiate exploration, which discloses,in turn, more information until a way to attain the solution is finally discovered.— Newell et al. [21]
+
+Research on human problem-solving suggests that people search through a combinatorial problemspace -a tree where the nodes represent partial solutions,and the branches correspond to operators that modify them [21,22].Which branch to take is determined by heuristics that help to navigate the problem-space and guide the problem-solver towards a solution. This perspective highlights two key shortcomings of existing approaches that use LMs to solve general problems: 1) Locally, they do not explore different continuations within a thought process-the branches of the tree.2) Globally, they do not incorporate any type of planning,lookahead,or backtracking to help evaluate these different options - the kind of heuristic-guided search that seems characteristic of human problem-solving.
+
+To address these shortcomings, we introduce Tree of Thoughts (ToT),a paradigm that allows LMs to explore multiple reasoning paths over thoughts (Figure 1(c)). ToT frames any problem as a search over a tree, where each node is a state $s = [ x , z _ { 1 \cdots i } ]$ representing a partial solution with the input and the sequence of thoughts so far. A specific instantiation of ToT involves answering four questions: 1. How to decompose the intermediate process into thought steps; 2. How to generate potential thoughts from each state; 3. How to heuristically evaluate states; 4. What search algorithm to use.
+
+1.Thought decomposition. While CoT samples thoughts coherently without explicit decomposition, ToT leverages problem properties to design and decompose intermediate thought steps.As Table 1 shows,depending on diferent problems,a thought could be a couple of words (Crosswords),a line of equation (Game of 24), or a whole paragraph of writing plan (Creative Writing). In general,a thought should be “small" enough so that LMs can generate promising and diverse samples (e.g. generating a whole book is usually too“big” to be coherent), yet“big” enough so that LMs can evaluate its prospect toward problem solving (e.g. generating one token is usually too “small" to evaluate).
+
+2.Thought generator $G ( p _ { \theta } , s , k )$ . Given a tree state $s = [ x , z _ { 1 } . . . i ]$ , we consider two strategies to generate $k$ candidates for the next thought step:
+
+(a) Sample i.i.d.thoughts from a CoT prompt (Creative Writing，Figure 4): $z ^ { ( j ) } \sim$ $p _ { \theta } ^ { C o \hat { T _ { ( z _ { i + 1 } | s } ) } } = p _ { \theta } ^ { \check { C } o T } ( z _ { i + 1 } | x , z _ { 1 \cdots i } ) \ ( \dot { j } = \dot { 1 } \cdots k )$ . This works better when the thought space is rich (e.g.each thought is a paragraph),and i.i.d.samples lead to diversity;   
+(b) Propose thoughts sequentially using a“propose prompt”(Game of 24, Figure 2; Crosswords, Figure 6): $[ z ^ { ( 1 ) } , \cdot \cdot \cdot , z ^ { ( k ) } ] \sim p _ { \theta } ^ { p r o p o s e } ( z _ { i + 1 } ^ { ( 1 \cdots k ) } \mid s )$ ]\~propose() . This works better when the thought space is more constrained (e.g.each thought is just a word or a line),so proposing different thoughts in the same context avoids duplication.
+
+3. State evaluator $V ( p _ { \theta } , S )$ . Given a frontier of different states,the state evaluator evaluates the progress they make towards solving the problem, serving as a heuristic for the search algorithm to determine which states to keep exploring and in which order. While heuristics are a standard approach to solving search problems, they are typically either programmed (e.g. DeepBlue [3]) or learned (e.g. AlphaGo [29]). We propose a third alternative,by using the LM to deliberately reason about states.When applicable,such a deliberate heuristic can be more flexible than programmed rules,and more sample-efficient than learned models. Similar to the thought generator, we consider two strategies to evaluate states either independently or together:
+
+(a) Value each state independently: $V ( p _ { \theta } , S ) ( s ) ~ \sim ~ p _ { \theta } ^ { v a l u e } ( v | s ) ~ \forall s ~ \in ~ S$ where a value prompt reasons about the state $s$ to generate a scalar value $v$ (e.g.1-10) or a classification (e.g.sure/likely/impossible) that could be heuristically turned into a value.The basis of such evaluative reasoning can vary across problems and thought steps. In this work, we explore evaluation via few lookahead simulations (e.g.quickly confirm that 5,5,14 can reach 24 via $5 + 5 + 1 4$ ,or “hot_l" can mean“inn”via filling“e”in“-") plus commonsense (e.g.1 2 3 are too smallto reach 24, or no word can start with “tzxc"). While the former might promote “good” states,the latter could help eliminate “bad” states. Such valuations do not need to be perfect,and only need to be approximately helpful for decision making. (b) Vote across states: $V ( p _ { \theta } , S ) ( s ) = \mathbb { 1 } [ s = s ^ { * } ]$ ， where a“good” state $s ^ { * } \sim p _ { \theta } ^ { v o t e } ( s ^ { * } | S )$ is voted out based on deliberately comparing different states in $S$ in a vote prompt. When problem success is harder to directly value (e.g. passage coherency), it is natural to to instead compare different partial solutions and vote for the most promising one. This is similar in spirit to a“step-wise” self-consistency strategy,i.e.cast “which state to explore” as a multi-choice QA,and use LM samples to vote for it.
+
+For both strategies, we could prompt the LM multiple times to aggregate the value or vote results to trade time/resource/cost for more faithful/robust heuristics.
+
+<table><tr><td>Algorithm1 ToT-BFS(x,pe,G,k,V,T,b)</td><td>Algorithm 2 ToT-DFS(s,t,pe,G,k,V,T,Uth)</td></tr><tr><td>Require: Input x,LM po, thought generator G(Require: Current state s, step t,LM po, thought &amp; size limit k, states evaluator V(),step limit T， generator G() and size limit k, states evaluator</td><td></td></tr><tr><td>breadth limit b. So←{x}</td><td>V(), step limit T, threshold Uth if t &gt; T then record output G(pe, s,1)</td></tr><tr><td>for t=1,..,T do</td><td>end if</td></tr><tr><td>St←{[s,z]|s∈ St-1,zt ∈G(p0,s,k)}</td><td>for s&#x27; ∈ G(pe,s,k) do&gt; sorted candidates</td></tr><tr><td>Vt ←V(po,St)</td><td>if V(pe,{s&#x27;})(s) &gt; Uthres then &gt; pruning</td></tr><tr><td>St ← arg maxscs&#x27;,|s|=b∑ses Vt(s)</td><td>DFS(s&#x27;,t+1)</td></tr><tr><td>end for return G(pe,arg maxs∈Sr Vr(s),1)</td><td>end if</td></tr></table>
+
+4. Search algorithm.Finally, within the ToT framework,one can plug and play different search algorithms depending on the tree structure. We explore two relatively simple search algorithms and leave more advanced ones (e.g. $\mathbf { A } ^ { * }$ [11],MCTS [2]) for future work:
+
+(a)Breadth-first search (BFS) (Algorithm 1) maintains a set of the $b$ most promising states per step. This is used for Game of 24 and Creative Writing where the tree depth is limit $( T \leq 3 )$ ,and initial thought steps can be evaluated and pruned to a small set $( b \leq 5 )$ ）   
+(b) Depth-first search (DFS) (Algorithm 2) explores the most promising state first, until the final output is reached $( t > T )$ ,or the state evaluator deems it impossible to solve the problem from the current $s$ $( V ( p _ { \theta } , \{ s \} ) ( s ) \le v _ { t h }$ for a value threshold $v _ { t h }$ ). In the latter case, the subtree from $s$ is pruned to trade exploration for exploitation. In both cases,DFS backtracks to the parent state of $s$ to continue exploration.
+
+Conceptually,ToT has several benefits as a method for general problem-solving with LMs: (1) Generality. IO,CoT, CoT-SC,and self-refinement can be seen as special cases of ToT(i.e. trees of limited depth and breadth; Figure 1). (2) Modularity. The base LM, as well as the thought decomposition, generation, evaluation, and search procedures can allbe varied independently. (3) Adaptability. Different problem properties,LM capabilities,and resource constraints can be accommodated. (4) Convenience.No extra training is needed, just a pre-trained LM is suficient. The next section will show how these conceptual benefits translate to strong empirical performance in diferent problems.
+
+# 4Experiments
+
+We propose three tasks that are hard even when sampling from the state-of-the-art language model. GPT-4 [23], using standard IO prompting or chain-of-thought (CoT) prompting. We show how deliberate search in trees of thoughts (ToT) produces beter results,and more importantly, interesting and promising new ways to use language models to solve problems requiring search or planning. Unless otherwise stated, we perform experiments using a Chat Completion mode GPT $\dot { 4 } ^ { 1 }$ with a sampling temperature of 0.7.
+
+<table><tr><td></td><td>Game of 24</td><td>Creative Writing</td><td>5x5 Crosswords</td></tr><tr><td>Input</td><td>4 numbers (4 9 10 13)</td><td>4 random sentences</td><td>10 clues (h1. presented;..)</td></tr><tr><td>Output</td><td>An equation to reach 24 (13-9)*(10-4)=24</td><td>A passage of 4 paragraphs ending in the 4 sentences</td><td>5x5 letters: SHOWN; WIRRA; AVAIL;..</td></tr><tr><td>Thoughts</td><td>3 intermediate equations (13-9=4 (left 4,4,10); 10- 4=6 (left 4,6); 4*6=24)</td><td>Ashort writingplan (1.Introduce a book that connects...)</td><td>Words to fill in for clues: (h1. shown; v5. naled; .)</td></tr><tr><td>#ToT steps</td><td>3</td><td>1</td><td>5-10 (variable)</td></tr></table>
+
+Table 1: Task overview. Input, output, thought examples are in blue.
+
+# 4.1 Game of 24
+
+Game of 24 is a mathematical reasoning challenge, where the goal is to use 4 numbers and basic arithmetic operations $( + - ^ { * } / )$ to obtain 24. For example, given input $^ { \bullet } 4 9 1 0 1 3 ^ { \prime \prime }$ , a solution output could be $\ ^ { \cdot } ( 1 0 - 4 ) \ ^ { * } \ ( 1 3 - 9 ) = 2 4 ^ { , 9 }$ ：
+
+![](images/1e51ca3d9ec5118d046f44d504919561fcccabdb24a756895c2162f29b531ffb.jpg)  
+Figure 2: ToT in a game of 24. The LM is prompted for (a) thought generation and (b) valuation.
+
+Task Setup. We scrape data from 4nums.com, which has 1,362 games that are sorted from easy to hard by human solving time,and use a subset of relatively hard games indexed 901-1,0Oo for testing. For each task, we consider the output as success if it is a valid equation that equals 24 and uses the input numbers each exactly once.We report the success rate across 1OO games as the metric.
+
+Baselines.We use a standard input-output (IO) prompt with 5 in-context examples.For chain-ofthought (CoT) prompting, we augment each input-output pair with 3 intermediate equations, each operating on two remaining numbers. For example, given input $^ { \dots } 4 9 1 0 1 3 ^ { \prime \prime }$ ,the thoughts could be $\cdot 1 3 - 9 = 4$ (left: 4 4 10); $1 0 - 4 = 6$ (left: $4 6$ ） $4 \ast 6 = 2 4$ (left: 24)". For each game, we sample IO and CoT prompting for 100 times for average performance. We also consider a CoT self-consistency baseline,which takes the majority output from $1 0 0 \mathrm { C o T }$ samples,and an iterative-refine approach on top of an IO sample for at most 10 iterations. At each iteration, the LM is conditioned on all previous history to “reflect on your mistakes and generate a refined answer”if the output is incorrect. Note that it uses groundtruth feedback signals about equation correctness.
+
+ToT Setup. To frame Game of 24 into ToT, it is natural to decompose the thoughts into 3 steps, each an intermediate equation. As shown in Figure 2(a), at each tree node, we exact the remaining numbers and prompt the LM to propose some possible next steps. The same “propose prompt” is used for all 3 thought steps,though it only has one example with 4 input numbers. We perform a breadth-first search (BFS) in ToT, where at each step we keep the best $b = 5$ candidates. To perform deliberate BFS in ToT,as shown in Figure 2(b), we prompt LM to evaluate each thought candidate as “sure/maybe/impossible” with regard to reaching 24. The aim is to promote correct partial solutions that can be verdicted within few lookahead trials,and eliminate impossible partial solutions based on “too big/small” commonsense,and keep the rest “maybe". We sample values 3 times for each thought.
+
+<table><tr><td>Method</td><td> Success</td></tr><tr><td>IO prompt</td><td>7.3%</td></tr><tr><td>CoT prompt</td><td>4.0%</td></tr><tr><td>CoT-SC (k=100) ToT (ours) (b=1)</td><td>9.0% 45%</td></tr><tr><td>ToT (ours) (b=5)</td><td>74%</td></tr><tr><td>IO + Refine (k=10)</td><td>27%</td></tr><tr><td>IO (best of 100)</td><td>33%</td></tr><tr><td>CoT (best of 100)</td><td>49%</td></tr></table>
+
+Table 2: Game of 24 Results.
+
+![](images/d9810d9d96bc205c2dc3f1adc0e2933a0adab5854989f8701a562674a3c12cef.jpg)  
+Figure 3: Game of 24 (a) scale analysis & (b) error analysis.
+
+Results. As shown in Table 2,IO, CoT,and CoT-SC prompting methods perform badly on the task, achieving only $7 . 3 \%$ $4 . 0 \%$ ,and $9 . 0 \%$ success rates.In contrast,ToT with a breadth of $b = 1$ already achieves a success rate of $4 5 \%$ ，while $b = 5$ achieves $7 4 \%$ .We also consider an oracle setup for IO/CoT, by calculating the success rate using best of $k$ samples ( $1 \leq k \leq 1 0 0 $ ). To compare IO/CoT (best of $\mathbf { k }$ ） with ToT,we consider calculating the tree nodes visited per task in ToT across $b = 1 \cdots 5$ and map the 5 success rates in Figure 3(a), treating IO/CoT (best of $k$ )asvisiting $k$ nodes in a bandit. Not surprisingly, CoT scales better than IO,and best of $1 0 0 \mathrm { C o T }$ samples achieve a success rate of $4 9 \%$ ,but still much worse than exploring more nodes in ToT $( b > 1 )$ ).
+
+Error analysis.Figure 3(b) breaks down at which step CoT and ToT samples fail the task, i.e.the thought (in CoT) or all $b$ thoughts (in ToT) are invalid or impossible to reach 24.Notably,around $60 \%$ of CoT samples already failed the task after generating the first step, or equivalently, the first three words (e.g. $\mathbf { \^ 6 4 + 9 ^ { 9 } } ,$ . This highlights the issues with direct left-to-right decoding.
+
+# 4.2 Creative writing
+
+Next, we invent a creative writing task where the input is 4 random sentences and the output should be a coherent passage with 4 paragraphs that end in the 4 input sentences respectively. Such a task is open-ended and exploratory,and challenges creative thinking as well as high-level planning.
+
+Task setup.We sample random sentences from randomwordgenerator.com to form 1OO inputs,and there is no groundtruth passage for each input constraint. As we find that GPT-4 can follow the input constraints most of the time, we focus on evaluating passage coherency in two ways: using a GPT-4 zero-shot prompt to provide a 1-10 scalar score, or using human judgments to compare pairs of outputs from different methods.For the former, we sample 5 scores and average them for each task output,and we find these 5 scores usually consistent, with a standard deviation of around 0.56 on average across outputs.For the latter, we employ a subset of the authors in a blind study to compare the coherency of CoT vs.ToT generated passage pairs, where the order of passges is random flipped over 100 inputs.
+
+Baselines. Given the creative nature of the task, both IO and CoT prompts are zero-shot.While the former prompts the LM to directly generate a coherent passage given input constraints, the later prompts the LM to first make a brief plan then write the passage,i.e.the plan serves as the intermediate thought step. We generate $1 0 ~ \mathrm { I O }$ and CoT samples per task.We also consider an iterative-refine $k \leq 5 ,$ method on top of a random IO sample for each task,where the LM is conditioned on input constraints and the last generated passage to decide if the passage is already “perfectly coherent”, and if not generate a refined one.
+
+ToT setup. We build a ToT with depth 2 (and only 1 intermediate thought step)—the LM first generates $k = 5$ plans and votes for the best one (Figure 4), then similarly generate $k = 5$ passages based on the best plan then vote for the best one.Here the breadth limit $b = 1$ ,as only one choice is kept per step. A simple zero-shot vote prompt ("analyze choices below, then conclude which is most promising for the instruction") is used to sample 5 votes at both steps.
+
+Results.Figure 5(a) shows average GPT-4 scores across 1OO tasks,where ToT(7.56) is deemed to generate more coherent passages than IO (6.19) and CoT (6.93) on average. While such an automatic metric might be noisy,Figure 5(b) confirms the finding by showing that humans prefer ToT over CoT in 41 out of 100 passage pairs, while only prefer CoT over ToT in 21 (other 38 pairs are found "similarly coherent"). Lastly,iterative-refine is more effective on this natural language task, where it improves IO coherency score from 6.19 to 7.67,and ToT coherency score from 7.56 to 7.91. We believe itcould be thought of as a third approach to thought generation in the ToT framework, where new thoughts can arise from refining old thoughts instead of i.i.d.or sequentially generated.
+
+![](images/86eb94dd2d0b516038e313736bb8eb03701d9d5e9a43ed7e3cb40ee5c83b22a3.jpg)  
+Figure 4: A step of deliberate search in a randomly picked Creative Writing task. Given the input, the LM samples 5 diferent plans,then votes 5 times to decide which plan is best. The majority choice is used to consequently write the output passage with the same sample-vote procedure.
+
+![](images/8f84c416cb87838aea30c4e3df231aa2034fc2a22f5efd1f291e764425310c93.jpg)  
+Figure 5: Creative Writing results.
+
+<table><tr><td>Method</td><td>Success Rate (%) Letter Word Game</td></tr><tr><td>10 CoT</td><td>38.7 14 0 40.6 15.6 1</td></tr><tr><td>ToT (ours)</td><td>78 60 20</td></tr><tr><td>+best state</td><td>82.4 67.5 35</td></tr><tr><td> -prune</td><td>65.4 41.5 5</td></tr><tr><td>-backtrack</td><td>54.6 20 5</td></tr></table>
+
+Table 3: Mini Crosswords results.
+
+# 4.3Mini crosswords
+
+In Game of 24 and Creative Writing,ToT is relatively shallow —at most 3 thought steps are needed to reach the final output. Here we explore $5 \times 5$ mini crosswords as a harder search problem involving natural language. Again, the goal is not just to solve the task, as more general crosswords can be readily solved with specialized NLP pipelines [34] that leverages large-scale retrieval instead of LM. Rather, we aim to explore the limit of LMas a general problem solver that explores its own thoughts and guides its own exploration with deliberate reasoning as heuristics.
+
+Task setup.We scrape data from GooBix, which contains 156 games of $5 \times 5$ mini crosswords. As we observe adjacent games contain similar clues,we use 2O games with indices $1 , 6 , \cdots , 9 1 , 9 6$ for testing,and games 136,141,146,151,156 for prompting. For each task, the input describes the 5 horizontal clues and 5 vertical clues,and the output should be a board of $5 \times 5 = 2 5$ letters to solve the crosswords. For evaluation, we consider three levels of success: the portion of correct letters (25 per game), words (10 per game), and games.
+
+Baselines. We provide 5 example input-output pairs in the IO prompt, and in the CoT prompt additionally include intermediate words in the order h1..5 then v1..5. We run each prompt for 10 samples and average the results.
+
+ToT setup. We leverage a depth-first search (Algorithm 2) that keeps exploring the most promising subsequent word clue until the state is no longer promising, then backtrack to the parent state to explore alternative thoughts.To make search tractable, subsequent thoughts are constrained not to change any filled words or letters,so that the ToT has at most 1O intermediate steps.For thought generation, at each state we translate all existing thoughts (e.g.“h2.motor; h1.tasks” for the state in Figure 6(a)) into letter constraints for remaining clues (e.g."v1.To heap: tm__-;.") and prompt a proposal prompt 5 times to come up with candidates for where and what to fill in the next word. Importantly, we also prompt the LM to give a confidence level for different thoughts,and aggregate these across proposals to obtain a sorted list of next thoughts to explore (Figure 6(a). For state evaluations, we similarly translate each state into leter constraints for remaining clues,then evaluate for each clue if it is possible to fillgiven the constraints. If any remaining clue is demed “impossible” to fill in (e.g.“v1. To heap: tm_s-"), then the exploration of the state's subtree is pruned and DFS backtracks to its parent to explore the next promising thought. We limit DFS search steps to 100,and simply render the deepest explored state (the first explored one if multiple) into the final output.
+
+![](images/b92ef17d9ff862acec28f259769b8b0a71bcd39ac3994319e0b060e8f88b3c7c.jpg)  
+Figure 6: In Mini Crosswords,(a) how thoughts are proposed and aggregated in a priority queue for depth-first search (DFS),and (b) how a state is evaluated based on the possibility of filling in each remaining word clue,and pruned if any remaining clue is deemed not possible to fill by the LM. Then DFS backtracks to the parent state and explore the next promising thought for clue.
+
+Results. As shown in Table 3, IO and CoT prompting methods perform poorly with a word-level success rate less than $1 6 \%$ ,while ToT significantly improves all metrics,achieving a word-level success rate of $6 0 \%$ and solving 4 out of 20 games. Such an improvement is not surprising, given IO and CoT lack mechanisms to try different clues,make changes to decisions,or backtrack.
+
+Oracle and ablation studies. When outputing from the oracle best DFS state (instead of the heuristically determined best state) per task, ToT performance is even higher and actually solves 7/20 games (Table 3,‘ $^ { + }$ best state"), indicating our simple output heuristics can be readily improved. Interestingly, sometimes when the crosswords game is actually solved, the state evaluator might still deem some words as “impossible" and prune - possibly because $5 \times 5$ crosswords by design have some rare or obselete words that GPT-4 cannot recognize2. Given the state evaluation as a pruning heuristic is imperfect, we also explore ablating the pruning, and find the performance generally worse (Table 3,“-prune"). However, it could actually find the correct solution for 4/20 games (though only outputing 1 via heuristic),3 of which are games ToT+pruning cannot solve within 100 steps. Thus, beter heuristics for DFS pruning are critical for problem solving in this case.Lastly, we confirm the importance of backtracking by running an ablation that keeps filing the most promising clue for at most 20 steps,allowing overwrites. This is similar to a“greedy”BFS search with breadth limit of $b = 1$ ,and performs poorly with a word level success of only $2 \dot { 0 } \%$ (Table 3,“-backtrack").
+
+# 5Related Work
+
+Planning and decision making. Smart planning and decision making are critical to achieving predefined goals. As they are trained on vast amount of world knowledge and human examples,LMs are known to have already absorbed rich commonsense that makes it possible to propose reasonable plans conditioned on problem seting and environmental states [12,42,37,13,35,41,40]. Our proposed ToT approach extends existing planning formulations by considering multiple potentially feasible plans simultaneously at each problem-solving step, and proceeding with the most promising ones. The integration between thought sampling and value feedback organically integrates planning and decision-making mechanisms,enabling effective search inside a solution tree. On the other hand, traditional decision-making procedures usually require training dedicated reward and policy models as in reinforcement learning (for example CHAI [33]), whereas we use the LM itself to provide the value estimates for decision making. RAP [9] is a concurrent work that treats language model reasoning as planning with its internal world model, and proposes a MCTS-based method similar to ToT.However,its tasks are simpler than ours,and its framework lacks the modularity to incorporate different tree search algorithms.
+
+Self-reflection. Using LLMs to assess the viability of their own predictions is becoming an increasingly important procedure in problem solving. [28,20,24] introduced the“self-reflection” mechanism, in which LMs provide feedback to their generation candidates. [4] improves LMs code generation accuracy by injecting feedback messages generated by the LM itself based on its code execution results. Similarly,[17] also introduces “critic”or review steps over the actions and states, deciding the next action to take in solving computer operation tasks. Another recent work very relevant to ours is “self-eval guided decoding” [39]. Similar to our method, self-eval decoding also follows a tree-search procedure with leaves sampled from stochastic beam search decoding, which are then evaluated by LLM itself with carefully prepared self-eval prompts. Their approach however, uses the PAL formulation [8] which represents thoughts as codes,which makes it difficult to tackle chalenging tasks like creative writing which we consider in this paper. Our Tree-of-Thought formulation is thus more versatile and handles challenging tasks on which GPT-4 only achieves very low accuracy with standard prompts.
+
+Program-guided LLM generation. Our proposal is also related to recent advancements that organize LM's behavior with systematic procedures [14,44,6,43] or symbolic program guidance.For example, Schlag et al.[27] embeds LMs in an algorithmic search procedure to help solve problems like question answering step-by-step, in which the search trees are expanded by relevant paragraphs that might provide answers. This approach however differs from ours in that trees are expanded by sampling external paragraphs instead of the LM's own thoughts,and there is no reflection or voting steps. Another approach, $\mathrm { L L M + P }$ [18], goes one step further and delegates the actual planning process to a classical planner.
+
+Classical search methods. Last but not least, our approach can be treated as a modern rendition of classical search methods for problem solving. For example it can be considered as a heuristic search algorithm like $\mathbf { A } ^ { * }$ [10], in which the heuristic at each search node is provided by the LM's selfassessment. From this perspective, our method is also related to NeuroLogic $\mathbf { A } ^ { * }$ esque decoding [19], which is inspired by $\mathbf { A } ^ { * }$ search but introduces look-ahead heuristics that are efficient for LMs to improve the beam-search or top-k sampling decoding. This method however is constrained to sentence generation tasks,whereas our framework are designed for complex, multi-step problem solving guarded by value feedback.
+
+# 6Discussion
+
+Limitations and future directions. Deliberate search such as ToT might not be necessary for many existing tasks that GPT-4 already excels at (see Appendix B.1),and as an initial step this work only explores three relatively simple tasks that challenges GPT-4 (see Appendix B.2 for some GPT-3.5 experiment results) and calls of beter search and planning abilities incorporated with LMs. However, as we begin to deploy LMs for more real-world decision making applications (e.g.coding, data analysis, robotics, etc.), more complex tasks could emerge and present new opportunities to study these research questions. Also, search methods like ToT requires more resources (e.g. GPT-4 API cost) than sampling methods in order to improve task performances, but the modular flexibility of ToT allows users to customize such performance-cost tradeoffs,and ongoing open-source efforts [32] should readily reduce such costs in the near future. More details about cost and effciency are in Appendix B.3. Lastly, this work focuses on using an off-the-shelf LM, and fine-tuning LMs using a ToT-style high-level counterfactual decision making (e.g.deliberating over potential choices for the next paragraph, instead of predicting the next token) might present opportunities to enhance the problem-solving capabilities of LMs.
+
+Conclusion. The associative “System 1” of LMs can be beneficially augmented by a“System $2 ^ { \circ }$ based on searching a tree of possible paths to the solution to a problem. The Tree of Thoughts framework provides a way to translate classical insights about problem-solving into actionable methods for contemporary LMs. At the same time,LMs address a weakness of these classical methods, providing a way to solve complex problems that are not easily formalized, such as creative writing. We see this intersection of LMs with classical approaches to AI as an exciting direction.
+
+# Broader Impact
+
+ToT is a framework that empowers LMs to more autonomously and intelligently make decisions and solve problems. While current tasks are limited to reasoning and search problems, future applications involving interaction with external environments or humans could bring potential danger, e.g.facilitating harmful uses of LMs. On the other hand, ToT also improves the interpretability of model decisions and the opportunity for human alignment, as the resulting representations are readable, high-level language reasoning instead of implicit, low-level token values.
+
+# Acknowledgements
+
+SY and KN acknowledge support from an Oracle Collaborative Research award and the National Science Foundation under Grant No. 2239363. Any opinions,findings,conclusions,or recommendations expressed in this material are those of the author(s)and do not necessarily reflect the views of the National Science Foundation. SY is also supported by the Harold W.Dodds Fellowship from Princeton.
+
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+
+# A Code, Prompts, Trajectories
+
+All code is available at https://github.com/princeton-nlp/tree-of-thought-llm.
+
+All prompts are available at https://github.com/princeton-nlp/tree-of-thought-llm/ tree/master/src/tot/prompts.
+
+Trajectories are available at https://github.com/princeton-nlp/tree-of-thought-llm/ tree/master/logs.
+
+# BAdditional Experiment Results
+
+Given the motivation of exploring and extending the capability frontier of language models, our experiments in the main paper have focused on a setup with the state-of-the-art language model (GPT-4),and three hard tasks invented to chalenge it. Here, we report additional experiments with weaker LLM or easier tasks,and discusscost and efficiency.
+
+<table><tr><td>GPT-4</td><td>GPT-3.5</td></tr><tr><td>10 6.19</td><td>4.47</td></tr><tr><td>CoT 6.93</td><td>5.16</td></tr><tr><td>ToT 7.56</td><td>6.62</td></tr></table>
+
+Table 5: Game of 24 with GPT-4 vs GPT-3.5.   
+
+<table><tr><td>GSM8K</td><td>StrategyQA</td></tr><tr><td>10 51</td><td>73</td></tr><tr><td>CoT 86</td><td>82</td></tr><tr><td>ToT 90</td><td>83</td></tr></table>
+
+Table 4:New tasks with zero-shot ToT and GPT-4.
+
+<table><tr><td></td><td>GPT-4</td><td>GPT-3.5</td></tr><tr><td>I0</td><td>7.3%</td><td>6%</td></tr><tr><td>CoT</td><td>4.0%</td><td>3%</td></tr><tr><td>ToT</td><td>74%</td><td>19%</td></tr></table>
+
+Table 6: Creative Writing with GPT-4 vs. GPT-3.5.
+
+# B.1Extension to new tasks (GSM8k, StrategyQA) with zero-shot ToT
+
+While more common NLP tasks might be too easy for GPT-4 and do not require ToT(which is why we considered harder new tasks), we believe applying ToT to new tasks could be straightforward. For example, we implemented a simple and generic zero-shot ToT-BFS similar to creative writing (sample 5 problem solving strategies then vote for the best one; then sample 5 solutions based on the best strategy then vote for the best one) for GSM8K and StrategyQA with few extra lines of code:
+
+# define the answer format of new tasks gsm8k_format $=$ ("the answer is n" where n is a number' strategyqa_format $=$ ‘either "the answer is yes" or "the answer is no"’ # define zero-shot io prompting standard_prompt $=$ ‘Answer the following question with {format}: {input}' # define thought format for zero-shot cot and zero-shot tot cot_prompt $\begin{array} { r l } { = } & { { } \hat { \pmb { \mathscr { \imath } } } \pmb { \mathscr { \imath } } \pmb { \mathscr { \imath } } \pmb { \mathscr { \imath } } } \end{array}$ Answer the following question: {input}
+
+Make a strategy then write. Your output should be of the following format:
+
+Strategy:
+
+Your strategy about how to answer the question.
+
+Answer:
+
+Your answer to the question.It should end with {format}. ，，，
+
+# define zero-shot voting used for zero-shot tot   
+vote_prompt $\begin{array} { r l } { \mathbf { \Sigma } } & { { } = \mathbf { \Sigma } ^ { \textit { \textbf { c } } \texttt { \textsf { c } } \mathcal { \epsilon } \textbf { \Lambda } } } \end{array}$ Given an instruction and several choices, decide which choice is most promising.   
+Analyze each choice in detail,then conclude in the last line "The best choice is {s}"，where s the integer id of the choice. ，，
+
+We evaluated on a subset of 100 random GSM8K test and StrategyQA dev questions. As shown in Table 4 and as expected,ToT improves over CoTon both tasks (but only slightly, given GPT-4 $+ \mathrm { C o T }$ is already very good on such tasks, and StrategyQA's botleneck is external knowledge, not reasoning). Considering computational costs, it is more suitable to try smaller LLMs $^ +$ ToT for traditional NLP tasks,or GPT $\cdot 4 + \mathrm { T o T }$ for hard tasks that challenge GPT $. 4 + \mathrm { C o T ' }$ reasoning.
+
+# B.2Extension to new LMs (GPT-3.5)
+
+To understand how ToT works with other LLMs, we also ran GPT-3.5-turbo for Creative Writing (Table 6) and Game of 24 (Table 5). On both tasks, $\mathbf { \partial } ^ { \cdot } \mathbf { \Phi } ^ { \cdot } \mathbf { T o T } > \mathbf { C o T } > \mathbf { I O } ^ { \prime }$ remains true for GPT-3.5. On Creative Writing,we find GPT $3 . 5 \mathrm { + T o T }$ outperform GPT $_ { 4 + 1 0 }$ ,and similar to GPT- $4 { + } \mathrm { C o T }$ ，which suggests ToT could also work well on weaker language models.
+
+On Game of 24 (we changed 1-shot proposal prompt to 3-shot to make it work), GPT- $. 3 . 5 +$ ToT's $19 \%$ is far worse than GPT- $^ { 4 + }$ ToT's $74 \%$ . To further understand the importance of generation vs. evaluation, we ran GPT-4 generation $+ \mathrm { G P T } { - 3 . 5 }$ evaluation $( 6 4 \% )$ and GPT-3.5 generation $^ +$ GPT-4 evaluation $( 3 1 \% )$ . This suggests the game's bottleneck is thought generation,and different generation/evaluation language models might attain decent results while reducing costs.
+
+# B.3Cost and effciency
+
+Running ToT requires significantly more computations than IO or CoT prompting. For example, in Game of 24 (Table 7 below), solving a problem with ToT requires $5 . 5 \mathrm { k }$ completion tokens, close to $1 0 0 \mathrm { C o T }$ trials ( $\mathrm { 6 . 7 k }$ tokens). But the performance of ToT is better than best of 1OO independent CoT trials.
+
+<table><tr><td>Game of 24</td><td>Generate/Prompt tokens</td><td>Cost per case</td><td> Success</td></tr><tr><td>IO (best of 100)</td><td>1.8k /1.0k</td><td>$0.13</td><td>33%</td></tr><tr><td>CoT (best of 100)</td><td>6.7k /2.2k</td><td>$0.47</td><td>49%</td></tr><tr><td>ToT</td><td>5.5k / 1.4k</td><td>$0.74</td><td>74%</td></tr></table>
+
+Table 7: Cost analysis on Game of 24.
+
+On Creative Writing (Table 8 below), we found ToT takes around ${ 5 } \mathrm { x }$ completion tokens and money cost, which is intuitive as $b = 5$ and most tokens are generated passages.
+
+<table><tr><td>Creative Writing</td><td>Generate/Prompt tokens</td><td>Cost per case</td></tr><tr><td>I0</td><td>0.9k /0.4k</td><td>$0.06</td></tr><tr><td>CoT</td><td>0.9k/0.4k</td><td>$0.07</td></tr><tr><td>ToT</td><td>4k /2.9k</td><td>$0.32</td></tr></table>
+
+Table 8: Cost analysis on Game of 24.
+
+So completing Game of 24 and Creative Writing's main ToT experiments cost around $0 . 7 4 \times 1 0 0 +$ $0 . 3 2 \times 1 0 0 = 1 0 6$ dollars.Crosswords’DFS experiments should be also within 1OO dollars.In general, cost and efficiency of ToT highly depend on the prompts and search algorithms used,and could require 5-1OO times more generated tokens than CoT. Some actionable insights:
+
+· We recommend using ToT on tasks requiring deliberate reasoning, on which CoT struggles. · Flexibility of ToT allows some performance-cost tradeoff, e.g., change beam size or vote number in BFS,few-shot vs. zero-shot prompting,GPT-3.5 vs. GPT-4,etc. One could configure the setup based on some resource constraints or performance goal. · There is much space for improving effciency, e.g., BFS could early stop when solution is found,or trim down beam size to when some thoughts are "impossible". · We believe that more computation is indeed required in order for the model to achieve stronger intelligence,and this should not become a blocking issue as in the long run, (opensource) LMs will become much cheaper and more efficient. It is also a great direction how to better train/finetune LMs for thought generation and/or evaluation.

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+    {
+        "type": "text",
+        "text": "Tree of Thoughts: Deliberate Problem Solving with Large Language Models ",
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+    {
+        "type": "text",
+        "text": "Shunyu Yao Princeton University ",
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+    {
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+        "text": "Dian Yu Google DeepMind ",
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+    {
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+        "text": "Jeffrey Zhao Google DeepMind ",
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+    {
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+    {
+        "type": "text",
+        "text": "Thomas L. Griffiths Princeton University ",
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Yuan Cao Google DeepMind ",
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+    {
+        "type": "text",
+        "text": "Karthik Narasimhan Princeton University ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "Language models are increasingly being deployed for general problem solving across a wide range of tasks,but are still confined to token-level, left-to-right decision-making processes during inference. This means they can fall short in tasks that require exploration, strategic lookahead, or where initial decisions play a pivotal role. To surmount these challenges,we introduce a new framework for language model inference,“Tree of Thoughts”(ToT),which generalizes over the popular“Chain of Thought” approach to prompting language models,and enables exploration over coherent units of text (\"thoughts\") that serve as intermediate steps toward problem solving. ToT allows LMs to perform deliberate decision making by considering multiple different reasoning paths and self-evaluating choices to decide the next course of action, as well as looking ahead or backtracking when necessary to make global choices. Our experiments show that ToT significantly enhances language models’ problem-solving abilities on three novel tasks requiring non-trivial planning or search: Game of 24, Creative Writing,and Mini Crosswords. For instance, in Game of 24, while GPT-4 with chain-of-thought prompting only solved $4 \\%$ of tasks,our method achieved a success rate of $74 \\%$ . Code repo with all prompts: https://github.com/princeton-nlp/tree-of-thought-llm. ",
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+    {
+        "type": "text",
+        "text": "1Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Originally designed to generate text, scaled-up versions of language models (LMs) such as GPT [25, 26,1,23] and PaLM [5] have been shown to be increasingly capable of performing an ever wider range of tasks requiring mathematical, symbolic, commonsense,and knowledge reasoning. It is perhaps surprising that underlying all this progress is still the original autoregressive mechanism for generating text, which makes token-level decisions one by one and in a left-to-right fashion. Is such a simple mechanism sufficient for a LM to be built toward a general problem solver? If not, what problems would challenge the current paradigm,and what should be alternative mechanisms? ",
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+        "text": "The literature on human cognition provides some clues to answer these questions. Research on “dual process” models suggests that people have two modes in which they engage with decisions - a fast, automatic,unconscious mode (\"System 1\") and a slow, deliberate, conscious mode (\"System 2\") [30,31,16,15]. These two modes have previously been connected to a variety of mathematical models used in machine learning. For example, research on reinforcement learning in humans and other animals has explored the circumstances under which they engage in associative “model free\" learning or more deliberative“model based” planning [7]. The simple associative token-level choices of LMs are also reminiscent of “System 1\",and thus might benefit from augmentation by a more deliberate “System $2 ^ { \\circ }$ planning process that (1) maintains and explores diverse alternatives for current choices instead of just picking one,and (2) evaluates its current status and actively looks ahead or backtracks to make more global decisions. ",
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+    {
+        "type": "image",
+        "img_path": "images/b7fb3f215b6ad763cd26c1185478d0e2a29d7cb659f6ab760a80492664d522bb.jpg",
+        "image_caption": [
+            "Figure 1: Schematic ilustrating Various approaches to problem solving with LLMs. Each rectangle box represents a thought, which is a coherent language sequence that serves as an intermediate step toward problem solving. See concrete examples of how thoughts are generated, evaluated, and searched in Figures 2,4,6. "
+        ],
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+    {
+        "type": "text",
+        "text": "To design such a planning process, we return to the origins of artificial intelligence (and cognitive science), drawing inspiration from the planning processes explored by Newell, Shaw,and Simon starting in the 195Os [21,22]. Newell and colleagues characterized problem solving [21] as search through a combinatorial problem space,represented as a tree.We thus propose the Tree of Thoughts (ToT) framework for general problem solving with language models. As Figure 1 illustrates, while existing methods (detailed below) sample continuous language sequences for problem solving, ToT actively maintains a tree of thoughts,where each thought is a coherent language sequence that serves as an intermediate step toward problem solving (Table 1). Such a high-level semantic unit allows the LM to self-evaluate the progress different intermediate thoughts make towards solving the problem through a deliberate reasoning process that is also instantiated in language (Figures 2,4,6). This implementation of search heuristics via LM self-evaluation and deliberation is novel,as previous search heuristics are either programmed or learned. Finally，we combine this language-based capability to generate and evaluate diverse thoughts with search algorithms,such as breadth-first search (BFS) or depth-first search (DFS), which allow systematic exploration of the tree of thoughts with lookahead and backtracking. ",
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+    {
+        "type": "text",
+        "text": "Empirically, we propose three new problems that challenge existing LM inference methods even with the state-of-the-art language model, GPT-4 [23]: Game of 24, Creative Writing,and Crosswords (Table 1). These tasks require deductive,mathematical, commonsense,lexical reasoning abilities, and a way to incorporate systematic planning or search. We show ToT obtains superior results on allthree tasks by being general and flexible enough to support different levels of thoughts,different ways to generate and evaluate thoughts,and diferent search algorithms that adapt to the nature of different problems. We also analyze how such choices affect model performances via systematic ablations and discuss future directions to better train and use LMs. ",
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+    {
+        "type": "text",
+        "text": "2Background ",
+        "text_level": 1,
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+        "text": "We first formalize some existing methods that use large language models for problem-solving, which our approach is inspired by and later compared with. We use $p _ { \\theta }$ to denote a pre-trained LM with parameters $\\theta$ ，and lowercase letters $x , y , z , s , \\cdots$ to denote a language sequence, i.e. $x =$ $( x [ 1 ] , \\hat { \\cdot } \\cdot \\cdot , x [ n ] )$ where each $x [ i ]$ is a token, so that $\\begin{array} { r } { p _ { \\theta } ( x ) = \\prod _ { i = 1 } ^ { n } p _ { \\theta } ( x [ i ] | x [ 1 . . . i ] ) } \\end{array}$ . We use uppercase letters $S , \\cdots$ to denote a collection of language sequences. ",
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+        "text": "Input-output (IO) prompting is the most common way to turn a problem input $x$ into output $y$ with LM: $y \\sim p _ { \\boldsymbol \\theta } ( y | \\mathrm { p r o m p t } _ { I O } ( x ) )$ ，where $\\mathsf { p r o m p t } _ { I O } ( x )$ wraps input $x$ with task instructions and/or few-shotinputoutputexamples.Forsimplicityetsde $p _ { \\theta } ^ { \\mathrm { p r o m p t } } ( \\mathsf { o u t p u t } \\mid \\mathsf { i n p u t } ) =$ $p _ { \\theta } ( \\mathrm { { o u t p u t } \\mid p r o m p t ( i n p u t ) } )$ , so that IO prompting can be formulated as $y \\sim p _ { \\theta } ^ { I O } ( y | x )$ ： ",
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+        "type": "text",
+        "text": "Chain-of-thought $\\mathbf { ( C o T ) }$ prompting [38] was proposed to address cases where the mapping of input $x$ to output $y$ is non-trivial (e.g. when $x$ is a math question and $y$ is the final numerical answer). The key idea is to introduce a chain of thoughts $z _ { 1 } , \\cdots , z _ { n }$ to bridge $x$ and $y$ ，where each $z _ { i }$ is a coherent language sequence that serves as a meaningful intermediate step toward problem solving (e.g. $z _ { i }$ could be an intermediate equation for math QA). To solve problems with CoT, each thought $\\boldsymbol { z } _ { i } \\sim p _ { \\theta } ^ { C o T } ( \\boldsymbol { z } _ { i } \\mid x , \\boldsymbol { z } _ { 1 \\cdots i - 1 } )$ is sampledsequentially,thentheoutput $y \\sim p _ { \\theta } ^ { C o T } ( y | x , z _ { 1 } . . . n )$ In practice, $[ z _ { 1 \\cdots n } , y ] \\sim p _ { \\theta } ^ { C o T } ( z _ { 1 \\cdots n } , y | x )$ is sampled asacontiuouslanguagesequenceadthe decomposition of thoughts (e.g.is each $z _ { i }$ a phrase, a sentence, or a paragraph) is left ambiguous. ",
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+    {
+        "type": "text",
+        "text": "Self-consistency with CoT (CoT-SC) [36] is an ensemble approach that samples $k$ i.i.d. chains of thought: $[ z _ { 1 \\cdots n } ^ { ( i ) } , y ^ { ( i ) } ] \\sim p _ { \\theta } ^ { C o T } ( z _ { 1 \\cdots n } , y | x )$ $( i = 1 \\cdots k )$ thenretusteostfrequentoutput arg $\\operatorname* { m a x } _ { y }$ # $\\{ i \\mid y ^ { ( i ) } = y \\}$ .CoT-SC improves upon CoT, because there are generally different thought processes for the same problem (e.g. different ways to prove the same theorem), and the output decision can be more faithful by exploring a richer set of thoughts. However, within each chain there is no local exploration of different thought steps,and the “most frequent” heuristic only applies when the output space is limited (e.g. multi-choice QA). ",
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+    {
+        "type": "text",
+        "text": "3Tree of Thoughts: Deliberate Problem Solving with LM ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A genuine problem-solving process involves the repeated use of available information to initiate exploration, which discloses,in turn, more information until a way to attain the solution is finally discovered.— Newell et al. [21] ",
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+    {
+        "type": "text",
+        "text": "Research on human problem-solving suggests that people search through a combinatorial problemspace -a tree where the nodes represent partial solutions,and the branches correspond to operators that modify them [21,22].Which branch to take is determined by heuristics that help to navigate the problem-space and guide the problem-solver towards a solution. This perspective highlights two key shortcomings of existing approaches that use LMs to solve general problems: 1) Locally, they do not explore different continuations within a thought process-the branches of the tree.2) Globally, they do not incorporate any type of planning,lookahead,or backtracking to help evaluate these different options - the kind of heuristic-guided search that seems characteristic of human problem-solving. ",
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+    {
+        "type": "text",
+        "text": "To address these shortcomings, we introduce Tree of Thoughts (ToT),a paradigm that allows LMs to explore multiple reasoning paths over thoughts (Figure 1(c)). ToT frames any problem as a search over a tree, where each node is a state $s = [ x , z _ { 1 \\cdots i } ]$ representing a partial solution with the input and the sequence of thoughts so far. A specific instantiation of ToT involves answering four questions: 1. How to decompose the intermediate process into thought steps; 2. How to generate potential thoughts from each state; 3. How to heuristically evaluate states; 4. What search algorithm to use. ",
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+    {
+        "type": "text",
+        "text": "1.Thought decomposition. While CoT samples thoughts coherently without explicit decomposition, ToT leverages problem properties to design and decompose intermediate thought steps.As Table 1 shows,depending on diferent problems,a thought could be a couple of words (Crosswords),a line of equation (Game of 24), or a whole paragraph of writing plan (Creative Writing). In general,a thought should be “small\" enough so that LMs can generate promising and diverse samples (e.g. generating a whole book is usually too“big” to be coherent), yet“big” enough so that LMs can evaluate its prospect toward problem solving (e.g. generating one token is usually too “small\" to evaluate). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.Thought generator $G ( p _ { \\theta } , s , k )$ . Given a tree state $s = [ x , z _ { 1 } . . . i ]$ , we consider two strategies to generate $k$ candidates for the next thought step: ",
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+    {
+        "type": "text",
+        "text": "(a) Sample i.i.d.thoughts from a CoT prompt (Creative Writing，Figure 4): $z ^ { ( j ) } \\sim$ $p _ { \\theta } ^ { C o \\hat { T _ { ( z _ { i + 1 } | s } ) } } = p _ { \\theta } ^ { \\check { C } o T } ( z _ { i + 1 } | x , z _ { 1 \\cdots i } ) \\ ( \\dot { j } = \\dot { 1 } \\cdots k )$ . This works better when the thought space is rich (e.g.each thought is a paragraph),and i.i.d.samples lead to diversity;   \n(b) Propose thoughts sequentially using a“propose prompt”(Game of 24, Figure 2; Crosswords, Figure 6): $[ z ^ { ( 1 ) } , \\cdot \\cdot \\cdot , z ^ { ( k ) } ] \\sim p _ { \\theta } ^ { p r o p o s e } ( z _ { i + 1 } ^ { ( 1 \\cdots k ) } \\mid s )$ ]\\~propose() . This works better when the thought space is more constrained (e.g.each thought is just a word or a line),so proposing different thoughts in the same context avoids duplication. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3. State evaluator $V ( p _ { \\theta } , S )$ . Given a frontier of different states,the state evaluator evaluates the progress they make towards solving the problem, serving as a heuristic for the search algorithm to determine which states to keep exploring and in which order. While heuristics are a standard approach to solving search problems, they are typically either programmed (e.g. DeepBlue [3]) or learned (e.g. AlphaGo [29]). We propose a third alternative,by using the LM to deliberately reason about states.When applicable,such a deliberate heuristic can be more flexible than programmed rules,and more sample-efficient than learned models. Similar to the thought generator, we consider two strategies to evaluate states either independently or together: ",
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+        "text": "(a) Value each state independently: $V ( p _ { \\theta } , S ) ( s ) ~ \\sim ~ p _ { \\theta } ^ { v a l u e } ( v | s ) ~ \\forall s ~ \\in ~ S$ where a value prompt reasons about the state $s$ to generate a scalar value $v$ (e.g.1-10) or a classification (e.g.sure/likely/impossible) that could be heuristically turned into a value.The basis of such evaluative reasoning can vary across problems and thought steps. In this work, we explore evaluation via few lookahead simulations (e.g.quickly confirm that 5,5,14 can reach 24 via $5 + 5 + 1 4$ ,or “hot_l\" can mean“inn”via filling“e”in“-\") plus commonsense (e.g.1 2 3 are too smallto reach 24, or no word can start with “tzxc\"). While the former might promote “good” states,the latter could help eliminate “bad” states. Such valuations do not need to be perfect,and only need to be approximately helpful for decision making. (b) Vote across states: $V ( p _ { \\theta } , S ) ( s ) = \\mathbb { 1 } [ s = s ^ { * } ]$ ， where a“good” state $s ^ { * } \\sim p _ { \\theta } ^ { v o t e } ( s ^ { * } | S )$ is voted out based on deliberately comparing different states in $S$ in a vote prompt. When problem success is harder to directly value (e.g. passage coherency), it is natural to to instead compare different partial solutions and vote for the most promising one. This is similar in spirit to a“step-wise” self-consistency strategy,i.e.cast “which state to explore” as a multi-choice QA,and use LM samples to vote for it. ",
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+    {
+        "type": "text",
+        "text": "For both strategies, we could prompt the LM multiple times to aggregate the value or vote results to trade time/resource/cost for more faithful/robust heuristics. ",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm1 ToT-BFS(x,pe,G,k,V,T,b)</td><td>Algorithm 2 ToT-DFS(s,t,pe,G,k,V,T,Uth)</td></tr><tr><td>Require: Input x,LM po, thought generator G(Require: Current state s, step t,LM po, thought &amp; size limit k, states evaluator V(),step limit T， generator G() and size limit k, states evaluator</td><td></td></tr><tr><td>breadth limit b. So←{x}</td><td>V(), step limit T, threshold Uth if t &gt; T then record output G(pe, s,1)</td></tr><tr><td>for t=1,..,T do</td><td>end if</td></tr><tr><td>St←{[s,z]|s∈ St-1,zt ∈G(p0,s,k)}</td><td>for s&#x27; ∈ G(pe,s,k) do&gt; sorted candidates</td></tr><tr><td>Vt ←V(po,St)</td><td>if V(pe,{s&#x27;})(s) &gt; Uthres then &gt; pruning</td></tr><tr><td>St ← arg maxscs&#x27;,|s|=b∑ses Vt(s)</td><td>DFS(s&#x27;,t+1)</td></tr><tr><td>end for return G(pe,arg maxs∈Sr Vr(s),1)</td><td>end if</td></tr></table>",
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+        "text": "4. Search algorithm.Finally, within the ToT framework,one can plug and play different search algorithms depending on the tree structure. We explore two relatively simple search algorithms and leave more advanced ones (e.g. $\\mathbf { A } ^ { * }$ [11],MCTS [2]) for future work: ",
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+        "type": "text",
+        "text": "(a)Breadth-first search (BFS) (Algorithm 1) maintains a set of the $b$ most promising states per step. This is used for Game of 24 and Creative Writing where the tree depth is limit $( T \\leq 3 )$ ,and initial thought steps can be evaluated and pruned to a small set $( b \\leq 5 )$ ）   \n(b) Depth-first search (DFS) (Algorithm 2) explores the most promising state first, until the final output is reached $( t > T )$ ,or the state evaluator deems it impossible to solve the problem from the current $s$ $( V ( p _ { \\theta } , \\{ s \\} ) ( s ) \\le v _ { t h }$ for a value threshold $v _ { t h }$ ). In the latter case, the subtree from $s$ is pruned to trade exploration for exploitation. In both cases,DFS backtracks to the parent state of $s$ to continue exploration. ",
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+        "type": "text",
+        "text": "Conceptually,ToT has several benefits as a method for general problem-solving with LMs: (1) Generality. IO,CoT, CoT-SC,and self-refinement can be seen as special cases of ToT(i.e. trees of limited depth and breadth; Figure 1). (2) Modularity. The base LM, as well as the thought decomposition, generation, evaluation, and search procedures can allbe varied independently. (3) Adaptability. Different problem properties,LM capabilities,and resource constraints can be accommodated. (4) Convenience.No extra training is needed, just a pre-trained LM is suficient. The next section will show how these conceptual benefits translate to strong empirical performance in diferent problems. ",
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+    {
+        "type": "text",
+        "text": "4Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We propose three tasks that are hard even when sampling from the state-of-the-art language model. GPT-4 [23], using standard IO prompting or chain-of-thought (CoT) prompting. We show how deliberate search in trees of thoughts (ToT) produces beter results,and more importantly, interesting and promising new ways to use language models to solve problems requiring search or planning. Unless otherwise stated, we perform experiments using a Chat Completion mode GPT $\\dot { 4 } ^ { 1 }$ with a sampling temperature of 0.7. ",
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+        "type": "table",
+        "img_path": "images/55fdc8a6c0e25deb3ab4399affa9ecb03455e27747f445bb654219a79bb7c56d.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Task overview. Input, output, thought examples are in blue. "
+        ],
+        "table_body": "<table><tr><td></td><td>Game of 24</td><td>Creative Writing</td><td>5x5 Crosswords</td></tr><tr><td>Input</td><td>4 numbers (4 9 10 13)</td><td>4 random sentences</td><td>10 clues (h1. presented;..)</td></tr><tr><td>Output</td><td>An equation to reach 24 (13-9)*(10-4)=24</td><td>A passage of 4 paragraphs ending in the 4 sentences</td><td>5x5 letters: SHOWN; WIRRA; AVAIL;..</td></tr><tr><td>Thoughts</td><td>3 intermediate equations (13-9=4 (left 4,4,10); 10- 4=6 (left 4,6); 4*6=24)</td><td>Ashort writingplan (1.Introduce a book that connects...)</td><td>Words to fill in for clues: (h1. shown; v5. naled; .)</td></tr><tr><td>#ToT steps</td><td>3</td><td>1</td><td>5-10 (variable)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.1 Game of 24 ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Game of 24 is a mathematical reasoning challenge, where the goal is to use 4 numbers and basic arithmetic operations $( + - ^ { * } / )$ to obtain 24. For example, given input $^ { \\bullet } 4 9 1 0 1 3 ^ { \\prime \\prime }$ , a solution output could be $\\ ^ { \\cdot } ( 1 0 - 4 ) \\ ^ { * } \\ ( 1 3 - 9 ) = 2 4 ^ { , 9 }$ ： ",
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+    {
+        "type": "image",
+        "img_path": "images/1e51ca3d9ec5118d046f44d504919561fcccabdb24a756895c2162f29b531ffb.jpg",
+        "image_caption": [
+            "Figure 2: ToT in a game of 24. The LM is prompted for (a) thought generation and (b) valuation. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Task Setup. We scrape data from 4nums.com, which has 1,362 games that are sorted from easy to hard by human solving time,and use a subset of relatively hard games indexed 901-1,0Oo for testing. For each task, we consider the output as success if it is a valid equation that equals 24 and uses the input numbers each exactly once.We report the success rate across 1OO games as the metric. ",
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+        "type": "text",
+        "text": "Baselines.We use a standard input-output (IO) prompt with 5 in-context examples.For chain-ofthought (CoT) prompting, we augment each input-output pair with 3 intermediate equations, each operating on two remaining numbers. For example, given input $^ { \\dots } 4 9 1 0 1 3 ^ { \\prime \\prime }$ ,the thoughts could be $\\cdot 1 3 - 9 = 4$ (left: 4 4 10); $1 0 - 4 = 6$ (left: $4 6$ ） $4 \\ast 6 = 2 4$ (left: 24)\". For each game, we sample IO and CoT prompting for 100 times for average performance. We also consider a CoT self-consistency baseline,which takes the majority output from $1 0 0 \\mathrm { C o T }$ samples,and an iterative-refine approach on top of an IO sample for at most 10 iterations. At each iteration, the LM is conditioned on all previous history to “reflect on your mistakes and generate a refined answer”if the output is incorrect. Note that it uses groundtruth feedback signals about equation correctness. ",
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+    {
+        "type": "text",
+        "text": "ToT Setup. To frame Game of 24 into ToT, it is natural to decompose the thoughts into 3 steps, each an intermediate equation. As shown in Figure 2(a), at each tree node, we exact the remaining numbers and prompt the LM to propose some possible next steps. The same “propose prompt” is used for all 3 thought steps,though it only has one example with 4 input numbers. We perform a breadth-first search (BFS) in ToT, where at each step we keep the best $b = 5$ candidates. To perform deliberate BFS in ToT,as shown in Figure 2(b), we prompt LM to evaluate each thought candidate as “sure/maybe/impossible” with regard to reaching 24. The aim is to promote correct partial solutions that can be verdicted within few lookahead trials,and eliminate impossible partial solutions based on “too big/small” commonsense,and keep the rest “maybe\". We sample values 3 times for each thought. ",
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+    {
+        "type": "table",
+        "img_path": "images/9a2197a2a3fc6e29a108809a2f1a35f388c5a0579c122de6e089084833d09991.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Game of 24 Results. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td> Success</td></tr><tr><td>IO prompt</td><td>7.3%</td></tr><tr><td>CoT prompt</td><td>4.0%</td></tr><tr><td>CoT-SC (k=100) ToT (ours) (b=1)</td><td>9.0% 45%</td></tr><tr><td>ToT (ours) (b=5)</td><td>74%</td></tr><tr><td>IO + Refine (k=10)</td><td>27%</td></tr><tr><td>IO (best of 100)</td><td>33%</td></tr><tr><td>CoT (best of 100)</td><td>49%</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/d9810d9d96bc205c2dc3f1adc0e2933a0adab5854989f8701a562674a3c12cef.jpg",
+        "image_caption": [
+            "Figure 3: Game of 24 (a) scale analysis & (b) error analysis. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Results. As shown in Table 2,IO, CoT,and CoT-SC prompting methods perform badly on the task, achieving only $7 . 3 \\%$ $4 . 0 \\%$ ,and $9 . 0 \\%$ success rates.In contrast,ToT with a breadth of $b = 1$ already achieves a success rate of $4 5 \\%$ ，while $b = 5$ achieves $7 4 \\%$ .We also consider an oracle setup for IO/CoT, by calculating the success rate using best of $k$ samples ( $1 \\leq k \\leq 1 0 0 $ ). To compare IO/CoT (best of $\\mathbf { k }$ ） with ToT,we consider calculating the tree nodes visited per task in ToT across $b = 1 \\cdots 5$ and map the 5 success rates in Figure 3(a), treating IO/CoT (best of $k$ )asvisiting $k$ nodes in a bandit. Not surprisingly, CoT scales better than IO,and best of $1 0 0 \\mathrm { C o T }$ samples achieve a success rate of $4 9 \\%$ ,but still much worse than exploring more nodes in ToT $( b > 1 )$ ). ",
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+    {
+        "type": "text",
+        "text": "Error analysis.Figure 3(b) breaks down at which step CoT and ToT samples fail the task, i.e.the thought (in CoT) or all $b$ thoughts (in ToT) are invalid or impossible to reach 24.Notably,around $60 \\%$ of CoT samples already failed the task after generating the first step, or equivalently, the first three words (e.g. $\\mathbf { \\^ 6 4 + 9 ^ { 9 } } ,$ . This highlights the issues with direct left-to-right decoding. ",
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+    {
+        "type": "text",
+        "text": "4.2 Creative writing ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Next, we invent a creative writing task where the input is 4 random sentences and the output should be a coherent passage with 4 paragraphs that end in the 4 input sentences respectively. Such a task is open-ended and exploratory,and challenges creative thinking as well as high-level planning. ",
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+        "type": "text",
+        "text": "Task setup.We sample random sentences from randomwordgenerator.com to form 1OO inputs,and there is no groundtruth passage for each input constraint. As we find that GPT-4 can follow the input constraints most of the time, we focus on evaluating passage coherency in two ways: using a GPT-4 zero-shot prompt to provide a 1-10 scalar score, or using human judgments to compare pairs of outputs from different methods.For the former, we sample 5 scores and average them for each task output,and we find these 5 scores usually consistent, with a standard deviation of around 0.56 on average across outputs.For the latter, we employ a subset of the authors in a blind study to compare the coherency of CoT vs.ToT generated passage pairs, where the order of passges is random flipped over 100 inputs. ",
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+        "type": "text",
+        "text": "Baselines. Given the creative nature of the task, both IO and CoT prompts are zero-shot.While the former prompts the LM to directly generate a coherent passage given input constraints, the later prompts the LM to first make a brief plan then write the passage,i.e.the plan serves as the intermediate thought step. We generate $1 0 ~ \\mathrm { I O }$ and CoT samples per task.We also consider an iterative-refine $k \\leq 5 ,$ method on top of a random IO sample for each task,where the LM is conditioned on input constraints and the last generated passage to decide if the passage is already “perfectly coherent”, and if not generate a refined one. ",
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+    {
+        "type": "text",
+        "text": "ToT setup. We build a ToT with depth 2 (and only 1 intermediate thought step)—the LM first generates $k = 5$ plans and votes for the best one (Figure 4), then similarly generate $k = 5$ passages based on the best plan then vote for the best one.Here the breadth limit $b = 1$ ,as only one choice is kept per step. A simple zero-shot vote prompt (\"analyze choices below, then conclude which is most promising for the instruction\") is used to sample 5 votes at both steps. ",
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+    {
+        "type": "text",
+        "text": "Results.Figure 5(a) shows average GPT-4 scores across 1OO tasks,where ToT(7.56) is deemed to generate more coherent passages than IO (6.19) and CoT (6.93) on average. While such an automatic metric might be noisy,Figure 5(b) confirms the finding by showing that humans prefer ToT over CoT in 41 out of 100 passage pairs, while only prefer CoT over ToT in 21 (other 38 pairs are found \"similarly coherent\"). Lastly,iterative-refine is more effective on this natural language task, where it improves IO coherency score from 6.19 to 7.67,and ToT coherency score from 7.56 to 7.91. We believe itcould be thought of as a third approach to thought generation in the ToT framework, where new thoughts can arise from refining old thoughts instead of i.i.d.or sequentially generated. ",
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+    {
+        "type": "image",
+        "img_path": "images/86eb94dd2d0b516038e313736bb8eb03701d9d5e9a43ed7e3cb40ee5c83b22a3.jpg",
+        "image_caption": [
+            "Figure 4: A step of deliberate search in a randomly picked Creative Writing task. Given the input, the LM samples 5 diferent plans,then votes 5 times to decide which plan is best. The majority choice is used to consequently write the output passage with the same sample-vote procedure. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/8f84c416cb87838aea30c4e3df231aa2034fc2a22f5efd1f291e764425310c93.jpg",
+        "image_caption": [
+            "Figure 5: Creative Writing results. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "table",
+        "img_path": "images/52f94457b8ac6236fc8a8ca62e71743461ac4aa19b8680c997068e1747357e4a.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Mini Crosswords results. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>Success Rate (%) Letter Word Game</td></tr><tr><td>10 CoT</td><td>38.7 14 0 40.6 15.6 1</td></tr><tr><td>ToT (ours)</td><td>78 60 20</td></tr><tr><td>+best state</td><td>82.4 67.5 35</td></tr><tr><td> -prune</td><td>65.4 41.5 5</td></tr><tr><td>-backtrack</td><td>54.6 20 5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.3Mini crosswords ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In Game of 24 and Creative Writing,ToT is relatively shallow —at most 3 thought steps are needed to reach the final output. Here we explore $5 \\times 5$ mini crosswords as a harder search problem involving natural language. Again, the goal is not just to solve the task, as more general crosswords can be readily solved with specialized NLP pipelines [34] that leverages large-scale retrieval instead of LM. Rather, we aim to explore the limit of LMas a general problem solver that explores its own thoughts and guides its own exploration with deliberate reasoning as heuristics. ",
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+    {
+        "type": "text",
+        "text": "Task setup.We scrape data from GooBix, which contains 156 games of $5 \\times 5$ mini crosswords. As we observe adjacent games contain similar clues,we use 2O games with indices $1 , 6 , \\cdots , 9 1 , 9 6$ for testing,and games 136,141,146,151,156 for prompting. For each task, the input describes the 5 horizontal clues and 5 vertical clues,and the output should be a board of $5 \\times 5 = 2 5$ letters to solve the crosswords. For evaluation, we consider three levels of success: the portion of correct letters (25 per game), words (10 per game), and games. ",
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+    {
+        "type": "text",
+        "text": "Baselines. We provide 5 example input-output pairs in the IO prompt, and in the CoT prompt additionally include intermediate words in the order h1..5 then v1..5. We run each prompt for 10 samples and average the results. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ToT setup. We leverage a depth-first search (Algorithm 2) that keeps exploring the most promising subsequent word clue until the state is no longer promising, then backtrack to the parent state to explore alternative thoughts.To make search tractable, subsequent thoughts are constrained not to change any filled words or letters,so that the ToT has at most 1O intermediate steps.For thought generation, at each state we translate all existing thoughts (e.g.“h2.motor; h1.tasks” for the state in Figure 6(a)) into letter constraints for remaining clues (e.g.\"v1.To heap: tm__-;.\") and prompt a proposal prompt 5 times to come up with candidates for where and what to fill in the next word. Importantly, we also prompt the LM to give a confidence level for different thoughts,and aggregate these across proposals to obtain a sorted list of next thoughts to explore (Figure 6(a). For state evaluations, we similarly translate each state into leter constraints for remaining clues,then evaluate for each clue if it is possible to fillgiven the constraints. If any remaining clue is demed “impossible” to fill in (e.g.“v1. To heap: tm_s-\"), then the exploration of the state's subtree is pruned and DFS backtracks to its parent to explore the next promising thought. We limit DFS search steps to 100,and simply render the deepest explored state (the first explored one if multiple) into the final output. ",
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+    {
+        "type": "image",
+        "img_path": "images/b92ef17d9ff862acec28f259769b8b0a71bcd39ac3994319e0b060e8f88b3c7c.jpg",
+        "image_caption": [
+            "Figure 6: In Mini Crosswords,(a) how thoughts are proposed and aggregated in a priority queue for depth-first search (DFS),and (b) how a state is evaluated based on the possibility of filling in each remaining word clue,and pruned if any remaining clue is deemed not possible to fill by the LM. Then DFS backtracks to the parent state and explore the next promising thought for clue. "
+        ],
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+        "text": "Results. As shown in Table 3, IO and CoT prompting methods perform poorly with a word-level success rate less than $1 6 \\%$ ,while ToT significantly improves all metrics,achieving a word-level success rate of $6 0 \\%$ and solving 4 out of 20 games. Such an improvement is not surprising, given IO and CoT lack mechanisms to try different clues,make changes to decisions,or backtrack. ",
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+        "text": "Oracle and ablation studies. When outputing from the oracle best DFS state (instead of the heuristically determined best state) per task, ToT performance is even higher and actually solves 7/20 games (Table 3,‘ $^ { + }$ best state\"), indicating our simple output heuristics can be readily improved. Interestingly, sometimes when the crosswords game is actually solved, the state evaluator might still deem some words as “impossible\" and prune - possibly because $5 \\times 5$ crosswords by design have some rare or obselete words that GPT-4 cannot recognize2. Given the state evaluation as a pruning heuristic is imperfect, we also explore ablating the pruning, and find the performance generally worse (Table 3,“-prune\"). However, it could actually find the correct solution for 4/20 games (though only outputing 1 via heuristic),3 of which are games ToT+pruning cannot solve within 100 steps. Thus, beter heuristics for DFS pruning are critical for problem solving in this case.Lastly, we confirm the importance of backtracking by running an ablation that keeps filing the most promising clue for at most 20 steps,allowing overwrites. This is similar to a“greedy”BFS search with breadth limit of $b = 1$ ,and performs poorly with a word level success of only $2 \\dot { 0 } \\%$ (Table 3,“-backtrack\"). ",
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+        "text": "5Related Work ",
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+        "text": "Planning and decision making. Smart planning and decision making are critical to achieving predefined goals. As they are trained on vast amount of world knowledge and human examples,LMs are known to have already absorbed rich commonsense that makes it possible to propose reasonable plans conditioned on problem seting and environmental states [12,42,37,13,35,41,40]. Our proposed ToT approach extends existing planning formulations by considering multiple potentially feasible plans simultaneously at each problem-solving step, and proceeding with the most promising ones. The integration between thought sampling and value feedback organically integrates planning and decision-making mechanisms,enabling effective search inside a solution tree. On the other hand, traditional decision-making procedures usually require training dedicated reward and policy models as in reinforcement learning (for example CHAI [33]), whereas we use the LM itself to provide the value estimates for decision making. RAP [9] is a concurrent work that treats language model reasoning as planning with its internal world model, and proposes a MCTS-based method similar to ToT.However,its tasks are simpler than ours,and its framework lacks the modularity to incorporate different tree search algorithms. ",
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+        "text": "",
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+        "type": "text",
+        "text": "Self-reflection. Using LLMs to assess the viability of their own predictions is becoming an increasingly important procedure in problem solving. [28,20,24] introduced the“self-reflection” mechanism, in which LMs provide feedback to their generation candidates. [4] improves LMs code generation accuracy by injecting feedback messages generated by the LM itself based on its code execution results. Similarly,[17] also introduces “critic”or review steps over the actions and states, deciding the next action to take in solving computer operation tasks. Another recent work very relevant to ours is “self-eval guided decoding” [39]. Similar to our method, self-eval decoding also follows a tree-search procedure with leaves sampled from stochastic beam search decoding, which are then evaluated by LLM itself with carefully prepared self-eval prompts. Their approach however, uses the PAL formulation [8] which represents thoughts as codes,which makes it difficult to tackle chalenging tasks like creative writing which we consider in this paper. Our Tree-of-Thought formulation is thus more versatile and handles challenging tasks on which GPT-4 only achieves very low accuracy with standard prompts. ",
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+        "type": "text",
+        "text": "Program-guided LLM generation. Our proposal is also related to recent advancements that organize LM's behavior with systematic procedures [14,44,6,43] or symbolic program guidance.For example, Schlag et al.[27] embeds LMs in an algorithmic search procedure to help solve problems like question answering step-by-step, in which the search trees are expanded by relevant paragraphs that might provide answers. This approach however differs from ours in that trees are expanded by sampling external paragraphs instead of the LM's own thoughts,and there is no reflection or voting steps. Another approach, $\\mathrm { L L M + P }$ [18], goes one step further and delegates the actual planning process to a classical planner. ",
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+        "type": "text",
+        "text": "Classical search methods. Last but not least, our approach can be treated as a modern rendition of classical search methods for problem solving. For example it can be considered as a heuristic search algorithm like $\\mathbf { A } ^ { * }$ [10], in which the heuristic at each search node is provided by the LM's selfassessment. From this perspective, our method is also related to NeuroLogic $\\mathbf { A } ^ { * }$ esque decoding [19], which is inspired by $\\mathbf { A } ^ { * }$ search but introduces look-ahead heuristics that are efficient for LMs to improve the beam-search or top-k sampling decoding. This method however is constrained to sentence generation tasks,whereas our framework are designed for complex, multi-step problem solving guarded by value feedback. ",
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+        "type": "text",
+        "text": "6Discussion ",
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+    {
+        "type": "text",
+        "text": "Limitations and future directions. Deliberate search such as ToT might not be necessary for many existing tasks that GPT-4 already excels at (see Appendix B.1),and as an initial step this work only explores three relatively simple tasks that challenges GPT-4 (see Appendix B.2 for some GPT-3.5 experiment results) and calls of beter search and planning abilities incorporated with LMs. However, as we begin to deploy LMs for more real-world decision making applications (e.g.coding, data analysis, robotics, etc.), more complex tasks could emerge and present new opportunities to study these research questions. Also, search methods like ToT requires more resources (e.g. GPT-4 API cost) than sampling methods in order to improve task performances, but the modular flexibility of ToT allows users to customize such performance-cost tradeoffs,and ongoing open-source efforts [32] should readily reduce such costs in the near future. More details about cost and effciency are in Appendix B.3. Lastly, this work focuses on using an off-the-shelf LM, and fine-tuning LMs using a ToT-style high-level counterfactual decision making (e.g.deliberating over potential choices for the next paragraph, instead of predicting the next token) might present opportunities to enhance the problem-solving capabilities of LMs. ",
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+    {
+        "type": "text",
+        "text": "Conclusion. The associative “System 1” of LMs can be beneficially augmented by a“System $2 ^ { \\circ }$ based on searching a tree of possible paths to the solution to a problem. The Tree of Thoughts framework provides a way to translate classical insights about problem-solving into actionable methods for contemporary LMs. At the same time,LMs address a weakness of these classical methods, providing a way to solve complex problems that are not easily formalized, such as creative writing. We see this intersection of LMs with classical approaches to AI as an exciting direction. ",
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+    {
+        "type": "text",
+        "text": "Broader Impact ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "ToT is a framework that empowers LMs to more autonomously and intelligently make decisions and solve problems. While current tasks are limited to reasoning and search problems, future applications involving interaction with external environments or humans could bring potential danger, e.g.facilitating harmful uses of LMs. On the other hand, ToT also improves the interpretability of model decisions and the opportunity for human alignment, as the resulting representations are readable, high-level language reasoning instead of implicit, low-level token values. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "SY and KN acknowledge support from an Oracle Collaborative Research award and the National Science Foundation under Grant No. 2239363. Any opinions,findings,conclusions,or recommendations expressed in this material are those of the author(s)and do not necessarily reflect the views of the National Science Foundation. SY is also supported by the Harold W.Dodds Fellowship from Princeton. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "text": "[1] T. Brown, B. Mann,N. Ryder, M. Subbiah, J. D. Kaplan, P. Dhariwal, A. Neelakantan, P. Shyam, G. Sastry, A. Askell, et al. Language models are few-shot learners. Advances in neural information processing systems, 33:1877-1901, 2020.   \n[2] C. Browne,E.J. Powley, D. Whitehouse, S. M. M. Lucas, P. I. Cowling, P. Rohlfshagen, S.Tavener,D.P.Liebana, S. Samothrakis,and S. Colton.A survey of monte carlo tree search methods. IEEE Transactions on Computational Intelligence and AI in Games, 4:1-43,2012.   \n[3] M. Campbell, A. J. Hoane Jr, and F.-h. Hsu. Deep blue. Artificial intelligence,134(1-2):57-83, 2002.   \n[4] X. Chen, M. Lin, N. Schärli, and D. Zhou. Teaching large language models to self-debug, 2023.   \n[5] A.Chowdhery, S. Narang,J. Devlin, M. Bosma, G. Mishra,A. Roberts,P. Barham, H.W. Chung, C. Sutton, S. Gehrmann, et al. Palm: Scaling language modeling with pathways. arXiv preprint arXiv:2204.02311,2022.   \n[6] A. Creswelland M. Shanahan. 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Huang, P. Abbeel, D. Pathak,and I. Mordatch. Language models as zero-shot planners: Extracting actionable knowledge for embodied agents, 2022.   \n[13] W. Huang,F. Xia, T. Xiao, H. Chan, J. Liang, P.Florence, A. Zeng, J. Tompson, I. Mordatch, Y. Chebotar, et al. Inner monologue: Embodied reasoning through planning with language models. arXiv preprint arXiv:2207.05608,2022.   \n[14] J. Jung, L. Qin, S. Weleck, F. Brahman, C. Bhagavatula, R. L. Bras,and Y. Choi. Maieutic prompting: Logically consistent reasoning with recursive explanations. arXiv preprint arXiv:2205.11822,2022.   \n[15] D. Kahneman. Thinking, fast and slow. Macmillan, 2011.   \n[16] D. Kahneman, S. Frederick, et al. Representativeness revisited: Attribute substitution in intuitive judgment. Heuristics and biases: The psychology of intuitive judgment, 49(49-81):74, 2002.   \n[17] G. Kim, P. Baldi, and S. McAleer. Language models can solve computer tasks, 2023.   \n[18] B. Liu, Y. Jiang, X. Zhang, Q. Liu, S. Zhang, J. Biswas,and P. Stone. Llm $+ \\mathsf { p }$ : Empowering large language models with optimal planning proficiency, 2023.   \n[19] X. Lu, S. Welleck, P. West, L. Jiang, J. Kasai, D. Khashabi,R. L. Bras,L. Qin, Y. Yu, R. Zellers, N. A. Smith,and Y. Choi. Neurologic a\\*esque decoding: Constrained text generation with lookahead heuristics. In North American Chapter of the Association for Computational Linguistics, 2021.   \n[20] A. Madaan, N. Tandon, P. Gupta, S. Hallinan,L. Gao, S. Wiegreffe,U. Alon, N. Dziri, S. Prabhumoye, Y. Yang, S. Welleck,B.P. Majumder, S. Gupta, A. Yazdanbakhsh, and P. Clark. Self-refine: Iterative refinement with self-feedback, 2023.   \n[21] A. Newell, J. C. Shaw, and H. A. Simon. Report on a general problem solving program. In IFIP congress, volume 256, page 64. Pittsburgh, PA,1959.   \n[22] A. Newell, H. A. Simon, et al. Human problem solving. Prentice-Hall,1972.   \n[23] OpenAI. Gpt-4 technical report. ArXiv,abs/2303.08774, 2023.   \n[24] D. Paul,M. Ismayilzada, M. Peyrard, B.Borges, A. Bosselut, R. West, and B.Faltings. Refiner: Reasoning feedback on intermediate representations,2023.   \n[25] A. Radford, K. Narasimhan,T. Salimans,I Sutskever, et al. Improving language understanding by generative pre-training. OpenAI blog, 2018.   \n[26] A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, I. Sutskever, et al. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9,2019.   \n[27] I. Schlag, S. Sukhbaatar, A. Celikyilmaz, W. tau Yih,J. Weston, J. Schmidhuber, and X. Li. Large language model programs, 2023.   \n[28] N. Shinn, B. Labash,and A. Gopinath. Reflexion: an autonomous agent with dynamic memory and self-reflection, 2023.   \n[29] D. Silver, J. Schritwieser, K. Simonyan, I. Antonoglou, A. Huang, A. Guez, T. Hubert,L. Baker, M.Lai, A. Bolton, et al. Mastering the game of go without human knowledge. nature, 550 (7676):354-359, 2017.   \n[30] S. A. Sloman. The empirical case for two systems of reasoning. Psychological bulletin,119(1): 3,1996.   \n[31] K.E. Stanovich. Who is rational? Studies ofindividual diferences in reasoning. Psychology Press, 1999.   \n[32] H. Touvron,T. Lavril, G. Izacard, X. Martinet, M.-A. Lachaux, T.Lacroix,B.Roziere, N. Goyal, E. Hambro,F. Azhar, et al. Llama: Open and efficient foundation language models. arXiv preprint arXiv:2302.13971,2023.   \n[33] S. Verma, J.Fu, S. Yang, and S. Levine. Chai: A chatbot ai for task-oriented dialogue with offline reinforcement learning. In Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pages 4471-4491, 2022.   \n[34] E. Wallace, N. Tomlin, A. Xu, K. Yang,E. Pathak, M. Ginsberg,and D. Klein. Automated crossword solving. arXiv preprint arXiv:2205.09665,2022.   \n[35] L. Wang, W. Xu, Y. Lan, Z. Hu, Y. Lan, R. K.-W. Lee, and E.-P. Lim. Plan-and-solve prompting: Improving zero-shot chain-of-thought reasoning by large language models, 2023.   \n[36] X. Wang, J. Wei, D. Schuurmans, Q.Le,E. Chi, and D. Zhou. Self-consistency improves chain of thought reasoning in language models. arXiv preprint arXiv:2203.11171, 2022.   \n[37] Z. Wang, S. Cai, A. Liu, X. Ma,and Y. Liang. Describe, explain, plan and select: Interactive planning with large language models enables open-world multi-task agents, 2023.   \n[38] J. Wei, X. Wang, D. Schuurmans,M. Bosma, E. Chi, Q.Le,and D. Zhou. Chain of thought prompting elicits reasoning in large language models. arXiv preprint arXiv:2201.11903, 2022.   \n[39] Y. Xie, K. Kawaguchi, Y. Zhao, X. Zhao, M.-Y. Kan, J. He,and Q. Xie. Decomposition enhances reasoning via self-evaluation guided decoding, 2023.   \n[40] S.Yang, O. Nachum, Y.Du,J. Wei, P. Abbeel,and D. Schuurmans. Foundation models for decision making: Problems, methods,and opportunities, 2023.   \n[41] S. Yao, J. Zhao,D. Yu,N. Du, I. Shafran, K. Narasimhan,and Y. Cao.ReAct: Synergizing reasoning and acting in language models. arXiv preprint arXiv:2210.03629,2022.   \n[42] S. Zhang, Z. Chen, Y. Shen, M. Ding, J. B. Tenenbaum, and C. Gan. Planning with large language models for code generation. In The Eleventh International Conference on Learning Representations,2023. URL https://openreview.net/forum?id=Lr8c00tYbfL.   \n[43] D. Zhou, N. Scharli,L. Hou, J. Wei, N. Scales, X. Wang,D. Schuurmans, C. Cui, O. Bousquet, Q.Le, et al. Least-to-most prompting enables complex reasoning in large language models. arXiv preprint arXiv:2205.10625,2022.   \n[44] X. Zhu, J. Wang, L. Zhang, Y. Zhang, R. Gan, J. Zhang,and Y. Yang. Solving math word problem via cooperative reasoning induced language models. arXiv preprint arXiv:2210.16257, 2022. ",
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+        "type": "text",
+        "text": "A Code, Prompts, Trajectories ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "All code is available at https://github.com/princeton-nlp/tree-of-thought-llm. ",
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+        "text": "All prompts are available at https://github.com/princeton-nlp/tree-of-thought-llm/ tree/master/src/tot/prompts. ",
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+        "text": "Trajectories are available at https://github.com/princeton-nlp/tree-of-thought-llm/ tree/master/logs. ",
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+        "type": "text",
+        "text": "BAdditional Experiment Results ",
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+        "type": "text",
+        "text": "Given the motivation of exploring and extending the capability frontier of language models, our experiments in the main paper have focused on a setup with the state-of-the-art language model (GPT-4),and three hard tasks invented to chalenge it. Here, we report additional experiments with weaker LLM or easier tasks,and discusscost and efficiency. ",
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+        "img_path": "images/3b9b6a44bcee8f0eec7eacc0dc8f33d6c324abf54a5f9c39dd22c3f41b6de0f6.jpg",
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+        "table_body": "<table><tr><td>GPT-4</td><td>GPT-3.5</td></tr><tr><td>10 6.19</td><td>4.47</td></tr><tr><td>CoT 6.93</td><td>5.16</td></tr><tr><td>ToT 7.56</td><td>6.62</td></tr></table>",
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+        "table_caption": [
+            "Table 5: Game of 24 with GPT-4 vs GPT-3.5. "
+        ],
+        "table_footnote": [
+            "Table 4:New tasks with zero-shot ToT and GPT-4. "
+        ],
+        "table_body": "<table><tr><td>GSM8K</td><td>StrategyQA</td></tr><tr><td>10 51</td><td>73</td></tr><tr><td>CoT 86</td><td>82</td></tr><tr><td>ToT 90</td><td>83</td></tr></table>",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 6: Creative Writing with GPT-4 vs. GPT-3.5. "
+        ],
+        "table_body": "<table><tr><td></td><td>GPT-4</td><td>GPT-3.5</td></tr><tr><td>I0</td><td>7.3%</td><td>6%</td></tr><tr><td>CoT</td><td>4.0%</td><td>3%</td></tr><tr><td>ToT</td><td>74%</td><td>19%</td></tr></table>",
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+        "type": "text",
+        "text": "B.1Extension to new tasks (GSM8k, StrategyQA) with zero-shot ToT ",
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+        "text": "While more common NLP tasks might be too easy for GPT-4 and do not require ToT(which is why we considered harder new tasks), we believe applying ToT to new tasks could be straightforward. For example, we implemented a simple and generic zero-shot ToT-BFS similar to creative writing (sample 5 problem solving strategies then vote for the best one; then sample 5 solutions based on the best strategy then vote for the best one) for GSM8K and StrategyQA with few extra lines of code: ",
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+        "text": "# define the answer format of new tasks gsm8k_format $=$ (\"the answer is n\" where n is a number' strategyqa_format $=$ ‘either \"the answer is yes\" or \"the answer is no\"’ # define zero-shot io prompting standard_prompt $=$ ‘Answer the following question with {format}: {input}' # define thought format for zero-shot cot and zero-shot tot cot_prompt $\\begin{array} { r l } { = } & { { } \\hat { \\pmb { \\mathscr { \\imath } } } \\pmb { \\mathscr { \\imath } } \\pmb { \\mathscr { \\imath } } \\pmb { \\mathscr { \\imath } } } \\end{array}$ Answer the following question: {input} ",
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            664,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Make a strategy then write. Your output should be of the following format: ",
+        "bbox": [
+            169,
+            705,
+            807,
+            718
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Strategy: ",
+        "bbox": [
+            173,
+            733,
+            251,
+            746
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Your strategy about how to answer the question. ",
+        "bbox": [
+            174,
+            747,
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+            760
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Answer: ",
+        "bbox": [
+            173,
+            775,
+            233,
+            786
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Your answer to the question.It should end with {format}. ，，， ",
+        "bbox": [
+            174,
+            787,
+            660,
+            810
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "# define zero-shot voting used for zero-shot tot   \nvote_prompt $\\begin{array} { r l } { \\mathbf { \\Sigma } } & { { } = \\mathbf { \\Sigma } ^ { \\textit { \\textbf { c } } \\texttt { \\textsf { c } } \\mathcal { \\epsilon } \\textbf { \\Lambda } } } \\end{array}$ Given an instruction and several choices, decide which choice is most promising.   \nAnalyze each choice in detail,then conclude in the last line \"The best choice is {s}\"，where s the integer id of the choice. ，， ",
+        "bbox": [
+            173,
+            829,
+            710,
+            909
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We evaluated on a subset of 100 random GSM8K test and StrategyQA dev questions. As shown in Table 4 and as expected,ToT improves over CoTon both tasks (but only slightly, given GPT-4 $+ \\mathrm { C o T }$ is already very good on such tasks, and StrategyQA's botleneck is external knowledge, not reasoning). Considering computational costs, it is more suitable to try smaller LLMs $^ +$ ToT for traditional NLP tasks,or GPT $\\cdot 4 + \\mathrm { T o T }$ for hard tasks that challenge GPT $. 4 + \\mathrm { C o T ' }$ reasoning. ",
+        "bbox": [
+            173,
+            90,
+            825,
+            161
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.2Extension to new LMs (GPT-3.5) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            176,
+            442,
+            190
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To understand how ToT works with other LLMs, we also ran GPT-3.5-turbo for Creative Writing (Table 6) and Game of 24 (Table 5). On both tasks, $\\mathbf { \\partial } ^ { \\cdot } \\mathbf { \\Phi } ^ { \\cdot } \\mathbf { T o T } > \\mathbf { C o T } > \\mathbf { I O } ^ { \\prime }$ remains true for GPT-3.5. On Creative Writing,we find GPT $3 . 5 \\mathrm { + T o T }$ outperform GPT $_ { 4 + 1 0 }$ ,and similar to GPT- $4 { + } \\mathrm { C o T }$ ，which suggests ToT could also work well on weaker language models. ",
+        "bbox": [
+            173,
+            202,
+            825,
+            257
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "On Game of 24 (we changed 1-shot proposal prompt to 3-shot to make it work), GPT- $. 3 . 5 +$ ToT's $19 \\%$ is far worse than GPT- $^ { 4 + }$ ToT's $74 \\%$ . To further understand the importance of generation vs. evaluation, we ran GPT-4 generation $+ \\mathrm { G P T } { - 3 . 5 }$ evaluation $( 6 4 \\% )$ and GPT-3.5 generation $^ +$ GPT-4 evaluation $( 3 1 \\% )$ . This suggests the game's bottleneck is thought generation,and different generation/evaluation language models might attain decent results while reducing costs. ",
+        "bbox": [
+            173,
+            263,
+            825,
+            333
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.3Cost and effciency ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            348,
+            346,
+            364
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Running ToT requires significantly more computations than IO or CoT prompting. For example, in Game of 24 (Table 7 below), solving a problem with ToT requires $5 . 5 \\mathrm { k }$ completion tokens, close to $1 0 0 \\mathrm { C o T }$ trials ( $\\mathrm { 6 . 7 k }$ tokens). But the performance of ToT is better than best of 1OO independent CoT trials. ",
+        "bbox": [
+            176,
+            373,
+            825,
+            429
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/4c52db47c33a87a31615eecf437ecdd9c6aad15cb0592c12e84802daef7a2891.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 7: Cost analysis on Game of 24. "
+        ],
+        "table_body": "<table><tr><td>Game of 24</td><td>Generate/Prompt tokens</td><td>Cost per case</td><td> Success</td></tr><tr><td>IO (best of 100)</td><td>1.8k /1.0k</td><td>$0.13</td><td>33%</td></tr><tr><td>CoT (best of 100)</td><td>6.7k /2.2k</td><td>$0.47</td><td>49%</td></tr><tr><td>ToT</td><td>5.5k / 1.4k</td><td>$0.74</td><td>74%</td></tr></table>",
+        "bbox": [
+            240,
+            439,
+            756,
+            508
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "On Creative Writing (Table 8 below), we found ToT takes around ${ 5 } \\mathrm { x }$ completion tokens and money cost, which is intuitive as $b = 5$ and most tokens are generated passages. ",
+        "bbox": [
+            173,
+            546,
+            821,
+            575
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/fa32962fbdab35b505509e0ee2f43b9fd6b6421f4a7a0a1616bc2fcc777b019a.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 8: Cost analysis on Game of 24. "
+        ],
+        "table_body": "<table><tr><td>Creative Writing</td><td>Generate/Prompt tokens</td><td>Cost per case</td></tr><tr><td>I0</td><td>0.9k /0.4k</td><td>$0.06</td></tr><tr><td>CoT</td><td>0.9k/0.4k</td><td>$0.07</td></tr><tr><td>ToT</td><td>4k /2.9k</td><td>$0.32</td></tr></table>",
+        "bbox": [
+            276,
+            585,
+            720,
+            656
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "So completing Game of 24 and Creative Writing's main ToT experiments cost around $0 . 7 4 \\times 1 0 0 +$ $0 . 3 2 \\times 1 0 0 = 1 0 6$ dollars.Crosswords’DFS experiments should be also within 1OO dollars.In general, cost and efficiency of ToT highly depend on the prompts and search algorithms used,and could require 5-1OO times more generated tokens than CoT. Some actionable insights: ",
+        "bbox": [
+            176,
+            690,
+            826,
+            747
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "· We recommend using ToT on tasks requiring deliberate reasoning, on which CoT struggles. · Flexibility of ToT allows some performance-cost tradeoff, e.g., change beam size or vote number in BFS,few-shot vs. zero-shot prompting,GPT-3.5 vs. GPT-4,etc. One could configure the setup based on some resource constraints or performance goal. · There is much space for improving effciency, e.g., BFS could early stop when solution is found,or trim down beam size to when some thoughts are \"impossible\". · We believe that more computation is indeed required in order for the model to achieve stronger intelligence,and this should not become a blocking issue as in the long run, (opensource) LMs will become much cheaper and more efficient. It is also a great direction how to better train/finetune LMs for thought generation and/or evaluation. ",
+        "bbox": [
+            215,
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+        ],
+        "page_idx": 13
+    }
+]

parse/dev/5btWTw1vcw1/5btWTw1vcw1.md

@@ -0,0 +1,350 @@
+# Trajectory balance: Improved credit assignment in GFlowNets
+
+Nikolay Malkin Mila, Université de Montréal Montréal, Québec, Canada
+
+Moksh Jain Mila, Université de Montréal Montréal, Québec, Canada
+
+Emmanuel Bengio Mila, McGill University, Recursion Montréal, Québec, Canada
+
+Chen Sun Mila, Université de Montréal Montréal, Québec, Canada
+
+Yoshua Bengio Mila, Université de Montréal Montréal, Québec, Canada {nikolay.malkin,moksh.jain,chen.sun,yoshua.bengio}@mila.quebec emmanuel.bengio@recursionpharma.com
+
+# Abstract
+
+Generative flow networks (GFlowNets) are a method for learning a stochastic policy for generating compositional objects, such as graphs or strings, from a given unnormalized density by sequences of actions, where many possible action sequences may lead to the same object. We find previously proposed learning objectives for GFlowNets, flow matching and detailed balance, which are analogous to temporal difference learning, to be prone to inefficient credit propagation across long action sequences. We thus propose a new learning objective for GFlowNets, trajectory balance, as a more efficient alternative to previously used objectives. We prove that any global minimizer of the trajectory balance objective can define a policy that samples exactly from the target distribution. In experiments on four distinct domains, we empirically demonstrate the benefits of the trajectory balance objective for GFlowNet convergence, diversity of generated samples, and robustness to long action sequences and large action spaces.
+
+# 1 Introduction
+
+Generative flow networks [GFlowNets; 3, 4] are models that exploit generalizable structure in an energy function $\mathcal { E }$ to amortize sampling from the corresponding probability density function on a space of compositional objects $\mathcal { X }$ , for example, graphs composed of nodes and edges. A GFlowNet learns a stochastic policy that generates such structured objects by producing a stochastic sequence of actions that incrementally modify a partial object (state), e.g., by adding an edge or a node to a graph, starting from a universal initial state (like an empty graph). A special ‘exit’ action signals that the construction of the object $x \in \mathcal { X }$ is completed, and the policy is trained so as to make the likelihood of generating $x$ proportional to the given unnormalized probability or reward $R ( x ) = e ^ { - \mathcal { E } ( x ) }$ .
+
+Like other models in deep reinforcement learning [RL; 25], GFlowNets are trained with a parametric policy that can be given desired inductive biases (e.g., a particular deep net architecture) and allows generalization to states not seen in training. Natural domains for applying GFlowNets are those where exact sampling is intractable and local exploration (MCMC) methods perform poorly, but diversity of samples is desired [3, 32, 14, 8]. For example, GFlowNets have been used [3] to generate graphical descriptions of molecules by incremental addition of simple building blocks, where the reward $R ( x )$ is the estimated strength of binding the constructed molecule to a protein target: the number of candidates grows rapidly with the molecule size and the reward has many separated modes. Like all RL models that iteratively sample action sequences for training, GFlowNets pose the learning challenges of exploration/exploitation and credit assignment, i.e., propagation of a reward signal over an action sequence [27, 2, 17]. The efficiency of credit assignment and training in GFlowNets is the focus of the present paper.
+
+The learning problem solved by GFlowNets also has two fundamental differences with the standard reward-maximization paradigm of RL. First, a GFlowNet aims to make the likelihood of reaching a terminating state proportional to the reward, not to concentrate it at a maximal-reward state. Thus, a GFlowNet must model the diversity in the target distribution, not only its dominant mode. Reward maximization in RL can be turned into sampling proportionally to the reward with appropriate entropy maximization regularization, if there is only one way to reach every state [3]. The second difference with reward-maximization in RL is indeed that the GFlowNet training objectives still lead to correct sampling even when multiple action sequences lead to the same terminating state. Note that the likelihood of reaching a state is the sum of likelihoods of all action sequences leading to it, and that the number of such paths may be exponential in their length.
+
+The set of all achievable sequences of actions and states can be conceptually organized in a directed graph $G = ( S , A )$ in which the vertices $s$ are states (some of them designated as terminal states, in bijection with $\mathcal { X }$ ) and the edges $u {  } v$ in $\mathcal { A }$ each correspond to applying an action while in a state $u \in S$ and landing in state $v$ . In [3], a GFlowNet is described by a nonnegative function on the edges, called the edge flow $F : \mathcal { A }  \mathbb { R } _ { \geq 0 }$ , where $F ( u {  } v )$ is an unnormalized likelihood of taking the action that modifies state $u$ to state $v$ . The GFlowNet policy samples the transition $u {  } v$ from state $u$ with probability $\scriptstyle { F ( u \to v ) / \sum _ { v ^ { \prime } } F ( u \to v ^ { \prime } ) }$ where the denominator sums over the outgoing edges from $u$ . By analogy with the classical notion of flows in networks [9], one can think of this flow like the amount of water flowing through an edge (like a pipe) or a state (like a tee, where pipes meet). It is shown that this GFlowNet policy samples $x$ proportionally to $R ( x )$ if $F$ satisfies a set of linear flow matching constraints (a conservation law: the sum of flows into a state should equal the sum of flows out of it). These constraints are converted into a temporal difference-like objective that can be optimized with respect to the parameters of a neural net that approximates $F$ . An alternative objective based on detailed balance constraints was proposed in [4]. These objectives, however, like temporal-difference learning, can suffer from slow credit assignment [27, 2, 17].
+
+The main contribution of this work (§3) is a new parametrization and objective for GFlowNets. This objective, which we call trajectory balance, is computed on sampled full action sequences (trajectories) from the initial state to a terminal state, unlike the flow matching and detailed balance objectives. We prove that global minimization of trajectory balance implies that the learned action policy samples proportionally to $R$ . We also empirically show that the trajectory balance objective accelerates training convergence relative to previously proposed objectives, improves the learned sampling policy with respect to metrics of diversity and divergence from the reward function, and allows learning GFlowNets that generate sequences far longer than was possible before. As a secondary contribution, we perform the first empirical validation of the detailed balance training objective. Comparative evaluation of the three GFlowNet objectives and non-GFlowNet baselines is performed on four domains illustrating different features of the reward landscape:
+
+• Hypergrid (§5.1), an illustrative synthetic environment with modes separated by wide troughs; • Molecule synthesis (§5.2), a practical graph generation problem, where the trajectory balance objective leads to significant computational speed-ups and more diverse generated candidates; • Sequence generation (§5.3), where we show the robustness of trajectory balance to large action spaces and long action sequences on synthetic data and real AMP sequence data.
+
+Since the initial appearance of this work on arXiv, several published papers and preprints have used trajectory balance and its generalizations successfully in various applications [32, 14, 8, 18].
+
+# 2 Preliminaries
+
+# 2.1 Markovian flows
+
+We give some essential definitions, following $\ S 2$ of [4]. Fix a directed acyclic graph $G = (  { \boldsymbol { S } } ,  { \boldsymbol { A } } )$ with state space $s$ and action space $\mathcal { A }$ . Let $s _ { 0 } \in S$ be the special initial (source) state, the only state with no incoming edges, and designate vertices with no outgoing edges as terminal (sinks) 1. We call the vertices states, the edges actions, the states reachable through outgoing edges from a state its children, and the sources of its incoming edges its parents.
+
+A complete trajectory is a sequence of transitions $\tau = ( s _ { 0 } {  } s _ { 1 } {  } \dots {  } s _ { n }$ ) going from the initial state $s _ { 0 }$ to a terminal state $s _ { n }$ with $( s _ { t } {  } s _ { t + 1 } ) \in \mathcal { A }$ for all $t$ . Let $\tau$ be the set of complete trajectories. A trajectory flow is a nonnegative function $F : \mathcal { T } {  } \mathbb { R } _ { \geq 0 }$ . With our water analogy, it could be the number of water molecules travelling along this path (the units don’t matter because the flow function can be scaled arbitrarily, since we normalize them to get probabilities). For any state $s$ , define the state flow $\begin{array} { r } { F ( s ) = \sum _ { s \in \tau } F ( \tau ) } \end{array}$ , and, for any edge $s {  } s ^ { \prime }$ , the edge flow
+
+$$
+F ( s \to s ^ { \prime } ) = \sum _ { \tau = ( \ldots \to s \to s ^ { \prime } \to \ldots ) } F ( \tau ) .
+$$
+
+As a consequence of this definition, the flow matching constraint (incoming flow $=$ outgoing flow) is satisfied for all states $s$ that are not initial or terminal:
+
+$$
+F ( s ) = \sum _ { ( s ^ { \prime \prime }  s ) \in A } F ( s ^ { \prime \prime } {  } s ) = \sum _ { ( s  s ^ { \prime } ) \in A } F ( s {  } s ^ { \prime } ) .
+$$
+
+A nontrivial (i.e., not identically zero) trajectory flow $F$ determines a distribution $P$ over trajectories,
+
+$$
+P ( \tau ) = \frac { 1 } { Z } F ( \tau ) , \quad Z = F ( s _ { 0 } ) = \sum _ { \tau \in \mathcal { T } } F ( \tau ) .
+$$
+
+The trajectory flow $F$ is Markovian if there exist action distributions $P _ { F } ( - | s )$ over the children of each nonterminal state $s$ such that the distribution $P$ has a factorization
+
+$$
+P ( \tau = ( s _ { 0 } {  } . . . {  } s _ { n } ) ) = \prod _ { t = 1 } ^ { n } P _ { F } ( s _ { t } | s _ { t - 1 } ) .
+$$
+
+Equivalently ([4], Prop. 3) there are distributions $P _ { B } ( - | s )$ over the parents of each noninitial state $s$ such that for any terminal $x$ ,
+
+$$
+P ( \tau = ( s _ { 0 }  . . .  s _ { n } ) | s _ { n } = x ) = \prod _ { t = 1 } ^ { n } P _ { B } ( s _ { t - 1 } | s _ { t } ) .
+$$
+
+If $F$ is a Markovian flow, then $P _ { F }$ and $P _ { B }$ can be computed in terms of state and edge flows:
+
+$$
+P _ { F } ( s ^ { \prime } | s ) = \frac { F ( s  s ^ { \prime } ) } { F ( s ) } , \quad P _ { B } ( s | s ^ { \prime } ) = \frac { F ( s  s ^ { \prime } ) } { F ( s ^ { \prime } ) } ,
+$$
+
+supposing denominators do not vanish. We call $P _ { F }$ and $P _ { B }$ the forward policy and backward policy corresponding to $F$ , respectively. These relations are summarized by the detailed balance constraint
+
+$$
+F ( s ) P _ { F } ( s ^ { \prime } | s ) = F ( s ^ { \prime } ) P _ { B } ( s | s ^ { \prime } ) .
+$$
+
+Uniqueness properties. A Markovian flow is uniquely determined by an edge flow, i.e., a nontrivial choice of nonnegative value on every edge satisfying the flow matching constraint (2). By Corollary 1 of [4], a Markovian flow is also uniquely determined by either of
+
+• a constant $Z = F ( s _ { 0 } ) > 0$ and a distribution $P _ { F } ( - | s )$ over children of every nonterminal state; or • a nontrivial choice of nonnegative state flows $F ( x )$ for every terminal state $x$ and a choice of distribution $P _ { B } ( - | s )$ over parents of every noninitial state.
+
+# 2.2 GFlowNets
+
+Suppose that a nontrivial nonnegative reward function $R : \mathcal { X } \to \mathbb { R } _ { \geq 0 }$ is given on the set of terminal states. GFlowNets [3] aim to approximate a Markovian flow $F$ on $\bar { G }$ such that
+
+$$
+F ( x ) = R ( x ) \quad \forall x \in { \mathcal { X } } .
+$$
+
+We adopt the broad definition that a GFlowNet is any learning algorithm consisting of:
+
+• a model capable of providing the initial state flow $Z = F ( s _ { 0 } )$ as well as the forward action distributions $P _ { F } ( - | \bar { s } )$ for any nonterminal state $s$ (and therefore, by the above, uniquely but possibly in an implicit way determining a Markovian flow $F$ );   
+• an objective function, such that if the model is capable of expressing any action distribution and the objective function is globally minimized, then the constraint (8) is satisfied for the corresponding Markovian flow $F$ .
+
+The forward policy of a GFlowNet can be used to sample trajectories from the corresponding Markovian flow $F$ by iteratively taking actions according to policy $\mathop { P _ { F } ( - | s ) }$ . If the objective function is globally minimized, then the likelihood of terminating at $x$ is proportional to $R ( x )$ .
+
+In general, an objective optimizing for (8) cannot be minimized directly because $F ( x )$ is a sum over all trajectories leading to $x$ , and computing it may not be practical. Therefore, two local objectives – flow matching and detailed balance – have previously been proposed.
+
+Flow matching objective [3]. A model $F _ { \theta } ( s , s ^ { \prime } )$ 2 with learnable parameters $\theta$ approximates the edge flows $F ( s {  } s ^ { \prime } )$ . The corresponding forward policy is given by $P _ { F } ( s ^ { \prime } | s ; \theta ) \overset { \sim } { \propto } F _ { \theta } ( s , s ^ { \prime } )$ (Eq. (6)). Denote the corresponding Markovian flow by $F _ { \theta }$ and distribution over trajectories by $P _ { \theta }$ . The parameters are trained to minimize the error in the flow matching constraint (2) for all noninitial and nonterminal nodes $s$ :
+
+$$
+{ \mathcal { L } } _ { \mathrm { F M } } ( s ) = ( \log { \frac { \sum _ { ( s ^ { \prime \prime }  s ) \in A } F _ { \theta } ( s ^ { \prime \prime } , s ) } { \sum _ { ( s  s ^ { \prime } ) \in A } F _ { \theta } ( s , s ^ { \prime } ) } } ) ^ { 2 }
+$$
+
+and a similar objective $\mathcal { L } _ { \mathrm { F M } } ^ { \prime }$ pushing the inflow at $x \in \mathcal { X }$ to equal $R ( x )$ at terminal nodes $x$ . This objective is optimized for nonterminal states $s$ and terminal states $x$ from trajectories sampled from a training policy $\pi _ { \theta }$ . Usually, $\pi _ { \theta }$ is chosen to be a tempered (higher temperature) version of $P _ { F } ( - | s , \theta )$ , which also helps exploration during training. The parameters are updated with stochastic gradient
+
+$$
+\mathbb { E } _ { \tau = ( s _ { 0 }  \dots  s _ { n } ) \sim \pi _ { \theta } } \nabla _ { \theta } \Bigg [ \sum _ { { t = 1 } } ^ { n - 1 } \mathcal { L } _ { \mathrm { F M } } ( s _ { t } ) + \mathcal { L } _ { \mathrm { F M } } ^ { \prime } ( s _ { n } ) \Bigg ] .
+$$
+
+As per Proposition 10 of [4], if the training policy $\pi _ { \theta }$ has full support, and a global minimum of the expected loss (9) over states on trajectories sampled from $\pi _ { \theta }$ is reached, then the GFlowNet samples from the target distribution (i.e., $F _ { \theta }$ satisfies (8)).
+
+Detailed balance objective [4]. A neural network model with parameters $\theta$ has input $s$ and three kinds of outputs: an estimated state flow $F _ { \theta } ( s )$ , an estimated distribution over children $P _ { F } ( - | s ; \theta )$ , and an estimated distribution over parents $P _ { B } ( - | s ; \theta )$ . The policy $P _ { F } ( - | - ; \theta )$ and the initial state flow $F _ { \theta } ( s _ { 0 } )$ uniquely determine a Markovian flow $F _ { \theta }$ , which is not necessarily compatible with the estimated backward policy $P _ { B } ( - | - ; \theta )$ . The error in the detailed balance constraint (7) is optimized on actions $( s {  } s ^ { \prime } )$ ) between nonterminal nodes seen along trajectories sampled from the training policy:
+
+$$
+\mathcal { L } _ { \mathrm { D B } } ( s , s ^ { \prime } ) = \left( \log \frac { F _ { \boldsymbol \theta } ( s ) P _ { F } ( s ^ { \prime } | s ; \boldsymbol \theta ) } { F _ { \boldsymbol \theta } ( s ^ { \prime } ) P _ { B } ( s | s ^ { \prime } ; \boldsymbol \theta ) } \right) ^ { 2 } ,
+$$
+
+and a similar constraint $\mathcal { L } _ { \mathrm { D B } } ^ { \prime } ( s , x )$ is optimized at actions leading to terminal nodes. Similarly to flow matching, the parameters are updated with stochastic gradient
+
+$$
+\mathbb { E } _ { ( s _ { 0 }  \ldots  s _ { n } ) \sim \pi _ { \theta } } \nabla _ { \theta } [ \sum _ { t = 1 } ^ { n - 1 } \mathcal { L } _ { \mathrm { D B } } ( s _ { t - 1 } , s _ { t } ) + \mathcal { L } _ { \mathrm { D B } } ^ { \prime } ( s _ { n - 1 } , s _ { n } ) ]
+$$
+
+along trajectories sampled from a training policy $\pi _ { \theta }$ . By Proposition 6 of [4], a global minimum of the expected detailed balance loss under a $\pi _ { \theta }$ with full support specifies a GFlowNet that samples from the target distribution, i.e., the flow $F _ { \theta }$ satisfies (8).
+
+<table><tr><td>inputReward functionR:X →R&gt;o,model and optimizer hyperparameters</td></tr><tr><td>1:Initialize models PF,Pb,Z with parameters 0</td></tr><tr><td>2: ：repeat</td></tr><tr><td>3: Sample trajectory T = (so-→.. → Sn) from policy PF(-|-;0) or a tempered version of it</td></tr><tr><td>4: θ ←θ-nVθLTB(τ) {gradient update on (14)}</td></tr><tr><td>5:</td></tr><tr><td>until convergence monitoring on running LTB(T)</td></tr></table>
+
+Remarks. In some problems, such as autoregressive sequence generation (§5.3), the directed graph $G$ is a tree, so each state has only one parent. In this case, $P _ { B }$ is trivial and the detailed balance objective reduces to the flow matching objective, which in turn can be shown to be equivalent to Soft Q-Learning [12, 5] with temperature $\alpha = 1$ , a uniform $q _ { \mathbf { a ^ { \prime } } }$ , and $\gamma = 1$ .
+
+# 3 Trajectory balance
+
+Let $F$ be a Markovian flow and $P$ the corresponding distribution over complete trajectories, defined by (3), and let $P _ { F }$ and $P _ { B }$ be forward and backward policies determined by $F$ . A direct algebraic manipulation of Eqs. (3,4,5) gives the trajectory balance constraint for any complete trajectory $\tau = ( s _ { 0 } {  } s _ { 1 } {  } \dots {  } s _ { n } = x _ { , }$ ):
+
+$$
+Z \prod _ { t = 1 } ^ { n } P _ { F } ( s _ { t } | s _ { t - 1 } ) = F ( x ) \prod _ { t = 1 } ^ { n } P _ { B } ( s _ { t - 1 } | s _ { t } ) ,
+$$
+
+where we have used that $\begin{array} { r } { P ( s _ { n } = x ) = \frac { F ( x ) } { Z } } \end{array}$ .
+
+As explained in $\ S \mathrm { A } . 2$ , the trajectory balance constraint (13) and the detailed balance constraint (7) are special cases of one general constraint, which has been studied as a training objective in [18].
+
+Trajectory balance as an objective. We propose to convert (13) into an objective to be optimized along trajectories sampled from a training policy. Suppose that a model with parameters $\theta$ outputs estimated forward policy $P _ { F } ( - | s ; \theta )$ and backward policy $P _ { B } ( - | s ; \theta )$ for states $s$ (just as for detailed balance above), as well as a global scalar $Z _ { \theta }$ estimating $F ( s _ { 0 } )$ . The scalar $Z _ { \theta }$ and forward policy $P _ { F } ( - | - ; \theta )$ uniquely determine an implicit Markovian flow $F _ { \theta }$ .
+
+For a trajectory $\tau = ( s _ { 0 } {  } s _ { 1 } {  } \dots {  } s _ { n } = x )$ ), define the trajectory loss
+
+$$
+\mathcal { L } _ { \mathrm { T B } } ( \tau ) = \left( \log \frac { Z _ { \theta } \prod _ { t = 1 } ^ { n } P _ { F } ( s _ { t } | s _ { t - 1 } ; \theta ) } { R ( x ) \prod _ { t = 1 } ^ { n } P _ { B } ( s _ { t - 1 } | s _ { t } ; \theta ) } \right) ^ { 2 } .
+$$
+
+If $\pi _ { \theta }$ is a training policy – usually that given by $P _ { F } ( - | - ; \theta )$ or a tempered version of it – then the trajectory loss is updated along trajectories sampled from $\pi _ { \theta }$ , i.e., with stochastic gradient
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \tau \sim \pi _ { \theta } } \nabla _ { \theta } \mathcal { L } _ { \mathrm { T B } } ( \tau ) . } \end{array}
+$$
+
+The full algorithm, with batch size of 1, is presented as Algorithm 1 and its correctness is guaranteed by the following.
+
+Proposition 1. Let $R$ be a positive reward function on $\mathcal { X }$ .
+
+(a) If $P _ { F } ( - | - ; \theta )$ , $P _ { B } ( - | - ; \theta )$ , and $Z _ { \theta }$ are the forward and backward policies and normalizing constant of a Markovian flow $F$ satisfying (8), then $\mathcal { L } _ { \mathrm { T B } } ( \tau ) = 0$ for all complete trajectories $\tau$   
+(b) Conversely, suppose that $\mathcal { L } _ { \mathrm { T B } } ( \tau ) = 0$ for all complete trajectories $\tau$ . Then the corresponding Markovian flow $F _ { \theta }$ satisfies (8), and $P _ { F } ( - | - ; \theta )$ samples proportionally to the reward.
+
+The proof is given in $\ S \mathrm { A . 1 }$ . In particular, if $\pi _ { \theta }$ has full support and $\mathbb { E } _ { \tau \sim \pi _ { \theta } } \mathcal { L } _ { \mathrm { T B } } ( \tau )$ is globally minimized over all forward and backward policies $( P _ { F } , P _ { B } )$ and normalizing constants $Z$ , then the corresponding Markovian flow $F _ { \theta }$ satisfies (8) and $P _ { F } ( - | - ; \theta )$ samples proportionally to the reward. (The positivity assumption on $R$ is necessary to avoid division by 0 in (14), but can be relaxed by introduction of smoothing constants, just as was done for the losses proposed in [3, 4].)
+
+Remarks. (1) As discussed in $\ S 2$ , in the case of auto-regressive generation, $G$ is a directed tree, where each $s \in S$ has a single parent state. In this case $P _ { B }$ is trivially $P _ { B } = 1$ , $\forall s \in S$ . We get
+
+$$
+\mathcal { L } _ { \mathrm { T B } } ( \tau ) = \left( \log \frac { Z _ { \theta } \prod _ { t = 1 } ^ { n } P _ { F } ( s _ { t } | s _ { t - 1 } ; \theta ) } { R ( x ) } \right) ^ { 2 }
+$$
+
+(2) We found it beneficial to parametrize $Z$ in the logarithmic domain $\log Z$ is the trainable parameter) and output logits for $P _ { F } ( - | s ; \theta )$ and $P _ { B } { \big ( } - | s ; \theta { \big ) }$ , so that all products in (14) become sums under the logarithm. This is consistent with the log-domain parametrization of flows in [3]. In addition, we found it helpful to set a higher learning rate for $Z$ than for the parameters of $P _ { F }$ and $P _ { B }$ . 3
+
+# 3.1 Canonical choice of reward-matching flow
+
+The constraint (8), in general, does not have a unique solution: if the underlying undirected graph of $G$ has cycles, there may be multiple Markovian flows whose corresponding action policies sample proportionally to the reward. However, by the uniqueness properties, for any choice of backward policy $P _ { B } ( - | - )$ , there is a unique flow satisfying (8), and thus a unique corresponding forward policy $P _ { F } ( - | - )$ for states with nonzero flow. (See Fig. 1.)
+
+In some settings, it may be beneficial to $\mathit { \Omega } \mathcal { f } x$ the backward policy $P _ { B }$ and train only the parameters giving $P _ { F }$ and $Z _ { \theta }$ . For example, it may difficult to construct a model that outputs a distribution over the parents of a given input state (e.g., for the molecule domain (§5.2), it is hard to force invariance to molecule isomorphism). A natural choice is to set $P _ { B } ( - | s )$ to be uniform over all the parents of a state $s$ , i.e., $P _ { B } ( - | \bar { s } ) = 1 / \# \{ s ^ { \prime } | ( s ^ { \prime } { \to } s ) \in \mathcal { A } \}$ .
+
+# 4 Related work
+
+Reinforcement learning. GFlowNets are trained to sample proportionally the reward rather than maximize it as usual in RL. However, on tree-structured DAGs (autoregressive generation) are equivalent to RL with appropriate entropy regularization or soft Q-learning and control as inference [5, 12, 13]. The experiments and discussion of [3] show how these methods can fail badly in the general DAG case well handled by GFlowNets. Signal propagation over sequences of several actions in trajectory balance is also related to losses used in RL computed on subtrajectories [22].
+
+Local exploration vs. amortized generalization to unseen modes. GFlowNets are also related to MCMC methods for sampling from unnormalized densities. While there has been work on accelerating or partially amortizing sampling from unnormalized densities over discrete spaces when exact sampling is intractable [11, 7], some of it domain- or problem-specific [30], GFlowNets treat the compositional structure in data as a learning problem (enabling generalization to unseen modes), not as a bias to build in to the sampler. Thus, the cost is amortized and borne by the learning of that structure through sampling, not through search at generation time.
+
+Variational inference. GFlowNets are connected with variational methods for fitting hierarchical generative models. The squared log-ratio loss proposed in [20] as a control variate in the optimization of an evidence lower bound can be seen as a special case of trajectory balance. See $\ S \mathrm { A } . 3$ for further discussion, in which we prove that on-policy optimization of trajectory balance is equivalent to minimization of a certain KL divergence. Two recent papers [31, 19] extend our observations.
+
+# 5 Experiments
+
+We evaluate the proposed trajectory balance objective against prior objectives for training GFlowNets as well as standard methods for learning policies that approximately sample objects proportionally to their rewards, like MCMC as well as against other RL techniques. Our experiments include the hypergrid and molecule synthesis tasks from [3] and two new tasks in which $G$ is a directed tree.
+
+# 5.1 Hypergrid environment
+
+In this subsection, we study a synthetic hypergrid environment introduced by [3]. This task is easier than others we study, but we include it for completeness, and because it allows us to illustrate some interesting behaviours.4
+
+In this environment, the nonterminal states $S ^ { \circ }$ form a $D$ -dimensional hypergrid with side length $H$ :
+
+$$
+\begin{array} { r } { \mathcal { S } ^ { \circ } = \{ ( s ^ { 1 } , \dotsc , s ^ { D } ) \mid s ^ { d } \in \{ 0 , 1 , \dotsc , H - 1 \} , d = 1 , \dotsc , D \} , } \end{array}
+$$
+
+and actions are operations of incrementing one coordinate in a state by 1 without exiting the grid. The initial state is $( 0 , \ldots , 0 )$ . For every nonterminal state $s$ , there is also a termination action that transitions to a corresponding terminal state $s ^ { \top }$ (cf. footnote 1). The reward at a terminal state $s ^ { \top } = ( s ^ { 1 } , \ldots , s ^ { d } ) ^ { \top }$ is given by
+
+$$
+R ( s ^ { \top } ) = R _ { 0 } + 0 . 5 \prod _ { d = 1 } ^ { D } \mathbb { I } \left[ \left| \frac { s ^ { d } } { H - 1 } - 0 . 5 \right| \in ( 0 . 2 5 , 0 . 5 ) \right] + 2 \prod _ { d = 1 } ^ { D } \mathbb { I } \left[ \left| \frac { s ^ { d } } { H - 1 } - 0 . 5 \right| \in ( 0 . 3 , 0 . 4 ) \right]
+$$
+
+where $\mathbb { I }$ is an indicator function and $R _ { 0 }$ is a constant controlling the difficulty of exploration. This reward has peaks of height $2 . 5 + R _ { 0 }$ near the corners of the hypergrid, surrounded by plateaux of height $0 . 5 + R _ { 0 }$ . These plateaux are separated by wide troughs with reward $R _ { 0 }$ . An illustration with $H = 8$ and $D = 2$ is shown in the left panel of Fig. 1. This environment evaluates the ability of a GFlowNet to generalize from visited states to infer the existence of yet-unvisited modes.
+
+We train GFlowNets with the detailed balance (DB) and trajectory balance (TB) objectives with different $H , D$ , and $R _ { 0 }$ , in addition to reproducing the flow matching (FM) experiments and nonGFlowNet baselines based upon [3]’s published code. Our GFlowNet policy model is a multilayer perceptron (MLP) that accepts as input a one-hot encoding of a state $s$ (with the goal of enabling generalization) and outputs logits of the forward and backward policies $P _ { F } ( - | - ; \theta )$ and $P _ { B } ( - | - ; \bar { \theta ) }$ (as well as the estimated state flow $F _ { \theta } ( s )$ in the case of DB). The forward policy, backward policy, and state flow models share all but the last weight matrix of the MLP. This is consistent with [3]’s model, where an identical architecture accepted $s$ as input and output estimated flows $F _ { \theta } ( s , s ^ { \prime } )$ for all children $s ^ { \prime }$ of $s$ . Details are given in $\ S \mathrm { B } . 1$ .
+
+We consider a 4-dimensional grid with $H = 8$ and and a 2-dimensional grid with $H = 6 4$ . The two grids have the same number of terminal states, but the 2-dimensional grid has longer expected trajectory lengths. For both grid sizes, we consider $R _ { 0 } = 0 . 1 , 0 . 0 1 , 0 . 0 0 1$ , with smaller $R _ { 0 }$ giving environments that are more difficult to explore due to the lower likelihood for models to cross the low-reward valley. For the models trained with DB and TB, we also explore the effect of fixing the backward policy to be uniform (§3.1).
+
+Results. In Fig. 2, we plot the evolution over the course of training of the $L _ { 1 }$ error between the true reward distribution (the reward $R ( x )$ normalized over all possible terminal states $x \in \mathcal { X }$ ) and the empirical distribution of the last $2 \cdot 1 0 ^ { 5 }$ visited states for all settings (which would have a probability of 0 on $x$ ’s not visited). Although convergence to the same stable minimum is achieved by all models and settings, DB and TB training tend to converge faster than FM, with a slight benefit of TB over detailed balance in the 4-D environment.
+
+Effect of uniform $P _ { B }$ . Note the difference in learning speed between models with fixed uniform backward policy $P _ { B }$ and models with learned $P _ { B }$ . As noted in $\ S 3 . 1$ , when $P _ { B }$ is fixed, there is a unique $P _ { F } ( - | - ; \theta )$ that globally minimizes the objective, and it may be approached slowly. However, if $P _ { B }$ and $P _ { F }$ are permitted to evolve jointly, they may more quickly approach one of the many optimal solutions. This is confirmed by the much faster convergence of models with learned $P _ { B }$ on the $6 4 \times 6 4$ grid. We have observed that, especially for large grid sizes, when $P _ { B }$ and $P _ { F }$ are both learned, the model has a bias towards first taking all actions in one coordinate direction, then proceeding in the other direction until terminating (as in the right panel of Fig. 1), perhaps because a constant distribution over two actions (‘continue to the right’ and ‘terminate’) can be modeled with higher precision over a large portion of the grid than the complex position-dependent distribution as shown in the centre panel of Fig. 1.
+
+![](images/ded6a0e6aeeb36a8fca4f2a53a0ac9b4780587b26edd97879125f61902ebe953.jpg)  
+Figure 1: Left: The reward function on an $8 \times 8$ grid environment (§5.1) with $R _ { 0 } = 0 . 1$ . Centre and right: Two forward action policies – with fixed uniform $P _ { B }$ and with a learned non-uniform $P _ { B }$ – that sample from this reward. The lengths of arrows pointing up and right from each state are proportional to the likelihoods of the corresponding actions under $P _ { F }$ , and the sizes of the red octagons are proportional to the termination action likelihoods.
+
+![](images/3aeacba67763413defdfe1bedbc1bc1d526cfa60de51d0701ad1a28e53b69088.jpg)  
+Figure 2: Empirical $L ^ { 1 }$ error between true and sampled state distributions on the grid environment with varying grid size and $R _ { 0 }$ . Mean and standard error over 5 seeds. The curves for PPO and MCMC baseline would lie outside the plot bounds.
+
+# 5.2 Small drug molecule synthesis
+
+Next, we consider the molecule generation task [30, 15, 16, 10, 24] introduced for GFlowNets in [3]. We extend [3]’s published code with an implementation of the TB and DB objectives.5
+
+The goal is to generate molecules, in the form of graphs, with a low binding energy to the 4JNC inhibitor of the sEH (soluble epoxide hydrolase) protein. The graphs generated are junction trees [15] of a vocabulary of building blocks. The reward is defined as the normalized negative binding energy as predicted by a proxy model, itself trained to predict energies computed via docking simulations [26]. The maximum trajectory length is 8, with the number of actions varying between around 100 and 2000 (the larger a molecule, the more possible additions exist), making $| \mathcal { X } |$ about $1 0 ^ { 1 6 }$ .
+
+Results. We plot in Fig. 3 (left and centre) the correlation of log-reward and log-sampling probability (the likelihood that a trajectory sampled from the learned policy terminates at $x$ ) for GFlowNets trained using TB, FM and DB. This correlation is significantly higher for models trained with TB. The points $x$ shown are from a fixed held-out set to which the GFlowNets do not have access in training. Note that a perfect model would have correlation 1, as $\log R ( x )$ and $\log p _ { \theta } ( x )$ would differ by a constant (equal to $\log Z ,$ ) that is independent of $x$ .
+
+In Fig. 3 (right) we plot the average pairwise Tanimoto similarity [1] for the 1000 highest-reward samples generated over the course of training. We see that TB consistently generates more diverse molecules than FM. These results showcase the benefits of TB, not only for faster temporal credit assignment, but for generalization and diversity. In addition, TB has up to $5 \times$ runtime speedup over FM as the enumeration of parents is not needed. See $\ S \mathrm { B } . 2$ for further discussion.
+
+# 5.3 Autoregressive sequence generation
+
+Finally, we evaluate the TB objective on the task of autoregressive sequence generation. In $\ S 5 . 3 . 1$ , we study the effect of trajectory length and action space size on the learning dynamics in GFlowNets. In $\ S 5 . 3 . 2$ , we consider the more realistic task of generating peptides (short protein sequences with anti-microbial properties) and evaluate GFlowNets against standard RL and MCMC baselines.
+
+![](images/2cbc6b95517804bfa1dbab6c862f71e4510105e6928bdd1ce93d4592dcfa915b.jpg)  
+Figure 3: Left, Centre: Pearson correlations between rewards and sampling probability. $\log p _ { \theta } ( x )$ is the log-likelihood that a trajectory sampled from the learned policy $P _ { F } \bar { ( } - | \bar { - } ; \theta )$ terminates at $x$ . Left: Scatter plot on a test set of $x$ ’s for the best hyperparameters of TB, FM, and DB. Centre: Violin plot of correlations for 16 hyperparameter settings and 3 seeds for each setting, showing TB being capable of fitting better. Right: Average pairwise Tanimoto similarity for the top 1000 samples generated by GFlowNets as training progresses. Lines are the average across runs, shaded regions the standard deviation. Models trained with TB have consistently lower similarity than those with FM, hence greater diversity. We hypothesize that the higher variance, in correlation and diversity, of TB relative to DB is related to high variance of the stochastic gradient; see [18].
+
+# 5.3.1 Bit sequences
+
+Task. The goal is to generate bit sequences of a fixed length $n = 1 2 0$ $\mathcal { X } = \{ 0 , 1 \} ^ { n } )$ , where the reward is designed to have modes at a given fixed set $M \subset \mathcal { X }$ that is unknown to the learner. The reward for a sequence $x$ is defined as $\begin{array} { r } { \bar { R ( x ) } = \exp ( - \operatorname* { m i n } _ { y \in M } d ( x , y ) ) } \end{array}$ , where $d$ is the edit distance. We describe the procedure to generate $M$ in $\ S \mathrm { B } . 3$ .
+
+For different integers $k$ dividing $n$ , we design action spaces for left-to-right generation of sequences in $\mathcal { X }$ , where a complete trajectory has $\frac { n } { k }$ actions and each action appends a $k$ -bit ‘word’ to the end of a partial sequence. A forward policy on this state space is the same an autoregressive sequence model over a vocabulary of size $2 ^ { k }$ . Varying $k$ while fixing $\mathcal { X }$ and $M$ allows us to study the effect of the tradeoff between trajectory lengths $\textstyle { \binom { n } { k } }$ and the action space sizes $( | V | = 2 ^ { k }$ ) without changing the underlying probabilistic modeling problem.
+
+We compare GFlowNets trained with the TB objective against GFlowNets trained with the FM objective (equivalent to DB and Soft Q-Learning in this case) and two non-GFlowNet baselines: A2C with Entropy Regularization [29, 21], Soft Actor-Critic [13, 6] and MARS [30]. We use a Transformer-based architecture [28] across all the methods. See $\ S \mathbf { B } . 3$ for details.
+
+To evaluate the methods we use (1) Spearman correlation between the probability of generating the sequence $p ( x ) = F ( x ) / Z$ and its reward $R ( x )$ on a test set sampled approximately uniformly over the possible values of the reward, (2) number of modes captured (number of reference sequences from $M$ for which a candidate within a distance $\delta$ has been generated). In our experiments, $n = 1 2 0$ , $| M | = 6 0$ , $k \in \{ 1 , 2 , 4 , 6 , 8 , 1 0 \}$ , and $\delta = 2 0$ .
+
+Results. Fig. 4 (left) presents the results for the Spearman correlation evaluation. We observe that GFlowNets trained with the TB objective learn policies that correlate best with the reward $R ( x )$ across all action spaces. In particular, we observe the effect of inefficient credit assignment in GFlowNets trained with FM, as the correlation improves with increasing $k$ , i.e., shorter trajectories. On the other hand, large action spaces also hurt GFlowNets trained with the FM objective, while the TB objective is robust to them. Additionally, we can observe in Fig. 4 (right) that for fixed $k$ , GFlowNets trained with TB discover more modes faster than other methods.
+
+# 5.3.2 Anti-Microbial Peptide (AMP) generation
+
+In this section, we consider the practical task of generating peptide sequences that have anti-microbial activity. The goal is to generate a protein sequence (where the vocabulary consists of 20 amino acids and a special end-of-sequence action), with maximum length 60. We take 6438 known AMP sequences and 9522 non-AMP sequences from the DBAASP database [23]. We then train a classifier on this dataset, using $2 0 \%$ of the data as a validation set. The probability output by this model for a sequence to be classified as an AMP is used as the reward $R ( x )$ in our experiments.
+
+Table 1: Results on the AMP generation task.   
+
+<table><tr><td></td><td>Top 100 Reward</td><td>Top 100Diversity</td></tr><tr><td>GFN-LTB</td><td>0.85 ± 0.03</td><td>18.35±1.65</td></tr><tr><td>GFN-LFM/LDB</td><td>0.78±0.05</td><td>12.61 ± 1.32</td></tr><tr><td>SAC</td><td>0.80±0.01</td><td>8.36±1.44</td></tr><tr><td>AAC-ER</td><td>0.79±0.02</td><td>7.32 ± 0.76</td></tr><tr><td>MCMC</td><td>0.75±0.02</td><td>12.56±1.45</td></tr></table>
+
+![](images/e419992c7749710eb2081ca5ec30da443d61590a54d3a58a98203d0418f75fb5.jpg)  
+Figure 4: Left: Spearman correlation of the sampling probability under different learned policies and reward on a test set, plotted against the number of bits $k$ in the symbols in $V$ in the bit sequence generation task. GFlowNets trained with trajectory balance learn policies that have the highest correlation with the reward $R ( x )$ and are robust to length and vocabulary size. Right: Number of modes discovered over the course of training on the bit sequence generation task with $k = 1$ . GFlowNets trained with trajectory balance discover more modes faster.
+
+The state and actions are designed just as in the previous experiment, with each action appending a symbol to the right of a state. We again compare TB and FM, as well as A2C with entropy regularization, SAC and MCMC as baselines. We again use Transformers for all the experiments on this task; see further details in Appendix B.4. We generate 2048 sequences from each method, and pick the top 100 sequences ranked by their reward $R ( x )$ . As metrics, we use the mean reward for these 100 sequences and the average pairwise edit distance among them as a measure of diversity.
+
+Results. We present the results in Table 1, where we observe that GFlowNets trained with TB outperform all baselines on both performance and diversity metrics.
+
+# 6 Discussion and conclusion
+
+We introduced a novel training loss for GFlowNets, trajectory balance (TB), which yields faster and better training than the previously proposed flow matching (FM) and detailed balance (DB) losses. We proved that this objective, when minimized, yields the desired GFlowNet property of sampling from the target distribution specified by an unnormalized reward function. This new loss was motivated by the observation that the FM and DB losses are local in the action sequence and may require many iterations for credit assignment to propagate to early actions: if a gradient update introduces a flow inconsistency at some state far from the initial state (such as when a novel high-reward state is sampled), the parent of this state must be visited before the update is propagated closer to the root, akin to the slow propagation of reward signals in temporal difference learning.
+
+We empirically found that TB discovered more modes of the energy function faster and was more robust than FM and DB to the exponential growth of the state space, due in part to the lengths of sequences and in part to the size of the action space. A factor to consider when interpreting our experimental results is that because we use a neural net rather than a tabular representation of policies, the early states’ transitions are informed by downstream credit assignment via parameter sharing. Early states also get many more visits because there are more possible states near the ends of sequences than near the initial state. Finally, TB trades off the advantage of immediately providing credit to early states with the disadvantage of relying on sampling of long trajectories and thus a potentially higher variance of the stochastic gradient. The high gradient variance is a possible limitation of TB in difficult environments, and ways to overcome it by interpolating between local and trajectory-level objectives have been studied in subsequent work [18].
+
+All in all, we found that trajectory balance is a superior training objective in a broad set of experiments, making it the default choice for future work on GFlowNets.
+
+# Acknowledgments
+
+This research was enabled in part by computational resources provided by Compute Canada. All authors are funded by their primary academic institution. We also acknowledge funding from CIFAR, Samsung, IBM, Microsoft, and the Banting Postdoctoral Fellowship.
+
+The authors are grateful to all the members of the Mila GFlowNet group, in particular to Dinghuai Zhang, for many fruitful research discussions, as well as to Yiheng Zhu for feedback on the published code. We also thank the anonymous reviewers for their comments.
+
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+[22] Nachum, O., Norouzi, M., Xu, K., and Schuurmans, D. Bridging the gap between value and policy based reinforcement learning. Neural Information Processing Systems (NeurIPS), 2017.   
+[23] Pirtskhalava, M., Amstrong, A. A., Grigolava, M., Chubinidze, M., Alimbarashvili, E., Vishnepolsky, B., Gabrielian, A., Rosenthal, A., Hurt, D. E., and Tartakovsky, M. Dbaasp v3: Database of antimicrobial/cytotoxic activity and structure of peptides as a resource for development of new therapeutics. Nucleic Acids Research, 49(D1):D288–D297, 2021.   
+[24] Shi, Y., Huang, Z., Feng, S., Zhong, H., Wang, W., and Sun, Y. Masked label prediction: Unified message passing model for semi-supervised classification. International Joint Conference on Artificial Intelligence (IJCAI), 2021.   
+[25] Sutton, R. S. and Barto, A. G. Reinforcement learning: An introduction. MIT Press, 2018.   
+[26] Trott, O. and Olson, A. J. AutoDock Vina: improving the speed and accuracy of docking with a new scoring function, efficient optimization, and multithreading. Journal of computational chemistry, 31(2):455–461, 2010.   
+[27] van Hasselt, H., Doron, Y., Strub, F., Hessel, M., Sonnerat, N., and Modayil, J. Deep reinforcement learning and the deadly triad. arXiv preprint 1812.02648, 2018.   
+[28] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., and Polosukhin, I. Attention is all you need. Neural Information Processing Systems (NeurIPS), 2017.   
+[29] Williams, R. J. and Peng, J. Function optimization using connectionist reinforcement learning algorithms. Connection Science, 3(3):241–268, 1991.   
+[30] Xie, Y., Shi, C., Zhou, H., Yang, Y., Zhang, W., Yu, Y., and Li, L. MARS: Markov molecular sampling for multi-objective drug discovery. International Conference on Learning Representations (ICLR), 2021.   
+[31] Zhang, D., Chen, R. T. Q., Malkin, N., , and Bengio, Y. Unifying generative models with GFlowNets. arXiv preprint 2209.02606, 2022.   
+[32] Zhang, D., Malkin, N., Liu, Z., Volokhova, A., Courville, A., and Bengio, Y. Generative flow networks for discrete probabilistic modeling. International Conference on Machine Learning (ICML), 2022.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] This is mainly a paper about theory and algorithms. Future work that applies these algorithms, in particular for domains where they can most immediately have an impact (e.g., molecule design for drug discovery), should consider the limitations and negative societal impacts of these applications.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] See previous.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] See $\ S \mathrm { A } . 1$
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] For the grid and molecule environments. [No] For the other environments.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See the Appendix.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See the Appendix.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See the Appendix.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See the relevant experiment sections.   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] References to the molecule and AMP sequence data are provided.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] No new data collection was done.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] Not relevant for the domains studied.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/5btWTw1vcw1/5btWTw1vcw1_content_list.json

@@ -0,0 +1,1577 @@
+[
+    {
+        "type": "text",
+        "text": "Trajectory balance: Improved credit assignment in GFlowNets ",
+        "text_level": 1,
+        "bbox": [
+            241,
+            122,
+            756,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Nikolay Malkin Mila, Université de Montréal Montréal, Québec, Canada ",
+        "bbox": [
+            259,
+            226,
+            454,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Moksh Jain Mila, Université de Montréal Montréal, Québec, Canada ",
+        "bbox": [
+            544,
+            226,
+            738,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Emmanuel Bengio Mila, McGill University, Recursion Montréal, Québec, Canada ",
+        "bbox": [
+            245,
+            289,
+            480,
+            332
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chen Sun Mila, Université de Montréal Montréal, Québec, Canada ",
+        "bbox": [
+            558,
+            289,
+            751,
+            332
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yoshua Bengio Mila, Université de Montréal Montréal, Québec, Canada {nikolay.malkin,moksh.jain,chen.sun,yoshua.bengio}@mila.quebec emmanuel.bengio@recursionpharma.com ",
+        "bbox": [
+            259,
+            352,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            457,
+            535,
+            474
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative flow networks (GFlowNets) are a method for learning a stochastic policy for generating compositional objects, such as graphs or strings, from a given unnormalized density by sequences of actions, where many possible action sequences may lead to the same object. We find previously proposed learning objectives for GFlowNets, flow matching and detailed balance, which are analogous to temporal difference learning, to be prone to inefficient credit propagation across long action sequences. We thus propose a new learning objective for GFlowNets, trajectory balance, as a more efficient alternative to previously used objectives. We prove that any global minimizer of the trajectory balance objective can define a policy that samples exactly from the target distribution. In experiments on four distinct domains, we empirically demonstrate the benefits of the trajectory balance objective for GFlowNet convergence, diversity of generated samples, and robustness to long action sequences and large action spaces. ",
+        "bbox": [
+            232,
+            489,
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+            669
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            694,
+            310,
+            712
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative flow networks [GFlowNets; 3, 4] are models that exploit generalizable structure in an energy function $\\mathcal { E }$ to amortize sampling from the corresponding probability density function on a space of compositional objects $\\mathcal { X }$ , for example, graphs composed of nodes and edges. A GFlowNet learns a stochastic policy that generates such structured objects by producing a stochastic sequence of actions that incrementally modify a partial object (state), e.g., by adding an edge or a node to a graph, starting from a universal initial state (like an empty graph). A special ‘exit’ action signals that the construction of the object $x \\in \\mathcal { X }$ is completed, and the policy is trained so as to make the likelihood of generating $x$ proportional to the given unnormalized probability or reward $R ( x ) = e ^ { - \\mathcal { E } ( x ) }$ . ",
+        "bbox": [
+            174,
+            726,
+            825,
+            838
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Like other models in deep reinforcement learning [RL; 25], GFlowNets are trained with a parametric policy that can be given desired inductive biases (e.g., a particular deep net architecture) and allows generalization to states not seen in training. Natural domains for applying GFlowNets are those where exact sampling is intractable and local exploration (MCMC) methods perform poorly, but diversity of samples is desired [3, 32, 14, 8]. For example, GFlowNets have been used [3] to generate graphical descriptions of molecules by incremental addition of simple building blocks, where the reward $R ( x )$ is the estimated strength of binding the constructed molecule to a protein target: the number of candidates grows rapidly with the molecule size and the reward has many separated modes. Like all RL models that iteratively sample action sequences for training, GFlowNets pose the learning challenges of exploration/exploitation and credit assignment, i.e., propagation of a reward signal over an action sequence [27, 2, 17]. The efficiency of credit assignment and training in GFlowNets is the focus of the present paper. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
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+            203
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The learning problem solved by GFlowNets also has two fundamental differences with the standard reward-maximization paradigm of RL. First, a GFlowNet aims to make the likelihood of reaching a terminating state proportional to the reward, not to concentrate it at a maximal-reward state. Thus, a GFlowNet must model the diversity in the target distribution, not only its dominant mode. Reward maximization in RL can be turned into sampling proportionally to the reward with appropriate entropy maximization regularization, if there is only one way to reach every state [3]. The second difference with reward-maximization in RL is indeed that the GFlowNet training objectives still lead to correct sampling even when multiple action sequences lead to the same terminating state. Note that the likelihood of reaching a state is the sum of likelihoods of all action sequences leading to it, and that the number of such paths may be exponential in their length. ",
+        "bbox": [
+            173,
+            208,
+            825,
+            347
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The set of all achievable sequences of actions and states can be conceptually organized in a directed graph $G = ( S , A )$ in which the vertices $s$ are states (some of them designated as terminal states, in bijection with $\\mathcal { X }$ ) and the edges $u {  } v$ in $\\mathcal { A }$ each correspond to applying an action while in a state $u \\in S$ and landing in state $v$ . In [3], a GFlowNet is described by a nonnegative function on the edges, called the edge flow $F : \\mathcal { A }  \\mathbb { R } _ { \\geq 0 }$ , where $F ( u {  } v )$ is an unnormalized likelihood of taking the action that modifies state $u$ to state $v$ . The GFlowNet policy samples the transition $u {  } v$ from state $u$ with probability $\\scriptstyle { F ( u \\to v ) / \\sum _ { v ^ { \\prime } } F ( u \\to v ^ { \\prime } ) }$ where the denominator sums over the outgoing edges from $u$ . By analogy with the classical notion of flows in networks [9], one can think of this flow like the amount of water flowing through an edge (like a pipe) or a state (like a tee, where pipes meet). It is shown that this GFlowNet policy samples $x$ proportionally to $R ( x )$ if $F$ satisfies a set of linear flow matching constraints (a conservation law: the sum of flows into a state should equal the sum of flows out of it). These constraints are converted into a temporal difference-like objective that can be optimized with respect to the parameters of a neural net that approximates $F$ . An alternative objective based on detailed balance constraints was proposed in [4]. These objectives, however, like temporal-difference learning, can suffer from slow credit assignment [27, 2, 17]. ",
+        "bbox": [
+            173,
+            354,
+            825,
+            560
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The main contribution of this work (§3) is a new parametrization and objective for GFlowNets. This objective, which we call trajectory balance, is computed on sampled full action sequences (trajectories) from the initial state to a terminal state, unlike the flow matching and detailed balance objectives. We prove that global minimization of trajectory balance implies that the learned action policy samples proportionally to $R$ . We also empirically show that the trajectory balance objective accelerates training convergence relative to previously proposed objectives, improves the learned sampling policy with respect to metrics of diversity and divergence from the reward function, and allows learning GFlowNets that generate sequences far longer than was possible before. As a secondary contribution, we perform the first empirical validation of the detailed balance training objective. Comparative evaluation of the three GFlowNet objectives and non-GFlowNet baselines is performed on four domains illustrating different features of the reward landscape: ",
+        "bbox": [
+            173,
+            565,
+            825,
+            718
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Hypergrid (§5.1), an illustrative synthetic environment with modes separated by wide troughs; • Molecule synthesis (§5.2), a practical graph generation problem, where the trajectory balance objective leads to significant computational speed-ups and more diverse generated candidates; • Sequence generation (§5.3), where we show the robustness of trajectory balance to large action spaces and long action sequences on synthetic data and real AMP sequence data. ",
+        "bbox": [
+            173,
+            724,
+            825,
+            801
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Since the initial appearance of this work on arXiv, several published papers and preprints have used trajectory balance and its generalizations successfully in various applications [32, 14, 8, 18]. ",
+        "bbox": [
+            173,
+            808,
+            823,
+            837
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            88,
+            318,
+            106
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Markovian flows ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            119,
+            330,
+            135
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We give some essential definitions, following $\\ S 2$ of [4]. Fix a directed acyclic graph $G = (  { \\boldsymbol { S } } ,  { \\boldsymbol { A } } )$ with state space $s$ and action space $\\mathcal { A }$ . Let $s _ { 0 } \\in S$ be the special initial (source) state, the only state with no incoming edges, and designate vertices with no outgoing edges as terminal (sinks) 1. We call the vertices states, the edges actions, the states reachable through outgoing edges from a state its children, and the sources of its incoming edges its parents. ",
+        "bbox": [
+            173,
+            145,
+            825,
+            215
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A complete trajectory is a sequence of transitions $\\tau = ( s _ { 0 } {  } s _ { 1 } {  } \\dots {  } s _ { n }$ ) going from the initial state $s _ { 0 }$ to a terminal state $s _ { n }$ with $( s _ { t } {  } s _ { t + 1 } ) \\in \\mathcal { A }$ for all $t$ . Let $\\tau$ be the set of complete trajectories. A trajectory flow is a nonnegative function $F : \\mathcal { T } {  } \\mathbb { R } _ { \\geq 0 }$ . With our water analogy, it could be the number of water molecules travelling along this path (the units don’t matter because the flow function can be scaled arbitrarily, since we normalize them to get probabilities). For any state $s$ , define the state flow $\\begin{array} { r } { F ( s ) = \\sum _ { s \\in \\tau } F ( \\tau ) } \\end{array}$ , and, for any edge $s {  } s ^ { \\prime }$ , the edge flow ",
+        "bbox": [
+            173,
+            220,
+            826,
+            306
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/66cb3c06362fe7cde1b234fa52360a6e3064e71830e8e6acdec04ee216c24849.jpg",
+        "text": "$$\nF ( s \\to s ^ { \\prime } ) = \\sum _ { \\tau = ( \\ldots \\to s \\to s ^ { \\prime } \\to \\ldots ) } F ( \\tau ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            313,
+            619,
+            349
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As a consequence of this definition, the flow matching constraint (incoming flow $=$ outgoing flow) is satisfied for all states $s$ that are not initial or terminal: ",
+        "bbox": [
+            174,
+            356,
+            823,
+            383
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bae4f5b893f94e142fd0a4ae3feccdd811d2088e1d08e8350f946dd6281c8fa9.jpg",
+        "text": "$$\nF ( s ) = \\sum _ { ( s ^ { \\prime \\prime }  s ) \\in A } F ( s ^ { \\prime \\prime } {  } s ) = \\sum _ { ( s  s ^ { \\prime } ) \\in A } F ( s {  } s ^ { \\prime } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            386,
+            669,
+            424
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A nontrivial (i.e., not identically zero) trajectory flow $F$ determines a distribution $P$ over trajectories, ",
+        "bbox": [
+            176,
+            429,
+            825,
+            445
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c837e2173c7bb4da762048c1a6680f1b0b91b6a94417ffcc6216ec13620f0d4b.jpg",
+        "text": "$$\nP ( \\tau ) = \\frac { 1 } { Z } F ( \\tau ) , \\quad Z = F ( s _ { 0 } ) = \\sum _ { \\tau \\in \\mathcal { T } } F ( \\tau ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            450,
+            650,
+            488
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The trajectory flow $F$ is Markovian if there exist action distributions $P _ { F } ( - | s )$ over the children of each nonterminal state $s$ such that the distribution $P$ has a factorization ",
+        "bbox": [
+            173,
+            494,
+            825,
+            523
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d238990c88b6dc59923ad266c0dbcc7bee469fa65a6b1dd48ced6a81152ad805.jpg",
+        "text": "$$\nP ( \\tau = ( s _ { 0 } {  } . . . {  } s _ { n } ) ) = \\prod _ { t = 1 } ^ { n } P _ { F } ( s _ { t } | s _ { t - 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            527,
+            647,
+            569
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Equivalently ([4], Prop. 3) there are distributions $P _ { B } ( - | s )$ over the parents of each noninitial state $s$ such that for any terminal $x$ , ",
+        "bbox": [
+            174,
+            575,
+            825,
+            604
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/750e55f330d02b6d00616cd48510c5234708954d044a3a141b140f25bf24009c.jpg",
+        "text": "$$\nP ( \\tau = ( s _ { 0 }  . . .  s _ { n } ) | s _ { n } = x ) = \\prod _ { t = 1 } ^ { n } P _ { B } ( s _ { t - 1 } | s _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            611,
+            673,
+            651
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "If $F$ is a Markovian flow, then $P _ { F }$ and $P _ { B }$ can be computed in terms of state and edge flows: ",
+        "bbox": [
+            173,
+            657,
+            779,
+            672
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6a149fb393912b07095241c4e936534224472c25846ba3ee886f0df9c7312c9e.jpg",
+        "text": "$$\nP _ { F } ( s ^ { \\prime } | s ) = \\frac { F ( s  s ^ { \\prime } ) } { F ( s ) } , \\quad P _ { B } ( s | s ^ { \\prime } ) = \\frac { F ( s  s ^ { \\prime } ) } { F ( s ^ { \\prime } ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            679,
+            658,
+            713
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "supposing denominators do not vanish. We call $P _ { F }$ and $P _ { B }$ the forward policy and backward policy corresponding to $F$ , respectively. These relations are summarized by the detailed balance constraint ",
+        "bbox": [
+            171,
+            718,
+            823,
+            747
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f9a2616a88bb5775c7fba793a3c86c3bd0ab8120e60ef0a00b8c5230f4a5b64f.jpg",
+        "text": "$$\nF ( s ) P _ { F } ( s ^ { \\prime } | s ) = F ( s ^ { \\prime } ) P _ { B } ( s | s ^ { \\prime } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            753,
+            607,
+            772
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Uniqueness properties. A Markovian flow is uniquely determined by an edge flow, i.e., a nontrivial choice of nonnegative value on every edge satisfying the flow matching constraint (2). By Corollary 1 of [4], a Markovian flow is also uniquely determined by either of ",
+        "bbox": [
+            173,
+            785,
+            825,
+            827
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• a constant $Z = F ( s _ { 0 } ) > 0$ and a distribution $P _ { F } ( - | s )$ over children of every nonterminal state; or • a nontrivial choice of nonnegative state flows $F ( x )$ for every terminal state $x$ and a choice of distribution $P _ { B } ( - | s )$ over parents of every noninitial state. ",
+        "bbox": [
+            174,
+            830,
+            825,
+            878
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 GFlowNets ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            290,
+            106
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Suppose that a nontrivial nonnegative reward function $R : \\mathcal { X } \\to \\mathbb { R } _ { \\geq 0 }$ is given on the set of terminal states. GFlowNets [3] aim to approximate a Markovian flow $F$ on $\\bar { G }$ such that ",
+        "bbox": [
+            173,
+            116,
+            825,
+            145
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f20f710b5af70409736bc2f77c27149d3b4d18b30ee6d0262aa58f280ca266ec.jpg",
+        "text": "$$\nF ( x ) = R ( x ) \\quad \\forall x \\in { \\mathcal { X } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
+            146,
+            583,
+            162
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We adopt the broad definition that a GFlowNet is any learning algorithm consisting of: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• a model capable of providing the initial state flow $Z = F ( s _ { 0 } )$ as well as the forward action distributions $P _ { F } ( - | \\bar { s } )$ for any nonterminal state $s$ (and therefore, by the above, uniquely but possibly in an implicit way determining a Markovian flow $F$ );   \n• an objective function, such that if the model is capable of expressing any action distribution and the objective function is globally minimized, then the constraint (8) is satisfied for the corresponding Markovian flow $F$ . ",
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+    {
+        "type": "text",
+        "text": "The forward policy of a GFlowNet can be used to sample trajectories from the corresponding Markovian flow $F$ by iteratively taking actions according to policy $\\mathop { P _ { F } ( - | s ) }$ . If the objective function is globally minimized, then the likelihood of terminating at $x$ is proportional to $R ( x )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In general, an objective optimizing for (8) cannot be minimized directly because $F ( x )$ is a sum over all trajectories leading to $x$ , and computing it may not be practical. Therefore, two local objectives – flow matching and detailed balance – have previously been proposed. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Flow matching objective [3]. A model $F _ { \\theta } ( s , s ^ { \\prime } )$ 2 with learnable parameters $\\theta$ approximates the edge flows $F ( s {  } s ^ { \\prime } )$ . The corresponding forward policy is given by $P _ { F } ( s ^ { \\prime } | s ; \\theta ) \\overset { \\sim } { \\propto } F _ { \\theta } ( s , s ^ { \\prime } )$ (Eq. (6)). Denote the corresponding Markovian flow by $F _ { \\theta }$ and distribution over trajectories by $P _ { \\theta }$ . The parameters are trained to minimize the error in the flow matching constraint (2) for all noninitial and nonterminal nodes $s$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/49be41daa672fbbede9863a5d7eabf573fde6d2614b90733593ff48dcfb361b9.jpg",
+        "text": "$$\n{ \\mathcal { L } } _ { \\mathrm { F M } } ( s ) = ( \\log { \\frac { \\sum _ { ( s ^ { \\prime \\prime }  s ) \\in A } F _ { \\theta } ( s ^ { \\prime \\prime } , s ) } { \\sum _ { ( s  s ^ { \\prime } ) \\in A } F _ { \\theta } ( s , s ^ { \\prime } ) } } ) ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and a similar objective $\\mathcal { L } _ { \\mathrm { F M } } ^ { \\prime }$ pushing the inflow at $x \\in \\mathcal { X }$ to equal $R ( x )$ at terminal nodes $x$ . This objective is optimized for nonterminal states $s$ and terminal states $x$ from trajectories sampled from a training policy $\\pi _ { \\theta }$ . Usually, $\\pi _ { \\theta }$ is chosen to be a tempered (higher temperature) version of $P _ { F } ( - | s , \\theta )$ , which also helps exploration during training. The parameters are updated with stochastic gradient ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aae07236965eebc9bb336c1d17cad30d3e3ad3e1c0c72d606e2be809b54bf94b.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\tau = ( s _ { 0 }  \\dots  s _ { n } ) \\sim \\pi _ { \\theta } } \\nabla _ { \\theta } \\Bigg [ \\sum _ { { t = 1 } } ^ { n - 1 } \\mathcal { L } _ { \\mathrm { F M } } ( s _ { t } ) + \\mathcal { L } _ { \\mathrm { F M } } ^ { \\prime } ( s _ { n } ) \\Bigg ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As per Proposition 10 of [4], if the training policy $\\pi _ { \\theta }$ has full support, and a global minimum of the expected loss (9) over states on trajectories sampled from $\\pi _ { \\theta }$ is reached, then the GFlowNet samples from the target distribution (i.e., $F _ { \\theta }$ satisfies (8)). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Detailed balance objective [4]. A neural network model with parameters $\\theta$ has input $s$ and three kinds of outputs: an estimated state flow $F _ { \\theta } ( s )$ , an estimated distribution over children $P _ { F } ( - | s ; \\theta )$ , and an estimated distribution over parents $P _ { B } ( - | s ; \\theta )$ . The policy $P _ { F } ( - | - ; \\theta )$ and the initial state flow $F _ { \\theta } ( s _ { 0 } )$ uniquely determine a Markovian flow $F _ { \\theta }$ , which is not necessarily compatible with the estimated backward policy $P _ { B } ( - | - ; \\theta )$ . The error in the detailed balance constraint (7) is optimized on actions $( s {  } s ^ { \\prime } )$ ) between nonterminal nodes seen along trajectories sampled from the training policy: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2449dd3d939fa1557db1a5ce5e0bc2a3844cb4a75fc38b9910a5b14389ce69e3.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { D B } } ( s , s ^ { \\prime } ) = \\left( \\log \\frac { F _ { \\boldsymbol \\theta } ( s ) P _ { F } ( s ^ { \\prime } | s ; \\boldsymbol \\theta ) } { F _ { \\boldsymbol \\theta } ( s ^ { \\prime } ) P _ { B } ( s | s ^ { \\prime } ; \\boldsymbol \\theta ) } \\right) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and a similar constraint $\\mathcal { L } _ { \\mathrm { D B } } ^ { \\prime } ( s , x )$ is optimized at actions leading to terminal nodes. Similarly to flow matching, the parameters are updated with stochastic gradient ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0b289784424e10b742870966724632038d49d2bb22cf5c3c42de00856950cae9.jpg",
+        "text": "$$\n\\mathbb { E } _ { ( s _ { 0 }  \\ldots  s _ { n } ) \\sim \\pi _ { \\theta } } \\nabla _ { \\theta } [ \\sum _ { t = 1 } ^ { n - 1 } \\mathcal { L } _ { \\mathrm { D B } } ( s _ { t - 1 } , s _ { t } ) + \\mathcal { L } _ { \\mathrm { D B } } ^ { \\prime } ( s _ { n - 1 } , s _ { n } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "along trajectories sampled from a training policy $\\pi _ { \\theta }$ . By Proposition 6 of [4], a global minimum of the expected detailed balance loss under a $\\pi _ { \\theta }$ with full support specifies a GFlowNet that samples from the target distribution, i.e., the flow $F _ { \\theta }$ satisfies (8). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/ac8118953d4e8112adc91eb11480db7cbb648fdae19aceb083f1dc46b09e700f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>inputReward functionR:X →R&gt;o,model and optimizer hyperparameters</td></tr><tr><td>1:Initialize models PF,Pb,Z with parameters 0</td></tr><tr><td>2: ：repeat</td></tr><tr><td>3: Sample trajectory T = (so-→.. → Sn) from policy PF(-|-;0) or a tempered version of it</td></tr><tr><td>4: θ ←θ-nVθLTB(τ) {gradient update on (14)}</td></tr><tr><td>5:</td></tr><tr><td>until convergence monitoring on running LTB(T)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remarks. In some problems, such as autoregressive sequence generation (§5.3), the directed graph $G$ is a tree, so each state has only one parent. In this case, $P _ { B }$ is trivial and the detailed balance objective reduces to the flow matching objective, which in turn can be shown to be equivalent to Soft Q-Learning [12, 5] with temperature $\\alpha = 1$ , a uniform $q _ { \\mathbf { a ^ { \\prime } } }$ , and $\\gamma = 1$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Trajectory balance ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let $F$ be a Markovian flow and $P$ the corresponding distribution over complete trajectories, defined by (3), and let $P _ { F }$ and $P _ { B }$ be forward and backward policies determined by $F$ . A direct algebraic manipulation of Eqs. (3,4,5) gives the trajectory balance constraint for any complete trajectory $\\tau = ( s _ { 0 } {  } s _ { 1 } {  } \\dots {  } s _ { n } = x _ { , }$ ): ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5ac647542579febf35c8b556ebe433299bc6851cd8e0a80708da747e81755d09.jpg",
+        "text": "$$\nZ \\prod _ { t = 1 } ^ { n } P _ { F } ( s _ { t } | s _ { t - 1 } ) = F ( x ) \\prod _ { t = 1 } ^ { n } P _ { B } ( s _ { t - 1 } | s _ { t } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where we have used that $\\begin{array} { r } { P ( s _ { n } = x ) = \\frac { F ( x ) } { Z } } \\end{array}$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As explained in $\\ S \\mathrm { A } . 2$ , the trajectory balance constraint (13) and the detailed balance constraint (7) are special cases of one general constraint, which has been studied as a training objective in [18]. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Trajectory balance as an objective. We propose to convert (13) into an objective to be optimized along trajectories sampled from a training policy. Suppose that a model with parameters $\\theta$ outputs estimated forward policy $P _ { F } ( - | s ; \\theta )$ and backward policy $P _ { B } ( - | s ; \\theta )$ for states $s$ (just as for detailed balance above), as well as a global scalar $Z _ { \\theta }$ estimating $F ( s _ { 0 } )$ . The scalar $Z _ { \\theta }$ and forward policy $P _ { F } ( - | - ; \\theta )$ uniquely determine an implicit Markovian flow $F _ { \\theta }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For a trajectory $\\tau = ( s _ { 0 } {  } s _ { 1 } {  } \\dots {  } s _ { n } = x )$ ), define the trajectory loss ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ee4666ac35d7b93c14069b566028332fee2bdf01123c4c37dee8c2849dbda8aa.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { T B } } ( \\tau ) = \\left( \\log \\frac { Z _ { \\theta } \\prod _ { t = 1 } ^ { n } P _ { F } ( s _ { t } | s _ { t - 1 } ; \\theta ) } { R ( x ) \\prod _ { t = 1 } ^ { n } P _ { B } ( s _ { t - 1 } | s _ { t } ; \\theta ) } \\right) ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "If $\\pi _ { \\theta }$ is a training policy – usually that given by $P _ { F } ( - | - ; \\theta )$ or a tempered version of it – then the trajectory loss is updated along trajectories sampled from $\\pi _ { \\theta }$ , i.e., with stochastic gradient ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb5d9cb44d6bf5852896fca3cca24e5b8633461e2849ff782def3b4b682b38a0.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { \\tau \\sim \\pi _ { \\theta } } \\nabla _ { \\theta } \\mathcal { L } _ { \\mathrm { T B } } ( \\tau ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The full algorithm, with batch size of 1, is presented as Algorithm 1 and its correctness is guaranteed by the following. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. Let $R$ be a positive reward function on $\\mathcal { X }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(a) If $P _ { F } ( - | - ; \\theta )$ , $P _ { B } ( - | - ; \\theta )$ , and $Z _ { \\theta }$ are the forward and backward policies and normalizing constant of a Markovian flow $F$ satisfying (8), then $\\mathcal { L } _ { \\mathrm { T B } } ( \\tau ) = 0$ for all complete trajectories $\\tau$   \n(b) Conversely, suppose that $\\mathcal { L } _ { \\mathrm { T B } } ( \\tau ) = 0$ for all complete trajectories $\\tau$ . Then the corresponding Markovian flow $F _ { \\theta }$ satisfies (8), and $P _ { F } ( - | - ; \\theta )$ samples proportionally to the reward. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The proof is given in $\\ S \\mathrm { A . 1 }$ . In particular, if $\\pi _ { \\theta }$ has full support and $\\mathbb { E } _ { \\tau \\sim \\pi _ { \\theta } } \\mathcal { L } _ { \\mathrm { T B } } ( \\tau )$ is globally minimized over all forward and backward policies $( P _ { F } , P _ { B } )$ and normalizing constants $Z$ , then the corresponding Markovian flow $F _ { \\theta }$ satisfies (8) and $P _ { F } ( - | - ; \\theta )$ samples proportionally to the reward. (The positivity assumption on $R$ is necessary to avoid division by 0 in (14), but can be relaxed by introduction of smoothing constants, just as was done for the losses proposed in [3, 4].) ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remarks. (1) As discussed in $\\ S 2$ , in the case of auto-regressive generation, $G$ is a directed tree, where each $s \\in S$ has a single parent state. In this case $P _ { B }$ is trivially $P _ { B } = 1$ , $\\forall s \\in S$ . We get ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c091813d0171c25feddeb7ee19e37df848b1d16311c5fb5ce16ae3307e74475c.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { T B } } ( \\tau ) = \\left( \\log \\frac { Z _ { \\theta } \\prod _ { t = 1 } ^ { n } P _ { F } ( s _ { t } | s _ { t - 1 } ; \\theta ) } { R ( x ) } \\right) ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(2) We found it beneficial to parametrize $Z$ in the logarithmic domain $\\log Z$ is the trainable parameter) and output logits for $P _ { F } ( - | s ; \\theta )$ and $P _ { B } { \\big ( } - | s ; \\theta { \\big ) }$ , so that all products in (14) become sums under the logarithm. This is consistent with the log-domain parametrization of flows in [3]. In addition, we found it helpful to set a higher learning rate for $Z$ than for the parameters of $P _ { F }$ and $P _ { B }$ . 3 ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.1 Canonical choice of reward-matching flow ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The constraint (8), in general, does not have a unique solution: if the underlying undirected graph of $G$ has cycles, there may be multiple Markovian flows whose corresponding action policies sample proportionally to the reward. However, by the uniqueness properties, for any choice of backward policy $P _ { B } ( - | - )$ , there is a unique flow satisfying (8), and thus a unique corresponding forward policy $P _ { F } ( - | - )$ for states with nonzero flow. (See Fig. 1.) ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In some settings, it may be beneficial to $\\mathit { \\Omega } \\mathcal { f } x$ the backward policy $P _ { B }$ and train only the parameters giving $P _ { F }$ and $Z _ { \\theta }$ . For example, it may difficult to construct a model that outputs a distribution over the parents of a given input state (e.g., for the molecule domain (§5.2), it is hard to force invariance to molecule isomorphism). A natural choice is to set $P _ { B } ( - | s )$ to be uniform over all the parents of a state $s$ , i.e., $P _ { B } ( - | \\bar { s } ) = 1 / \\# \\{ s ^ { \\prime } | ( s ^ { \\prime } { \\to } s ) \\in \\mathcal { A } \\}$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Related work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning. GFlowNets are trained to sample proportionally the reward rather than maximize it as usual in RL. However, on tree-structured DAGs (autoregressive generation) are equivalent to RL with appropriate entropy regularization or soft Q-learning and control as inference [5, 12, 13]. The experiments and discussion of [3] show how these methods can fail badly in the general DAG case well handled by GFlowNets. Signal propagation over sequences of several actions in trajectory balance is also related to losses used in RL computed on subtrajectories [22]. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Local exploration vs. amortized generalization to unseen modes. GFlowNets are also related to MCMC methods for sampling from unnormalized densities. While there has been work on accelerating or partially amortizing sampling from unnormalized densities over discrete spaces when exact sampling is intractable [11, 7], some of it domain- or problem-specific [30], GFlowNets treat the compositional structure in data as a learning problem (enabling generalization to unseen modes), not as a bias to build in to the sampler. Thus, the cost is amortized and borne by the learning of that structure through sampling, not through search at generation time. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Variational inference. GFlowNets are connected with variational methods for fitting hierarchical generative models. The squared log-ratio loss proposed in [20] as a control variate in the optimization of an evidence lower bound can be seen as a special case of trajectory balance. See $\\ S \\mathrm { A } . 3$ for further discussion, in which we prove that on-policy optimization of trajectory balance is equivalent to minimization of a certain KL divergence. Two recent papers [31, 19] extend our observations. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the proposed trajectory balance objective against prior objectives for training GFlowNets as well as standard methods for learning policies that approximately sample objects proportionally to their rewards, like MCMC as well as against other RL techniques. Our experiments include the hypergrid and molecule synthesis tasks from [3] and two new tasks in which $G$ is a directed tree. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 Hypergrid environment ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we study a synthetic hypergrid environment introduced by [3]. This task is easier than others we study, but we include it for completeness, and because it allows us to illustrate some interesting behaviours.4 ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this environment, the nonterminal states $S ^ { \\circ }$ form a $D$ -dimensional hypergrid with side length $H$ : ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0904750fcdafa993b49d82c05084258cf5138032d38f5c2a3481ec9d6b2552d8.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { S } ^ { \\circ } = \\{ ( s ^ { 1 } , \\dotsc , s ^ { D } ) \\mid s ^ { d } \\in \\{ 0 , 1 , \\dotsc , H - 1 \\} , d = 1 , \\dotsc , D \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "and actions are operations of incrementing one coordinate in a state by 1 without exiting the grid. The initial state is $( 0 , \\ldots , 0 )$ . For every nonterminal state $s$ , there is also a termination action that transitions to a corresponding terminal state $s ^ { \\top }$ (cf. footnote 1). The reward at a terminal state $s ^ { \\top } = ( s ^ { 1 } , \\ldots , s ^ { d } ) ^ { \\top }$ is given by ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c086255c7700e2c221e16e94d809b70950cd5f47cdf55fc4b8c8bcc177144200.jpg",
+        "text": "$$\nR ( s ^ { \\top } ) = R _ { 0 } + 0 . 5 \\prod _ { d = 1 } ^ { D } \\mathbb { I } \\left[ \\left| \\frac { s ^ { d } } { H - 1 } - 0 . 5 \\right| \\in ( 0 . 2 5 , 0 . 5 ) \\right] + 2 \\prod _ { d = 1 } ^ { D } \\mathbb { I } \\left[ \\left| \\frac { s ^ { d } } { H - 1 } - 0 . 5 \\right| \\in ( 0 . 3 , 0 . 4 ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            287,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbb { I }$ is an indicator function and $R _ { 0 }$ is a constant controlling the difficulty of exploration. This reward has peaks of height $2 . 5 + R _ { 0 }$ near the corners of the hypergrid, surrounded by plateaux of height $0 . 5 + R _ { 0 }$ . These plateaux are separated by wide troughs with reward $R _ { 0 }$ . An illustration with $H = 8$ and $D = 2$ is shown in the left panel of Fig. 1. This environment evaluates the ability of a GFlowNet to generalize from visited states to infer the existence of yet-unvisited modes. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We train GFlowNets with the detailed balance (DB) and trajectory balance (TB) objectives with different $H , D$ , and $R _ { 0 }$ , in addition to reproducing the flow matching (FM) experiments and nonGFlowNet baselines based upon [3]’s published code. Our GFlowNet policy model is a multilayer perceptron (MLP) that accepts as input a one-hot encoding of a state $s$ (with the goal of enabling generalization) and outputs logits of the forward and backward policies $P _ { F } ( - | - ; \\theta )$ and $P _ { B } ( - | - ; \\bar { \\theta ) }$ (as well as the estimated state flow $F _ { \\theta } ( s )$ in the case of DB). The forward policy, backward policy, and state flow models share all but the last weight matrix of the MLP. This is consistent with [3]’s model, where an identical architecture accepted $s$ as input and output estimated flows $F _ { \\theta } ( s , s ^ { \\prime } )$ for all children $s ^ { \\prime }$ of $s$ . Details are given in $\\ S \\mathrm { B } . 1$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We consider a 4-dimensional grid with $H = 8$ and and a 2-dimensional grid with $H = 6 4$ . The two grids have the same number of terminal states, but the 2-dimensional grid has longer expected trajectory lengths. For both grid sizes, we consider $R _ { 0 } = 0 . 1 , 0 . 0 1 , 0 . 0 0 1$ , with smaller $R _ { 0 }$ giving environments that are more difficult to explore due to the lower likelihood for models to cross the low-reward valley. For the models trained with DB and TB, we also explore the effect of fixing the backward policy to be uniform (§3.1). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results. In Fig. 2, we plot the evolution over the course of training of the $L _ { 1 }$ error between the true reward distribution (the reward $R ( x )$ normalized over all possible terminal states $x \\in \\mathcal { X }$ ) and the empirical distribution of the last $2 \\cdot 1 0 ^ { 5 }$ visited states for all settings (which would have a probability of 0 on $x$ ’s not visited). Although convergence to the same stable minimum is achieved by all models and settings, DB and TB training tend to converge faster than FM, with a slight benefit of TB over detailed balance in the 4-D environment. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Effect of uniform $P _ { B }$ . Note the difference in learning speed between models with fixed uniform backward policy $P _ { B }$ and models with learned $P _ { B }$ . As noted in $\\ S 3 . 1$ , when $P _ { B }$ is fixed, there is a unique $P _ { F } ( - | - ; \\theta )$ that globally minimizes the objective, and it may be approached slowly. However, if $P _ { B }$ and $P _ { F }$ are permitted to evolve jointly, they may more quickly approach one of the many optimal solutions. This is confirmed by the much faster convergence of models with learned $P _ { B }$ on the $6 4 \\times 6 4$ grid. We have observed that, especially for large grid sizes, when $P _ { B }$ and $P _ { F }$ are both learned, the model has a bias towards first taking all actions in one coordinate direction, then proceeding in the other direction until terminating (as in the right panel of Fig. 1), perhaps because a constant distribution over two actions (‘continue to the right’ and ‘terminate’) can be modeled with higher precision over a large portion of the grid than the complex position-dependent distribution as shown in the centre panel of Fig. 1. ",
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+    {
+        "type": "image",
+        "img_path": "images/ded6a0e6aeeb36a8fca4f2a53a0ac9b4780587b26edd97879125f61902ebe953.jpg",
+        "image_caption": [
+            "Figure 1: Left: The reward function on an $8 \\times 8$ grid environment (§5.1) with $R _ { 0 } = 0 . 1$ . Centre and right: Two forward action policies – with fixed uniform $P _ { B }$ and with a learned non-uniform $P _ { B }$ – that sample from this reward. The lengths of arrows pointing up and right from each state are proportional to the likelihoods of the corresponding actions under $P _ { F }$ , and the sizes of the red octagons are proportional to the termination action likelihoods. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/3aeacba67763413defdfe1bedbc1bc1d526cfa60de51d0701ad1a28e53b69088.jpg",
+        "image_caption": [
+            "Figure 2: Empirical $L ^ { 1 }$ error between true and sampled state distributions on the grid environment with varying grid size and $R _ { 0 }$ . Mean and standard error over 5 seeds. The curves for PPO and MCMC baseline would lie outside the plot bounds. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5.2 Small drug molecule synthesis ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Next, we consider the molecule generation task [30, 15, 16, 10, 24] introduced for GFlowNets in [3]. We extend [3]’s published code with an implementation of the TB and DB objectives.5 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The goal is to generate molecules, in the form of graphs, with a low binding energy to the 4JNC inhibitor of the sEH (soluble epoxide hydrolase) protein. The graphs generated are junction trees [15] of a vocabulary of building blocks. The reward is defined as the normalized negative binding energy as predicted by a proxy model, itself trained to predict energies computed via docking simulations [26]. The maximum trajectory length is 8, with the number of actions varying between around 100 and 2000 (the larger a molecule, the more possible additions exist), making $| \\mathcal { X } |$ about $1 0 ^ { 1 6 }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results. We plot in Fig. 3 (left and centre) the correlation of log-reward and log-sampling probability (the likelihood that a trajectory sampled from the learned policy terminates at $x$ ) for GFlowNets trained using TB, FM and DB. This correlation is significantly higher for models trained with TB. The points $x$ shown are from a fixed held-out set to which the GFlowNets do not have access in training. Note that a perfect model would have correlation 1, as $\\log R ( x )$ and $\\log p _ { \\theta } ( x )$ would differ by a constant (equal to $\\log Z ,$ ) that is independent of $x$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Fig. 3 (right) we plot the average pairwise Tanimoto similarity [1] for the 1000 highest-reward samples generated over the course of training. We see that TB consistently generates more diverse molecules than FM. These results showcase the benefits of TB, not only for faster temporal credit assignment, but for generalization and diversity. In addition, TB has up to $5 \\times$ runtime speedup over FM as the enumeration of parents is not needed. See $\\ S \\mathrm { B } . 2$ for further discussion. ",
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+    },
+    {
+        "type": "text",
+        "text": "5.3 Autoregressive sequence generation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Finally, we evaluate the TB objective on the task of autoregressive sequence generation. In $\\ S 5 . 3 . 1$ , we study the effect of trajectory length and action space size on the learning dynamics in GFlowNets. In $\\ S 5 . 3 . 2$ , we consider the more realistic task of generating peptides (short protein sequences with anti-microbial properties) and evaluate GFlowNets against standard RL and MCMC baselines. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/2cbc6b95517804bfa1dbab6c862f71e4510105e6928bdd1ce93d4592dcfa915b.jpg",
+        "image_caption": [
+            "Figure 3: Left, Centre: Pearson correlations between rewards and sampling probability. $\\log p _ { \\theta } ( x )$ is the log-likelihood that a trajectory sampled from the learned policy $P _ { F } \\bar { ( } - | \\bar { - } ; \\theta )$ terminates at $x$ . Left: Scatter plot on a test set of $x$ ’s for the best hyperparameters of TB, FM, and DB. Centre: Violin plot of correlations for 16 hyperparameter settings and 3 seeds for each setting, showing TB being capable of fitting better. Right: Average pairwise Tanimoto similarity for the top 1000 samples generated by GFlowNets as training progresses. Lines are the average across runs, shaded regions the standard deviation. Models trained with TB have consistently lower similarity than those with FM, hence greater diversity. We hypothesize that the higher variance, in correlation and diversity, of TB relative to DB is related to high variance of the stochastic gradient; see [18]. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.3.1 Bit sequences ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Task. The goal is to generate bit sequences of a fixed length $n = 1 2 0$ $\\mathcal { X } = \\{ 0 , 1 \\} ^ { n } )$ , where the reward is designed to have modes at a given fixed set $M \\subset \\mathcal { X }$ that is unknown to the learner. The reward for a sequence $x$ is defined as $\\begin{array} { r } { \\bar { R ( x ) } = \\exp ( - \\operatorname* { m i n } _ { y \\in M } d ( x , y ) ) } \\end{array}$ , where $d$ is the edit distance. We describe the procedure to generate $M$ in $\\ S \\mathrm { B } . 3$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For different integers $k$ dividing $n$ , we design action spaces for left-to-right generation of sequences in $\\mathcal { X }$ , where a complete trajectory has $\\frac { n } { k }$ actions and each action appends a $k$ -bit ‘word’ to the end of a partial sequence. A forward policy on this state space is the same an autoregressive sequence model over a vocabulary of size $2 ^ { k }$ . Varying $k$ while fixing $\\mathcal { X }$ and $M$ allows us to study the effect of the tradeoff between trajectory lengths $\\textstyle { \\binom { n } { k } }$ and the action space sizes $( | V | = 2 ^ { k }$ ) without changing the underlying probabilistic modeling problem. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We compare GFlowNets trained with the TB objective against GFlowNets trained with the FM objective (equivalent to DB and Soft Q-Learning in this case) and two non-GFlowNet baselines: A2C with Entropy Regularization [29, 21], Soft Actor-Critic [13, 6] and MARS [30]. We use a Transformer-based architecture [28] across all the methods. See $\\ S \\mathbf { B } . 3$ for details. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To evaluate the methods we use (1) Spearman correlation between the probability of generating the sequence $p ( x ) = F ( x ) / Z$ and its reward $R ( x )$ on a test set sampled approximately uniformly over the possible values of the reward, (2) number of modes captured (number of reference sequences from $M$ for which a candidate within a distance $\\delta$ has been generated). In our experiments, $n = 1 2 0$ , $| M | = 6 0$ , $k \\in \\{ 1 , 2 , 4 , 6 , 8 , 1 0 \\}$ , and $\\delta = 2 0$ . ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Results. Fig. 4 (left) presents the results for the Spearman correlation evaluation. We observe that GFlowNets trained with the TB objective learn policies that correlate best with the reward $R ( x )$ across all action spaces. In particular, we observe the effect of inefficient credit assignment in GFlowNets trained with FM, as the correlation improves with increasing $k$ , i.e., shorter trajectories. On the other hand, large action spaces also hurt GFlowNets trained with the FM objective, while the TB objective is robust to them. Additionally, we can observe in Fig. 4 (right) that for fixed $k$ , GFlowNets trained with TB discover more modes faster than other methods. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.3.2 Anti-Microbial Peptide (AMP) generation ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section, we consider the practical task of generating peptide sequences that have anti-microbial activity. The goal is to generate a protein sequence (where the vocabulary consists of 20 amino acids and a special end-of-sequence action), with maximum length 60. We take 6438 known AMP sequences and 9522 non-AMP sequences from the DBAASP database [23]. We then train a classifier on this dataset, using $2 0 \\%$ of the data as a validation set. The probability output by this model for a sequence to be classified as an AMP is used as the reward $R ( x )$ in our experiments. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/e64b75eb03bc20e669181919a948144ab595a89968fc0d1404e6e03537ef6b4d.jpg",
+        "table_caption": [
+            "Table 1: Results on the AMP generation task. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Top 100 Reward</td><td>Top 100Diversity</td></tr><tr><td>GFN-LTB</td><td>0.85 ± 0.03</td><td>18.35±1.65</td></tr><tr><td>GFN-LFM/LDB</td><td>0.78±0.05</td><td>12.61 ± 1.32</td></tr><tr><td>SAC</td><td>0.80±0.01</td><td>8.36±1.44</td></tr><tr><td>AAC-ER</td><td>0.79±0.02</td><td>7.32 ± 0.76</td></tr><tr><td>MCMC</td><td>0.75±0.02</td><td>12.56±1.45</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/e419992c7749710eb2081ca5ec30da443d61590a54d3a58a98203d0418f75fb5.jpg",
+        "image_caption": [
+            "Figure 4: Left: Spearman correlation of the sampling probability under different learned policies and reward on a test set, plotted against the number of bits $k$ in the symbols in $V$ in the bit sequence generation task. GFlowNets trained with trajectory balance learn policies that have the highest correlation with the reward $R ( x )$ and are robust to length and vocabulary size. Right: Number of modes discovered over the course of training on the bit sequence generation task with $k = 1$ . GFlowNets trained with trajectory balance discover more modes faster. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The state and actions are designed just as in the previous experiment, with each action appending a symbol to the right of a state. We again compare TB and FM, as well as A2C with entropy regularization, SAC and MCMC as baselines. We again use Transformers for all the experiments on this task; see further details in Appendix B.4. We generate 2048 sequences from each method, and pick the top 100 sequences ranked by their reward $R ( x )$ . As metrics, we use the mean reward for these 100 sequences and the average pairwise edit distance among them as a measure of diversity. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Results. We present the results in Table 1, where we observe that GFlowNets trained with TB outperform all baselines on both performance and diversity metrics. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 Discussion and conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We introduced a novel training loss for GFlowNets, trajectory balance (TB), which yields faster and better training than the previously proposed flow matching (FM) and detailed balance (DB) losses. We proved that this objective, when minimized, yields the desired GFlowNet property of sampling from the target distribution specified by an unnormalized reward function. This new loss was motivated by the observation that the FM and DB losses are local in the action sequence and may require many iterations for credit assignment to propagate to early actions: if a gradient update introduces a flow inconsistency at some state far from the initial state (such as when a novel high-reward state is sampled), the parent of this state must be visited before the update is propagated closer to the root, akin to the slow propagation of reward signals in temporal difference learning. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We empirically found that TB discovered more modes of the energy function faster and was more robust than FM and DB to the exponential growth of the state space, due in part to the lengths of sequences and in part to the size of the action space. A factor to consider when interpreting our experimental results is that because we use a neural net rather than a tabular representation of policies, the early states’ transitions are informed by downstream credit assignment via parameter sharing. Early states also get many more visits because there are more possible states near the ends of sequences than near the initial state. Finally, TB trades off the advantage of immediately providing credit to early states with the disadvantage of relying on sampling of long trajectories and thus a potentially higher variance of the stochastic gradient. The high gradient variance is a possible limitation of TB in difficult environments, and ways to overcome it by interpolating between local and trajectory-level objectives have been studied in subsequent work [18]. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "All in all, we found that trajectory balance is a superior training objective in a broad set of experiments, making it the default choice for future work on GFlowNets. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            89,
+            330,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This research was enabled in part by computational resources provided by Compute Canada. All authors are funded by their primary academic institution. We also acknowledge funding from CIFAR, Samsung, IBM, Microsoft, and the Banting Postdoctoral Fellowship. ",
+        "bbox": [
+            176,
+            121,
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+            162
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The authors are grateful to all the members of the Mila GFlowNet group, in particular to Dinghuai Zhang, for many fruitful research discussions, as well as to Yiheng Zhu for feedback on the published code. We also thank the anonymous reviewers for their comments. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            231,
+            266,
+            247
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[1] Bender, A. and Glen, R. C. Molecular similarity: a key technique in molecular informatics. Organic & biomolecular chemistry, 2(22):3204–3218, 2004. [2] Bengio, E., Pineau, J., and Precup, D. Interference and generalization in temporal difference learning. International Conference on Machine Learning (ICML), 2020.   \n[3] Bengio, E., Jain, M., Korablyov, M., Precup, D., and Bengio, Y. Flow network based generative models for non-iterative diverse candidate generation. Neural Information Processing Systems (NeurIPS), 2021.   \n[4] Bengio, Y., Lahlou, S., Deleu, T., Hu, E., Tiwari, M., and Bengio, E. GFlowNet foundations. arXiv preprint 2111.09266, 2021.   \n[5] Buesing, L., Heess, N., and Weber, T. Approximate inference in discrete distributions with Monte Carlo tree search and value functions. Artificial Intelligence and Statistics (AISTATS), 2019.   \n[6] Christodoulou, P. 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Fragment based drug design: from experimental to computational approaches. Current medicinal chemistry, 19(30):5128–5147, 2012.   \n[17] Kumar, A., Agarwal, R., Ghosh, D., and Levine, S. Implicit under-parameterization inhibits dataefficient deep reinforcement learning. International Conference on Learning Representations (ICLR), 2021.   \n[18] Madan, K., Rector-Brooks, J., Korablyov, M., Bengio, E., Jain, M., Nica, A., Bosc, T., Bengio, Y., and Malkin, N. Learning GFlowNets from partial episodes for improved convergence and stability. arXiv preprint 2209.12782, 2022.   \n[19] Malkin, N., Lahlou, S., Deleu, T., Ji, X., Hu, E., Everett, K., Zhang, D., and Bengio, Y. GFlowNets and variational inference. arXiv preprint 2210.00580, 2022.   \n[20] Mnih, A. and Gregor, K. Neural variational inference and learning in belief networks. International Conference on Machine Learning (ICML), 2014.   \n[21] Mnih, V., Badia, A. P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., and Kavukcuoglu, K. Asynchronous methods for deep reinforcement learning. International Conference on Machine Learning (ICML), 2016.   \n[22] Nachum, O., Norouzi, M., Xu, K., and Schuurmans, D. Bridging the gap between value and policy based reinforcement learning. Neural Information Processing Systems (NeurIPS), 2017.   \n[23] Pirtskhalava, M., Amstrong, A. A., Grigolava, M., Chubinidze, M., Alimbarashvili, E., Vishnepolsky, B., Gabrielian, A., Rosenthal, A., Hurt, D. E., and Tartakovsky, M. Dbaasp v3: Database of antimicrobial/cytotoxic activity and structure of peptides as a resource for development of new therapeutics. Nucleic Acids Research, 49(D1):D288–D297, 2021.   \n[24] Shi, Y., Huang, Z., Feng, S., Zhong, H., Wang, W., and Sun, Y. Masked label prediction: Unified message passing model for semi-supervised classification. International Joint Conference on Artificial Intelligence (IJCAI), 2021.   \n[25] Sutton, R. S. and Barto, A. G. Reinforcement learning: An introduction. MIT Press, 2018.   \n[26] Trott, O. and Olson, A. J. AutoDock Vina: improving the speed and accuracy of docking with a new scoring function, efficient optimization, and multithreading. Journal of computational chemistry, 31(2):455–461, 2010.   \n[27] van Hasselt, H., Doron, Y., Strub, F., Hessel, M., Sonnerat, N., and Modayil, J. Deep reinforcement learning and the deadly triad. arXiv preprint 1812.02648, 2018.   \n[28] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., and Polosukhin, I. Attention is all you need. Neural Information Processing Systems (NeurIPS), 2017.   \n[29] Williams, R. J. and Peng, J. Function optimization using connectionist reinforcement learning algorithms. Connection Science, 3(3):241–268, 1991.   \n[30] Xie, Y., Shi, C., Zhou, H., Yang, Y., Zhang, W., Yu, Y., and Li, L. MARS: Markov molecular sampling for multi-objective drug discovery. International Conference on Learning Representations (ICLR), 2021.   \n[31] Zhang, D., Chen, R. T. Q., Malkin, N., , and Bengio, Y. Unifying generative models with GFlowNets. arXiv preprint 2209.02606, 2022.   \n[32] Zhang, D., Malkin, N., Liu, Z., Volokhova, A., Courville, A., and Bengio, Y. Generative flow networks for discrete probabilistic modeling. International Conference on Machine Learning (ICML), 2022. ",
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parse/dev/5haAJAcofjc/5haAJAcofjc.md

@@ -0,0 +1,371 @@
+# General Cutting Planes for Bound-Propagation-Based Neural Network Verification
+
+Huan Zhang\*,1 Shiqi Wang\*,2 Kaidi Xu\*,3   
+Linyi Li4 Bo Li4 Suman Jana2 Cho-Jui Hsieh5 J. Zico Kolter1,6   
+2Columbia University 3Drexel University 4UIUC 5UCLA 6Bosch Center for AI   
+huan@huan-zhang.com sw3215@columbia.edu kx46@drexel.edu   
+linyi2@illinois.edu lbo@illinois.edu suman@cs.columbia.edu chohsieh@cs.ucla.edu zkolter@cs.cmu.edu
+
+\* Equal Contribution
+
+# Abstract
+
+Bound propagation methods, when combined with branch and bound, are among the most effective methods to formally verify properties of deep neural networks such as correctness, robustness, and safety. However, existing works cannot handle the general form of cutting plane constraints widely accepted in traditional solvers, which are crucial for strengthening verifiers with tightened convex relaxations. In this paper, we generalize the bound propagation procedure to allow the addition of arbitrary cutting plane constraints, including those involving relaxed integer variables that do not appear in existing bound propagation formulations. Our generalized bound propagation method, GCP-CROWN, opens up the opportunity to apply general cutting plane methods for neural network verification while benefiting from the efficiency and GPU acceleration of bound propagation methods. As a case study, we investigate the use of cutting planes generated by off-the-shelf mixed integer programming (MIP) solver. We find that MIP solvers can generate high-quality cutting planes for strengthening bound-propagation-based verifiers using our new formulation. Since the branching-focused bound propagation procedure and the cutting-plane-focused MIP solver can run in parallel utilizing different types of hardware (GPUs and CPUs), their combination can quickly explore a large number of branches with strong cutting planes, leading to strong verification performance. Experiments demonstrate that our method is the first verifier that can completely solve the oval20 benchmark and verify twice as many instances on the oval21 benchmark compared to the best tool in VNN-COMP 2021, and also noticeably outperforms state-of-the-art verifiers on a wide range of benchmarks. GCP-CROWN is part of the $\alpha , \beta$ -CROWN verifier, the VNN-COMP 2022 winner. Code is available at http://PaperCode.cc/GCP-CROWN.
+
+# 1 Introduction
+
+Neural network (NN) verification aims to formally prove or disprove certain properties (e.g., correctness and safety properties) of a NN under a certain set of inputs. These methods can provide worst-case performance guarantees of a NN, and have been applied to mission-critical applications that involve neural networks, such as automatic aircraft control [31, 4], learning-enabled cyber-physical systems [54], and NN based algorithms in an operating system [51].
+
+The NN verification problem is generally NP-complete [30]. For piece-wise linear networks, it can be encoded as a mixed integer programming (MIP) [53] problem with the non-linear ReLU neurons described by binary variables. Thus, fundamentally, the NN verification problem can be solved using the branch and bound (BaB) [10] method similar to generic MIP solvers, by branching some binary variables and relaxing the rest into a convex problem such as linear programming (LP) to obtain bounds on the objective. Although early neural network verifiers relied on off-the-shelf CPU-based LP solvers [36, 9] for bounding in BaB, LP solvers do not scale well to large NNs. Thus, many recent verifiers are instead based on efficient and GPU-accelerated algorithms customized to NN verification, such as bound propagation methods [60, 57], Lagrangian decomposition methods [8, 17] and others [16, 11]. Bound propagation methods, presented in a few different formulations [58, 18, 56, 61, 50, 25], empower state-of-the-art NN verifiers such as $\alpha , \beta$ -CROWN [61, 60, 57] and VeriNet [3], and can achieve two to three orders of magnitudes speedup compared to solving the NN verification problem using an off-the-shelf solver directly [57], especially on large networks.
+
+Despite the success of existing NN verifiers, we experimentally find that state-of-the-art NN verifiers may timeout on certain hard instances which a generic MIP solver can solve relatively quickly, sometimes even without branching. Compared to an MIP solver, a crucial factor missing in most scalable NN verifiers is the ability to efficiently generate and solve general cutting planes (or “cuts”). In generic MIP solvers, cutting planes are essential to strengthen the convex relaxation, so that much less branching is required. Advanced cutting planes are among the most important factors in modern MIP solvers [6]; they can strengthen the convex relaxation without removing any valid integer solution from the MIP formulation. In the setting of NN verification, cutting planes reflects complex intra-layer and inter-layer dependencies between multiple neurons, which cannot be easily captured by existing bound propagation methods with single neuron relaxations [45]. This motivates us to seek the combination of efficient bound propagation method with effective cutting planes to further increase the power of NN verifiers.
+
+A few key factors make the inclusion of general cutting planes in NN verifiers quite challenging. First, existing efficient bound propagation frameworks such as CROWN [61] and $\beta$ -CROWN [57] cannot solve general cutting plane constraints that may involve variables across different layers in the MIP formulation. Particularly, these frameworks do not explicitly include the integer variables in the MIP formulation that are crucial when encoding many classical strong cutting planes, such as Gomory cuts and mixed integer rounding (MIR) cuts. Furthermore, although some existing works [47, 52, 40] enhanced the basic convex relaxation used in NN verification (such as the Planet relaxation [19]), these enhanced relaxations involve only one or a few neurons in a single layer or two adjacent layers, and are not general enough. In addition, an LP solver is often required to handle these additional cutting plane constraints [40], for which the efficient and GPU-accelerated bound propagation cannot be used, so the use of these tighter relaxations may not always bring improvements.
+
+In this paper, we achieve major progress in using general cutting planes in bound propagation based NN verifiers. To mitigate the challenge of efficiently solving general cuts, our first contribution is to generalize existing bound propagation methods to their most general form, enabling constraints involving variables from neurons of any layer as well as integer variables that encode the status of a ReLU neuron. This allows us to consider any cuts during bound propagation without relying on a slow LP solver, and opens up the opportunity for using advanced cutting plane techniques efficiently for the NN verification problem. Our second contribution involves combining a cutting-plane-focused, off-the-shelf MIP solver with our GPU-accelerated, branching-focused bound propagation method capable of handling general cuts. We entirely disable branching in the MIP solver and use it only for generating high quality cutting planes not restricting to neurons within adjacent layers. Although an MIP solver often cannot verify large neural networks, we find that they can generate high quality cutting planes within a short time, significantly helping bound propagation to achieve better bounds.
+
+Our experiments show that general cutting planes can bring significant improvements to NN verifiers: we are the first verifier that completely solves all instances in the oval20 benchmark, with an average time of less than 5 seconds per instance; on the even harder oval21 benchmark in VNNCOMP 2021 [3], we can verify twice as many instances compared to the competition winner. We also outperform existing state-of-the-art bound-propagation-based methods including those using multi-neuron relaxations [20] (a limited form of cutting planes).
+
+# 2 Background
+
+The NN verification problem We consider the verification problem for an $L$ -layer ReLU-based Neural Network (NN) with inputs $\pmb { \hat { x } } ^ { ( 0 ) } : = \pmb { x } \in \mathbb { R } ^ { d _ { 0 } }$ , weights $\mathbf W ^ { ( i ) } \in \mathbb R ^ { d _ { i } \times \dot { d } _ { i - 1 } }$ , and biases $\mathbf { b } ^ { ( i ) } \in \mathbb { R } ^ { d _ { i } }$ $( i \in \{ 1 , \cdots , L \} )$ . We can get the NN outputs $f ( \pmb { x } ) = \pmb { x } ^ { ( L ) } \in \mathbb { R } ^ { d _ { L } }$ by sequentially propagating the input $_ { \textbf { \em x } }$ through affine layers with $\begin{array} { r } { \mathbf { \Delta } \mathbf { x } ^ { ( i ) } = \mathbf { W } ^ { ( i ) } \hat { \mathbf { x } } ^ { ( i - 1 ) } + \mathbf { b } ^ { ( i ) } } \end{array}$ and ReLU layer with $\hat { \pmb x } ^ { ( i ) } = \mathrm { R e L U } ( { \pmb x } ^ { ( i ) } )$ . We also let scalars $\hat { x } _ { j } ^ { ( i ) }$ and $x _ { j } ^ { ( i ) }$ denote the post-activation and pre-activation, respectively, of $j$ -th ReLU neuron in $i$ -th layer. Throughout the paper, we use bold symbols to denote vectors (e.g., $\pmb { x } ^ { ( i ) }$ ) and regular symbols to denote scalars (e.g., $\overline { { x _ { j } ^ { ( i ) } } }$ is the $j$ -th element of $\pmb { x } ^ { ( i ) }$ ). We use the shorthand $[ N ]$ to denote $\{ 1 , \cdots , N \}$ , and $\mathbf { W } _ { : , j } ^ { ( i ) }$ is the $j$ -th column of $\mathbf { W } ^ { ( i ) }$ .
+
+Commonly, the input $_ { \textbf { \em x } }$ is bounded within a perturbation set $\mathcal { C }$ (such as an $\ell _ { p }$ norm ball) and the verification specification defines a property of the output $f ( { \pmb x } )$ that should hold for any ${ \textbf { \em x } } \in { \mathcal { C } }$ , e.g., whether the true label’s logit $f _ { y } ( { \pmb x } )$ will be always larger than another label’s logit $f _ { j } ( { \pmb x } )$ , (i.e., checking if $f _ { y } ( { \pmb x } ) - f _ { j } ( { \pmb x } )$ is always positive). Since we can append the verification specification (such as a linear function on neural network output) as an additional layer of the network, canonically, the NN verification problem requires one to solve the following one-dimensional $\begin{array} { r } { \ d _ { d _ { L } } = 1 } \end{array}$ ) optimization objective on $f ( { \pmb x } )$ :
+
+$$
+f ^ { * } = \operatorname* { m i n } _ { \mathbf { x } } f ( \mathbf { x } ) , \forall \mathbf { x } \in \mathcal { C }
+$$
+
+with the relevant property defined to be proven if the optimal solution $f ^ { * } \geq 0$ . Throughout this work, we consider the $\ell _ { \infty }$ norm ball $\mathcal { C } : = \{ \pmb { x } : \| \pmb { x } - \pmb { x } _ { 0 } \| _ { \infty } \le \epsilon \}$ where $\scriptstyle { \mathbf { { \mathit { x } } } } _ { 0 }$ is a predefined constant (e.g., a clean input image), although it is possible to extend to other norms or specifications [43, 59].
+
+The MIP and LP formulation for NN verification The mixed integer programming (MIP) formulation is the root of many NN verification algorithms. This formulation uses binary variables $\mathbf { z }$ to encode the non-linear ReLU neurons to make the non-convex optimization problem (1) tractable. Additionally, we assume that we know sound pre-activation bounds $l ^ { ( i ) } \le \bar { \mathbf { x } ^ { ( i ) } } \le \mathbf { u } ^ { ( i ) }$ for ${ \pmb x } \in \mathcal { C }$ which can be obtained via cheap bound propagation methods such as IBP [23] or CROWN [61]. Then ReLU neurons for each layer $i$ can be classified into three classes [58], namely “active” $( \mathcal { I } ^ { + ( i ) } )$ , “inactive” $( \mathcal { I } ^ { - ( i ) } )$ and “unstable” $( \mathcal { T } ^ { ( i ) } )$ neurons, respectively:
+
+$$
+\begin{array} { r } { \mathcal { Z } ^ { + ( i ) } : = \{ j : l _ { j } ^ { ( i ) } \geq 0 \} ; \quad \mathcal { Z } ^ { - ( i ) } : = \{ j : u _ { j } ^ { ( i ) } \leq 0 \} ; \quad \mathcal { Z } ^ { ( i ) } : = \{ j : l _ { j } ^ { ( i ) } \leq 0 , u _ { j } ^ { ( i ) } \geq 0 \} } \end{array}
+$$
+
+Based on the definition of ReLU, activate and inactive neurons are linear functions, so only unstable neurons require binary encoding. The MIP formulation of (1) is:
+
+$$
+\begin{array} { r l } & { \begin{array} { r l } & { f ^ { * } = \underset { x , x , x } { \mathrm { m i n } } \ f ( x ) \ } \\ & { x ^ { ( i ) } = \underset { y \in \Omega } { \mathrm { m i n } } \ f ( x ) \ } \\ & { x ^ { ( i ) } = \underset { x ^ { ( i ) } } { \mathrm { m i n } } \ f ( x ) + \underset { y \in \Omega } { \mathrm { m i n } } \ f ( x ) = x ^ { ( i ) } ; \quad \hat { w } ^ { ( i ) } = \mathrm { m } ; \quad x \in \mathcal { L } ; } \end{array} } \\ & { \begin{array} { r l } & { x ^ { ( i ) } = 0 ; \quad j \in L ^ { ( i ) } , \ \dotsc } \\ & { \hat { v } _ { j } ^ { ( i ) } \geq 0 ; \quad j \in L ^ { ( i ) } , \ \dotsc } \end{array} } \\ & { \begin{array} { r l } & { \hat { v } _ { j } ^ { ( i ) } \geq x _ { j } ^ { ( i ) } ; \quad j \in L ^ { ( i ) } , i \in [ L - 1 ] } \\ & { \hat { v } _ { j } ^ { ( i ) } \leq u _ { j } ^ { ( i ) } z _ { j } ^ { ( i ) } ; \quad j \in L ^ { ( i ) } , i \in [ L - 1 ] } \\ & { \hat { v } _ { j } ^ { ( i ) } \leq u _ { j } ^ { ( i ) } z _ { j } ^ { ( i ) } ; \quad j \in L ^ { ( i ) } , i \in [ L - 1 ] } \end{array} } \\ & { \begin{array} { r l } & { \hat { v } _ { j } ^ { ( i ) } \leq x _ { j } ^ { ( i ) } ; \quad j ( 1 - z _ { j } ^ { ( i ) } ) ; \quad j \in \mathcal { I } ^ { ( i ) } , \ \dotsc ~ \mathcal { I } ^ { ( i ) } , \ \dotsc | L - 1 | } \\ & { \hat { v } _ { j } ^ { ( i ) } \in \{ 0 , 1 \} ; \quad j \in L ^ { ( i ) } , i \in [ L - 1 ] } \\ & { \hat { v } _ { j } ^ { ( i ) } \in \frac { \mathcal { I } } { 2 } ; \quad j \in \mathcal { I } ^ { ( i ) } , \ \dotsc | L - 1 | } \end{array} } \end{array}
+$$
+
+Since a MIP problem is slow or intractable to solve, it is commonly relaxed as a When the integer variables are relaxed to continuous ones, we obtain the $L P$ relaxation of NN verification problem:
+
+$$
+\begin{array} { c } { f _ { \mathrm { L P } } ^ { * } = \displaystyle \operatorname* { m i n } _ { x , \hat { x } , \mathbf { z } } f ( x ) } \\ { ( 3 ) , ( 2 ) , ( 4 ) , ( 5 ) , ( 6 ) , ( 7 ) , ( 9 ) , ( 1 0 ) , \quad 0 \leq z _ { j } ^ { ( i ) } \leq 1 ; \quad j \in \mathbb { Z } ^ { ( i ) } , i \in [ L - 1 ] } \end{array}
+$$
+
+The ReLU constraints involving $z$ is often projected out, leading to the well-known Planet relaxation used in many NN verifiers, replacing (6), (7) and (8) with a single constraint to get an equivalent LP:
+
+$$
+\begin{array} { l } { f _ { \mathrm { L P } } ^ { * } = \displaystyle \operatorname* { m i n } _ { \mathbf { x } , \hat { x } , \mathbf { z } } f ( \mathbf { x } ) } \\ { \displaystyle \hat { x } _ { j } ^ { ( i ) } \leq \frac { u _ { j } ^ { ( i ) } } { u _ { j } ^ { ( i ) } - l _ { j } ^ { ( i ) } } ( x _ { j } ^ { ( i ) } - l _ { j } ^ { ( i ) } ) ; j \in \mathbb { Z } ^ { ( i ) } , i \in [ L - 1 ] } \end{array}
+$$
+
+Due to the relaxations, the objective of the LP formulation is always a lower bound of the MIP formulation: $f _ { \mathrm { L P } } ^ { * } \leq f ^ { * }$ . A verifier using this formulation is incomplete: if $f _ { \mathrm { L P } } ^ { * } \geq 0$ , then $f ^ { * } \geq 0$ and the property is verified; otherwise, we cannot conclude the sign of $f ^ { * }$ so the verifier must return “unknown”. Branch and bound can be used to improve the lower bound and achieve completeness [10, 57]. However, in this paper, we work on an orthogonal direction of strengthening the LP formulation by adding cutting planes to obtain larger bounds.
+
+Bound propagation methods Instead of solving the LP formulation directly using a LP solver, bound propagation methods aims to quickly give a lower bound for $f _ { \mathrm { L P } } ^ { * }$ . For example, CROWN [61] and $\beta$ -CROWN [57] propagate a sound linear lower bound backwards for $f _ { L } ( \pmb { x } )$ with respect to each intermediate layer. For example, suppose we know
+
+$$
+\operatorname* { m i n } _ { \pmb { x } \in \mathcal { C } } f _ { L } ( \pmb { x } ) \geq \operatorname* { m i n } _ { \pmb { x } \in \mathcal { C } } \mathbf { a } ^ { ( i ) ^ { \top } } \hat { \pmb { x } } ^ { ( i ) } + c ^ { ( i ) }
+$$
+
+With $i = L - 1$ the above is trivially hold with $\mathbf { a } ^ { ( L - 1 ) } \mathbf { \Psi } = \mathbf { W } ^ { ( L ) }$ , $c ^ { ( L - 1 ) } \ = \ \mathbf { b } ^ { ( L ) }$ . In bound propagation methods, a propagation rule propagates an inequality (13) through a previous layer $\hat { \mathbf { \pmb { x } } } ^ { ( i ) } : = \mathrm { R e L U } ( \mathbf { W } ^ { ( i ) } \hat { \mathbf { \ x { x } } } ^ { ( i - 1 ) } + \bar { \mathbf { b } } ^ { ( i ) } )$ to obtain a sound inequality with respect to $\hat { \pmb x } ^ { ( i - 1 ) }$ :
+
+$$
+\operatorname* { m i n } _ { \pmb { x } \in \mathcal { C } } f _ { L } ( \pmb { x } ) \geq \operatorname* { m i n } _ { \pmb { x } \in \mathcal { C } } \mathbf { a } ^ { ( i - 1 ) ^ { \top } } \hat { \pmb { x } } ^ { ( i - 1 ) } + c ^ { ( i - 1 ) }
+$$
+
+Here $\mathbf { a } ^ { ( i - 1 ) }$ , $c ^ { ( i - 1 ) }$ can be calculated in close-form via $\mathbf { a } ^ { ( i ) } , c ^ { ( i ) } , \mathbf { W } ^ { ( i ) } , \mathbf { b } ^ { ( i ) } , l ^ { ( i ) }$ and $\mathbf { \boldsymbol { \mathbf { \mathit { u } } } } ^ { ( i ) }$ such that the bound still holds (see Lemma 1 in [57]). Applying the procedure repeatedly will eventually reach the input layer:
+
+$$
+\operatorname* { m i n } _ { \pmb { x } \in \mathcal { C } } f _ { L } ( \pmb { x } ) \geq \operatorname* { m i n } _ { \pmb { x } \in \mathcal { C } } \mathbf { a } ^ { ( 0 ) ^ { \top } } \pmb { x } + c ^ { ( 0 ) }
+$$
+
+The minimization on linear layer can be solved easily when $\mathcal { C }$ is a $\ell _ { p }$ norm ball to obtain a valid lower bound of $f ^ { * }$ . Since the bounds propagate layer-by-layer, this process can be implemented efficiently on GPUs [59] without relying on a slow LP solver, which greatly improves the scalability and solving time. Additionally, it is often used to obtain intermediate layer bounds $\mathbf { \chi } _ { l ^ { ( i ) } }$ and and $\mathbf { \boldsymbol { u } } ^ { ( i ) }$ required for the MIP formulation (6)(7), by treating each $x _ { j } ^ { ( i ) }$ as the output neuron. The bound propagation rule can either be derived in primal space [61], dual space [58] or abstract interpretations [50]. In Sec.3.1, we will discuss the our bound propagation procedure with general cutting plane constraints.
+
+# 3 Neural Network Verification with General Cutting Planes
+
+# 3.1 GCP-CROWN: General Cutting Planes in Bound Propagation
+
+In this section, we generalize existing bound propagation method to handle general cutting plane constraints. Our goal is to derive a bound propagation rule similar to CROWN and $\beta$ -CROWN discussed in Section 2, however considering additional constraints among any variables within the LP relaxation. To achieve this, we first derive the dual problem of the LP; inspired by the dual formulation, we derive the bound prorogation rule in a layer by layer manner that takes all cutting plane constraints into consideration. The derivation process is inspired by [58, 45] and [57].
+
+LP relaxation with cutting planes. In this section, we derive the bound propagation procedure under the presence of general cutting plane constraints. A cutting plane is a constraint involving any variables $\mathbf { \boldsymbol { x } } ^ { ( i ) }$ (pre-activation), $\hat { \pmb x } ^ { ( i ) }$ (post-activation), $\mathbf { z } ^ { ( i ) }$ (ReLU indicators) from any layer $i$ :
+
+$$
+\sum _ { i = 1 } ^ { L - 1 } \left( { h ^ { { ( i ) } ^ { \top } } } { { \pmb x } ^ { ( i ) } } + { { \pmb g } ^ { ( i ) } } ^ { \top } { \hat { \pmb x } ^ { ( i ) } } + { { \pmb q } ^ { ( i ) } } ^ { \top } { { \pmb z } ^ { ( i ) } } \right) \le d
+$$
+
+Here $\mathbf { \Lambda } _ { h } ( i ) , \mathbf { \Lambda } _ { g } ( i )$ and $\pmb q ^ { ( i ) }$ are coefficients for this cut constraint. The difference between a valid cutting plane and an arbitrary constraint is that a valid cutting plane should not remove any valid integer solution from the MIP formulation. Our new bound propagation procedure can work for any constraints, although in this work we focus on studying the impacts of cutting planes. When there are $N$ cutting planes, we write them in a matrix form:
+
+$$
+\sum _ { i = 1 } ^ { L - 1 } \left( H ^ { ( i ) } \pmb { x } ^ { ( i ) } + \pmb { G } ^ { ( i ) } \hat { \pmb { x } } ^ { ( i ) } + \pmb { Q } ^ { ( i ) } \mathbf { z } ^ { ( i ) } \right) \le d
+$$
+
+where $\pmb { H } ^ { ( i ) } , \pmb { G } ^ { ( i ) } , \pmb { Q } ^ { ( i ) } \in \mathbb { R } ^ { N \times d _ { i } }$ . The LP relaxation with all cutting planes is:
+
+$$
+\begin{array} { r l r } & { } & { f _ { \mathrm { L P - c u t } } ^ { * } = \displaystyle \operatorname* { m i n } _ { x , \hat { x } , \mathbf { z } } f ( x ) } \\ & { } & { \mathrm { t . ~ } ( 3 ) , ( 2 ) , ( 4 ) , ( 5 ) , ( 6 ) , ( 7 ) , ( 9 ) , ( 1 0 ) , \quad 0 \le z _ { j } ^ { ( i ) } \le 1 ; \quad j \in \mathcal { Z } ^ { ( i ) } , i \in [ L - 1 ] } \\ & { } & { \displaystyle \sum _ { i = 1 } ^ { L - 1 } \left( H ^ { ( i ) } { \pmb x } ^ { ( i ) } + { \pmb G } ^ { ( i ) } \hat { \pmb x } ^ { ( i ) } + { \pmb Q } ^ { ( i ) } { \pmb z } ^ { ( i ) } \right) \le d } \end{array}
+$$
+
+Since an additional constraint is added, $f _ { \mathrm { L P - c u t } } ^ { \ast } \geq f _ { \mathrm { L P } } ^ { \ast }$ and we get closer to $f ^ { * }$ . Unlike the original LP where each constraint only contains variables from two consecutive layers, our general cutting plane constraint may involve any variable from any layer in a single constraint.
+
+The dual problem with cutting planes We first show the dual problem for the above LP. The dual problem we consider here is different from existing works in two ways: first, we have constraints with integer variables to support potential cutting planes on $\mathbf { z }$ . Additionally, we have cutting planes constraints that may involve variables in any layer, so the dual in previous works such as [58] cannot be directly reused. Our dual problem is given below (derivation details in Appendix A):
+
+$$
+\begin{array} { r l } & { f _ { \mathrm { L P - c u t } } ^ { * } = \underset { \gamma , \mu \geq 0 , \tau \geq 0 } { \mathrm { m a x } } - \epsilon \Vert \pmb { \nu } ^ { ( 1 ) \top } \mathbf { W } ^ { ( 1 ) } \pmb { x } _ { 0 } \Vert _ { 1 } - \beta ^ { \top } \pmb { d } - \displaystyle \sum _ { i = 1 } ^ { L } \pmb { \nu } ^ { ( i ) \top } \mathbf { b } ^ { ( i ) } } \\ & { \qquad + \displaystyle \sum _ { i = 1 } ^ { L - 1 } \sum _ { j \in \mathcal { D } ^ { ( i ) } } \left[ \pi _ { j } ^ { ( i ) } l _ { j } ^ { ( i ) } - \mathrm { R e L U } ( u _ { j } ^ { ( i ) } \gamma _ { j } ^ { ( i ) } + l _ { j } ^ { ( i ) } \pi _ { j } ^ { ( i ) } - \beta ^ { \top } \pmb { Q } _ { : , j } ^ { ( i ) } ) \right] } \end{array}
+$$
+
+s.t. $\pmb { \nu } ^ { ( L ) } = - 1$ ; and for each $i \in [ L - 1 ]$
+
+$$
+\begin{array} { r l } & { \nu _ { j } ^ { ( i ) } = \nu ^ { ( i + 1 ) \top } \mathbf { W } _ { : , j } ^ { ( i + 1 ) } - \beta ^ { \top } ( H _ { : , j } ^ { ( i ) } + { \pmb { G } } _ { : , j } ^ { ( i ) } ) ; ~ j \in { \mathbb { Z } } ^ { + ( i ) } } \\ & { \nu _ { j } ^ { ( i ) } = - \beta ^ { \top } H _ { : , j } ^ { ( i ) } ; ~ j \in { \mathbb { Z } } ^ { - ( i ) } } \end{array}
+$$
+
+and for each $j \in \mathcal { T } ^ { \left( i \right) }$ the two equalities below hold:
+
+$$
+\begin{array} { r } { \nu _ { j } ^ { ( i ) } = \pi _ { j } ^ { ( i ) } - \tau _ { j } ^ { ( i ) } - \beta ^ { \top } H _ { : , j } ^ { ( i ) } ; \quad \left( \pi _ { j } ^ { ( i ) } + \gamma _ { j } ^ { ( i ) } \right) - \left( \mu _ { j } ^ { ( i ) } + \tau _ { j } ^ { ( i ) } \right) = \nu ^ { ( i + 1 ) \top } \mathbf { W } _ { : , j } ^ { ( i + 1 ) } - \beta ^ { \top } G _ { : , j } ^ { ( i ) } } \end{array}
+$$
+
+Instead of solving the dual problem exactly, we use it to obtain a lower bound of $f _ { \mathrm { L P - c u t } } ^ { * }$ . Intuitively, due to the definition of due problem, any valid setting of dual variables leads to a lower bound of $f _ { \mathrm { L P - c u t } } ^ { * }$ . Informally, starting from $\nu ^ { ( L ) } = - 1$ , by applying the constraints in this dual formulation, we can compute $\bar { \nu ^ { ( L - 1 ) } } , \bar { \nu ^ { ( L - 2 ) } } , \cdot \cdot \cdot$ until $\nu ^ { ( 1 ) }$ . The final objective is a function of $\nu ^ { ( i ) }$ , $i \in [ N ]$ and other dual variables. Precisely, our GCP-CROWN bound propagation procedure with general cutting plane constraint is presented in the theorem below (proof in Appendix A):
+
+Theorem 3.1 (Bound propagation with general cutting planes). Given any optimizable parameters $0 \leq \alpha _ { j } ^ { ( i ) } \leq 1$ ) ≤ 1 and β ≥ 0, f ∗LP-cut is lower bounded by the following objective function, π(i)j is $a$ function of $Q _ { : , j } ^ { ( i ) }$ :
+
+$$
+g ( \alpha , \beta ) = - \epsilon \| \pmb { \nu } ^ { ( 1 ) \top } \mathbf { W } ^ { ( 1 ) } \pmb { x } _ { 0 } \| _ { 1 } - \sum _ { i = 1 } ^ { L } \pmb { \nu } ^ { ( i ) \top } \mathbf { b } ^ { ( i ) } - \beta ^ { \top } d + \sum _ { i = 1 } ^ { L - 1 } \sum _ { j \in \mathbb { Z } ^ { ( i ) } } h _ { j } ^ { ( i ) } ( \beta )
+$$
+
+where variables $\nu ^ { ( i ) }$ are obtained by propagating $\pmb { \nu } ^ { ( L ) } = - 1$ throughout all $i \in [ L - 1 ]$
+
+$$
+\begin{array} { r l } & { \nu _ { j } ^ { ( i ) } = \pmb { \nu } ^ { ( i + 1 ) \top } \mathbf { W } _ { : , j } ^ { ( i + 1 ) } - \beta ^ { \top } ( \pmb { H } _ { : , j } ^ { ( i ) } + \pmb { G } _ { : , j } ^ { ( i ) } ) , \ j \in \mathbb { Z } ^ { + ( i ) } } \\ & { \nu _ { j } ^ { ( i ) } = - \beta ^ { \top } \pmb { H } _ { : , j } ^ { ( i ) } , \ j \in \mathbb { Z } ^ { - ( i ) } } \\ & { \nu _ { j } ^ { ( i ) } = \pi _ { j } ^ { ( i ) } - \alpha _ { j } ^ { ( i ) } [ \hat { \nu } _ { j } ^ { ( i ) } ] _ { - } - \beta ^ { \top } \pmb { H } _ { : , j } ^ { ( i ) } , \ j \in \mathbb { Z } ^ { ( i ) } } \end{array}
+$$
+
+Here νˆ(i)j , πj $\pi _ { j } ^ { ( i ) ^ { * } }$ and $h _ { j } ^ { ( i ) } ( \beta )$ are defined for each unstable neuron $j \in \mathcal { T } ^ { ( i ) }$ .
+
+$$
+\begin{array} { r l } & { \hat { \nu } _ { j } ^ { ( i ) } : = \pmb { \nu } ^ { ( i + 1 ) \top } \mathbf { W } _ { : , j } ^ { ( i + 1 ) } - \pmb { \beta } ^ { \top } \pmb { G } _ { : , j } ^ { ( i ) } } \\ & { \pi _ { j } ^ { ( i ) ^ { * } } = \operatorname* { m a x } \left( \operatorname* { m i n } \left( \frac { u _ { j } ^ { ( i ) } [ \hat { \nu } _ { j } ^ { ( i ) } ] _ { + } + \beta ^ { \top } \pmb { Q } _ { : , j } ^ { ( i ) } } { u _ { j } ^ { ( i ) } - l _ { j } ^ { ( i ) } } , [ \hat { \nu } _ { j } ^ { ( i ) } ] _ { + } \right) , 0 \right) } \end{array}
+$$
+
+$$
+\begin{array} { r } { h _ { j } ^ { ( i ) } ( \beta ) = \left\{ \begin{array} { l l } { l _ { j } ^ { ( i ) } \pi _ { j } ^ { ( i ) ^ { * } } } & { i f ~ l _ { j } ^ { ( i ) } [ \hat { \nu } _ { j } ^ { ( i ) } ] + \leq \beta ^ { \top } Q _ { : , j } ^ { ( i ) } \leq u _ { j } ^ { ( i ) } [ \hat { \nu } _ { j } ^ { ( i ) } ] + } \\ { 0 } & { i f ~ \beta ^ { \top } Q _ { : , j } ^ { ( i ) } \geq u _ { j } ^ { ( i ) } [ \hat { \nu } _ { j } ^ { ( i ) } ] + } \\ { \beta ^ { \top } Q _ { : , j } ^ { ( i ) } } & { i f ~ \beta ^ { \top } Q _ { : , j } ^ { ( i ) } \leq l _ { j } ^ { ( i ) } [ \hat { \nu } _ { j } ^ { ( i ) } ] + } \end{array} \right. } \end{array}
+$$
+
+$$
+\pi _ { j } ^ { ( i ) } { } ^ { * } i s a f u n c t i o n o f Q _ { : , j } ^ { ( i ) }
+$$
+
+Based on Theorem 3.1, to obtain an lower bound of $f _ { \mathrm { L P - c u t } } ^ { * }$ , we start with any valid setting of $0 \leq \alpha \leq 1$ and $\beta \geq 0$ and $\pmb { \nu } ^ { ( L ) } = - 1$ . According to the bound propagation rule, we can compute each $\nu ^ { ( i ) }$ , $i \in [ L - 1 ]$ , in a layer by layer manner. Then objective $g ( \alpha , \beta )$ can be evaluated based on all $\nu ^ { ( i ) }$ to give an lower bound of $f _ { \mathrm { L P - c u t } } ^ { * }$ . Since any valid setting of $_ { \pmb { \alpha } }$ and $\beta$ lead to a valid lower bound, we can optimize $_ { \pmb { \alpha } }$ and $\beta$ using gradient ascent in a similar manner as in [60, 57] to tighten this lower bound. The entire procedure can also run on GPU for great acceleration.
+
+Connection to Convex Outer Adversarial Polytope In convex outer adversarial polytope [58], a bound propagation rule was developed in a similar manner in the dual space without considering cutting plane constraints, and is a special case of ours. We denote their bound propagation objective function as $g _ { \mathrm { W K } }$ which also contains optimizable parameters $\pmb { \alpha } _ { \mathrm { W K } }$ .
+
+Proposition 3.2. Given the same input $_ { \textbf { \em x } }$ , perturbation set $\mathcal { C }$ , network weights, and $N$ cutting plane constraints,
+
+$$
+\operatorname* { m a x } _ { \alpha , \beta } g ( \alpha , \beta ) \geq \operatorname* { m a x } _ { \alpha _ { W K } } g _ { W K } ( \alpha _ { W K } )
+$$
+
+Proof. In Theorem 3.1, when all $\beta$ are set to 0, then $\begin{array} { r } { \pi _ { j } ^ { ( i ) ^ { * } } = \frac { u _ { j } ^ { ( i ) } [ \hat { \nu } _ { j } ^ { ( i ) } ] _ { + } } { u _ { j } ^ { ( i ) } - l _ { j } ^ { ( i ) } } } \end{array}$ and $h _ { j } ^ { ( i ) } ( \beta ) = \pi _ { j } ^ { ( i ) ^ { * } } l _ { j } ^ { ( i ) }$ we recover exactly the same bound propagation equations as in [58]. However, since we allow the addition of cutting plane methods and we can maximize over the additional parameter $\beta$ , the objective given by our bound propagation is always at least as good as $g _ { \mathrm { W K } }$ . □
+
+Connection to CROWN-like bound propagation methods CROWN [61] and $\alpha$ -CROWN [60] use the same bound propagation rule as [58] so Proposition 3.2 also applies, although they were derived from primal space without explicitly formulating the problem as a LP. Salman et al. [45] showed that many other bound propagation methods [50, 55] are equivalent to or weaker than [58]. Recently, Wang et al. [57] extends CROWN to $\beta$ -CROWN to handle split constraints (e.g., $x _ { j } ^ { ( i ) } \geq 0 \mathrm { ~ }$ ). It can be seen a special case as GCP-CROWN where all $\pmb { H }$ , $G$ and $Q$ matrices are zeros except:
+
+$$
+\begin{array} { r } { { \pmb { H } } _ { j , j } ^ { ( i ) } = 1 , j \in { \pmb { \mathbb { Z } } } ^ { - ( i ) } \qquad \mathrm { f o r ~ } x _ { j } ^ { ( i ) } \leq 0 \mathrm { ~ s p l i t } ; \qquad { \pmb { H } } _ { j , j } ^ { ( i ) } = - 1 , j \in { \pmb { \mathbb { Z } } } ^ { + ( i ) } \qquad \mathrm { f o r ~ } x _ { j } ^ { ( i ) } \geq 0 \mathrm { ~ s p l i t } ; } \end{array}
+$$
+
+In addition, [20] encodes multi-neuron relaxations using sparse $\pmb { H } ^ { ( i ) }$ and $\pmb { G } ^ { ( i ) }$ and each cut contains a small number of neurons involving $\pmb { x } ^ { ( i ) }$ and $\hat { \pmb x } ^ { ( i ) }$ for the same layer $i$ . Wang et al. [57] derived bound propagation rules from both the dual LP and the primal space with a Lagrangian without LP. However, in our case, it is not intuitive to derive bound propagation without LP due to the potential cutting planes on relaxed integer variables $\mathbf { z }$ , which do not appear without the explicit LP formulation. Furthermore, although we derived cutting planes for bound propagation methods, it is technically also possible to derive them using other bounding frameworks such as Lagrangian decomposition [8].
+
+# 3.2 Branch-and-bound with GCP-CROWN and MIP Solver Generated Cuts
+
+To build a complete NN verifier, we follow the popular branch-and-bound (BaB) procedure [10, 9] in state-of-the-art NN verifiers with GPU accelerated bound propagation method [8, 60, 16, 57], and our GCP-CROWN is used as the bounding procedure in BaB. We refer the readers to Appendix B for a more detailed background on branch-and-bound. Having the efficient bound propagation procedure with general cutting plane constraints, we now need to find a good set of general cutting planes $\mathbf { \Delta } H ^ { ( i ) } , \mathbf { \bar { G } } ^ { ( i ) } , Q ^ { ( i ) }$ to accelerate NN verification. Since GCP-CROWN can adopt any cutting planes, to fully exploit its power, we propose to use off-the-shelf MIP solvers to generate cutting planes and create an NN verifier combining GPU-accelerated bound propagation with strong cuts generated by a MIP solver. We make the bound propagation on GPU and the MIP solver on CPU run in parallel with cuts added on-the-fly, so the original strong performance of bound-propagation-based
+
+NN verifier will never be affected by a potentially slow MIP solver. This allows us to make full use of available computing resource $( \mathbf { G P U } + \mathbf { C P U } )$ . The architecture of our verifier is shown in Fig 1.
+
+![](images/be66ad93eae4ffd6c779a04aaa073551bff0b8166adeef541cf2c903ad3342c5.jpg)  
+Figure 1: Overview of our cutting-planeenhanced, fully-parallelized NN verifier.
+
+MIP solvers for cutting plane generation Generic MIP solvers such as cplex[28] and gurobi[24] also apply a branch-and-bound strategy, conceptually similar to state-of-the-art NN verifiers. They often tend to be slower than specialized NN verifiers because MIP solvers rely on slower bounding procedures (e.g., Simplex or barrier method) and cannot apply an GPU-accelerated method such as bound propagation or Lagrangian decomposition. However, we still find that MIP solvers are a strong baseline when combined with tight intermediate layer bounds. For example, in the oval21 benchmark in Table 2, $\alpha$ - CROWN+MIP (MIP solver combined with tight intermediate layer bound computed by $\alpha$ -CROWN) is able to solve 4 more instances compared to all other tools in the competition. Our investigation found that $\alpha$ -CROWN+MIP explores much less branches than other state-of-the-art branch-and-bound based NN verifiers, however before branching starts, the MIP solver produces very effective cutting planes that can often verify an instance with little branching. MIP solvers is able to discover sophisticated cutting planes involving several NN layers reflecting complex correlations among neurons, while existing
+
+NN verifiers only exploit specific forms of constraints within same layer or between adjacent layers [52, 47, 40]. This motivates us to combine the strong cutting planes generated by MIP solvers with GPU-accelerated bound-propagation-based BaB.
+
+In this work, we use cplex as the MIP solver for cutting plane generation since gurobi cannot export cuts. We entirely disable all branching features in cplex and make it focus on finding cutting planes only. These cutting planes are generated only for the root node of the BaB search tree so they are sound for any subdomains with neuron splits in BaB. We conduct branching using our generalized bound propagation procedure in Section (3.1) with the cutting planes generated by cplex.
+
+Fully-parallelized NN verifier design We design our verifier as shown in Figure 1. After parsing the NN under verification, we launch a separate process to encode the NN verification problem as MIP problem and start multiple MIP solvers, one for each target label to be verified. At the same time, the main NN verifier process executes branch-and-bound without waiting for the MIP solver process. In each iteration of branch and bound, we query the MIP solving processes and fetch any newly generated cutting planes. If any cutting planes are produced, they are added as $\pmb { H } ^ { ( i ) } , \pmb { G } ^ { ( i ) } , \pmb { Q } ^ { ( i ) }$ in GCP-CROWN and tighten the bounds for subsequent branching and bounding. If no cutting planes are produced, GCP-CROWN reduces to $\beta$ -CROWN [57]. Since our verifier is based on strengthening the bounds in $\beta$ -CROWN with sound cutting planes, it is also sound and complete.
+
+Adjustments to existing branch and bound method. We implement GCP-CROWN into the $\alpha \beta$ -CROWN verifier [61, 59, 57], the winning verifier in VNN-COMP 2021 [3], as the backbone for BaB with bound propagation. To better exploit the power of cutting planes under a fully parallel and asynchronous design, we made a key changes to the BaB procedure. When the number of BaB subdomains are greater than batch size, we rank the subdomains by their lower bounds and choose the easiest domains with largest lower bounds first to verify with GCP-CROWN, unlike most existing verifiers which solve the worst domains first. We use such a order because the MIP solver generates cutting planes incrementally. Solving these easier subdomains tend to require no or fewer cutting planes, so we solve them at earlier stages where cutting planes have not been generated or are relatively weak. On the other hand, if we split worst subdomains first, the number of subdomains will grow quickly, and it can take a long time to verify these domains when stronger cuts become available later. Under a similar rationale, when verifying a multi-class model and BaB needs to verify each target class label one by one, we start BaB from the easiest label first $\scriptstyle { \alpha }$ -CROWN bound closest to 0), allowing the MIP solver to run longer for harder labels, generating stronger cuts for harder labels.
+
+Table 1: Average runtime and average number of branches on oval20 benchmarks with 100 properties per model. Timeout is set to 3,600 seconds (consistent with other literature results). GCP-CROWN is the only method that can completely solve all instances ( $0 \%$ timeout) and the average time per-instance is less than 5 seconds on all three networks.   
+
+<table><tr><td></td><td colspan="3">CIFAR-10 Base</td><td colspan="3">CIFAR-10 Wide</td><td colspan="3">CIFAR-10 Deep</td></tr><tr><td>Method</td><td>time(s)</td><td>branches</td><td>%timeout</td><td>time(s)</td><td>branches</td><td>%timeout</td><td>time(s)</td><td>branches</td><td>%timeout</td></tr><tr><td>MIPplanet [19]</td><td>2849.69</td><td></td><td>68.00</td><td>2417.53</td><td></td><td>46.00</td><td>2302.25</td><td>=</td><td>40.00</td></tr><tr><td>BaBSR[9]</td><td>2367.78</td><td>1020.55</td><td>36.00</td><td>2871.14</td><td>812.65</td><td>49.00</td><td>2750.75</td><td>401.28</td><td>39.00</td></tr><tr><td>GNN-online [36]</td><td>1794.85</td><td>565.13</td><td>33.00</td><td>1367.38</td><td>372.74</td><td>15.00</td><td>1055.33</td><td>131.85</td><td>4.00</td></tr><tr><td>BDD+BaBSR[8]</td><td>807.91</td><td>195480.14</td><td>20.00</td><td>505.65</td><td>74203.11</td><td>10.00</td><td>266.28</td><td>12722.74</td><td>4.00</td></tr><tr><td>Fast-and-Complete [60]</td><td>695.01</td><td>119522.65</td><td>17.00</td><td>495.88</td><td>80519.85</td><td>9.00</td><td>105.64</td><td>2455.11</td><td>1.00</td></tr><tr><td>OVAL (BDD+ GNN)*[8,36]</td><td>662.17</td><td>67938.38</td><td>16.00</td><td>280.38</td><td>17895.94</td><td>6.00</td><td>94.69</td><td>1990.34</td><td>1.00</td></tr><tr><td>A.set BaBSR [16]</td><td>381.78</td><td>12004.60</td><td>7.00</td><td>165.91</td><td>2233.10</td><td>3.00</td><td>190.28</td><td>2491.55</td><td>2.00</td></tr><tr><td>BigM+A.set BaBSR[16]</td><td>390.44</td><td>11938.75</td><td>7.00</td><td>172.65</td><td>4050.59</td><td>3.00</td><td>177.22</td><td>3275.25</td><td>2.00</td></tr><tr><td>BaDNB (BDD+ FSB)[17]</td><td>309.29</td><td>38239.04</td><td>7.00</td><td>165.53</td><td>11214.44</td><td>4.00</td><td>10.50</td><td>368.16</td><td>0.00</td></tr><tr><td>ERAN*[47,48,50,49]</td><td>805.94</td><td>=</td><td>5.00</td><td>632.20</td><td>=</td><td>9.00</td><td>545.72</td><td>=</td><td>0.00</td></tr><tr><td>β-CROWN [57]</td><td>118.23</td><td>208018.21</td><td>3.00</td><td>78.32</td><td>116912.57</td><td>2.00</td><td>5.69</td><td>41.12</td><td>0.00</td></tr><tr><td>α-CROWN+MIP†</td><td>335.50</td><td>8523.37</td><td>3.00</td><td>203.87</td><td>2029.60</td><td>0.00</td><td>76.90</td><td>1364.24</td><td>0.00</td></tr><tr><td>GCP-CROWNwith MIP cuts</td><td>4.07</td><td>2580.53</td><td>0.00</td><td>3.02</td><td>2095.18</td><td>0.00</td><td>3.87</td><td>110.92</td><td>0.00</td></tr></table>
+
+\* Results from VNN-COMP 2020 report [34]. † A new baseline proposed and evaluated in this work, not presented in previous papers.
+
+![](images/d64805e01369070c43248bdca03ffe2117803236cb2480df39cb80f777e1065b.jpg)  
+Figure 2: Percentage of solved properties on the oval20 benchmark vs. running time (timeout 1 hour).
+
+# 4 Experiments
+
+We now evaluate our verifier, GCP-CROWN with MIP cuts, on a few popular verification benchmarks. Since our verifier uses a MIP solver in our pipeline, we also include a new baseline, $\alpha$ -CROWN $^ +$ MIP, which uses gurobi as the MIP solver with the tightest possible intermediate layer bounds from $\alpha$ -CROWN [59]. We use the same branch and bound algorithm as in $\beta$ -CROWN and we use filtered smart branching (FSB) [16] as the branching heuristic in all experiments. Without cutting planes, GCP-CROWN becomes vanilla $\beta$ -CROWN as we share the same code base as $\beta$ -CROWN. We include model information and detailed setup for our experiments in Appendix C. GCP-CROWN has been integrated into the $\alpha , \beta$ -CROWN (alpha-beta-CROWN) verifier, and the instructions to reproduce results in this paper are available at http://PaperCode.cc/GCP-CROWN.
+
+Results on the oval20 benchmark in VNN-COMP 2020. oval20 is a popular benchmark consistently used in huge amount of NN verifiers and it perfectly reflects the progress of NN verifiers. We include literature results for many baselines in Table 1. We are the only verifier that can completely solve all three models without any timeout. Our average runtime is significantly lower compared to the time of baselines because we have no timeout (counted as 3600s), and our slowest instance only takes about a few minutes while easy ones only take a few seconds, as shown in Figure 2. Additionally, we often use less number of branches compared to the state-of-the-art verifier, $\beta$ -CROWN, since our strong cutting planes help us to eliminate many hard to solve subdomains in BaB. Furthermore, we highlight that $\alpha$ -CROWN+MIP also achieves a low timeout rate, although it is much slower than our bound propagation based approach combined with cuts from a MIP solver.
+
+Results on VNN-COMP 2021 benchmarks. Among the eight scoring benchmarks in VNN-COMP 2021 [3], only two (oval21 and cifar10-resnet) are most suitable for the evaluation of this work. Among other benchmarks, acasxu and nn4sys have low input dimensionality and require input space branching rather than ReLU branching; verivital, mnistfc, and eran benchmarks consist of small MLP networks that can be solved directly by MIP solvers; marabou contains mostly adversarial examples, making it a good benchmark for falsifiers rather than verifiers. We present our results in Table 2. Besides results from 6 VNN-COMP 2021 participants, we also include two additional baselines, $\alpha$ $_ { \mathcal { X } - \mathrm { C R O W N } + \mathrm { M I P } }$ (same as in Table 1), and MN-BaB [20], a recently proposed branch and bound framework with multi-neuron relaxations [39], which can be viewed as a restricted form of cutting planes. On the oval21 benchmark, OVAL, VeriNet and $\alpha \beta$ -CROWN are the best performing tools, verified 11 out of 27 instances, while we can verify twice more instances (22 out of 27) on this benchmark. On the cifar10-resnet benchmark, our verifier also solves the most number of instances and achieves the lowest average time. In fact, $\alpha$ -CROWN $^ +$ MIP is also a strong baseline, solving 4 more instances than all competition participants, showing the importance of strong cutting planes. Our GCP-CROWN with MIP cuts combines the benefits of fast bound propagation on GPU with the strong cutting planes generated by a MIP solver and achieves the best performance. We present a more detailed analysis on the cutting planes used in this benchmark in Appendix C.2.
+
+Table 2: VNN-COMP 2021 benchmarks: oval21 and cifar10-resnet. Results marked with VNN-COMP are from publicly available benchmark data on VNN-COMP 2021 Github. “-” indicates unsupported model.   
+
+<table><tr><td colspan="4">oval21 (30 properties;PGD upper bound 27)</td><td colspan="3">cifar10-resnet (72 properties)</td></tr><tr><td>Method</td><td>time(s)</td><td># verified</td><td>%timeout</td><td>time(s)</td><td># verified</td><td>%timeout</td></tr><tr><td>nnenum*[2,4]</td><td>630.06</td><td>2</td><td>86.66</td><td></td><td>=</td><td></td></tr><tr><td>Marabou*[31]</td><td>429.13</td><td>5</td><td>73.33</td><td>157.70</td><td>39</td><td>45.83</td></tr><tr><td>ERAN*[40,38]</td><td>233.84</td><td>6</td><td>70.00</td><td>129.48</td><td>43</td><td>40.28</td></tr><tr><td>OVAL*[17,16]</td><td>393.14</td><td>11</td><td>53.33</td><td></td><td>-</td><td></td></tr><tr><td>VeriNet*[25,26]</td><td>414.61</td><td>11</td><td>53.33</td><td>105.91</td><td>48</td><td>33.33</td></tr><tr><td>α,β-CROWN* [61,60,57]</td><td>395.32</td><td>11</td><td>53.33</td><td>99.87</td><td>58</td><td>19.44</td></tr><tr><td>MN-BaB [20]</td><td>435.46</td><td>10</td><td>56.66</td><td>1</td><td>1</td><td>-</td></tr><tr><td>Venus2t [7,33]</td><td>386.71</td><td>17</td><td>33.33</td><td>1</td><td>-</td><td>-</td></tr><tr><td>α-CROWN+MIP</td><td>301.23</td><td>15</td><td>40.00</td><td>125.48</td><td>46</td><td>36.11</td></tr><tr><td>GCP-CROWN with MIP cuts</td><td>145.26</td><td>23</td><td>13.33</td><td>53.49</td><td>63</td><td>12.5</td></tr></table>
+
+\* Results from VNN-COMP 2021 report [3]. $^ \dagger$ We use the latest code of Venus2 in the vnncomp branch, committed on Jul 18, 2022. Older versions cannot run these convolutional networks.
+
+Table 3: Verified accuracy $( \% )$ and avg. per-example verification time (s) on 7 models from SDP-FO [15].   
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Model ∈=0.3 ande=2/255</td><td colspan="2">SDP-FO[15]* Verified%</td><td colspan="2">PRIMA [40] Ver.%</td><td colspan="2">β-CROWN [57] Ver.%</td><td colspan="2">MN-BaB [20] Ver.%Time(s)</td><td colspan="2">Venus2 [7,33] Ver.%</td><td colspan="2">α-CROWN+MIP Ver.%</td><td colspan="2">GCP-CROWN Ver.% Time(s)</td><td rowspan="2">Upper bound</td></tr><tr><td></td><td>Time (s)</td><td></td><td>Time(s)</td><td></td><td>Time(s)</td><td></td><td></td><td></td><td>Time(s)</td><td></td><td>Time(s)</td><td></td><td></td></tr><tr><td rowspan="6">MNIST</td><td>CNN-A-Adv</td><td>43.4</td><td>&gt;20h</td><td>44.5</td><td>135.9</td><td>70.5</td><td>21.1</td><td>-</td><td></td><td>35.5</td><td>148.4</td><td>56.5</td><td>224.3</td><td>72.0</td><td>19.9 57.8</td><td>76.5</td></tr><tr><td>CNN-B-Adv</td><td>32.8</td><td>&gt;25h</td><td>38.0</td><td>343.6</td><td>46.5</td><td>32.2</td><td>-</td><td></td><td>-</td><td></td><td>27.0</td><td>360.6</td><td>48.5</td><td></td><td>65.0</td></tr><tr><td>CNN-B-Adv-4</td><td>46.0</td><td>&gt;25h</td><td>53.5</td><td>43.8</td><td>54.0</td><td>11.6</td><td>-</td><td></td><td>-</td><td></td><td>52.5</td><td>129.5</td><td>59.0</td><td>21.5</td><td>63.5</td></tr><tr><td>CNN-A-Adv</td><td>39.6</td><td>&gt;25h</td><td>41.5</td><td>4.8</td><td>44.0</td><td>5.8</td><td>42.5</td><td>68.3</td><td>47.5</td><td>26.0</td><td>46.0</td><td>63.1</td><td>48.5</td><td>9.8</td><td>50.0</td></tr><tr><td>CNN-A-Adv-4</td><td>40.0</td><td>&gt;25h</td><td>45.0</td><td>4.9</td><td>46.0</td><td>5.6</td><td>46.0</td><td>37.7</td><td>47.5</td><td>13.1</td><td>48.5</td><td>16.4</td><td>48.5</td><td>5.7</td><td>49.5</td></tr><tr><td>CNN-A-Mix</td><td>39.6</td><td>&gt;25h</td><td>37.5</td><td>34.3</td><td>41.5</td><td>49.6</td><td>35.0</td><td>140.3</td><td>33.5</td><td>72.4</td><td>32.5</td><td>231.3</td><td>47.5</td><td>29.2</td><td>53.0</td></tr><tr><td></td><td>CNN-A-Mix-4</td><td>47.8</td><td>&gt;25h</td><td>48.5</td><td>7.0</td><td>50.5</td><td>5.9</td><td>49.0</td><td>70.9</td><td>49.0</td><td>37.3</td><td>52.5</td><td>77.7</td><td>55.5</td><td>12.4</td><td>57.5</td></tr></table>
+
+\* We run α-CROWN+MIP and MN-BaB with 600s timeout threshold for all models. “-” indicates that we could not run a model due to unsupported model structure or other errors. We run our GCP-CROWN with MIP cuts with a shorter 200s timeout for all models and it achieves better verified accuracy than all other baselines. Other results are reported from [57].
+
+The oval21 benchmark was also included as part of VNN-COMP 2022, concluded in July 2022, with a different sample of 30 instances. GCP-CROWN is part of the winning tool in VNN-COMP 2022, $\alpha , \beta$ -CROWN, which verified 25 out of 30 instances in this benchmark, outperforming the second place tool (MN-BaB [20] with multi-neuron relaxations) with 19 verified instances by a large margin. More results on VNN-COMP 2022 can be found in these slides1.
+
+Results on SDP-FO benchmarks. We further evaluate our method on the SDP-FO benchmarks in [15, 57]. This benchmark contains 7 mostly adversarially trained MNIST and CIFAR models with 200 instances each, which are hard for many existing verifiers. Beyond the baselines reported in [57], we also include two additional baselines, $\alpha { \mathrm { - } } \mathbf { C R O W N + M I P }$ (same as in Table 1) and MN-BaB [20] (same as in Table 2). Table 3 shows that our method improves the percentage of verified images (“verified accuracy”) on all models compared to state-of-the-art verifiers, further closing the gap between verified accuracy and empirical robust accuracy obtained by PGD attack (reported as “upper bound” in Table 3).
+
+# 5 Related Work
+
+Cutting plane method is a classic technique to strengthen the convex relaxation of an integer programming problem. Generic cutting planes such as Gomory’s cut [21, 22], Chvátal–Gomory cut [12], implied bound cut [27], lift-and-project [35], reformulation-linearization techniques [46] and mixed integer rounding cuts [41, 37] can be applied to almost any LP relaxed problems, and problem specific cutting planes such as Knapsack cut [14], Flow-cover cut [42] and Clique cut [29] require specific problem structures. Modern MIP solvers typically uses a branch-and-cut strategy, which tends to generate a large number of cuts before starting the next iteration of branching, and solve the LP relaxation of the MIP problem with cutting planes with an exact method such as the Simplex method. Our GCP-CROWN is a specialized solver for the NN verification problem, which can quickly obtain a lower bound of the LP relaxation with cutting planes specially for the NN verification problem.
+
+The verification of piece-wise linear NNs can be formulated as a MIP problem, so early works [30, 31, 53] solve an integer or combinatorial formulation directly. For efficiency reasons, most recent works use a convex relaxation such as linear relaxation [19, 58] or semidefinite relaxation [44, 15]. Salman et al. [45] discussed the limitation of many convex relaxation based NN verification algorithms and coined the term “convex relaxation barrier”, specifically for the popular single-neuron “Planet” relaxation [19]. Several works developed novel techniques to break this barrier. [47] added constraints that depends on the aggregation of multiple neurons, and these constraints were passed to a LP solver. [40] enhanced the multi-neuron formulation of [47] to obtain tighter relaxations. [1] studied stronger convex relaxations for a ReLU neuron after an affine layer, and [52] constructed efficient algorithms based on this relaxation for incomplete verification. [16] extended the formulation in [1] to a dual form and combined it with branch and bound to achieve completeness. [7] proposed specialized cuts by considering neuron dependencies and solve them using a MIP solver. [20] combined the multi-neuron relaxation [40] with branch and bound. Although these works can be seen as a special form of cutting planes, they mostly focused on enhancing the relaxation for several neurons within a single layer or two adjacent layers. GCP-CROWN can efficiently handle general cutting plane constraints with neurons from any layers in a bound propagation manner, and the cutting planes we find from a MIP solver can be seen as tighter convex relaxations encoding multi-neuron and multi-layer correlations.
+
+# 6 Conclusion
+
+In this paper, we propose GCP-CROWN, an efficient and GPU-accelerated bound propagation method for NN verification capable of handling any cutting plane constraints. We combine GCP-CROWN with branch and bound and high quality cutting planes generated by a MIP solver to tighten the convex relaxation for NN verification. The combination of fast bound propagation and strong cutting planes lead to state-of-the-art verification performance on multiple benchmarks. Our work opens up a great opportunity for studying more efficient and powerful cutting planes for NN verification.
+
+Limitations of this work Our work generalizes existing bound propagation methods that can handle only simple constraints (such as neuron split constraints in $\beta$ -CROWN [57]) to general constraints, and we share a few common limitations as in previous works [10, 60, 16]: the branch-and-bound procces and bound propagation procedure are developed on ReLU networks, and it can be non-trivial to extend it to neural networks with non-piecewise-linear operations. In addition, we currently directly use cutting planes generated by a generic MIP solver, and there might exist stronger and faster cutting plane methods that can exploit the structure of the neural network verification problem. We hope these limitations can be addressed in future works.
+
+Potential Negative Societal Impact Our work focuses on formally proving desired properties of a neural network under investigation such as safety and robustness, which is an important direction of trustworthy machine learning and has overall positive societal impact. Since our verifier is a complete verifier, it might be possible to use it to find weakness of a neural network and guide adversarial attacks. However, we believe that formally characterizing a model’s behavior and potenti al weakness is important for building robust models and preventing real-world malicious attack.
+
+Funding Disclosure This work is partially supported by the NSF grant No.1910100, NSF CNS No.2046726, NSF IIS No.2008173, NSF IIS No.2048280 and the Alfred P. Sloan Foundation. Huan Zhang is supported by a grant from the Bosch Center for Artificial Intelligence. Suman Jana acknowledges the NSF CAREER award.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
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+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
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+3. If you ran experiments...
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+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A] Not applicable, our verification results typically do not change after running multi times.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
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+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
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+5. If you used crowdsourcing or conducted research with human subjects...
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+# MOTIF-BASED ROTO-TRANSLATION INVARIANTTRANSFORMER FOR MOLECULAR PROPERTY PRE-DICTION IN 3D SPACE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recent studies use geometric deep learning to represent molecules and predict properties. However, they are computationally expensive in capturing long-range dependencies and ignore the non-uniformity of interatomic distances. More importantly, few of them consider injecting the biochemical structure knowledge such as functional groups into model architectures. To overcome such issues, we introduce Molformer, a variant of the Transformer for molecular representations that exploits both semantic motifs and 3D spatial information. Specifically, Molformer extracts motifs based on functional groups and learns customized embeddings to store the semantic meanings of those informative substructures. In order to fully employ 3D geometry, we adopt a convolutional position encoding to achieve roto-translation invariance, a multi-scale self-attention mechanism to capture local fine-grained patterns with increasing contextual scales, and an attentive farthest point sampling algorithm to attain the molecular representation. We validate Molformer across several domains in quantum chemistry, physiology, and biophysics. Our experiments show better or competitive performance in those datasets. Our work provides a promising way to amalgamate 3D geometric information and make better usage of informative substructures in representing molecules.
+
+# 1 INTRODUCTION
+
+Spatial structures are among the most crucial factors to decide molecular properties and understand their principles of action in the physical world. For example, 3D structures of proteins provide valuable information for inferring biological interventions, such as structure-based drug development and targeted mutagenesis (Senior et al., 2020; Jumper et al., 2021; Baek et al., 2021). In chemistry, zeolites show obvious differences in separation properties caused by subtle changes in their 3D geometric compositions (Chai et al., 2020; Pfriem et al., 2021). Apart from that, in the pharmaceutical industry, the same compounds can have different 3D structures, resulting in different solubility (Zhang et al., 2017). To sum up, capturing 3D spatial structures is essential to accurately forecast molecular properties. Based on these facts, researchers have studied molecular representation learning techniques (Rao et al., 2019) to include 3D spatial information (Zhavoronkov et al., 2019).
+
+The dominant 3D molecular models are Graph Neural Networks (GNNs) and 3D Convolutional Neural Networks (3DCNNs) (Derevyanko et al., 2018; Pagès et al., 2019; Townshend et al., 2019). GNNs create edges by using either chemical bonds or finding the neighbors of each node within a distance cutoff (Zhang et al., 2020b). They encode pairwise connectivity of atoms and require running multiple hops for an atom to reach to another. 3DCNNs encode translational and permutational symmetries, but need to stack deep layers to build direct connections between distant regions, incurring significant computational costs. In contrast, Transformers rely on the self-attention mechanism to capture long-term dependencies in parallel (Hernández & Amigó, 2021). Meanwhile, Equivariant Neural Networks (ENNs) (Thomas et al., 2018) have emerged as a new class of methods, where geometric transformations of their inputs lead to well-defined transformations of outputs. Some ENNs adopt Transformers as the backbone but fail to surmount the intrinsic drawbacks of this architecture, including its insensibility to local patterns among non-uniformly distancing atoms and its inefficiency to aggregate atom features. Some other Transformer-based methods have been proposed to fuse distance and graph neighbourhood information (Maziarka et al., 2020; 2021). However, they take no consideration of employing motifs, which are frequently-occurring substructures in molecules and can be leveraged to uncover global graph properties.
+
+In this work, we present the Molformer on the basis of all preceding analysis. For the sake of injecting chemical domain knowledge, we construct a motif-template vocabulary based on functional groups and adopt trainable motif embeddings to maintain the semantic meanings of those essential substructures. Then with both motifs and atoms as input, Molformer operates on a fully-connected graph with direct connections between remote regions (Velickovi ˇ c et al., 2017; Joshi, 2020), which ´ reduces computational burden of multi-hop GNNs and stacked 3DCNNs. However, this characteristic limits Molformer’s capacity in exploiting local structures and leads to poor generalization in unseen cases (Qi et al., 2017). Therefore, we propose a Multi-scale Self-Attention (MSA) module to recognize fine-grained patterns from neighborhoods. Moreover, we introduce a roto-translation invariant Convolutional Position Encoding (CPE) to depict position relationships among atoms and their adjacencies. After that, to retain a comprehensive representation of the entire molecule, we propose an Attentive Farthest Point Sampling (AFPS) module that selects important atoms with the assistance of the attention score map.
+
+To summarize, our contributions are as follows:
+
+• To the best of our knowledge, we are the foremost to incorporate motifs with knowledge of functional groups into a Transformer architecture for 3D molecular representation learning. • We propose a novel MSA to extract local patterns, a roto-translation invariant CPE method to encode relative distance at a linear computational time cost, and a simple yet effective downsampling algorithm to gather molecular representations. • We show significant improvements on several benchmarks in three domains. Code and all datasets are available at https://github.com/smiles724/Molformer.
+
+# 2 PRELIMINARIES
+
+Problem Definition. A molecule ${ \cal S } = ( { \cal E } , { \cal P } )$ has $N$ atoms and $C$ atom classes, where ${ \pmb { { \cal E } } } =$ $\{ e _ { 1 } , . . . , e _ { N } \} \in \mathbb { R } ^ { N \times C }$ contains the one-hot atom representations and $P = \{ p _ { 1 } , . . . , p _ { N } \} \in \mathbb { R } ^ { N \times 3 }$ contains the 3D coordinates of each atom. Each one-hot $e _ { i }$ can be converted to a dense vector $\pmb { x } _ { i } = e _ { i } \pmb { W } ^ { E }$ , with $\pmb { x } _ { i } \in \mathbb { R } ^ { d _ { m o d e l } }$ and $W ^ { E } \in \mathbb { R } ^ { C \times d _ { m o d e l } }$ being the embedding matrix. The 3D coordinates of the atom $i$ is a three-dimensional vector $\pmb { p } _ { i } = [ p _ { i } ^ { x } , \bar { p } _ { i } ^ { y } , p _ { i } ^ { z } ]$ . A representation learning model $f$ acts on $_ { s }$ , obtaining its representation $r = f ( S )$ . Then $\mathbfit { \Delta } \mathbf { r }$ is forwarded to a prediction model $g$ and attain the prediction of a biochemical property ${ \hat { y } } = g ( \pmb { r } )$ .
+
+Self-attention Mechanism. The Transformer (Vaswani et al., 2017) has become very successful due to its core component, self-attention. Given a set of input features $\{ \pmb { x } _ { i } \} _ { i = 1 , . . . , N }$ , the standard dot-product attention layer is as the following:
+
+$$
+q _ { i } = f _ { Q } ( { x } _ { i } ) , k _ { i } = f _ { K } ( { x } _ { i } ) , v _ { i } = f _ { V } ( { x } _ { i } ) , a _ { i j } = q _ { i } k _ { j } ^ { T } / \sqrt { d _ { k } } , z _ { i } = \sum _ { j = 1 } ^ { N } \sigma ( a _ { i j } ) v _ { j }
+$$
+
+where $\left\{ f _ { Q } , f _ { K } , f _ { V } \right\}$ are embedding transformations, and $\{ q _ { i } , k _ { i } , v _ { i } \}$ are respectively the query, key, and value vectors with the same dimension $d _ { k }$ . $a _ { i j }$ is the attention that the token $i$ pays to the token $j$ . $\sigma$ denotes the Softmax function and $z _ { i }$ is the output embedding of the token $i$ . This formula conforms to a non-local network (Wang et al., 2018), indicating its inability to capture fine-grained patterns in a local context.
+
+Position Encoding. Self-attention is invariant to permutation of the input (Dufter et al., 2021), and position encoding ensures that the Transformer will reveal positional information. Position encoding methods can be either based on absolute positions or relative distances. The former takes the raw position information as input and is sensitive to spatial transformations. The latter manipulates the attention score by incorporating relative distances (Guo et al., 2020a; Pan et al., 2021):√ $a _ { i j } = { \bf q } _ { i } { \bf k } _ { j } ^ { T } / \sqrt { d _ { k } } + f _ { \mathrm { P E } } ( \bar { p _ { i } } - p _ { j } )$ , where $\bar { f } _ { \mathrm { P E } } ( \cdot )$ is the position encoding function and is translation invariant. The rotation invariance can be further accomplished by taking a L2-norm $| | p _ { i } - p _ { j } | | _ { 2 }$ (Chen et al., 2019b).
+
+![](images/4aa7cb0818a54b520d22d500d8474338b6f0b8c1a3cb8942bfd66a2211b458c5.jpg)  
+Figure 1: The overall architecture of our Molformer. FFN stands for a feed-forward network. Local features are shown in purple and orange; yellow corresponds to a global feature.
+
+# 3 MOLFORMER
+
+Molformer is based on the architecture of Transformer but adopts several significantly different and novel components (see Figure 1). First, a vocabulary of motif templates is constructed on the basis of functional groups and we extract all available motifs from each molecule. Then both atoms and motifs acquire their corresponding embeddings and are forwarded into $L$ feature learning blocks. Each block consists of a convolutional position encoding, a multi-scale self-attention, and a feedforward network. After that, an attentive subsampling method is utilized to adaptively aggregate the molecular presentation, which is later fed into a predictor to forecast properties in a broad range of downstream tasks.
+
+# 3.1 TRAINABLE MOTIF-BASED EMBEDDING
+
+Motifs are frequently-occurring substructure patterns as well as the building blocks of complex molecular structures. They usually maintain semantic meanings and have great expressiveness of the biochemical characteristics of the whole molecule (Zhang et al., 2020a). In the chemical community, researchers have developed a set of standard criterion to recognize motifs with essential functionalities in molecules (Milo et al., 2002). Despite that, few of prior studies directly incorporate those informative motifs into their model architectures. To fill this gap, we define a series of momentous substructures using external domain knowledge, and introduce a trainable motif embeddings method to fully exploit them in our Molformer.
+
+To begin with, all motifs are first extracted according to the motif vocabulary, which is built by functional groups. Practically, we rely on RDKit (Landrum, 2013) to draw them from the SMILES (Weininger, 1988) representation of each molecule. We assume $M$ motifs $\{ m _ { 1 } , . . . , m _ { M } \}$ are detected in the molecule $\pmb { S }$ , and each motif $m _ { i }$ contains a certain number of at least two atoms. Then we regard each kind of motif as a new type of token and append them to the input. Therefore, the input for our Molformer becomes $\{ \pmb { x } _ { 1 } , . . . , \pmb { x } _ { N } , \pmb { x } _ { m _ { 1 } } , . . . , \pmb { x } _ { m _ { M } } \}$ , where $\mathbf { \boldsymbol { x } } _ { m _ { i } }$ is obtained through an learnable embedding matrix $W ^ { M } \in \mathbb { R } ^ { C ^ { \prime } \times d _ { m o d e l } }$ and $C ^ { \prime }$ denotes the number of motif categories. As for the position of each motif, we adopt a weighted sum of the 3D coordinates of its component atoms as pmi = Pxi∈mi ( $\begin{array} { r } { \dot { p _ { m _ { i } } } = \sum _ { x _ { i } \in m _ { i } } \ ( \frac { w _ { i } } { \sum _ { x _ { i } \in m _ { i } } w _ { i } } ) \cdot \dot { p _ { i } } } \end{array}$ wi ) · pi, where wi are the atomic weights.
+
+Our approach requires the model to automatically learn a customized embedding for each motif template through backpropagations, which follows a data-driven pattern. In some data-sufficient tasks, its greatest potential can be unlocked and those motif embeddings can be well trained. Nevertheless, in the case of few-shot learning or small datasets, each category of motif might only appear rare times. Those embeddings are not fully tuned and can be extremely biased and noisy, which will do little helps to the ultimate property prediction.
+
+# 3.2 CONVOLUTIONAL POSITION ENCODING
+
+To enable roto-translation invariance and take fully advantage of geometric information, instead of adding a term of $f _ { \mathrm { P E } } ( \pmb { p } _ { i } - \pmb { p } _ { j } )$ , we propose a CPE that applies a convolutional operation to the interatomic distance $\pmb { D } \in \mathbb { R } ^ { N \times N }$ :
+
+$$
+{ \cal A } _ { \mathrm { c o v } } = \mathrm { C o n v } _ { 2 d } ( D ) \odot { \cal A } ,
+$$
+
+where $\pmb { A } = [ a _ { i , j } ] _ { i , j = 1 , \cdots N } \in \mathbb { R } ^ { N \times N }$ is the attention matrix, $\mathrm { C o n v } _ { 2 d } ( \cdot )$ denotes a 2D shallow convolutional network with a kernel size of $1 \times 1$ , and $\odot$ is the element-wise product. With multi-headed self-attention, ${ \cal A } _ { \mathrm { c o v } }$ is expanded in the sense that $A _ { \mathrm { c o v } } \in \mathbb { R } ^ { H \times N \times N }$ , and $\mathrm { C o n v } _ { 2 d } ( \cdot )$ has $H$ output channels. The CPE method induces ${ \mathrm { O } } ( N )$ convolution operations on each atom and can drastically reduce training time when the number of atoms is very large ( $\mathrm { { W u } }$ et al., 2021).
+
+# 3.3 MULTI-SCALE SELF-ATTENTION
+
+The self-attention mechanism in the Transformer is good at capturing global data patterns but ignores local context (Guo et al., 2020a). Exploiting local context has proven to be important for 3D spatial data such as 3D point clouds (Qi et al., 2017). Therefore, we impose a distance-based constraint in self-attention in order to extract multi-scaled patterns from both local and global contexts.
+
+Guo et al. (2020b) propose to use integer-based distance to limit attention to local word neighbors, which cannot be used in molecules. This is because different types of molecules have different densities and molecules of the same type have different spatial regularity, which results in the nonuniformity of interatomic distances. Normally, small molecules have a mean interatomic distance of $1 { - } 2 \textup { \AA }$ (Angstrom, $1 0 ^ { - 1 0 } m _ { , }$ ), which is denser than large molecules like proteins with approximately $5 \mathrm { ~ \AA ~ }$ on average. To address that, we design a new multi-scale methodology to robustly capture details. Specifically, we mask atoms beyond a certain distance $\tau _ { s }$ (a real number as opposed to an integer in Guo et al. (2020b)) at each scale $s$ . We denote $d _ { i j } = | | { \pmb p } _ { i } - { \pmb p } _ { j } | | _ { 2 }$ as the Euclidean distance between the $i$ -th and $j$ -th atom. The attention calculation is modified as:
+
+$$
+a _ { i j } ^ { \tau _ { s } } = \frac { q _ { i } \pmb { k } _ { j } ^ { T } \cdot \mathbf { 1 } _ { \{ d _ { i j } < \tau _ { s } \} } } { \sqrt { d _ { k } } } , \ : z _ { i } ^ { \tau _ { s } } = \sum _ { j = 1 } ^ { N } \sigma ( a _ { i j } ^ { \tau _ { s } } ) \pmb { v } _ { j } ,
+$$
+
+where $\mathbf { 1 } _ { \{ d _ { i j } < \tau _ { s } \} }$ is the indicator function. For small molecules, Equation 3 can be complementally combined with Equation 2. Then features extracted from $S$ different scales $\{ \tau _ { s } \} _ { s = 1 , \dots , S }$ as well as the informative global feature are concatenated together to form a multi-scale representation, denoted by $\pmb { z } _ { i } ^ { \prime } = \pmb { z } _ { i } ^ { \mathcal { T } _ { 1 } } \oplus . . . \oplus \pmb { z } _ { i } ^ { \tau s } \oplus \pmb { z } _ { i } ^ { g l o b a l } \in \mathbb { R } ^ { ( S + 1 ) d _ { k } }$ . After that, $ { \boldsymbol { z } } _ { i } ^ { \prime }$ is forwarded into a multi-layer perceptron to be compressed as $z _ { i } ^ { \prime \prime }$ with the original dimension $d _ { k }$ .
+
+# 3.4 ATTENTIVE FARTHEST POINT SAMPLING
+
+After having the atom embeddings $\{ z _ { i } ^ { \prime \prime } \} _ { i = 1 , \dots , N }$ , we study how to obtain the molecular representation $\pmb { r }$ . For GNNs, several readout functions such as set2set (Vinyals et al., 2015) and GGNN (Gilmer et al., 2017) are invented. For Transformer architectures, one way is via a virtual atom.
+
+<table><tr><td colspan="2">Algorithm 1 Pseudocode of Attentive Farthest Point Sampling</td></tr><tr><td colspan="2">Input: The attention score matrix A ∈ RN×N, a Euclidean distance matrix D ∈ RN ×N. Output: K sampled points.</td></tr><tr><td>1: A←∑iAij ∈RN</td><td>&gt; sum up the attention matrix along rows</td></tr><tr><td>2:D←D∈RNxN</td><td>&gt; normalize the distance matrix</td></tr><tr><td>3:P= {𝑥#},M={1,2,..,N} 4:while length(P)&lt;k do</td><td></td></tr><tr><td>5: Xnew = argmax (min Dij + eAi)</td><td>&gt; pick up the atom that maximize the objective</td></tr><tr><td>iEM j∈p 6: P.append(xnew), M.remove(x new)</td><td></td></tr><tr><td>7: return P</td><td></td></tr></table>
+
+Though as Ying et al. (2021) state, it significantly improves the performance of existing models in the leaderboard of Open Graph Benchmark (Hu et al., 2020), this way concentrates more on close adjacent atoms and less on distant ones, and can lead to inadvertent over-smoothing of information propagation (Ishiguro et al., 2019). Besides, it is difficult to locate a virtual node in 3D space and build connections to existing atoms. The other way selects a subset of atoms via a downsampling algorithm named Farthest Point Search (FPS), but it ignores atomic differences and has sensitivity to outlier points (Pan et al., 2021) as well as uncontrollable randomness. To address these issues, we propose a new algorithm named AFPS. It aims to sample atoms by not merely spatial distances, but also their significance in terms of attention scores.
+
+Specifically, we choose the virtual atom $x _ { \# }$ as the starting point and initialize two lists $\mathcal { P } = \{ x _ { \# } \}$ and $\mathcal { M } = \mathrm { \bar { \{ } } 1  , . . . , N \}$ to store remaining candidate points. Then the process begins with the attention score matrix $\pmb { A } \in \mathbb { R } ^ { N \times N }$ and the interatomic distance matrix $\pmb { { \cal D } } \in \mathbb { R } ^ { N \times N }$ . It can be easily proved that each row of $\pmb { A }$ sums up to 1 after the Sof tmax operation along columns, i.e. $\begin{array} { r } { \sum _ { j } { \cal A } _ { i j } \dot { = } 1 } \end{array}$ for $\forall i \in [ N ]$ . In order to obtain the importance of each atom in the self-attention computation, we accumulate $\pmb { A }$ along rows and get $\begin{array} { r } { \tilde { \pmb { A } } = \sum _ { i } \pmb { A } _ { i j } \in \mathbb { R } ^ { N } } \end{array}$ . Besides, we adopt the min-max normalization to rescale the distance matrix $_ D$ into values between 0 and 1, and obtain $\begin{array} { r } { \tilde { D } = \frac { D - \operatorname* { m i n } D } { \operatorname* { m a x } D - \operatorname* { m i n } D } } \end{array}$ .
+
+After the above preprocess, we repeatedly move a point $x _ { n e w }$ from $\mathcal { M }$ to $\mathcal { P }$ , which ensures that $x _ { n e w }$ is as far from $\mathcal { P }$ as possible by maximizing $\tilde { D } _ { i j }$ and also plays a crucial role in attention computation by maximizing ${ \tilde { A } } _ { i }$ . Mathematically, the AFPS aims to achieve the following objective:
+
+$$
+\operatorname* { m a x } \sum _ { i \in \mathcal { M } } ( \operatorname* { m i n } _ { j \in \mathcal { P } \setminus \{ i \} } \tilde { D } _ { i j } + \epsilon \tilde { A } _ { i } )
+$$
+
+where $\epsilon$ is a hyperparameter to balance those two different goals. This process is repeated until $\mathcal { P }$ has reached $K$ points. Algorithm 1 provides a greedy approximation solution to solve this AFPS optimization objective for sake of computational efficiency.
+
+After that, sampled features $\{ z _ { i } ^ { \prime \prime } \} _ { i \in P }$ are gathered by a Global Average Pooling layer (Lin et al., 2013) to attain the molecular representation $\pmb { r } \in \mathbb { R } ^ { d _ { k } }$ .
+
+Remarkably, our proposed AFPS has considerable difference and superiority over a body of previous hierarchical learning approaches (Eismann et al., 2020; 2021). Their subsampling operations are mainly designed for protein complexities, which have more uniform structures than small molecules. To be specific, they hierarchically use alpha carbons as the intermediate set of points and aggregate information at the level of those carbons for the entire complex. However, the structures of small molecules have no such a stable paradigm, and we provide a universal methodology to adaptively subsample atoms without any prior assumptions on the atom arrangement.
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETUP
+
+We conduct extensive experiments on both small and large molecules (proteins) with various targets, including quantum chemistry, physiology, and biophysics. Table 1 summarises information of benchmark datasets, such as the number of tasks and task types, the number of molecules and atom classes, the minimum and maximum number of atoms, and the density (mean interatomic distances) of all molecules.
+
+Table 1: Key statistics of datasets from three different categories.   
+
+<table><tr><td>Category</td><td>Dataset</td><td>Tasks</td><td>Task Type</td><td>Molecules</td><td>Atom Class</td><td>Min. Atoms</td><td>Max. Atoms</td><td>Density （A</td><td>Metric</td></tr><tr><td rowspan="3">Quantum Chemistry</td><td>QM7</td><td>1</td><td>regression</td><td>7,160</td><td>5</td><td>4</td><td>23</td><td>2.91</td><td>MAE</td></tr><tr><td>QM8</td><td>12</td><td>regression</td><td>21,786</td><td>5</td><td>3</td><td>26</td><td>1.54</td><td>MAE</td></tr><tr><td>QM9</td><td>12</td><td>regression</td><td>133,885</td><td>5</td><td>3</td><td>28</td><td>1.61</td><td>MAE</td></tr><tr><td rowspan="2">Physiology</td><td>BBBP</td><td>1</td><td>classification</td><td>2.039</td><td>13</td><td>2</td><td>132</td><td>2.64</td><td>ROC-AUC</td></tr><tr><td>ClinTox</td><td>2</td><td>classification</td><td>1,478</td><td>27</td><td>1</td><td>136</td><td>2.83</td><td>ROC-AUC</td></tr><tr><td rowspan="2">Biophysics</td><td>PDBind1</td><td>1</td><td>regression</td><td>11,908</td><td>23</td><td>115</td><td>1,085</td><td>5.89</td><td>RMSE</td></tr><tr><td>BACE</td><td>1</td><td>classification</td><td>1,513</td><td>8</td><td>10</td><td>73</td><td>3.24</td><td>ROC-AUC</td></tr></table>
+
+Datasets. We test Molformer on a series of small molecule datasets, containing QM7 (Blum & Reymond, 2009), QM8 (Ramakrishnan et al., 2015), QM9 (Ramakrishnan et al., 2014), BBBP (Martins et al., 2012), ClinTox (Gayvert et al., 2016), and BACE (Subramanian et al., 2016) 2. QM7 is a subset of GDB-13 and composed of 7K molecules with up to 5 heavy atom types. QM8 and QM9 are subsets of GDB-17 with $2 2 \mathrm { k }$ molecules and 133K molecule respectively.
+
+Additionally, we also inspect Molformer’s ability of learning mutual relations between proteins and molecules on the PDBbind dataset (Wang et al., 2005). We follow Townshend et al. (2020) and split protein-ligand complexes by protein sequence identity at $30 \%$ . As for the target, we predict $p S = - \log ( S )$ , where $S$ is the binding affinity in Molar unit. In addition, we only use the pocket of each protein and put pocket-ligand pairs together as the input.
+
+For QM9, we use the exact train/validation/test split as Townshend et al. (2020). For PDBbind, $90 \%$ of the data is used for training and the rest is divided equally between validation and test like Chen et al. (2019c). For others, we adopt the scaffold splitting method with a ratio of 8:1:1 for train/validation/test as Rong et al. (2020). More implementing details can be found in Appendix A.1
+
+Baselines For small molecules, we compare our approach with a number of state-of-the-art baselines. TF_Robust (Ramsundar et al., 2015) takes molecular fingerprints as the input. GraphConv (Kipf & Welling, 2016), Weave (Kearnes et al., 2016), MPNN (Gilmer et al., 2017), Schnet (Schütt et al., 2018), MEGNet (Chen et al., 2019c), DMPNN (Yang et al., 2019), MGCN (Lu et al., 2019), AttentiveFP (Xiong et al., 2019), DimeNet $^ { + + }$ (Klicpera et al., 2020), SphereNet (Liu et al., 2021), and SpinConv (Shuaibi et al., 2021) are all graph convolutional models. Graph Transformer (Chen et al., 2019a), MAT (Maziarka et al., 2020), R-MAT (Maziarka et al., 2021), SE(3)- Transformer (Fuchs et al., 2020), and LieTransformer (Hutchinson et al., 2021) are Transformerbased models.
+
+For PDBbind, we choose six baselines. DeepDTA (Öztürk et al., 2018) and DeepAffinity (Karimi et al., 2019) take in pairs of ligand and protein SMILES as input. Cormorant (Anderson et al., 2019) is an ENN that represents each atom by its absolute 3D coordinates. Schnet, 3DCNN and 3DGCN (Townshend et al., 2020) are 3D molecular representation methods.
+
+# 4.2 RESULTS ON DOWNSTREAM TASKS
+
+Molecules. Table 2 and Table 3 document the overall results of Molformer and baselines on small molecules datasets, where best performance is marked bold and the second best is underlined for clear comparison. It can be discovered that Molformer achieves the lowest MAE of 11.6 on QM7 and 0.009 on QM8, beating several strong baselines including DMPNN and Graph Transformer. While not all state-of-the-art on QM9, Molformer offers competitive performance in 5 property regression tasks, which do not require thermochemical energy subtractions. Particularly, we outperforms all Transformer-based ENNs, including SE(3)-Transformer and LieTransformer. In classification problems, we surpass all non-pretrained methods and are only inferior to the pretrained GROVE. This accords to the fact that datasets with fewer samples can gain large improvements through the self-supervised pretraining (Rong et al., 2020).
+
+Table 2: The performance comparison. For regression tasks including QM7 and QM8, lower is better. For classification tasks including BBBP, ClinTox, and Bace, higher is better. The methods in purple are pretrained methods.   
+
+<table><tr><td>Method</td><td>QM7</td><td>QM8</td><td>BBBP</td><td>ClinTox</td><td>BACE</td></tr><tr><td>TF-Robust (Ramsundar et al., 2015)</td><td>120.6</td><td>0.024</td><td>0.860</td><td>0.765</td><td>0.824</td></tr><tr><td>GraphConv (Kipf &amp; Welling,2016)</td><td>118.9</td><td>0.021</td><td>0.877</td><td>0.845</td><td>0.854</td></tr><tr><td>Weave (Kearnes et al.,2016)</td><td>94.7</td><td>0.022</td><td>0.837</td><td>0.823</td><td>0.791</td></tr><tr><td>MPNN (Gilmer et al., 2017)</td><td>113.0</td><td>0.015</td><td>0.913</td><td>0.879</td><td>0.815</td></tr><tr><td>Schnet (Schuitt et al., 2018)</td><td>74.2</td><td>0.020</td><td>0.847</td><td>0.717</td><td>0.750</td></tr><tr><td>DMPNN (Yang et al.,2019)</td><td>105.8</td><td>0.014</td><td>0.919</td><td>0.897</td><td>0.852</td></tr><tr><td>MGCN (Lu et al., 2019)</td><td>77.6</td><td>0.022</td><td>0.850</td><td>0.634</td><td>0.734</td></tr><tr><td>Attentive FP (Xiong et al., 2019)</td><td>126.7</td><td>0.028</td><td>0.908</td><td>0.933</td><td>0.863</td></tr><tr><td>Graph Transformer (Chen et al.,2019a)</td><td>47.8</td><td>0.010</td><td>0.913</td><td>-</td><td>0.880</td></tr><tr><td>MAT (Maziarka et al.,2020)</td><td>102.8</td><td>1</td><td>0.728</td><td>-</td><td>0.846</td></tr><tr><td>R-MAT (Maziarka et al., 2021)</td><td>68.6</td><td>=</td><td>0.746</td><td>=</td><td>0.871</td></tr><tr><td>GROVElarge (Rong et al., 2020)</td><td>89.4</td><td>0.017</td><td>0.911</td><td>0.884</td><td>0.858</td></tr><tr><td> GROVElarge (Rong et al., 2020)</td><td>72.6</td><td>0.012</td><td>0.940</td><td>0.944</td><td>0.894</td></tr><tr><td>Molformer</td><td>11.5</td><td>0.009</td><td>0.926</td><td>0.941</td><td>0.884</td></tr></table>
+
+Table 3: Comparison of MAE on QM9. The methods in orange are Transformer-based methods.   
+
+<table><tr><td>Target (Unit)</td><td>€HOMO (eV)</td><td>∈LUMo (eV)</td><td>△e(eV)</td><td>μ(D)</td><td>a (bohr3)</td></tr><tr><td>MPNN (Gilmer et al., 2017)</td><td>.043</td><td>.037</td><td>.069</td><td>.030</td><td>.092</td></tr><tr><td>Schnet (Schuitt et al., 2018)</td><td>.041</td><td>.034</td><td>.063</td><td>.033</td><td>.235</td></tr><tr><td>MEGNet ful (Chen et al., 2019c)</td><td>.038</td><td>.031</td><td>.061</td><td>.040</td><td>.083</td></tr><tr><td>DimeNet++ (Klicpera et al.,2020)</td><td>.024</td><td>.019</td><td>.032</td><td>.029</td><td>.043</td></tr><tr><td>SphereNet (Liu et al., 2021)</td><td>.024</td><td>.019</td><td>.032</td><td>.026</td><td>.047</td></tr><tr><td>SpinConv (Shuaibi et al.,2021)</td><td>.026</td><td>.022</td><td>.047</td><td>.027</td><td>.058</td></tr><tr><td> SE(3)-Transformer (Fuchs et al., 2020)</td><td>.035</td><td>.033</td><td>.053</td><td>.051</td><td>.142</td></tr><tr><td>LieTransformer-SE(3) (Hutchinson et al., 2021)</td><td>.033</td><td>.029</td><td>.052</td><td>.061</td><td>.104</td></tr><tr><td>Molformer</td><td>.021</td><td>.026</td><td>.039</td><td>.045</td><td>.086</td></tr></table>
+
+Protein. Table 4 reports the Root-Mean-Squared Deviation (RMSD), the Pearson correlation $( R _ { p } )$ , and the Spearman correlation $( R _ { s } )$ on PDBbind. Molformer achieves the lowest RMSD among all baselines and the best Pearson and Spearman correlations. As Wu et al. (2018) claim, appropriate featurizations which contains pertinent information is significant for PDBbind. However, an important observation in our work is that deep learning approaches with the full exploitation of 3D geometric information can perform better than conventional methods like DeepDTA and DeepAffinity, which use a set of physicochemical descriptors but ignore 3D structures.
+
+# 5 ABLATION STUDY AND DISCUSSION
+
+# 5.1 WHAT ARE THE EFFECTS OF EACH COMPONENT
+
+We investigate the effectiveness of different modules of our Molformer in Table 5. It can be observed that CPE substantially boosts model’s performance compared with the naive method that immediately adds 3D coordinates as the atom input feature. In addition, AFPS is found to produce better predictions than the control group, which utilizes the virtual node as the molecular representation. Moreover, MSA significantly reduces RMSD from 17.6 to 11.6 on QM7, but its improvements in QM8 are much smaller. This phenomenon indicates that MSA is an appropriate way to alleviate the problem of inadequate training in small datasets. It endows Molformer with capability to extract local features by regulating the scope of self-attention. However, as the data size gets larger and larger, Molformer does not require the assistance of MSA to abstract local patterns, since the parameters of CPE is properly trained. What’s more, the trainable motif-level embedding leads to a MAE decrease of 2.1 in QM7 and a RMSD drop of 0.011 in PDBbind, indicating its effectiveness in both small molecules and proteins.
+
+Table 4: Comparison of RMSD, $R _ { p }$ , and $R _ { s }$ on PDBbind.   
+
+<table><tr><td>Method</td><td>Geometry</td><td>RMSD</td><td>Rp</td><td>Rs</td></tr><tr><td>DeepDTA (Ozturk et al., 2018)</td><td>Non-3D</td><td>1.565</td><td>0.573</td><td>0.574</td></tr><tr><td>DeepAffinity (Karimi et al.,2019)</td><td>Non-3D</td><td>1.893</td><td>0.415</td><td>0.426</td></tr><tr><td>Schnet (Schutt et al., 2018)</td><td>3D</td><td>1.892</td><td>0.601</td><td>=</td></tr><tr><td>Cormorant (Anderson et al., 2019)</td><td>3D</td><td>1.429</td><td>0.541</td><td>0.532</td></tr><tr><td>3DCNN (Townshend et al., 2020)</td><td>3D</td><td>1.520</td><td>0.558</td><td>0.556</td></tr><tr><td>3DGCN (Townshend et al., 2020)</td><td>3D</td><td>1.963</td><td>0.581</td><td>0.647</td></tr><tr><td>Molformer</td><td>3D</td><td>1.417</td><td>0.623</td><td>0.651</td></tr></table>
+
+Table 5: Effects of each module on QM7, QM8 and PDBbind (RMSD). ME stands for the trainable motif embedding method.   
+
+<table><tr><td></td><td>CPE</td><td>AFPS</td><td>MSA</td><td>ME</td><td>QM7</td><td>QM8</td><td>PDBbind</td></tr><tr><td></td><td>=</td><td>=</td><td></td><td>-</td><td>63.2</td><td>0.0205</td><td>1.925</td></tr><tr><td></td><td></td><td>=</td><td></td><td>=</td><td>17.6</td><td>0.0104</td><td>1.489</td></tr><tr><td></td><td></td><td>√</td><td></td><td>1</td><td>17.0</td><td>0.0103</td><td>1.455</td></tr><tr><td></td><td></td><td></td><td></td><td>=</td><td>11.6</td><td>0.0098</td><td>1.423</td></tr><tr><td>１２３４５７６</td><td></td><td></td><td></td><td>F</td><td>15.2</td><td></td><td>1.443</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>13.7</td><td>0.0099</td><td>1.428</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>11.5</td><td>-</td><td>1.417</td></tr></table>
+
+# 5.2 HOW USEFUL IS THE TRAINABLE MOTIF-BASED EMBEDDINGS?
+
+How to determine motifs are critical and crucial to our proposed trainable motif-based embeddings. In organic chemistry, a functional group is a substituent or moiety in a molecule that causes the molecule’s characteristic chemical reactions. The same functional group will undergo the same or similar chemical reactions regardless of the rest of the molecule’s composition (Smith, 2020). Therefore, we define motifs on the basis of functional groups and explore the contribution of four different categories. Specifically, we consider four common functional groups, including groups that contain only carbon and hydrogen (Hydrocarbons), groups that contain halogen (Haloalkanes), groups that contain oxygen, and groups that contain nitrogen (see the left part in Figure 2). The ablations (see the right part in Figure 2) demonstrate that Molformer can gain improvements from all sorts of motifs, where Hydrocarbons and Haloalkanes are the most and the least effective kinds, respectively. This is in line with the fact that Hydrocarbons occur most frequently in organic molecules. Moreover, our model achieves the best performance when all categories of the motifs are integrated, implying a promising direction to discover more effective motifs.
+
+# 6 RELATED WORKS
+
+# 6.1 3D MOLECULAR REPRESENTATION
+
+Deep learning has been widely applied to predict molecular properties during past decades. Small molecules are usually represented as lower-dimensional representations such as 1D linear sequence, including amino acid sequences and SMILES (Weininger, 1988), or 2D chemical bond graphs. In spite of that, more evidence indicates that 3D space structures lead to better modelling and superior performance. 3D models becomes a popular way to capture these complex geometries in a variety of bio-molecular applications using CNNs (Anand-Achim et al., 2021; Jiménez et al., 2018) and GNNs (Cho & Choi, 2018). Nonetheless, aforementioned methods have hardly been extended to the self-attention mechanism that is proven to be good at grabbing contextual feature (Tang et al., 2018) and long-range dependencies (Vaswani et al., 2017).
+
+![](images/31bbc953eb710da1bc428151f6917eb692cbf586d7e6684c00a0fbd0e0035c7a.jpg)  
+Figure 2: The left is the four different categories of motifs that we apply in Molformer based on functional groups. The right is the ablation study of those groups in QM7 and BBBP.
+
+Attempts have been undertaken to address that issue throughout Transformers. Initially, molecules are in the form of SMILES to obtain corresponding representations (Honda et al., 2019; Pesciullesi et al., 2020; Morris et al., 2020; Rao et al., 2021) and conduct pretraining (Chithrananda et al., 2020). Some researchers combine the characteristics of GNN and Transformer to solve generative tasks (Ingraham et al., 2019) or fulfill equivariance (Fuchs et al., 2020).
+
+# 6.2 MOTIF-BASED METHOD
+
+Motifs have been proven to benefit many tasks from exploratory analysis to transfer learning (Henderson et al., 2012). Various algorithms have been proposed to exploit motifs for contrastive learning (Zhang et al., 2020a), self-supervised pretraining (Rong et al., 2020; Zhang et al., 2021), and generation (Jin et al., 2020). However, none of previous work tries to embody those informative motifs in their model architectures.
+
+# 7 CONCLUSION
+
+In this study, we present a universal neural architecture, Molformer, for 3D molecular representations. Our model extracts motifs with semantic meanings from each molecule based on functional groups and learn customized embeddings to facilitate property predictions. Moreover, it adopts a convolutional position encoding method to make a full use of spatial information and augments the self-attention mechanism with multiplicate scales to catch local features. Furthermore, a simple but efficient downsampling algorithm is introduced to better accumulate representations of an entire molecule. Our experiments show the superiority of our model on various scientific domains.
+
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+
+# A EXPERIMENTAL SETUP
+
+A.1 EXPERIMENTAL DETAILS
+
+Molformer Architecture. A standard Molformer has 6 multi-scale self-attention layers, and each layer has 3 scales and 8 heads. Normally, scales are set by $\tau = [ \frac { \rho } { 2 } , \rho , 2 \rho ]$ , where $\rho$ is the density of each corresponding dataset. The number of selected atoms $K$ and the weight ratio $\epsilon$ in AFPS is set as 4 and 0.1, respectively. We use ReLU as the activation function and a dropout rate of 0.1 for all layers. The input embedding size is 512 and the hidden size for FFN is 2048.
+
+For BBBP and ClinTox, we use Molformer with 2 multi-scale self-attention layers with 4 heads. The scales are 0.8, 1.6, and $3 . 0 \mathring \mathrm { A }$ . The dropout rate is 0.2 and 0.6 for BBBP and ClinTox, respectively. For BACE, we use a standard Molformer but with a dropout rate of 0.2.
+
+Training Details. We use Pytorch (Paszke et al., 2019) to implement Molformer and data parallelism in two GeForce RTX 3090. An Adam (Kingma & Ba, 2014) optimizer is used and a lambda scheduler is enforced to adjust it. We apply no weight decay there. Each model is trained with 300 epochs, except for PDBbind where we solely train the model for 30 epochs. For QM7 and QM8, we use a batch size of 64 and a learning rate of $\mathrm { 1 0 ^ { - 4 } }$ . For QM9, we use a batch size of 256 and a learning rate of $1 0 ^ { - 3 }$ . All hyper-parameters are tuned based on validation sets. For all molecular datasets, we impose no limitation on the input length and normalise the values of each regression task by mean and the standard deviation of the training set. We used grid search to tune the hyper-parameters of our model and baselines based on the validation dataset.
+
+Motif Generation. We adopt RDKit (Landrum, 2013) to search motifs. However, QM8 and QM9 do not provide SMILES representations but only 3D coordinates, thus we cannot pull out motifs from these datasets. As for PDBbind, we only extract motifs of small molecules and leave out motifs in proteins.
+
+# B ADDITIONAL EXPERIMENTAL RESULTS
+
+# B.1 CONFORMATION CLASSIFICATION
+
+Task and Data. In order to explore the influence of multiple conformations, we introduce a new task, conformation classification, to evaluate model’s capacity to differentiate molecules with various low-energy conformations. We use the recent GEOM-QM9 (Axelrod & Gomez-Bombarelli, 2020) experiments. More specifically, GEOM-QM9 is an extension to QM9 dataset. It contains multiple conformations for most molecules, while the original QM9 only contains one.
+
+We randomly draw 1000 different molecules from GEOM-QM9, each with 20 different conformations. Models are required to distinguish the molecular type given different conformations. We take a half of each molecular conformations as the training set and another half as the test split. Since it is a multi-class classification problem with 1000 classes, we compute the micro-average and macro-average ROC-AUC as well as the accuracy for numerical evaluations.
+
+Results. Molformer achieves a micro-average and macro-average ROC-AUC of 1.0 and 1.0, and an accuracy of 0.999. This indicates strong robustness of our model against different spatial conformations of molecules.
+
+# B.2 AFPS VS. FPS.
+
+To have a vivid understanding of the atom sampling algorithm, we conducted a case study on a random crystal (see Figure 3). Points selected by FPS are randomized and exclude vital atoms like the heavy metal Nickel (Ni). With the adoption of AFPS, sampled points include Ni and Nitrogen (N) besides that they keep remote distances from each other. Moreover, FPS integrates too many features of trivial atoms like Hydrogen $\mathrm { ( H ) }$ while misses out key atoms, which will significantly smooth the molecular representations and lead to poor predictions. This illustrative example firmly shows the effectiveness of our AFPS to offset disadvantages of the conventional FPS in 3D molecular representation.
+
+![](images/52db107ed4cdb54efeadb60117f2a9a2423059ba34e09a70a0640e07dd25a710.jpg)  
+Figure 3: Sampled points using FPS and AFPS. We do not show dummy nodes there.

parse/dev/6Dz7RiRiMFd/6Dz7RiRiMFd_content_list.json

@@ -0,0 +1,2021 @@
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+    {
+        "type": "text",
+        "text": "MOTIF-BASED ROTO-TRANSLATION INVARIANTTRANSFORMER FOR MOLECULAR PROPERTY PRE-DICTION IN 3D SPACE",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent studies use geometric deep learning to represent molecules and predict properties. However, they are computationally expensive in capturing long-range dependencies and ignore the non-uniformity of interatomic distances. More importantly, few of them consider injecting the biochemical structure knowledge such as functional groups into model architectures. To overcome such issues, we introduce Molformer, a variant of the Transformer for molecular representations that exploits both semantic motifs and 3D spatial information. Specifically, Molformer extracts motifs based on functional groups and learns customized embeddings to store the semantic meanings of those informative substructures. In order to fully employ 3D geometry, we adopt a convolutional position encoding to achieve roto-translation invariance, a multi-scale self-attention mechanism to capture local fine-grained patterns with increasing contextual scales, and an attentive farthest point sampling algorithm to attain the molecular representation. We validate Molformer across several domains in quantum chemistry, physiology, and biophysics. Our experiments show better or competitive performance in those datasets. Our work provides a promising way to amalgamate 3D geometric information and make better usage of informative substructures in representing molecules. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Spatial structures are among the most crucial factors to decide molecular properties and understand their principles of action in the physical world. For example, 3D structures of proteins provide valuable information for inferring biological interventions, such as structure-based drug development and targeted mutagenesis (Senior et al., 2020; Jumper et al., 2021; Baek et al., 2021). In chemistry, zeolites show obvious differences in separation properties caused by subtle changes in their 3D geometric compositions (Chai et al., 2020; Pfriem et al., 2021). Apart from that, in the pharmaceutical industry, the same compounds can have different 3D structures, resulting in different solubility (Zhang et al., 2017). To sum up, capturing 3D spatial structures is essential to accurately forecast molecular properties. Based on these facts, researchers have studied molecular representation learning techniques (Rao et al., 2019) to include 3D spatial information (Zhavoronkov et al., 2019). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The dominant 3D molecular models are Graph Neural Networks (GNNs) and 3D Convolutional Neural Networks (3DCNNs) (Derevyanko et al., 2018; Pagès et al., 2019; Townshend et al., 2019). GNNs create edges by using either chemical bonds or finding the neighbors of each node within a distance cutoff (Zhang et al., 2020b). They encode pairwise connectivity of atoms and require running multiple hops for an atom to reach to another. 3DCNNs encode translational and permutational symmetries, but need to stack deep layers to build direct connections between distant regions, incurring significant computational costs. In contrast, Transformers rely on the self-attention mechanism to capture long-term dependencies in parallel (Hernández & Amigó, 2021). Meanwhile, Equivariant Neural Networks (ENNs) (Thomas et al., 2018) have emerged as a new class of methods, where geometric transformations of their inputs lead to well-defined transformations of outputs. Some ENNs adopt Transformers as the backbone but fail to surmount the intrinsic drawbacks of this architecture, including its insensibility to local patterns among non-uniformly distancing atoms and its inefficiency to aggregate atom features. Some other Transformer-based methods have been proposed to fuse distance and graph neighbourhood information (Maziarka et al., 2020; 2021). However, they take no consideration of employing motifs, which are frequently-occurring substructures in molecules and can be leveraged to uncover global graph properties. ",
+        "bbox": [
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+    {
+        "type": "text",
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+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "In this work, we present the Molformer on the basis of all preceding analysis. For the sake of injecting chemical domain knowledge, we construct a motif-template vocabulary based on functional groups and adopt trainable motif embeddings to maintain the semantic meanings of those essential substructures. Then with both motifs and atoms as input, Molformer operates on a fully-connected graph with direct connections between remote regions (Velickovi ˇ c et al., 2017; Joshi, 2020), which ´ reduces computational burden of multi-hop GNNs and stacked 3DCNNs. However, this characteristic limits Molformer’s capacity in exploiting local structures and leads to poor generalization in unseen cases (Qi et al., 2017). Therefore, we propose a Multi-scale Self-Attention (MSA) module to recognize fine-grained patterns from neighborhoods. Moreover, we introduce a roto-translation invariant Convolutional Position Encoding (CPE) to depict position relationships among atoms and their adjacencies. After that, to retain a comprehensive representation of the entire molecule, we propose an Attentive Farthest Point Sampling (AFPS) module that selects important atoms with the assistance of the attention score map. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "To summarize, our contributions are as follows: ",
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+    {
+        "type": "text",
+        "text": "• To the best of our knowledge, we are the foremost to incorporate motifs with knowledge of functional groups into a Transformer architecture for 3D molecular representation learning. • We propose a novel MSA to extract local patterns, a roto-translation invariant CPE method to encode relative distance at a linear computational time cost, and a simple yet effective downsampling algorithm to gather molecular representations. • We show significant improvements on several benchmarks in three domains. Code and all datasets are available at https://github.com/smiles724/Molformer. ",
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+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Problem Definition. A molecule ${ \\cal S } = ( { \\cal E } , { \\cal P } )$ has $N$ atoms and $C$ atom classes, where ${ \\pmb { { \\cal E } } } =$ $\\{ e _ { 1 } , . . . , e _ { N } \\} \\in \\mathbb { R } ^ { N \\times C }$ contains the one-hot atom representations and $P = \\{ p _ { 1 } , . . . , p _ { N } \\} \\in \\mathbb { R } ^ { N \\times 3 }$ contains the 3D coordinates of each atom. Each one-hot $e _ { i }$ can be converted to a dense vector $\\pmb { x } _ { i } = e _ { i } \\pmb { W } ^ { E }$ , with $\\pmb { x } _ { i } \\in \\mathbb { R } ^ { d _ { m o d e l } }$ and $W ^ { E } \\in \\mathbb { R } ^ { C \\times d _ { m o d e l } }$ being the embedding matrix. The 3D coordinates of the atom $i$ is a three-dimensional vector $\\pmb { p } _ { i } = [ p _ { i } ^ { x } , \\bar { p } _ { i } ^ { y } , p _ { i } ^ { z } ]$ . A representation learning model $f$ acts on $_ { s }$ , obtaining its representation $r = f ( S )$ . Then $\\mathbfit { \\Delta } \\mathbf { r }$ is forwarded to a prediction model $g$ and attain the prediction of a biochemical property ${ \\hat { y } } = g ( \\pmb { r } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Self-attention Mechanism. The Transformer (Vaswani et al., 2017) has become very successful due to its core component, self-attention. Given a set of input features $\\{ \\pmb { x } _ { i } \\} _ { i = 1 , . . . , N }$ , the standard dot-product attention layer is as the following: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8b0114712972f33308aed4ece4b8969c68d51a45a75e17c9315577a9f01a8955.jpg",
+        "text": "$$\nq _ { i } = f _ { Q } ( { x } _ { i } ) , k _ { i } = f _ { K } ( { x } _ { i } ) , v _ { i } = f _ { V } ( { x } _ { i } ) , a _ { i j } = q _ { i } k _ { j } ^ { T } / \\sqrt { d _ { k } } , z _ { i } = \\sum _ { j = 1 } ^ { N } \\sigma ( a _ { i j } ) v _ { j }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "where $\\left\\{ f _ { Q } , f _ { K } , f _ { V } \\right\\}$ are embedding transformations, and $\\{ q _ { i } , k _ { i } , v _ { i } \\}$ are respectively the query, key, and value vectors with the same dimension $d _ { k }$ . $a _ { i j }$ is the attention that the token $i$ pays to the token $j$ . $\\sigma$ denotes the Softmax function and $z _ { i }$ is the output embedding of the token $i$ . This formula conforms to a non-local network (Wang et al., 2018), indicating its inability to capture fine-grained patterns in a local context. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Position Encoding. Self-attention is invariant to permutation of the input (Dufter et al., 2021), and position encoding ensures that the Transformer will reveal positional information. Position encoding methods can be either based on absolute positions or relative distances. The former takes the raw position information as input and is sensitive to spatial transformations. The latter manipulates the attention score by incorporating relative distances (Guo et al., 2020a; Pan et al., 2021):√ $a _ { i j } = { \\bf q } _ { i } { \\bf k } _ { j } ^ { T } / \\sqrt { d _ { k } } + f _ { \\mathrm { P E } } ( \\bar { p _ { i } } - p _ { j } )$ , where $\\bar { f } _ { \\mathrm { P E } } ( \\cdot )$ is the position encoding function and is translation invariant. The rotation invariance can be further accomplished by taking a L2-norm $| | p _ { i } - p _ { j } | | _ { 2 }$ (Chen et al., 2019b). ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/4aa7cb0818a54b520d22d500d8474338b6f0b8c1a3cb8942bfd66a2211b458c5.jpg",
+        "image_caption": [
+            "Figure 1: The overall architecture of our Molformer. FFN stands for a feed-forward network. Local features are shown in purple and orange; yellow corresponds to a global feature. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "3 MOLFORMER ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Molformer is based on the architecture of Transformer but adopts several significantly different and novel components (see Figure 1). First, a vocabulary of motif templates is constructed on the basis of functional groups and we extract all available motifs from each molecule. Then both atoms and motifs acquire their corresponding embeddings and are forwarded into $L$ feature learning blocks. Each block consists of a convolutional position encoding, a multi-scale self-attention, and a feedforward network. After that, an attentive subsampling method is utilized to adaptively aggregate the molecular presentation, which is later fed into a predictor to forecast properties in a broad range of downstream tasks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1 TRAINABLE MOTIF-BASED EMBEDDING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Motifs are frequently-occurring substructure patterns as well as the building blocks of complex molecular structures. They usually maintain semantic meanings and have great expressiveness of the biochemical characteristics of the whole molecule (Zhang et al., 2020a). In the chemical community, researchers have developed a set of standard criterion to recognize motifs with essential functionalities in molecules (Milo et al., 2002). Despite that, few of prior studies directly incorporate those informative motifs into their model architectures. To fill this gap, we define a series of momentous substructures using external domain knowledge, and introduce a trainable motif embeddings method to fully exploit them in our Molformer. ",
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+        "type": "text",
+        "text": "To begin with, all motifs are first extracted according to the motif vocabulary, which is built by functional groups. Practically, we rely on RDKit (Landrum, 2013) to draw them from the SMILES (Weininger, 1988) representation of each molecule. We assume $M$ motifs $\\{ m _ { 1 } , . . . , m _ { M } \\}$ are detected in the molecule $\\pmb { S }$ , and each motif $m _ { i }$ contains a certain number of at least two atoms. Then we regard each kind of motif as a new type of token and append them to the input. Therefore, the input for our Molformer becomes $\\{ \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { N } , \\pmb { x } _ { m _ { 1 } } , . . . , \\pmb { x } _ { m _ { M } } \\}$ , where $\\mathbf { \\boldsymbol { x } } _ { m _ { i } }$ is obtained through an learnable embedding matrix $W ^ { M } \\in \\mathbb { R } ^ { C ^ { \\prime } \\times d _ { m o d e l } }$ and $C ^ { \\prime }$ denotes the number of motif categories. As for the position of each motif, we adopt a weighted sum of the 3D coordinates of its component atoms as pmi = Pxi∈mi ( $\\begin{array} { r } { \\dot { p _ { m _ { i } } } = \\sum _ { x _ { i } \\in m _ { i } } \\ ( \\frac { w _ { i } } { \\sum _ { x _ { i } \\in m _ { i } } w _ { i } } ) \\cdot \\dot { p _ { i } } } \\end{array}$ wi ) · pi, where wi are the atomic weights. ",
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+    {
+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Our approach requires the model to automatically learn a customized embedding for each motif template through backpropagations, which follows a data-driven pattern. In some data-sufficient tasks, its greatest potential can be unlocked and those motif embeddings can be well trained. Nevertheless, in the case of few-shot learning or small datasets, each category of motif might only appear rare times. Those embeddings are not fully tuned and can be extremely biased and noisy, which will do little helps to the ultimate property prediction. ",
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+    {
+        "type": "text",
+        "text": "3.2 CONVOLUTIONAL POSITION ENCODING ",
+        "text_level": 1,
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+        "text": "To enable roto-translation invariance and take fully advantage of geometric information, instead of adding a term of $f _ { \\mathrm { P E } } ( \\pmb { p } _ { i } - \\pmb { p } _ { j } )$ , we propose a CPE that applies a convolutional operation to the interatomic distance $\\pmb { D } \\in \\mathbb { R } ^ { N \\times N }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/7bb7dfec8437fd666448ca33ec45be901a869e54310be7398cdd504565c731a1.jpg",
+        "text": "$$\n{ \\cal A } _ { \\mathrm { c o v } } = \\mathrm { C o n v } _ { 2 d } ( D ) \\odot { \\cal A } ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { A } = [ a _ { i , j } ] _ { i , j = 1 , \\cdots N } \\in \\mathbb { R } ^ { N \\times N }$ is the attention matrix, $\\mathrm { C o n v } _ { 2 d } ( \\cdot )$ denotes a 2D shallow convolutional network with a kernel size of $1 \\times 1$ , and $\\odot$ is the element-wise product. With multi-headed self-attention, ${ \\cal A } _ { \\mathrm { c o v } }$ is expanded in the sense that $A _ { \\mathrm { c o v } } \\in \\mathbb { R } ^ { H \\times N \\times N }$ , and $\\mathrm { C o n v } _ { 2 d } ( \\cdot )$ has $H$ output channels. The CPE method induces ${ \\mathrm { O } } ( N )$ convolution operations on each atom and can drastically reduce training time when the number of atoms is very large ( $\\mathrm { { W u } }$ et al., 2021). ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 MULTI-SCALE SELF-ATTENTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The self-attention mechanism in the Transformer is good at capturing global data patterns but ignores local context (Guo et al., 2020a). Exploiting local context has proven to be important for 3D spatial data such as 3D point clouds (Qi et al., 2017). Therefore, we impose a distance-based constraint in self-attention in order to extract multi-scaled patterns from both local and global contexts. ",
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+    {
+        "type": "text",
+        "text": "Guo et al. (2020b) propose to use integer-based distance to limit attention to local word neighbors, which cannot be used in molecules. This is because different types of molecules have different densities and molecules of the same type have different spatial regularity, which results in the nonuniformity of interatomic distances. Normally, small molecules have a mean interatomic distance of $1 { - } 2 \\textup { \\AA }$ (Angstrom, $1 0 ^ { - 1 0 } m _ { , }$ ), which is denser than large molecules like proteins with approximately $5 \\mathrm { ~ \\AA ~ }$ on average. To address that, we design a new multi-scale methodology to robustly capture details. Specifically, we mask atoms beyond a certain distance $\\tau _ { s }$ (a real number as opposed to an integer in Guo et al. (2020b)) at each scale $s$ . We denote $d _ { i j } = | | { \\pmb p } _ { i } - { \\pmb p } _ { j } | | _ { 2 }$ as the Euclidean distance between the $i$ -th and $j$ -th atom. The attention calculation is modified as: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/0590cf90c0ef900d2e98a82e3b4afb2058df645058a2d8715712ff29cda64a13.jpg",
+        "text": "$$\na _ { i j } ^ { \\tau _ { s } } = \\frac { q _ { i } \\pmb { k } _ { j } ^ { T } \\cdot \\mathbf { 1 } _ { \\{ d _ { i j } < \\tau _ { s } \\} } } { \\sqrt { d _ { k } } } , \\ : z _ { i } ^ { \\tau _ { s } } = \\sum _ { j = 1 } ^ { N } \\sigma ( a _ { i j } ^ { \\tau _ { s } } ) \\pmb { v } _ { j } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { 1 } _ { \\{ d _ { i j } < \\tau _ { s } \\} }$ is the indicator function. For small molecules, Equation 3 can be complementally combined with Equation 2. Then features extracted from $S$ different scales $\\{ \\tau _ { s } \\} _ { s = 1 , \\dots , S }$ as well as the informative global feature are concatenated together to form a multi-scale representation, denoted by $\\pmb { z } _ { i } ^ { \\prime } = \\pmb { z } _ { i } ^ { \\mathcal { T } _ { 1 } } \\oplus . . . \\oplus \\pmb { z } _ { i } ^ { \\tau s } \\oplus \\pmb { z } _ { i } ^ { g l o b a l } \\in \\mathbb { R } ^ { ( S + 1 ) d _ { k } }$ . After that, $ { \\boldsymbol { z } } _ { i } ^ { \\prime }$ is forwarded into a multi-layer perceptron to be compressed as $z _ { i } ^ { \\prime \\prime }$ with the original dimension $d _ { k }$ . ",
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+    {
+        "type": "text",
+        "text": "3.4 ATTENTIVE FARTHEST POINT SAMPLING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "After having the atom embeddings $\\{ z _ { i } ^ { \\prime \\prime } \\} _ { i = 1 , \\dots , N }$ , we study how to obtain the molecular representation $\\pmb { r }$ . For GNNs, several readout functions such as set2set (Vinyals et al., 2015) and GGNN (Gilmer et al., 2017) are invented. For Transformer architectures, one way is via a virtual atom. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/9f80bf0a6b5b6d88334ab47ae1cb9cb49518e9348f83928178364d8a89dc335a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Algorithm 1 Pseudocode of Attentive Farthest Point Sampling</td></tr><tr><td colspan=\"2\">Input: The attention score matrix A ∈ RN×N, a Euclidean distance matrix D ∈ RN ×N. Output: K sampled points.</td></tr><tr><td>1: A←∑iAij ∈RN</td><td>&gt; sum up the attention matrix along rows</td></tr><tr><td>2:D←D∈RNxN</td><td>&gt; normalize the distance matrix</td></tr><tr><td>3:P= {𝑥#},M={1,2,..,N} 4:while length(P)&lt;k do</td><td></td></tr><tr><td>5: Xnew = argmax (min Dij + eAi)</td><td>&gt; pick up the atom that maximize the objective</td></tr><tr><td>iEM j∈p 6: P.append(xnew), M.remove(x new)</td><td></td></tr><tr><td>7: return P</td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Though as Ying et al. (2021) state, it significantly improves the performance of existing models in the leaderboard of Open Graph Benchmark (Hu et al., 2020), this way concentrates more on close adjacent atoms and less on distant ones, and can lead to inadvertent over-smoothing of information propagation (Ishiguro et al., 2019). Besides, it is difficult to locate a virtual node in 3D space and build connections to existing atoms. The other way selects a subset of atoms via a downsampling algorithm named Farthest Point Search (FPS), but it ignores atomic differences and has sensitivity to outlier points (Pan et al., 2021) as well as uncontrollable randomness. To address these issues, we propose a new algorithm named AFPS. It aims to sample atoms by not merely spatial distances, but also their significance in terms of attention scores. ",
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+    {
+        "type": "text",
+        "text": "Specifically, we choose the virtual atom $x _ { \\# }$ as the starting point and initialize two lists $\\mathcal { P } = \\{ x _ { \\# } \\}$ and $\\mathcal { M } = \\mathrm { \\bar { \\{ } } 1  , . . . , N \\}$ to store remaining candidate points. Then the process begins with the attention score matrix $\\pmb { A } \\in \\mathbb { R } ^ { N \\times N }$ and the interatomic distance matrix $\\pmb { { \\cal D } } \\in \\mathbb { R } ^ { N \\times N }$ . It can be easily proved that each row of $\\pmb { A }$ sums up to 1 after the Sof tmax operation along columns, i.e. $\\begin{array} { r } { \\sum _ { j } { \\cal A } _ { i j } \\dot { = } 1 } \\end{array}$ for $\\forall i \\in [ N ]$ . In order to obtain the importance of each atom in the self-attention computation, we accumulate $\\pmb { A }$ along rows and get $\\begin{array} { r } { \\tilde { \\pmb { A } } = \\sum _ { i } \\pmb { A } _ { i j } \\in \\mathbb { R } ^ { N } } \\end{array}$ . Besides, we adopt the min-max normalization to rescale the distance matrix $_ D$ into values between 0 and 1, and obtain $\\begin{array} { r } { \\tilde { D } = \\frac { D - \\operatorname* { m i n } D } { \\operatorname* { m a x } D - \\operatorname* { m i n } D } } \\end{array}$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "After the above preprocess, we repeatedly move a point $x _ { n e w }$ from $\\mathcal { M }$ to $\\mathcal { P }$ , which ensures that $x _ { n e w }$ is as far from $\\mathcal { P }$ as possible by maximizing $\\tilde { D } _ { i j }$ and also plays a crucial role in attention computation by maximizing ${ \\tilde { A } } _ { i }$ . Mathematically, the AFPS aims to achieve the following objective: ",
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+    {
+        "type": "equation",
+        "img_path": "images/5bebab609212893cd350637d80df50a0d61eea27d05a7d8861e2b42855eeb86f.jpg",
+        "text": "$$\n\\operatorname* { m a x } \\sum _ { i \\in \\mathcal { M } } ( \\operatorname* { m i n } _ { j \\in \\mathcal { P } \\setminus \\{ i \\} } \\tilde { D } _ { i j } + \\epsilon \\tilde { A } _ { i } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\epsilon$ is a hyperparameter to balance those two different goals. This process is repeated until $\\mathcal { P }$ has reached $K$ points. Algorithm 1 provides a greedy approximation solution to solve this AFPS optimization objective for sake of computational efficiency. ",
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+        "type": "text",
+        "text": "After that, sampled features $\\{ z _ { i } ^ { \\prime \\prime } \\} _ { i \\in P }$ are gathered by a Global Average Pooling layer (Lin et al., 2013) to attain the molecular representation $\\pmb { r } \\in \\mathbb { R } ^ { d _ { k } }$ . ",
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+    {
+        "type": "text",
+        "text": "Remarkably, our proposed AFPS has considerable difference and superiority over a body of previous hierarchical learning approaches (Eismann et al., 2020; 2021). Their subsampling operations are mainly designed for protein complexities, which have more uniform structures than small molecules. To be specific, they hierarchically use alpha carbons as the intermediate set of points and aggregate information at the level of those carbons for the entire complex. However, the structures of small molecules have no such a stable paradigm, and we provide a universal methodology to adaptively subsample atoms without any prior assumptions on the atom arrangement. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We conduct extensive experiments on both small and large molecules (proteins) with various targets, including quantum chemistry, physiology, and biophysics. Table 1 summarises information of benchmark datasets, such as the number of tasks and task types, the number of molecules and atom classes, the minimum and maximum number of atoms, and the density (mean interatomic distances) of all molecules. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "table",
+        "img_path": "images/27df12b6017c772c5c378b7d49a1e8de70b0e12ee63e398fefc008408141e1f4.jpg",
+        "table_caption": [
+            "Table 1: Key statistics of datasets from three different categories. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Category</td><td>Dataset</td><td>Tasks</td><td>Task Type</td><td>Molecules</td><td>Atom Class</td><td>Min. Atoms</td><td>Max. Atoms</td><td>Density （A</td><td>Metric</td></tr><tr><td rowspan=\"3\">Quantum Chemistry</td><td>QM7</td><td>1</td><td>regression</td><td>7,160</td><td>5</td><td>4</td><td>23</td><td>2.91</td><td>MAE</td></tr><tr><td>QM8</td><td>12</td><td>regression</td><td>21,786</td><td>5</td><td>3</td><td>26</td><td>1.54</td><td>MAE</td></tr><tr><td>QM9</td><td>12</td><td>regression</td><td>133,885</td><td>5</td><td>3</td><td>28</td><td>1.61</td><td>MAE</td></tr><tr><td rowspan=\"2\">Physiology</td><td>BBBP</td><td>1</td><td>classification</td><td>2.039</td><td>13</td><td>2</td><td>132</td><td>2.64</td><td>ROC-AUC</td></tr><tr><td>ClinTox</td><td>2</td><td>classification</td><td>1,478</td><td>27</td><td>1</td><td>136</td><td>2.83</td><td>ROC-AUC</td></tr><tr><td rowspan=\"2\">Biophysics</td><td>PDBind1</td><td>1</td><td>regression</td><td>11,908</td><td>23</td><td>115</td><td>1,085</td><td>5.89</td><td>RMSE</td></tr><tr><td>BACE</td><td>1</td><td>classification</td><td>1,513</td><td>8</td><td>10</td><td>73</td><td>3.24</td><td>ROC-AUC</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Datasets. We test Molformer on a series of small molecule datasets, containing QM7 (Blum & Reymond, 2009), QM8 (Ramakrishnan et al., 2015), QM9 (Ramakrishnan et al., 2014), BBBP (Martins et al., 2012), ClinTox (Gayvert et al., 2016), and BACE (Subramanian et al., 2016) 2. QM7 is a subset of GDB-13 and composed of 7K molecules with up to 5 heavy atom types. QM8 and QM9 are subsets of GDB-17 with $2 2 \\mathrm { k }$ molecules and 133K molecule respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Additionally, we also inspect Molformer’s ability of learning mutual relations between proteins and molecules on the PDBbind dataset (Wang et al., 2005). We follow Townshend et al. (2020) and split protein-ligand complexes by protein sequence identity at $30 \\%$ . As for the target, we predict $p S = - \\log ( S )$ , where $S$ is the binding affinity in Molar unit. In addition, we only use the pocket of each protein and put pocket-ligand pairs together as the input. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For QM9, we use the exact train/validation/test split as Townshend et al. (2020). For PDBbind, $90 \\%$ of the data is used for training and the rest is divided equally between validation and test like Chen et al. (2019c). For others, we adopt the scaffold splitting method with a ratio of 8:1:1 for train/validation/test as Rong et al. (2020). More implementing details can be found in Appendix A.1 ",
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+    {
+        "type": "text",
+        "text": "Baselines For small molecules, we compare our approach with a number of state-of-the-art baselines. TF_Robust (Ramsundar et al., 2015) takes molecular fingerprints as the input. GraphConv (Kipf & Welling, 2016), Weave (Kearnes et al., 2016), MPNN (Gilmer et al., 2017), Schnet (Schütt et al., 2018), MEGNet (Chen et al., 2019c), DMPNN (Yang et al., 2019), MGCN (Lu et al., 2019), AttentiveFP (Xiong et al., 2019), DimeNet $^ { + + }$ (Klicpera et al., 2020), SphereNet (Liu et al., 2021), and SpinConv (Shuaibi et al., 2021) are all graph convolutional models. Graph Transformer (Chen et al., 2019a), MAT (Maziarka et al., 2020), R-MAT (Maziarka et al., 2021), SE(3)- Transformer (Fuchs et al., 2020), and LieTransformer (Hutchinson et al., 2021) are Transformerbased models. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For PDBbind, we choose six baselines. DeepDTA (Öztürk et al., 2018) and DeepAffinity (Karimi et al., 2019) take in pairs of ligand and protein SMILES as input. Cormorant (Anderson et al., 2019) is an ENN that represents each atom by its absolute 3D coordinates. Schnet, 3DCNN and 3DGCN (Townshend et al., 2020) are 3D molecular representation methods. ",
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+    {
+        "type": "text",
+        "text": "4.2 RESULTS ON DOWNSTREAM TASKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Molecules. Table 2 and Table 3 document the overall results of Molformer and baselines on small molecules datasets, where best performance is marked bold and the second best is underlined for clear comparison. It can be discovered that Molformer achieves the lowest MAE of 11.6 on QM7 and 0.009 on QM8, beating several strong baselines including DMPNN and Graph Transformer. While not all state-of-the-art on QM9, Molformer offers competitive performance in 5 property regression tasks, which do not require thermochemical energy subtractions. Particularly, we outperforms all Transformer-based ENNs, including SE(3)-Transformer and LieTransformer. In classification problems, we surpass all non-pretrained methods and are only inferior to the pretrained GROVE. This accords to the fact that datasets with fewer samples can gain large improvements through the self-supervised pretraining (Rong et al., 2020). ",
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+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/845ff915f6cd7686ed17926601dbbd0e3d7dfa78809328dbd91718b03602283d.jpg",
+        "table_caption": [
+            "Table 2: The performance comparison. For regression tasks including QM7 and QM8, lower is better. For classification tasks including BBBP, ClinTox, and Bace, higher is better. The methods in purple are pretrained methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>QM7</td><td>QM8</td><td>BBBP</td><td>ClinTox</td><td>BACE</td></tr><tr><td>TF-Robust (Ramsundar et al., 2015)</td><td>120.6</td><td>0.024</td><td>0.860</td><td>0.765</td><td>0.824</td></tr><tr><td>GraphConv (Kipf &amp; Welling,2016)</td><td>118.9</td><td>0.021</td><td>0.877</td><td>0.845</td><td>0.854</td></tr><tr><td>Weave (Kearnes et al.,2016)</td><td>94.7</td><td>0.022</td><td>0.837</td><td>0.823</td><td>0.791</td></tr><tr><td>MPNN (Gilmer et al., 2017)</td><td>113.0</td><td>0.015</td><td>0.913</td><td>0.879</td><td>0.815</td></tr><tr><td>Schnet (Schuitt et al., 2018)</td><td>74.2</td><td>0.020</td><td>0.847</td><td>0.717</td><td>0.750</td></tr><tr><td>DMPNN (Yang et al.,2019)</td><td>105.8</td><td>0.014</td><td>0.919</td><td>0.897</td><td>0.852</td></tr><tr><td>MGCN (Lu et al., 2019)</td><td>77.6</td><td>0.022</td><td>0.850</td><td>0.634</td><td>0.734</td></tr><tr><td>Attentive FP (Xiong et al., 2019)</td><td>126.7</td><td>0.028</td><td>0.908</td><td>0.933</td><td>0.863</td></tr><tr><td>Graph Transformer (Chen et al.,2019a)</td><td>47.8</td><td>0.010</td><td>0.913</td><td>-</td><td>0.880</td></tr><tr><td>MAT (Maziarka et al.,2020)</td><td>102.8</td><td>1</td><td>0.728</td><td>-</td><td>0.846</td></tr><tr><td>R-MAT (Maziarka et al., 2021)</td><td>68.6</td><td>=</td><td>0.746</td><td>=</td><td>0.871</td></tr><tr><td>GROVElarge (Rong et al., 2020)</td><td>89.4</td><td>0.017</td><td>0.911</td><td>0.884</td><td>0.858</td></tr><tr><td> GROVElarge (Rong et al., 2020)</td><td>72.6</td><td>0.012</td><td>0.940</td><td>0.944</td><td>0.894</td></tr><tr><td>Molformer</td><td>11.5</td><td>0.009</td><td>0.926</td><td>0.941</td><td>0.884</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Comparison of MAE on QM9. The methods in orange are Transformer-based methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Target (Unit)</td><td>€HOMO (eV)</td><td>∈LUMo (eV)</td><td>△e(eV)</td><td>μ(D)</td><td>a (bohr3)</td></tr><tr><td>MPNN (Gilmer et al., 2017)</td><td>.043</td><td>.037</td><td>.069</td><td>.030</td><td>.092</td></tr><tr><td>Schnet (Schuitt et al., 2018)</td><td>.041</td><td>.034</td><td>.063</td><td>.033</td><td>.235</td></tr><tr><td>MEGNet ful (Chen et al., 2019c)</td><td>.038</td><td>.031</td><td>.061</td><td>.040</td><td>.083</td></tr><tr><td>DimeNet++ (Klicpera et al.,2020)</td><td>.024</td><td>.019</td><td>.032</td><td>.029</td><td>.043</td></tr><tr><td>SphereNet (Liu et al., 2021)</td><td>.024</td><td>.019</td><td>.032</td><td>.026</td><td>.047</td></tr><tr><td>SpinConv (Shuaibi et al.,2021)</td><td>.026</td><td>.022</td><td>.047</td><td>.027</td><td>.058</td></tr><tr><td> SE(3)-Transformer (Fuchs et al., 2020)</td><td>.035</td><td>.033</td><td>.053</td><td>.051</td><td>.142</td></tr><tr><td>LieTransformer-SE(3) (Hutchinson et al., 2021)</td><td>.033</td><td>.029</td><td>.052</td><td>.061</td><td>.104</td></tr><tr><td>Molformer</td><td>.021</td><td>.026</td><td>.039</td><td>.045</td><td>.086</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Protein. Table 4 reports the Root-Mean-Squared Deviation (RMSD), the Pearson correlation $( R _ { p } )$ , and the Spearman correlation $( R _ { s } )$ on PDBbind. Molformer achieves the lowest RMSD among all baselines and the best Pearson and Spearman correlations. As Wu et al. (2018) claim, appropriate featurizations which contains pertinent information is significant for PDBbind. However, an important observation in our work is that deep learning approaches with the full exploitation of 3D geometric information can perform better than conventional methods like DeepDTA and DeepAffinity, which use a set of physicochemical descriptors but ignore 3D structures. ",
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+    {
+        "type": "text",
+        "text": "5 ABLATION STUDY AND DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 WHAT ARE THE EFFECTS OF EACH COMPONENT",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We investigate the effectiveness of different modules of our Molformer in Table 5. It can be observed that CPE substantially boosts model’s performance compared with the naive method that immediately adds 3D coordinates as the atom input feature. In addition, AFPS is found to produce better predictions than the control group, which utilizes the virtual node as the molecular representation. Moreover, MSA significantly reduces RMSD from 17.6 to 11.6 on QM7, but its improvements in QM8 are much smaller. This phenomenon indicates that MSA is an appropriate way to alleviate the problem of inadequate training in small datasets. It endows Molformer with capability to extract local features by regulating the scope of self-attention. However, as the data size gets larger and larger, Molformer does not require the assistance of MSA to abstract local patterns, since the parameters of CPE is properly trained. What’s more, the trainable motif-level embedding leads to a MAE decrease of 2.1 in QM7 and a RMSD drop of 0.011 in PDBbind, indicating its effectiveness in both small molecules and proteins. ",
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+        "type": "table",
+        "img_path": "images/59d8798450786392002fd059e9aee5edd6342bcb69edc150d0797e7066bf5317.jpg",
+        "table_caption": [
+            "Table 4: Comparison of RMSD, $R _ { p }$ , and $R _ { s }$ on PDBbind. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Geometry</td><td>RMSD</td><td>Rp</td><td>Rs</td></tr><tr><td>DeepDTA (Ozturk et al., 2018)</td><td>Non-3D</td><td>1.565</td><td>0.573</td><td>0.574</td></tr><tr><td>DeepAffinity (Karimi et al.,2019)</td><td>Non-3D</td><td>1.893</td><td>0.415</td><td>0.426</td></tr><tr><td>Schnet (Schutt et al., 2018)</td><td>3D</td><td>1.892</td><td>0.601</td><td>=</td></tr><tr><td>Cormorant (Anderson et al., 2019)</td><td>3D</td><td>1.429</td><td>0.541</td><td>0.532</td></tr><tr><td>3DCNN (Townshend et al., 2020)</td><td>3D</td><td>1.520</td><td>0.558</td><td>0.556</td></tr><tr><td>3DGCN (Townshend et al., 2020)</td><td>3D</td><td>1.963</td><td>0.581</td><td>0.647</td></tr><tr><td>Molformer</td><td>3D</td><td>1.417</td><td>0.623</td><td>0.651</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/026011deb3a0bcc3ebb2ccbd515b53b2f6f03672c6738cb8b228dad02211729e.jpg",
+        "table_caption": [
+            "Table 5: Effects of each module on QM7, QM8 and PDBbind (RMSD). ME stands for the trainable motif embedding method. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CPE</td><td>AFPS</td><td>MSA</td><td>ME</td><td>QM7</td><td>QM8</td><td>PDBbind</td></tr><tr><td></td><td>=</td><td>=</td><td></td><td>-</td><td>63.2</td><td>0.0205</td><td>1.925</td></tr><tr><td></td><td></td><td>=</td><td></td><td>=</td><td>17.6</td><td>0.0104</td><td>1.489</td></tr><tr><td></td><td></td><td>√</td><td></td><td>1</td><td>17.0</td><td>0.0103</td><td>1.455</td></tr><tr><td></td><td></td><td></td><td></td><td>=</td><td>11.6</td><td>0.0098</td><td>1.423</td></tr><tr><td>１２３４５７６</td><td></td><td></td><td></td><td>F</td><td>15.2</td><td></td><td>1.443</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>13.7</td><td>0.0099</td><td>1.428</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>11.5</td><td>-</td><td>1.417</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.2 HOW USEFUL IS THE TRAINABLE MOTIF-BASED EMBEDDINGS? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "How to determine motifs are critical and crucial to our proposed trainable motif-based embeddings. In organic chemistry, a functional group is a substituent or moiety in a molecule that causes the molecule’s characteristic chemical reactions. The same functional group will undergo the same or similar chemical reactions regardless of the rest of the molecule’s composition (Smith, 2020). Therefore, we define motifs on the basis of functional groups and explore the contribution of four different categories. Specifically, we consider four common functional groups, including groups that contain only carbon and hydrogen (Hydrocarbons), groups that contain halogen (Haloalkanes), groups that contain oxygen, and groups that contain nitrogen (see the left part in Figure 2). The ablations (see the right part in Figure 2) demonstrate that Molformer can gain improvements from all sorts of motifs, where Hydrocarbons and Haloalkanes are the most and the least effective kinds, respectively. This is in line with the fact that Hydrocarbons occur most frequently in organic molecules. Moreover, our model achieves the best performance when all categories of the motifs are integrated, implying a promising direction to discover more effective motifs. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6.1 3D MOLECULAR REPRESENTATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Deep learning has been widely applied to predict molecular properties during past decades. Small molecules are usually represented as lower-dimensional representations such as 1D linear sequence, including amino acid sequences and SMILES (Weininger, 1988), or 2D chemical bond graphs. In spite of that, more evidence indicates that 3D space structures lead to better modelling and superior performance. 3D models becomes a popular way to capture these complex geometries in a variety of bio-molecular applications using CNNs (Anand-Achim et al., 2021; Jiménez et al., 2018) and GNNs (Cho & Choi, 2018). Nonetheless, aforementioned methods have hardly been extended to the self-attention mechanism that is proven to be good at grabbing contextual feature (Tang et al., 2018) and long-range dependencies (Vaswani et al., 2017). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/31bbc953eb710da1bc428151f6917eb692cbf586d7e6684c00a0fbd0e0035c7a.jpg",
+        "image_caption": [
+            "Figure 2: The left is the four different categories of motifs that we apply in Molformer based on functional groups. The right is the ablation study of those groups in QM7 and BBBP. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Attempts have been undertaken to address that issue throughout Transformers. Initially, molecules are in the form of SMILES to obtain corresponding representations (Honda et al., 2019; Pesciullesi et al., 2020; Morris et al., 2020; Rao et al., 2021) and conduct pretraining (Chithrananda et al., 2020). Some researchers combine the characteristics of GNN and Transformer to solve generative tasks (Ingraham et al., 2019) or fulfill equivariance (Fuchs et al., 2020). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6.2 MOTIF-BASED METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Motifs have been proven to benefit many tasks from exploratory analysis to transfer learning (Henderson et al., 2012). Various algorithms have been proposed to exploit motifs for contrastive learning (Zhang et al., 2020a), self-supervised pretraining (Rong et al., 2020; Zhang et al., 2021), and generation (Jin et al., 2020). However, none of previous work tries to embody those informative motifs in their model architectures. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this study, we present a universal neural architecture, Molformer, for 3D molecular representations. Our model extracts motifs with semantic meanings from each molecule based on functional groups and learn customized embeddings to facilitate property predictions. Moreover, it adopts a convolutional position encoding method to make a full use of spatial information and augments the self-attention mechanism with multiplicate scales to catch local features. Furthermore, a simple but efficient downsampling algorithm is introduced to better accumulate representations of an entire molecule. Our experiments show the superiority of our model on various scientific domains. ",
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+    },
+    {
+        "type": "text",
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+        "text": "Chengxuan Ying, Tianle Cai, Shengjie Luo, Shuxin Zheng, Guolin Ke, Di He, Yanming Shen, and Tie-Yan Liu. Do transformers really perform bad for graph representation? arXiv preprint arXiv:2106.05234, 2021. ",
+        "bbox": [
+            171,
+            491,
+            825,
+            535
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Haitao Zhang, Gye Won Han, Alexander Batyuk, Andrii Ishchenko, Kate L White, Nilkanth Patel, Anastasiia Sadybekov, Beata Zamlynny, Michael T Rudd, Kaspar Hollenstein, et al. Structural basis for selectivity and diversity in angiotensin ii receptors. Nature, 544(7650):327–332, 2017. ",
+        "bbox": [
+            173,
+            542,
+            825,
+            587
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Shichang Zhang, Ziniu Hu, Arjun Subramonian, and Yizhou Sun. Motif-driven contrastive learning of graph representations. arXiv preprint arXiv:2012.12533, 2020a. ",
+        "bbox": [
+            173,
+            594,
+            823,
+            625
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Shuo Zhang, Yang Liu, and Lei Xie. Molecular mechanics-driven graph neural network with multiplex graph for molecular structures. arXiv preprint arXiv:2011.07457, 2020b. ",
+        "bbox": [
+            174,
+            632,
+            820,
+            662
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Zaixi Zhang, Qi Liu, Hao Wang, Chengqiang Lu, and Chee-Kong Lee. Motif-based graph selfsupervised learning for molecular property prediction. arXiv preprint arXiv:2110.00987, 2021. ",
+        "bbox": [
+            173,
+            670,
+            821,
+            700
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Alex Zhavoronkov, Yan A Ivanenkov, Alex Aliper, Mark S Veselov, Vladimir A Aladinskiy, Anastasiya V Aladinskaya, Victor A Terentiev, Daniil A Polykovskiy, Maksim D Kuznetsov, Arip Asadulaev, et al. Deep learning enables rapid identification of potent ddr1 kinase inhibitors. Nature biotechnology, 37(9):1038–1040, 2019. ",
+        "bbox": [
+            174,
+            708,
+            825,
+            765
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            401,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.1 EXPERIMENTAL DETAILS ",
+        "bbox": [
+            176,
+            132,
+            395,
+            147
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Molformer Architecture. A standard Molformer has 6 multi-scale self-attention layers, and each layer has 3 scales and 8 heads. Normally, scales are set by $\\tau = [ \\frac { \\rho } { 2 } , \\rho , 2 \\rho ]$ , where $\\rho$ is the density of each corresponding dataset. The number of selected atoms $K$ and the weight ratio $\\epsilon$ in AFPS is set as 4 and 0.1, respectively. We use ReLU as the activation function and a dropout rate of 0.1 for all layers. The input embedding size is 512 and the hidden size for FFN is 2048. ",
+        "bbox": [
+            174,
+            159,
+            825,
+            228
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For BBBP and ClinTox, we use Molformer with 2 multi-scale self-attention layers with 4 heads. The scales are 0.8, 1.6, and $3 . 0 \\mathring \\mathrm { A }$ . The dropout rate is 0.2 and 0.6 for BBBP and ClinTox, respectively. For BACE, we use a standard Molformer but with a dropout rate of 0.2. ",
+        "bbox": [
+            174,
+            234,
+            825,
+            279
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Training Details. We use Pytorch (Paszke et al., 2019) to implement Molformer and data parallelism in two GeForce RTX 3090. An Adam (Kingma & Ba, 2014) optimizer is used and a lambda scheduler is enforced to adjust it. We apply no weight decay there. Each model is trained with 300 epochs, except for PDBbind where we solely train the model for 30 epochs. For QM7 and QM8, we use a batch size of 64 and a learning rate of $\\mathrm { 1 0 ^ { - 4 } }$ . For QM9, we use a batch size of 256 and a learning rate of $1 0 ^ { - 3 }$ . All hyper-parameters are tuned based on validation sets. For all molecular datasets, we impose no limitation on the input length and normalise the values of each regression task by mean and the standard deviation of the training set. We used grid search to tune the hyper-parameters of our model and baselines based on the validation dataset. ",
+        "bbox": [
+            174,
+            294,
+            825,
+            419
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Motif Generation. We adopt RDKit (Landrum, 2013) to search motifs. However, QM8 and QM9 do not provide SMILES representations but only 3D coordinates, thus we cannot pull out motifs from these datasets. As for PDBbind, we only extract motifs of small molecules and leave out motifs in proteins. ",
+        "bbox": [
+            174,
+            434,
+            823,
+            489
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B ADDITIONAL EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            510,
+            537,
+            525
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.1 CONFORMATION CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            540,
+            452,
+            554
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Task and Data. In order to explore the influence of multiple conformations, we introduce a new task, conformation classification, to evaluate model’s capacity to differentiate molecules with various low-energy conformations. We use the recent GEOM-QM9 (Axelrod & Gomez-Bombarelli, 2020) experiments. More specifically, GEOM-QM9 is an extension to QM9 dataset. It contains multiple conformations for most molecules, while the original QM9 only contains one. ",
+        "bbox": [
+            174,
+            565,
+            823,
+            636
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We randomly draw 1000 different molecules from GEOM-QM9, each with 20 different conformations. Models are required to distinguish the molecular type given different conformations. We take a half of each molecular conformations as the training set and another half as the test split. Since it is a multi-class classification problem with 1000 classes, we compute the micro-average and macro-average ROC-AUC as well as the accuracy for numerical evaluations. ",
+        "bbox": [
+            174,
+            643,
+            823,
+            713
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Results. Molformer achieves a micro-average and macro-average ROC-AUC of 1.0 and 1.0, and an accuracy of 0.999. This indicates strong robustness of our model against different spatial conformations of molecules. ",
+        "bbox": [
+            176,
+            728,
+            821,
+            770
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.2 AFPS VS. FPS. ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            786,
+            325,
+            800
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To have a vivid understanding of the atom sampling algorithm, we conducted a case study on a random crystal (see Figure 3). Points selected by FPS are randomized and exclude vital atoms like the heavy metal Nickel (Ni). With the adoption of AFPS, sampled points include Ni and Nitrogen (N) besides that they keep remote distances from each other. Moreover, FPS integrates too many features of trivial atoms like Hydrogen $\\mathrm { ( H ) }$ while misses out key atoms, which will significantly smooth the molecular representations and lead to poor predictions. This illustrative example firmly shows the effectiveness of our AFPS to offset disadvantages of the conventional FPS in 3D molecular representation. ",
+        "bbox": [
+            174,
+            811,
+            823,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/52db107ed4cdb54efeadb60117f2a9a2423059ba34e09a70a0640e07dd25a710.jpg",
+        "image_caption": [
+            "Figure 3: Sampled points using FPS and AFPS. We do not show dummy nodes there. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
+            392,
+            758,
+            599
+        ],
+        "page_idx": 15
+    }
+]

parse/dev/9Hrka5PA7LW/9Hrka5PA7LW.md

@@ -0,0 +1,317 @@
+# REPRESENTATIONAL CONTINUITY FOR UNSUPERVISED CONTINUAL LEARNING
+
+Divyam Madaan1∗ Jaehong Yoon2,3 † Yuanchun $\mathbf { L i } ^ { 5 , 6 }$ Yunxin Liu5,6 Sung Ju Hwang2,4 New York University1 KAIST2 Microsoft Research3 AITRICS4 Institute for AI Industry Research (AIR)5 Tsinghua University6 divyam.madaan@nyu.edu, {jaehong.yoon,sjhwang82}@kaist.ac.kr liyuanchun@air.tsinghua.edu.cn, liuyunxin@air.tsinghua.edu.cn
+
+# ABSTRACT
+
+Continual learning (CL) aims to learn a sequence of tasks without forgetting the previously acquired knowledge. However, recent CL advances are restricted to supervised continual learning (SCL) scenarios. Consequently, they are not scalable to real-world applications where the data distribution is often biased and unannotated. In this work, we focus on unsupervised continual learning (UCL), where we learn the feature representations on an unlabelled sequence of tasks and show that reliance on annotated data is not necessary for continual learning. We conduct a systematic study analyzing the learned feature representations and show that unsupervised visual representations are surprisingly more robust to catastrophic forgetting, consistently achieve better performance, and generalize better to out-ofdistribution tasks than SCL. Furthermore, we find that UCL achieves a smoother loss landscape through qualitative analysis of the learned representations and learns meaningful feature representations. Additionally, we propose Lifelong Unsupervised Mixup (LUMP), a simple yet effective technique that interpolates between the current task and previous tasks’ instances to alleviate catastrophic forgetting for unsupervised representations. We release our code online.
+
+# 1 INTRODUCTION
+
+Recently continual learning (Thrun, 1995) has gained a lot of attention in the deep learning community due to its ability to continually learn on a sequence of non-stationary tasks (Kumar & Daume III, 2012; Li & Hoiem, 2016) and close proximity to the human learning process (Flesch et al., 2018). However, the inability of the learner to prevent forgetting of the knowledge learnt from the previous tasks has been a long-standing problem (McCloskey & Cohen, 1989; Goodfellow et al., 2013). To address this problem, a large body of methods (Rusu et al., 2016; Zenke et al., 2017; Yoon et al., 2018; Li et al., 2019; Aljundi et al., 2019; Buzzega et al., 2020) have been proposed; however, all these methods focus on the supervised learning paradigm, but obtaining high-quality labels is expensive and often impractical in real-world scenarios. In contrast, CL for unsupervised representation learning has received limited attention in the community. Although Rao et al. (2019) instantiated a continual unsupervised representation learning framework (CURL), it is not scalable for high-resolution tasks, as it is composed of MLP encoders/decoders and a simple MoG generative replay. This is evident in their limited empirical evaluation using digit-based gray-scale datasets.
+
+Meanwhile, a set of directions have shown huge potential to tackle the representation learning problem without labels (He et al., 2020; Chen et al., 2020a; Grill et al., 2020; Chen et al., 2020b; Chen & He, 2021; Zbontar et al., 2021) by aligning contrastive pairs of training instances or maximizing the similarity between two augmented views of each image. However, a common assumption for existing methods is the availability of a large amount of unbiased and unlabelled datasets to learn the feature representations. We argue that this assumption is not realistic for most of the real-time applications, including self-driving cars (Bojarski et al., 2016), medical applications (Kelly et al., 2019) and conversational agents (Li et al., 2020). The collected datasets are often limited in size during the initial training phase (Finn et al., 2017), and datasets/tasks change continuously with time.
+
+![](images/b5965a3ad31159cd64ec531fe9195a659aa834a208272019739bfb0a7aa74ef6.jpg)  
+Figure 1: Illustration of supervised and unsupervised continual learning. The objective of SCL is to learn the ability to classify labeled images in the current task while preserving the past tasks’ knowledge, where the tasks are non-iid to each other. On the other hand, UCL aims to learn the representation of images without the presence of labels and the model learns general-purpose representations during sequential training.
+
+To accommodate such continuous shifts in data distributions, representation learning models need to increment the knowledge without losing the representations learned in the past. With this motivation, we attempt to bridge the gap between unsupervised representation learning and continual learning to address the challenge of learning the representations on a sequence of tasks. Specifically, we focus on unsupervised continual learning (UCL), where the goal of the continual learner is to learn the representations from a stream of unlabelled data instances without forgetting (see Figure 1). To this end, we extend various existing SCL strategies to the unsupervised continual learning framework and analyze the performance of current state-of-the-art representation learning techniques: SimSiam (Chen & He, 2021) and BarlowTwins (Zbontar et al., 2021) for UCL. Surprisingly, we observe that the unsupervised representations are comparatively more robust to catastrophic forgetting across all datasets and simply finetuning on the sequence of tasks can outperform various state-of-the-art continual learning alternatives. Furthermore, we show that UCL generalize better to various out of distribution tasks and outperforms SCL for few-shot training scenarios (Section 5.2).
+
+We demystify the robustness of unsupervised representations by investigating the feature similarity, measured by centered kernel alignment (CKA) (Kornblith et al., 2019) between two independent UCL and SCL models and between UCL and SCL models. We notice that two unsupervised model representations have a relatively high feature similarity compared to two supervised representations. Furthermore, in all cases, two models have high similarity in lower layers indicating that they learn similar low-level features. Further, we measure the $\ell _ { 2 }$ distance between model parameters (Neyshabur et al., 2020) and visually compare the feature representations learned by different SCL and UCL strategies. We observe that UCL obtains human perceptual feature patterns for previous tasks, demonstrating their effectiveness to alleviate catastrophic forgetting (Section 5.3). We conjecture that this is due to their characteristic ability to learn general-purpose features (Doersch et al., 2020), which makes them transfer better and comparatively more robust to catastrophic forgetting.
+
+To gain further insights, we visualize the loss landscape (Li et al., 2018) of the UCL and SCL models and observe that UCL obtains a flatter and smoother loss landscape than SCL. Additionally, we propose a simple yet effective technique coined Lifelong Unsupervised Mixup (LUMP), which utilizes mixup (Zhang et al., 2018) for unlabelled training instances. In particular, LUMP interpolates between the current task examples and examples from previous instances to minimize catastrophic forgetting. We emphasize that LUMP is easy to implement, does not require additional hyperparameters, and simply trains on the interpolated instances. To this end, LUMP requires little, or no modification to existing rehearsal-based methods effectively minimizes catastrophic forgetting even with uniformly selecting the examples from replay buffer. We show that LUMP with UCL outperforms the state-ofthe-art supervised continual learning methods across multiple experimental settings with significantly lower catastrophic forgetting. In summary, our contributions are as follows:
+
+• We attempt to bridge the gap between continual learning and representation learning and tackle the two crucial problems of continual learning with unlabelled data and representation learning on a sequence of tasks. • Systematic quantitative analysis shows that UCL achieves better performance over SCL with significantly lower catastrophic forgetting on Sequential CIFAR-10, CIFAR-100, and Tiny-ImageNet. Additionally, we evaluate out-of-distribution tasks and few-shot training demonstrating the expressive power of unsupervised representations. • We provide visualization of the representations and loss landscapes, which show that UCL learns discriminative, human perceptual patterns and achieves a flatter and smoother loss landscape. Furthermore, we propose Lifelong Unsupervised Mixup (LUMP) for UCL, which effectively alleviates catastrophic forgetting and provides better qualitative interpretations.
+
+# 2 RELATED WORK
+
+Continual learning. We can partition the existing continual learning methods into three categories. The regularization approaches (Li & Hoiem, 2016; Zenke et al., 2017; Schwarz et al., 2018; Ahn et al., 2019) impose a regularization constraint to the objective that mitigates catastrophic forgetting. The architectural approaches (Rusu et al., 2016; Yoon et al., 2018; Li et al., 2019) avoid this problem by including task-specific parameters and allowing the expansion of the network during continual learning. The rehearsal approaches (Rebuffi et al., 2017; Rolnick et al., 2019; Aljundi et al., 2019) allocate a small memory buffer to store and replay the examples from the previous task. However, all these methods are confined to supervised learning, which limits their application in real-life problems. Rao et al. (2019); Smith et al. (2021) tackled the problem of continual unsupervised representation learning; however, their methods are restricted to simple low-resolution tasks and not scalable to large-scale continual learning datasets.
+
+Representational learning. A large number of works have addressed the unsupervised learning problem in the standard machine learning framework. Specifically, contrastive learning frameworks (He et al., 2020; Chen et al., 2020a; Grill et al., 2020; Chen et al., 2020b;c) that learn the representations by measuring the similarities of positive and negative pairs have gained a lot of attention in the community. However, all these methods require large batches and negative sample pairs, which restrict the scalability of these networks. Chen & He (2021) tackled these limitations and proposed SimSiam, that use standard Siamese networks (Bromley et al., 1994) with the stop-gradient operation to prevent the collapsing of Siamese networks to a constant. Recently, Zbontar et al. (2021) formulated an objective that pushes the cross-correlation matrix between the embeddings of distorted versions of a sample closer to the identity matrix. However, all these methods assume the presence of large datasets for pre-training, which is impractical in real-world applications. In contrast, we tackle the problem of incremental representational learning and learn the representations sequentially while maximizing task adaptation and minimizing catastrophic forgetting.
+
+# 3 PRELIMINARIES
+
+# 3.1 PROBLEM SETUP
+
+We consider the continual learning setting, where we learn on a continuum of data consisting of $T$ $\mathcal { T } _ { 1 : T } = ( \mathcal { T } _ { 1 } \dots \mathcal { T } _ { T } )$ In superonding , where taskwithdistri nsists a task descriptorexamples. Each input-ion. Let us consider a $\tau \in \{ 1 \ldots T \}$ $\mathcal { D } _ { \tau } = \{ ( { \boldsymbol { x } } _ { i , \tau } , y _ { i , \tau } ) _ { i = 1 } ^ { n _ { \tau } } \}$ $n _ { \tau }$ $( { \bf x } _ { i , \tau } , \bar { y _ { i , \tau } } ) \in \mathcal { X } _ { \tau } \times \mathcal { Y } _ { \tau }$ $( \mathcal { X } _ { \tau } , \mathcal { Y } _ { \tau } )$ network $f _ { \Theta } : \mathcal { X } _ { \tau } \to \mathbb { R } ^ { D }$ parametrized by $\mathbf { \Theta } \Theta = \{ \pmb { w } _ { l } \} _ { l = 1 } ^ { l = L }$ , where $\mathbb { R } ^ { D }$ and $L$ denote $D$ -dimensional embedding space and number of layers respectively. The classifier is denoted by $h _ { \psi } : \mathbb { R } ^ { D }  \mathcal { V } _ { \tau }$ . The network error using cross entropy loss (CE) for SCL with finetuning can be formally defined as:
+
+$$
+\mathcal { L } _ { \mathrm { S C L } } ^ { \mathrm { F I N E T U N E } } = \mathrm { C E } \left( h _ { \psi } \left( f _ { \Theta } \left( \pmb { x } _ { i , \tau } \right) , \tau \right) , y _ { i , \tau } \right) .
+$$
+
+In this work, we assume the absence of label supervision during training and focus on unsupervised continual learning. In particular, each task consists of $\mathcal { U } _ { \tau } \bar { = } \{ ( { \pmb x } _ { i , \tau } ) _ { i = 1 } ^ { n _ { \tau } } \}$ , $\mathbf { \boldsymbol { x } } _ { i , \tau } ~ \in ~ \mathcal { X } _ { \tau }$ with $n _ { \tau }$ examples. Our aim is to learn the representations $f _ { \Theta } : \mathcal { X } _ { \tau } \to \mathbb { R } ^ { D }$ on a sequence of tasks while preserving the knowledge of the previous tasks. We introduce the representation learning framework and propose LUMP in Section 4 to learn unsupervised representations while effectively mitigating catastrophic forgetting.
+
+# 3.2 LEARNING PROTOCOL AND EVALUATION METRICS
+
+Currently, the traditional continual learning strategies follow the standard training protocol, where we learn the network representations $f _ { \Theta } : \mathcal { X } _ { \tau } \to \mathcal { Y } _ { \tau }$ on a sequence of tasks. In contrast, our goal is to learn the feature representations $f _ { \Theta } : \mathcal { X } _ { \tau } \to \mathbb { R } ^ { D }$ , so we follow a two-step learning protocol to obtain the model predictions. First, we pre-train the representations on a sequence of tasks $T _ { 1 : T } = ( \tau _ { \cdot \cdot \cdot } \mathcal { T } _ { T } )$ to obtain the representations. Next, we evaluate the quality of our pre-trained representations using a $\mathbf { K }$ -nearest neighbor (KNN) classifier (Wu et al., 2018) following the setup in Chen et al. (2020a); Chen & He (2021); Zbontar et al. (2021).
+
+To validate knowledge transfer of the learned representations, we adopt the metrics from the SCL literature (Chaudhry et al., 2019b; Mirzadeh et al., 2020). Let $\boldsymbol { a } _ { \tau , i }$ denote the test accuracy of task $i$ after learning task $\mathcal { T } _ { \tau }$ using a KNN on frozen pre-trained representations on task $\mathcal { T } _ { \tau }$ . More formally, we can define the metrics to evaluate the continually learned representations as follow:
+
+1. Average accuracy is the average test accuracy of all the tasks completed until the continual learning of task τ : Aτ = 1τ Pτi=1 aτ,i 2. Average Forgetting is the average performance decrease of each task between its maximum accuracy and accuracy at the completion of training: $\begin{array} { r } { F = \frac { 1 } { T - 1 } \sum _ { i = 1 } ^ { T - 1 } \operatorname* { m a x } _ { \tau \in \{ 1 , \dots , T \} } \left( a _ { \tau , i } - a _ { T , i } \right) } \end{array}$
+
+# 4 UNSUPERVISED CONTINUAL LEARNING
+
+# 4.1 CONTINUOUS REPRESENTATION LEARNING WITH SEQUENTIAL TASKS
+
+To learn feature representations, contrastive learning (Chen et al., 2020a;b; He et al., 2020) maximizes the similarity of representations between the images of the same views (positive pairs) and minimizes the similarity between images of different views (negative pairs). However, these methods require large batches, negative sample pairs (Chen et al., 2020a;b), or architectural modifications (He et al., 2020; Chen et al., 2020c), or non-differentiable operators (Caron et al., 2020), which makes their application difficult for continual learning scenarios. In this work, we focus on SimSiam (Chen & He, 2021) and BarlowTwins (Zbontar et al., 2021), which tackle these limitations and achieve state-of-the-art performance on standard representation learning benchmarks.
+
+SimSiam (Chen & He, 2021) uses a variant of Siamese networks (Bromley et al., 1994) for learning input data representations. It consists of an encoder network $f _ { \Theta }$ , which is composed of a backbone network, and is shared across a projection MLP and prediction MLP head $h ( \cdot )$ . Specifically, SimSiam minimizes the cosine-similarity between the output vectors of the projector and the predictor MLP across two different augmentations for an image. Initially, we consider FINETUNE, which is a a naive CL baseline that minimizes the cosine-similarity between the projector output $( z _ { i , \tau } ^ { 1 } = f _ { \Theta } ( x _ { i , \tau } ^ { 1 } ) )$ and the predictor output $( p _ { i , \tau } ^ { 2 } = h ( f _ { \Theta } ( x _ { i , \tau } ^ { 2 } ) )$ on a sequence of tasks as follows:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \mathrm { U C L } } ^ { \mathrm { F I N E T U N E } } = \displaystyle \frac { 1 } { 2 } D ( p _ { i , \tau } ^ { 1 } , \mathrm { s t o p g r a d } ( z _ { i , \tau } ^ { 2 } ) ) + \frac { 1 } { 2 } D ( p _ { i , \tau } ^ { 2 } , \mathrm { s t o p g r a d } ( z _ { i , \tau } ^ { 1 } ) ) , } \\ { \mathrm { w h e r e } D ( p _ { i , \tau } ^ { 1 } , z _ { i , \tau } ^ { 2 } ) = - \frac { p _ { i , \tau } ^ { 1 } } { \| p _ { i , \tau } ^ { 2 } \| _ { 2 } } \cdot \frac { z _ { i , \tau } ^ { 2 } } { \| z _ { i , \tau } ^ { 2 } \| _ { 2 } } , } \end{array}
+$$
+
+$x _ { i , \tau } ^ { 1 }$ nd -n $x _ { i , \tau } ^ { 2 }$ are two randomly augmented views of an input example  Note that, the stopgrad is a crucial component in Si $x _ { i , \tau } \in \mathcal { T } _ { \tau }$ and preve $\lVert \cdot \rVert _ { 2 }$ denotese trivial $\ell _ { 2 }$ solutions obtained by Siamese networks. Due to its simplicity and effectiveness, we chose Simsiam to analyze the performance of unsupervised representations for continual learning.
+
+BarlowTwins (Zbontar et al., 2021) minimizes the redundancy between the embedding vector components of the distorted versions of an instance while conserving the maximum information inspired from Barlow (1961). In particular, the objective function eliminates the SimSiam stopgrad component and instead makes the cross-correlation matrix computed between the outputs of two identical networks closer to the identity matrix. Let $\mathcal { C }$ be the cross-correlation matrix between the outputs of two Siamese branches along the batch dimension and $Z _ { 1 }$ and $Z _ { 2 }$ denote the batch embeddings of the distorted views for all images of a batch from the current task $( x _ { \tau } \in \mathcal { U } _ { \tau }$ ). The objective function for UCL with finetuning and BarlowTwins can then be defined as:
+
+$$
+\mathcal { L } _ { \mathrm { U C L } } ^ { \mathrm { F I N E T U N E } } = \sum _ { i } ( 1 - \mathcal { C } _ { i i } ) ^ { 2 } + \ \lambda \cdot \sum _ { i } \sum _ { j \neq i } \mathcal { C } _ { i j } ^ { 2 } , \mathrm { w h e r e } \ \mathcal { C } _ { i j } = \frac { \sum _ { B } z _ { B , i } ^ { 1 } z _ { B , j } ^ { 2 } } { \sqrt { \sum _ { B } { ( z _ { B , i } ^ { 1 } ) } ^ { 2 } } \sqrt { \sum _ { B } { ( z _ { B , j } ^ { 2 } ) } ^ { 2 } } } .
+$$
+
+$\lambda$ is a positive constant trading off the importance of the invariance and redundancy reduction terms of the loss, $i$ and $j$ denote the network’s output vector dimensions. Similar to SimSiam, BarlowTwins is simple, easy to implement, and can be applied to existing continual learning strategies with little or no modification.
+
+Learning feature representations from labelled instances on a sequence of tasks has been long studied in continual learning. However, the majority of these learning strategies are not directly applicable to UCL. To compare with the regularization-based strategies, we extend Synaptic Intelligence (SI) (Zenke et al., 2017) to UCL and consider the online per-synapse consolidation during the entire training trajectory of the unsupervised representations. For architectural-based strategies, we investigate Progressive Neural Networks (PNN) (Rusu et al., 2016) and learn the feature representations progressively using the representations learning frameworks proposed in Section 4.1.
+
+We also formulate Dark Experience Replay (DER) (Buzzega et al., 2020) for UCL. DER for SCL alleviates catastrophic forgetting by matching the network logits across a sequence of tasks during the optimization trajectory. Notably, the loss for SCL-DER can be defined as follow:
+
+$$
+\mathcal { L } _ { \mathrm { S C L } } ^ { \mathrm { D E R } } = \mathcal { L } _ { \mathrm { S C L } } ^ { \mathrm { F I N E T U N E } } + ~ \alpha \cdot \mathbb { E } _ { ( \boldsymbol { x } , \boldsymbol { p } ) \sim \mathcal { M } } \big [ \| \mathrm { s o f t m a x } ( \boldsymbol { p } ) - \mathrm { s o f t m a x } ( h _ { \psi } ( \boldsymbol { x } _ { i , \tau } ) ) \| _ { 2 } ^ { 2 } \big ] ,
+$$
+
+where p = hψτ (x), LFINETSCL $\mathcal { L } _ { \mathrm { S C L } } ^ { \mathrm { F I N E T U N E } }$ denotes the cross-entropy loss on the current task (see Equation (1)) and random examples are selected using reservoir sampling from the replay-buffer $\mathcal { M }$ . Since, we do not have access to the labels for UCL, we cannot minimize the aforementioned objective.
+
+Instead, we utilize the output of the projected output by the backbone network to preserve the knowledge of the past tasks over the entire training trajectory. In particular, DER for UCL consists of a combination of two terms. The first term learns the representations using SimSiam from Equation (2) or BarlowTwins from Equation (3) and the second term minimizes the Euclidean distance between the projected outputs to minimize catastrophic forgetting. More formally, UCL-DER minimizes the following loss:
+
+$$
+\mathcal { L } _ { \mathrm { U C L } } ^ { \mathrm { D E R } } = \mathcal { L } _ { \mathrm { U C L } } ^ { \mathrm { F I N E T U N E } } + \ \alpha \cdot \mathbb { E } _ { ( x ) \sim \mathcal { M } } \big [ \| f _ { \Theta _ { \tau } } ( x ) - f _ { \Theta } ( x _ { i , \tau } ) \| _ { 2 } ^ { 2 } \big ]
+$$
+
+However, the performance of the rehearsal-based methods is sensitive to the choice of $\alpha$ and often requires supervised training setup, task identities, and boundaries. To tackle this issue, we propose Lifelong Unsupervised Mixup in the subsequent subsection, which interpolates between the current and past task instances to mitigate catastrophic forgetting effectively.
+
+# 4.3 LIFELONG UNSUPERVISED MIXUP
+
+The standard Mixup (Zhang et al., 2018) training constructs virtual training examples based on the principle of Vicinal Risk Minimization . In particular, let $( x _ { i } , y _ { i } )$ and $( x _ { j } , y _ { j } )$ denote two random feature-target pairs sampled from the training data distribution and let $( \tilde { x } , \tilde { y } )$ denote the interpolated feature-target pair in the vicinity of these examples; mixup then minimizes the following objective:
+
+$$
+\begin{array} { r l } & { \mathcal { L } ^ { \mathrm { M x U P } } ( \tilde { x } , \tilde { y } ) = \mathrm { C E } \left( h _ { \psi } \left( f _ { \Theta } \left( \tilde { x } \right) \right) , \tilde { y } \right) , } \\ & { \quad \quad \mathrm { w h e r e } \tilde { x } = \lambda \cdot x _ { i } + \left( 1 - \lambda \right) \cdot x _ { j } \mathrm { a n d } \tilde { y } = \lambda \cdot y _ { i } + \left( 1 - \lambda \right) \cdot y _ { j } . } \end{array}
+$$
+
+$\lambda \sim \operatorname { B e t a } ( \alpha , \alpha )$ , for $\alpha \in ( 0 , \infty )$ . In this work, we focus on lifelong self-supervised learning and propose Lifelong Unsupervised Mixup (LUMP) that utilizes mixup for UCL by incorporating the instances stored in the replay-buffer from the previous tasks into the vicinal distribution. In particular, LUMP interpolates between the examples of the current task $( x _ { i , \tau } ) \in \mathcal { U } _ { \tau }$ and random examples selected using uniform sampling from the replay buffer, which encourages the model to behave linearly across a sequence of tasks. More formally, LUMP minimizes the objective in Equation (2) and Equation (3) on the following interpolated instances $\tilde { x } _ { i , \tau }$ for the current task $\tau$ :
+
+$$
+\boldsymbol { \tilde { x } } _ { i , \tau } = \lambda \cdot \boldsymbol { x } _ { i , \tau } + ( 1 - \lambda ) \cdot \boldsymbol { x } _ { j , \boldsymbol { M } } ,
+$$
+
+where $x _ { j , \mathcal { M } } \sim \mathcal { M }$ denotes the example selected using uniform sampling from replay buffer $\mathcal { M }$ . The interpolated examples not only augments the past tasks’ instances in the replay buffer but also approximates a regularized loss minimization (Zhang et al., 2021). During UCL, LUMP enhances the robustness of learned representation by revisiting the attributes of the past task that are similar to the current task. Recently, Kim et al. (2020); Lee et al. (2021); Verma et al. (2021); Shen et al. (2022) also employed mixup for contrastive learning. Our work is different from these existing works in that our objective is different, and we focus on unsupervised continual learning. To this end, LUMP successively mitigates catastrophic forgetting and learns discriminative & human-perceptual features over the current state-of-the-art SCL strategies (see Table 1 and Figure 4).
+
+# 5 EXPERIMENTS
+
+# 5.1 EXPERIMENTAL SETUP
+
+Baselines. We compare with multiple supervised and unsupervised continual learning baselines across different categories of continual learning methods.
+
+1. Supervised continual learning. FINETUNE is a vanilla supervised learning method trained on a sequence of tasks without regularization or episodic memory and MULTITASK optimizes the model on complete data. For regularization-based CL methods, we compare against SI (Zenke et al., 2017) and AGEM (Chaudhry et al., 2019a). We include PNN (Rusu et al., 2016) for architecture-based methods. Lastly, we consider GSS (Aljundi et al., 2019) that populates the replay-buffer using solid-angle minimization and DER (Buzzega et al., 2020) matches the network logits sampled through the optimization trajectory for rehearsal during continual learning.
+
+2. Unsupervised continual learning. We consider the unsupervised variants of various SCL baselines to show the utility of the unsupervised representations for sequential learning. Specifically, we use SIMSIAM (Chen & He, 2021) and BARLOWTWINS (Zbontar et al., 2021), which are the state-of-the-art representational learning techniques for learning the unsupervised continual representations. We compare with FINETUNE and MULTITASK following the supervised learning baselines, and SI (Zenke et al., 2017), PNN (Rusu et al., 2016) for unsupervised regularization and architecture CL methods respectively. For rehearsal-based method, we compare with the UCL variant of DER (Buzzega et al., 2020) described in Section 4.2
+
+Datasets. We compare the performance of SCL and UCL on various continual learning benchmarks using single-head ResNet-18 (He et al., 2016) architecture. Split CIFAR-10 (Krizhevsky, 2012) consists of two random classes out of the ten classes for each task. Split CIFAR-100 (Krizhevsky, 2012) consists of five random classes out of the 100 classes for each task. Split Tiny-ImageNet is a variant of the ImageNet dataset (Deng et al., 2009) containing five random classes out of the 100 classes for each task with the images sized $6 4 \times 6 4$ pixels.
+
+Training and evaluation setup. We follow the hyperparameter setup of Buzzega et al. (2020) for all the SCL strategies and tune them for the UCL representation learning strategies. All the learned representations are evaluated with KNN classifier (Wu et al., 2018) across three independent runs. Further, we use the hyper-parameters obtained by SimSiam for training UCL strategies with BarlowTwins to analyze the sensitivity of UCL to hyper-parameters and for a fair comparison between different methods. We train all the UCL methods for 200 epochs and evaluate with the KNN classifier (Wu et al., 2018). We provide the hyper-parameters in detail in Table A.5.
+
+# 5.2 QUANTITATIVE RESULTS
+
+Evaluation on SimSiam. Table 1 shows the evaluation results for supervised and unsupervised representations learnt by SimSiam (Chen & He, 2021) across various continual learning strategies. In all cases, continual learning with unsupervised representations achieves significantly better performance than supervised representations with substantially lower forgetting. For instance, SI with UCL obtains better performance and $6 8 \%$ , $5 4 \%$ , and $4 4 \%$ lower forgetting relative to the best-performing SCL strategy on Split CIFAR-10, Split CIFAR-100, and Split Tiny-ImageNet, respectively. Surprisingly, FINETUNE with UCL achieves higher performance and significantly lower forgetting in comparison to all SCL strategies except DER. Furthermore, LUMP improves upon the UCL strategies: $2 . 8 \%$ and $5 . 9 \%$ relative increase in accuracy and $1 5 \%$ and $5 7 . 1 \%$ relative decrease in forgetting on Split CIFAR-100 and Split Tiny-ImageNet, respectively.
+
+Evaluation on BarlowTwins. To verify that unsupervised representations are indeed more robust to catastrophic forgetting, we train BarlowTwins (Zbontar et al., 2021) on a sequence of tasks. We notice that the representations learned with BarlowTwins substantially improve the accuracy and forgetting over SCL: $7 1 . 4 \%$ , $6 9 . 7 \%$ and $7 3 . 2 \%$ decrease in forgetting with FINETUNE on Split CIFAR-10, Split CIFAR-100 and Split Tiny-ImageNet respectively. Similarly, we observe that SI, and DER are more robust to catastrophic forgetting; however, PNN underperforms on complicated tasks since feature accumulation using adaptor modules is insufficient to construct useful representations for current task adaptation. Interestingly, representations learnt with BarlowTwins achieve lower forgetting for FINETUNE, DER and LUMP than SimSiam with comparable accuracy across all the datasets.
+
+Table 1: Accuracy and forgetting of the learnt representations on Split CIFAR-10, Split CIFAR-100 and Split Tiny-ImageNet on Resnet-18 architecture with KNN classifier (Wu et al., 2018). All the values are measured by computing mean and standard deviation across three trials. The best and second-best results are highlighted in bold and underline respectively.   
+
+<table><tr><td>METHOD</td><td colspan="2">SPLIT CIFAR-10</td><td colspan="2">SPLIT CIFAR-100</td><td colspan="2">SPLIT TINY-IMAGENET</td></tr><tr><td colspan="7">ACCURACY FORGETTING ACCURACY FORGETTING ACCURACY FORGETTING</td></tr><tr><td colspan="7">SUPERVISED CONTINUAL LEARNING</td></tr><tr><td>FINETUNE</td><td>82.87 (± 0.47)</td><td>14.26 (± 0.52)</td><td>61.08 (± 0.04)</td><td>31.23 (± 0.41)</td><td>53.10 (± 1.37)</td><td>33.15 (± 1.22)</td></tr><tr><td>PNN (Rusu et al., 2016)</td><td>82.74 (± 2.12)</td><td></td><td>66.05 (±0.86)</td><td></td><td>64.38 (± 0.92)</td><td></td></tr><tr><td>SI (Zenke et al., 2017)</td><td>85.18 (± 0.65)</td><td>11.39 (± 0.77)</td><td>63.58 (± 0.37)</td><td>27.98 (± 0.34)</td><td>44.96 (± 2.41)</td><td>26.29 (± 1.40)</td></tr><tr><td>A-GEM (Chaudhry et al., 2019a)</td><td>82.41 (± 1.24)</td><td>13.82 (± 1.27)</td><td>59.81 (± 1.07)</td><td>30.08 (± 0.91)</td><td>60.45 (± 0.24)</td><td>24.94 (± 1.24)</td></tr><tr><td>Gss (Aljundi et al.,2019)</td><td>89.49 (± 1.75)</td><td>7.50 (± 1.52)</td><td>70.78 (± 1.67)</td><td>21.28 (± 1.52)</td><td>70.96 (± 0.72)</td><td>14.76 (± 1.22)</td></tr><tr><td>DER (Buzzega et al., 2020)</td><td>91.35 (± 0.46)</td><td>5.65 (± 0.35)</td><td>79.52 (± 1.88)</td><td>12.80 (± 1.47)</td><td>68.03 (±0.85)</td><td>17.74 (± 0.65)</td></tr><tr><td>MULTITASK</td><td>97.77 (± 0.15)</td><td></td><td>93.89 (±0.78)</td><td></td><td>91.79 (± 0.46)</td><td></td></tr><tr><td colspan="7">UNSUPERVISED CONTINUAL LEARNING</td></tr><tr><td colspan="7">FINETUNE PNN (Rusu et al., 2016)</td></tr><tr><td></td><td>90.11 (±0.12) 90.93 (± 0.22)</td><td>5.42 (±0.08)</td><td>75.42 (± 0.78)</td><td>10.19 (± 0.37)</td><td>71.07 (± 0.20)</td><td>9.48 (±0.56)</td></tr><tr><td>SIISIIN SI (Zenke et al., 2017)</td><td>92.75 (± 0.06)</td><td></td><td>66.58 (± 1.00)</td><td>5.54 (± 1.30)</td><td>62.15 (± 1.35) 72.34 (±0.42)</td><td></td></tr><tr><td>DER (Buzzega et al.,2020)</td><td>91.22 (± 0.30)</td><td>1.81 (± 0.21)</td><td>80.08 (± 1.30)</td><td></td><td>71.90 (± 1.44)</td><td>8.26 (± 0.64)</td></tr><tr><td>LUMP</td><td>91.00 (± 0.40)</td><td>4.63 (±0.26) 2.92 (± 0.53)</td><td>77.27 (± 0.30)</td><td>9.31 (± 0.09)</td><td></td><td>8.36 (± 2.06)</td></tr><tr><td></td><td></td><td></td><td>82.30 (± 1.35)</td><td>4.71 (± 1.52)</td><td>76.66 (± 2.39)</td><td>3.54 (± 1.04)</td></tr><tr><td>MULTITASK</td><td>95.76 (± 0.08)</td><td></td><td>86.31 (±0.38)</td><td></td><td>82.89 (± 0.49)</td><td></td></tr><tr><td colspan="7">FINETUNE</td></tr><tr><td>PNN (Rusu et al., 2016)</td><td>87.72 (± 0.32)</td><td>4.08 (± 0.56)</td><td>71.97 (± 0.54)</td><td>9.45 (± 1.01)</td><td>66.28 (± 1.23)</td><td>8.89 (±0.66)</td></tr><tr><td>SI (Zenke et al., 2017)</td><td>87.52 (± 0.33)</td><td></td><td>57.93 (± 2.98)</td><td></td><td>48.70 (± 2.59)</td><td></td></tr><tr><td>DER (Buzzega et al., 2020)</td><td>90.21 (± 0.08) 88.67 (± 0.24)</td><td>2.03 (±0.22)</td><td>75.04 (± 0.63)</td><td>7.43 (± 0.67)</td><td>56.96 (± 1.48)</td><td>17.04 (± 0.89)</td></tr><tr><td>LUMP</td><td></td><td>2.41 (± 0.26)</td><td>73.48 (± 0.53)</td><td>7.98 (± 0.29)</td><td>68.56 (± 1.47)</td><td>7.87 (± 0.44)</td></tr><tr><td>PPITSIITTIS</td><td>90.31 (± 0.30)</td><td>1.13 (± 0.18)</td><td>80.24 (± 1.04)</td><td>3.53 (± 0.83)</td><td>72.17 (± 0.89)</td><td>2.43 (± 1.00)</td></tr><tr><td>MULTITASK</td><td>95.48 (± 0.14)</td><td></td><td>87.16 (± 0.52)</td><td></td><td>82.42 (± 0.74)</td><td></td></tr><tr><td colspan="7">accuracy overdata size (h hrieteigreeere forgetting overdata size SCL-FT UCL-FT SCL-DER SCL-SI UCL-SI UCL-FT SCL-DER LUMP 10 UCL-SI LUMP 0.5</td></tr></table>
+
+Figure 2: Evaluation on Few-shot training for Split CIFAR-100 across different number of training instances per task. The results are measured across three independent trials.   
+Figure 3: CKA Feature similarity between two independent UCL models (red), two independent SCL models (blue), and UCL and SCL model (green) for different strategies on Split CIFAR-100 test distribution.
+
+Evaluation on Few-shot training. Figure 2 compares the effect of few-shot training on UCL and SCL, where each task has a limited number of training instances. Specifically, we conduct the experimental evaluation using 100, 200, 500, and 2500 training instances for each task in split CIFAR-100 dataset. Surprisingly, we observe that the gap in average accuracy between SCL and UCL methods widens with a decrease in the number of training instances. Note that UCL decreases the accuracy by $1 5 . 7 8 \% p$ on average with lower forgetting when the number of training instances decreases from 2500 to 100; whereas, SCL obtains a severe $3 2 . 2 1 \% p$ deterioration in accuracy. We conjecture that this is an outcome of the discriminative feature embeddings learned by UCL, which discriminates all the images in the dataset and captures more than class-specific information as also observed in Doersch et al. (2020). Furthermore, LUMP improves the performance over all the baselines with a significant margin across all few-shot experiments.
+
+Evaluation on OOD datasets. We evaluate the learnt representations on various out-of-distribution (OOD) datasets in Table 2 to measure their generalization to unseen data distributions. In particular, we conduct the OOD evaluation on MNIST (LeCun, 1998), Fashion-MNIST (FMNIST) (Xiao et al., 2017), SVHN (Netzer et al., 2011), CIFAR-10 and CIFAR-100 (Krizhevsky, 2012) using a KNN classifier (Wu et al., 2018). We observe that unsupervised representations outperform the supervised representations in all cases across all the datasets. In particular, the UCL representations learned with Simsiam, and SI on Split-CIFAR-10 improves the absolute performance over the best-performing SCL strategy by $4 . 5 8 \%$ , $6 . 0 9 \%$ , $1 5 . 2 6 \%$ , and $1 7 . 0 7 \%$ on MNIST, FMNIST, SVHN, and CIFAR-100 respectively. Further, LUMP trained on Split-CIFAR-100 outperforms SI across all datasets and obtains comparable performance with Split CIFAR-10 dataset.
+
+Table 2: Comparison of accuracy on out of distribution datasets using a KNN classifier (Wu et al., 2018) on pretrained SCL and UCL representations. We consider MNIST (LeCun, 1998), Fashion-MNIST (FMNIST) (Xiao et al., 2017), SVHN (Netzer et al., 2011) as out of distribution for Split CIFAR-100 and Split CIFAR-10. All the values are measured by computing mean and standard deviation across three trials. The best and second-best results are highlighted in bold and underline respectively.   
+
+<table><tr><td>IN-CLASS</td><td colspan="4">SPLIT CIFAR-10</td><td colspan="4">SPLIT CIFAR-100</td></tr><tr><td>OUT-OF-CLASS</td><td>MNIST</td><td>FMNIST</td><td>SVHN</td><td>CIFAR-100</td><td>MNIST</td><td>FMNIST</td><td>SVHN</td><td>CIFAR-10</td></tr><tr><td colspan="9">SUPERVISED CONTINUAL LEARNING</td></tr><tr><td>FINETUNE</td><td>86.42 (± 1.11)</td><td>74.47 (±0.84)4</td><td>41.00 (±0.85)</td><td>17.42 (± 0.96)</td><td>75.02 (±3.97)</td><td>62.37 (± 3.20)</td><td>38.05 (±0.73)</td><td>39.18 (± 0.83)</td></tr><tr><td>SI (Zenke et al., 2017)</td><td>87.08 (± 0.79)</td><td>76.41 (± 0.81)</td><td>42.62 (± 1.31)</td><td>19.14 (± 0.91)</td><td>79.96 (± 2.63)</td><td>63.71 (± 1.36)</td><td>40.92 (± 1.64)</td><td>40.41 (± 1.71)</td></tr><tr><td>A-GEM (Chaudhry et al., 2019a)</td><td>86.07 (± 1.94)</td><td>74.74(± 3.21)</td><td>37.77 (± 3.49)</td><td>16.11 (± 0.38)</td><td>77.56 (± 3.21)</td><td>64.16 (± 2.29)</td><td>37.48 (± 1.73)</td><td>37.91 (± 1.33)</td></tr><tr><td>Gss (Aljundi et al., 2019)</td><td>70.36 (± 3.54)</td><td>69.20 (± 2.51)</td><td>33.11 (± 2.26)</td><td>18.21 (± 0.39)</td><td>76.54 (± 0.46)</td><td>65.31 (± 1.72)</td><td>35.72 (± 2.37)</td><td>49.41 (± 1.81)</td></tr><tr><td>DER (Buzzega et al., 2020)</td><td>80.32 (± 1.91)</td><td>70.49 (± 1.54)</td><td>41.48 (± 2.76)</td><td>17.72 (± 0.25)</td><td>87.71 (± 2.23)</td><td>75.97 (± 1.29)</td><td>50.26 (± 0.95)</td><td>59.07 (± 1.06)</td></tr><tr><td>MULTITASK</td><td></td><td>88.79 (± 1.13) 79.50 (±0.52)41.26(± 1.95)</td><td></td><td>27.68 (±0.66)</td><td>92.29 (± 3.37)</td><td>86.12 (± 1.87)</td><td>54.94 (± 1.77)</td><td>54.04 (± 3.68)</td></tr><tr><td colspan="9">UNSUPERVISED CONTINUAL LEARNING</td></tr><tr><td>FINETUNE</td><td>89.23 (± 0.99)</td><td>80.05 (±0.34)</td><td>49.66 (±0.81)</td><td>34.52 (± 0.12)</td><td>85.99 (±0.86)</td><td>76.90 (± 0.11)</td><td>50.09 (± 1.41)</td><td>57.15 (± 0.96)</td></tr><tr><td>SI (Zenke et al., 2017)</td><td>93.72 (± 0.58)</td><td>82.50 (± 0.51)</td><td>57.88 (±0.16)</td><td>36.21 (±0.69)</td><td>91.50 (± 1.26)</td><td>80.57 (±0.93)</td><td>54.07 (± 2.73)</td><td>60.55 (± 2.54)</td></tr><tr><td>DER (Buzzega et al., 2020)</td><td>88.35 (±0.82)</td><td>79.33 (± 0.62)</td><td>48.83 (± 0.55))</td><td>30.68 (± 0.36)</td><td>87.96 (± 2.04)</td><td>76.21 (± 0.63)</td><td>47.70 (± 0.94)</td><td>56.26 (± 0.16)</td></tr><tr><td>WAISNIS LUMP</td><td>91.03 (± 0.22)</td><td>80.78 (±0.88)</td><td>45.18 (± 1.57)</td><td>31.17 (± 1.83)</td><td>91.76 (± 1.17)</td><td>81.61 (± 0.45)</td><td>50.13 (±0.71)</td><td>63.00 (±0.53)</td></tr><tr><td>MULTITASK</td><td>90.69 (± 0.13)</td><td>80.65 (±0.42)</td><td>47.67 (± 0.45)</td><td>39.55 (± 0.18)</td><td>90.35 (±0.24)</td><td>81.11 (± 1.86)</td><td>52.20 (± 0.61)</td><td>70.19 (± 0.15)</td></tr><tr><td>FINETUNE</td><td>86.86 (± 1.62)</td><td>78.37 (± 0.74)</td><td>44.64 (± 2.39)</td><td>28.03 (±0.52)</td><td>76.08 (± 2.86)</td><td>76.82 (±0.83)</td><td>42.95 (±0.90)</td><td>53.12 (± 0.13)</td></tr><tr><td>SI (Zenke et al., 2017)</td><td>90.31 (± 0.69)</td><td>80.58 (± 0.68)</td><td>49.18 (± 0.51)</td><td>31.80 (± 0.4)</td><td>85.24 (± 0.99)</td><td>78.82 (± 0.67)</td><td>45.18 (± 1.37)</td><td>53.99 (± 0.56)</td></tr><tr><td>DER (Buzzega et al.,2020)</td><td>85.15 (± 2.19)</td><td>77.96 (± 0.59)</td><td>45.68 (± 0.93)</td><td>27.83 (± 0.86)</td><td>78.08 (± 1.95)</td><td>76.67 (±0.68)</td><td>44.58 (± 1.01)</td><td>53.24 (±0.82)</td></tr><tr><td>LUMP</td><td>88.73 (± 0.54)</td><td>81.69 (± 0.45)</td><td>51.53(± 0.41)</td><td>31.53 (± 0.36)</td><td>90.22 (± 1.39)</td><td>81.28 (± 0.91)</td><td>50.24 (± 0.95)</td><td>60.76 (± 0.87)</td></tr><tr><td>PBPIIIITTIS MULTITASK</td><td>88.63 (± 1.38)</td><td>79.49 (± 0.29)</td><td>49.24 (± 2.44)</td><td>36.33 (±0.29)</td><td>86.98 (± 1.70)</td><td>79.40 (± 1.10)</td><td>50.19 (±0.81)</td><td>49.50 (± 0.38)</td></tr></table>
+
+# 5.3 QUALITATIVE ANALYSIS
+
+Similarity in feature and parameter space. We analyze the similarity between the representations learnt between (i) Two independent UCL models, (ii) Two independent SCL models (iii) SCL and UCL models using centered kernel alignment (CKA) (Kornblith et al., 2019) in Figure 3, which provides a score between 0 and 1 measuring the similarity between a pair of hidden representations. For two representations Θ1 : X → Rd1 and Θ2 : X → Rd1 , CKA(Θ1, Θ2) = ||Cov(Θ1(x),Θ2(x))||2F||Cov(Θ1(x))||F ·||Cov(Θ2(x))||F , where covariances are with respect to the test distribution. Additionally, we measure the $\ell _ { 2 }$ distance (Neyshabur et al., 2020) between the parameters of two independent UCL models (see Table 3) and two independent SCL models (see Table 4). First, we observe that the representations learned by two independent UCL methods have a high feature similarity and lower $\ell _ { 2 }$ distance compared to the two independent SCL methods, demonstrating UCL representations’ robustness. Second, we note that the representations between any two independent models are highly similar in the lower layers indicating that they learn similar high-level features, including edges and shapes; however, the features are dissimilar for the higher modules. Lastly, we see that the representations between a UCL and SCL model are similar in the lower layers but diverge in the higher layers across all CL strategies.
+
+Visualization of feature space. Next, we visualize the learned features to dissect further the representations learned by UCL and SCL strategies. Figure 4 shows the visualization of the latent feature maps for tasks $\mathcal { T } _ { 0 }$ and $\mathcal { T } _ { 1 3 }$ after the completion of continual learning. For $\mathcal { T } _ { 0 }$ , we observe that the SCL methods are prone to catastrophic forgetting, as the features appear noisy and do not have coherent patterns. In contrast, the features learned by UCL strategies are perceptually relevant and robust to catastrophic forgetting, with LUMP learning the most distinctive features. Similar to $\mathcal { T } _ { 0 }$ , we observe that the UCL features are more relevant and distinguishable than SCL for $\mathcal { T } _ { 1 3 }$ . Note that we randomly selected the examples and feature maps for all visualizations.
+
+Loss landscape visualization. To gain further insights, we visualize the loss landscape of task $\mathcal { T } _ { 0 }$ after the completion of training on task $\mathcal { T } _ { 0 }$ and $\mathcal { T } _ { 1 9 }$ for various UCL and SCL strategies in Figure 5. We measure the cross-entropy loss for all methods with a randomly initialized linear classifier for a fair evaluation of two different directions. We use the visualization tool from Li et al. (2018) that searches the task loss surface by repeatedly adding random perturbations to model weights. We observe that the loss landscape after $\mathcal { T } _ { 0 }$ looks quite similar across all the strategies since the forgetting does not exist yet. However, after training $\mathcal { T } _ { 1 9 }$ , there is a clear difference with the UCL strategies obtaining a flatter and smoother loss landscape because UCL methods are more stable and robust to the forgetting, which hurts the loss landscapes of past tasks for SCL. It is important to observe that LUMP obtains a smoother landscape than other UCL strategies, demonstrating its effectiveness. We defer further analyses for feature and loss landscape visualization to Appendix A.2.
+
+Table 3: $\ell _ { 2 }$ distance between UCL parameters after completion of training.   
+
+<table><tr><td>MODEL</td><td>FINETUNE</td><td>S1</td><td>DER</td><td>MULTITASK</td></tr><tr><td>FINETUNE</td><td>60.00 (± 1.70)</td><td></td><td></td><td></td></tr><tr><td>SI</td><td>76.46 (± 0.48)</td><td>92.35 (± 0.61)</td><td></td><td></td></tr><tr><td>DER</td><td>55.60 (± 1.42)</td><td>75.54 (± 0.97)</td><td>48.76 (± 1.54)</td><td></td></tr><tr><td>MULTITASK</td><td>61.32 (± 0.59)</td><td>79.95 (± 0.40)</td><td>57.90 (± 0.86)</td><td>61.42 (± 0.78)</td></tr></table>
+
+Table 4: $\ell _ { 2 }$ distance between SCL paraneters after completion of training.   
+
+<table><tr><td>MODEL</td><td>FINETUNE</td><td>SI</td><td>DER</td><td>MULTITASK</td></tr><tr><td>FINETUNE</td><td>183.31 (± 0.10)</td><td></td><td></td><td></td></tr><tr><td>SI</td><td>206.16 (± 0.28)</td><td>226.05 (± 0.13)</td><td></td><td></td></tr><tr><td>DER</td><td>202.61 (± 0.46)</td><td>224.78 (± 0.75)</td><td>219.06 (± 0.27)</td><td></td></tr><tr><td>MULTITASK</td><td>258.12 (± 0.26)</td><td>277.30 (± 0.69)</td><td>271.48 (± 0.45)</td><td>314.84 (± 0.92)</td></tr></table>
+
+![](images/c3ac687f242d718cd3c73bee19640019de4ed74e1f83c4ef3f892aa16f6a4dc8.jpg)  
+Figure 4: Visualization of feature maps for the second block representations learnt by SCL and UCL strategies (with Simsiam) for ResNet-18 architecture after the completion of CL for Split CIFAR-100 dataset $( n = 2 0 )$ ).
+
+![](images/9462ac68652082e8aaec80aa8da988e820a12909990da7aa46015a8992027101.jpg)  
+Figure 5: Loss landscape visualization of $\mathcal { T } _ { 0 }$ after the completion of training on task $\mathcal { T } _ { 0 }$ (top) and $\mathcal { T } _ { 1 9 }$ (bottom) for Split CIFAR-100 dataset on ResNet-18 architecture. We use Simsiam for UCL methods.
+
+# 6 DISCUSSION AND CONCLUSION
+
+This work attempts to bridge the gap between unsupervised representation learning and continual learning. In particular, we establish the following findings for unsupervised continual learning.
+
+Surpassing supervised continual learning. Our empirical evaluation across various CL strategies and datasets shows that UCL representations are more robust to catastrophic forgetting than SCL representations. Furthermore, we notice that UCL generalizes better to OOD tasks and achieves stronger performance on few-shot learning tasks. We propose Lifelong unsupervised mixup (LUMP), which interpolates the unsupervised instances between the current task and past task and obtains higher performance with lower catastrophic forgetting across a wide range of tasks.
+
+Dissecting the learned representations. We conduct a systematic analysis to understand the differences between the representations learned by UCL and SCL strategies. By investigating the similarity between the representations, we observe that UCL and SCL strategies have high similarities in the lower layers but are dissimilar in the higher layers. We also show that UCL representations learn coherent and discriminative patterns and smoother loss landscape than SCL.
+
+Limitations and future work. In this work, we do not consider the high-resolution tasks for CL. We intend to evaluate the forgetting of the learnt representations on ImageNet (Deng et al., 2009) in future work, since UCL shows lower catastrophic forgetting and representation learning has made significant progress on ImageNet over the past years. In follow-up work, we intend to conduct further analysis to understand the behavior of UCL and develop sophisticated methods to continually learn unsupervised representations under various setups, such as class-incremental or task-agnostic CL.
+
+# ACKNOWLEDGEMENTS
+
+We thank the anonymous reviewers for their insightful comments and suggestions. This work was supported by Microsoft Research Asia, the Engineering Research Center Program through the National Research Foundation of Korea (NRF) funded by the Korean Government MSIT (NRF2018R1A5A1059921), Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program (KAIST) and 2021-0-01696). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies.
+
+# AUTHOR CONTRIBUTIONS
+
+Divyam Madaan conceived of the presented idea, developed the experimental framework, carried out OOD evaluation, CKA visualization and took the lead in writing the manuscript. Jaehong Yoon performed the hyperparameter search, carried out the visualization of loss landscape and feature maps and performed the few-shot training analysis. Yuanchun Li, Yunxin Liu, and Sung Ju Hwang supervised the project.
+
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+
+# A SUPPLEMENTARY MATERIAL
+
+Organization. In the supplementary material, we provide the implementation details followed by the hyper-parameter configurations in Appendix A.1. Further, we show the other experiments we conducted and additional visualizations and results in Appendix A.2.
+
+# A.1 EXPERIMENTAL DETAILS
+
+Implementations. We use the DER (Buzzega et al., 2020) open-source codebase1 for all the experiments. In particular, we reproduce all their experimental results for supervised continual learning and use various models with their set of hyper-parameters as our baselines. We follow the original representations for $\mathrm { S i m } \mathrm { S i a m } ^ { 2 }$ and BarlowTwins3 for unsupervised continual learning. We verify our implementation by reproducing the reported results on CIFAR-10 in the original paper, where we train the representations on the complete CIFAR-10 dataset and evaluate on the test-set using KNN classifier (Wu et al., 2018). In particular, (Wu et al., 2018) stores the features for each instance in the task-level training set in a discrete memory bank. The optimal feature-level embeddings are then learned by instance-level discrimination, which maximally scatters the features of the training samples. Following prior works in representation learning, we use the task-level training set without any augmentation in the task-incremental setup for the supervised and unsupervised KNN evaluation.
+
+Hyperparameter configurations. We use the tuned hyper-parameters reported by Buzzega et al. (2020) for all the SCL experiments. On the other hand, we tune the hyper-parameters for continual learning strategies for UCL. We provide the hyper-parameters setup for UCL for different datasets in Table A.5. We train all the UCL methods with a batch size of 256 for 200 epochs, while training the SCL methods with a batch size of 32 for 50 epochs following Buzzega et al. (2020). We observed that training the SCL methods further lead to a degredation in performance for all the methods. We use the same set of augmentations for both SCL and UCL except that we use RandomResizedCrop with scale in [0.2, 1.0] for UCL (Wu et al., 2018; Chen & He, 2021) and RandomCrop for SCL. For rehearsal-based methods, we use the buffer size 200 for Split CIFAR-10, Split CIFAR-100 and 256 for Split Tiny-ImageNet dataset. We use a learning rate of 0.03 for SGD optimizer with weight decay 5e-4 and momentum 0.9.
+
+Table A.5: Hyperparameter configurations for all the datasets on ResNet-18 architecture.   
+
+<table><tr><td>METHOD</td><td>SPLIT CIFAR-10</td><td>SPLIT CIFAR-100</td><td>SEQ.TINY-IMAGENET</td></tr><tr><td>S1</td><td>c : 100 m:1</td><td>c : 0.1 m:1</td><td>c : 0.01 m:1</td></tr><tr><td>PNN</td><td>wd : 64</td><td>wd : 12</td><td>wd : 8</td></tr><tr><td>DER</td><td>α : 0.1</td><td>α : 0.1</td><td>α : 0.01</td></tr><tr><td>LUMP</td><td>入: 0.1</td><td>入: 0.1</td><td>入: 0.4</td></tr></table>
+
+# A.2 ADDITIONAL EXPERIMENTS
+
+We provide additional loss landscape on Split CIFAR-100 in Figure A.6 and Figure A.7, Figure A.8 show the second and third block feature visualizations on Split CIFAR-100 respectively. Figure A.9 shows the feature visualizations for Split Tiny-ImageNet on ResNet-18 architecture.
+
+![](images/69dfb7f41d6d34dd4ca483cb846c642db74f91d3e8a8f0a208134f58e0d8b215.jpg)  
+Figure A.6: Loss landscape visualization of $\mathcal { T } _ { 0 }$ after the completion of training on task $\mathcal { T } _ { 0 } , \mathcal { T } _ { 1 7 } , \mathcal { T } _ { 1 8 }$ , and $\mathcal { T } _ { 1 9 }$ for Split CIFAR-100 dataset on ResNet-18 architecture. We use Simsiam for UCL methods.
+
+![](images/59887c9b6bc8aced71be07cae33f26a0069fe68cfc5e774efeb90cee9dbe05bd.jpg)  
+Figure A.7: Visualization of feature maps for the second block representations learnt by SCL and UCL strategies (with Simsiam) for Resnet-18 architecture after the completion of continual learning for Split CIFAR-100 dataset $n = 2 0$ ). The accuracy is the mean across three runs for the corresponding task.
+
+![](images/ea5a16a4f7f0ee8d4c02d67c79a880ba9151dc89a3655d440e628d31abcc528e.jpg)  
+Figure A.8: Visualization of feature maps for the third block representations learnt by SCL and UCL strategies (with Simsiam) for Resnet-18 architecture after the completion of continual learning for Split CIFAR-100 dataset $n = 2 0$ ). The accuracy is the mean across three runs for the corresponding task.
+
+![](images/3b14374517bbb4507f4f16d6e77d6f2933fb6e36c554d2969976955b539a8d1a.jpg)  
+Figure A.9: Visualization of feature maps for the second block representations learnt by SCL and UCL strategies (with Simsiam) for Resnet-18 architecture after the completion of continual learning for Split Tiny-ImageNet dataset $n = 2 0$ ). The accuracy is the mean across three runs for the corresponding task.

parse/dev/9Hrka5PA7LW/9Hrka5PA7LW_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "REPRESENTATIONAL CONTINUITY FOR UNSUPERVISED CONTINUAL LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            651,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Divyam Madaan1∗ Jaehong Yoon2,3 † Yuanchun $\\mathbf { L i } ^ { 5 , 6 }$ Yunxin Liu5,6 Sung Ju Hwang2,4 New York University1 KAIST2 Microsoft Research3 AITRICS4 Institute for AI Industry Research (AIR)5 Tsinghua University6 divyam.madaan@nyu.edu, {jaehong.yoon,sjhwang82}@kaist.ac.kr liyuanchun@air.tsinghua.edu.cn, liuyunxin@air.tsinghua.edu.cn ",
+        "bbox": [
+            184,
+            167,
+            818,
+            241
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            277,
+            544,
+            292
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Continual learning (CL) aims to learn a sequence of tasks without forgetting the previously acquired knowledge. However, recent CL advances are restricted to supervised continual learning (SCL) scenarios. Consequently, they are not scalable to real-world applications where the data distribution is often biased and unannotated. In this work, we focus on unsupervised continual learning (UCL), where we learn the feature representations on an unlabelled sequence of tasks and show that reliance on annotated data is not necessary for continual learning. We conduct a systematic study analyzing the learned feature representations and show that unsupervised visual representations are surprisingly more robust to catastrophic forgetting, consistently achieve better performance, and generalize better to out-ofdistribution tasks than SCL. Furthermore, we find that UCL achieves a smoother loss landscape through qualitative analysis of the learned representations and learns meaningful feature representations. Additionally, we propose Lifelong Unsupervised Mixup (LUMP), a simple yet effective technique that interpolates between the current task and previous tasks’ instances to alleviate catastrophic forgetting for unsupervised representations. We release our code online. ",
+        "bbox": [
+            233,
+            309,
+            766,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            556,
+            336,
+            571
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently continual learning (Thrun, 1995) has gained a lot of attention in the deep learning community due to its ability to continually learn on a sequence of non-stationary tasks (Kumar & Daume III, 2012; Li & Hoiem, 2016) and close proximity to the human learning process (Flesch et al., 2018). However, the inability of the learner to prevent forgetting of the knowledge learnt from the previous tasks has been a long-standing problem (McCloskey & Cohen, 1989; Goodfellow et al., 2013). To address this problem, a large body of methods (Rusu et al., 2016; Zenke et al., 2017; Yoon et al., 2018; Li et al., 2019; Aljundi et al., 2019; Buzzega et al., 2020) have been proposed; however, all these methods focus on the supervised learning paradigm, but obtaining high-quality labels is expensive and often impractical in real-world scenarios. In contrast, CL for unsupervised representation learning has received limited attention in the community. Although Rao et al. (2019) instantiated a continual unsupervised representation learning framework (CURL), it is not scalable for high-resolution tasks, as it is composed of MLP encoders/decoders and a simple MoG generative replay. This is evident in their limited empirical evaluation using digit-based gray-scale datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Meanwhile, a set of directions have shown huge potential to tackle the representation learning problem without labels (He et al., 2020; Chen et al., 2020a; Grill et al., 2020; Chen et al., 2020b; Chen & He, 2021; Zbontar et al., 2021) by aligning contrastive pairs of training instances or maximizing the similarity between two augmented views of each image. However, a common assumption for existing methods is the availability of a large amount of unbiased and unlabelled datasets to learn the feature representations. We argue that this assumption is not realistic for most of the real-time applications, including self-driving cars (Bojarski et al., 2016), medical applications (Kelly et al., 2019) and conversational agents (Li et al., 2020). The collected datasets are often limited in size during the initial training phase (Finn et al., 2017), and datasets/tasks change continuously with time. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "image",
+        "img_path": "images/b5965a3ad31159cd64ec531fe9195a659aa834a208272019739bfb0a7aa74ef6.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of supervised and unsupervised continual learning. The objective of SCL is to learn the ability to classify labeled images in the current task while preserving the past tasks’ knowledge, where the tasks are non-iid to each other. On the other hand, UCL aims to learn the representation of images without the presence of labels and the model learns general-purpose representations during sequential training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To accommodate such continuous shifts in data distributions, representation learning models need to increment the knowledge without losing the representations learned in the past. With this motivation, we attempt to bridge the gap between unsupervised representation learning and continual learning to address the challenge of learning the representations on a sequence of tasks. Specifically, we focus on unsupervised continual learning (UCL), where the goal of the continual learner is to learn the representations from a stream of unlabelled data instances without forgetting (see Figure 1). To this end, we extend various existing SCL strategies to the unsupervised continual learning framework and analyze the performance of current state-of-the-art representation learning techniques: SimSiam (Chen & He, 2021) and BarlowTwins (Zbontar et al., 2021) for UCL. Surprisingly, we observe that the unsupervised representations are comparatively more robust to catastrophic forgetting across all datasets and simply finetuning on the sequence of tasks can outperform various state-of-the-art continual learning alternatives. Furthermore, we show that UCL generalize better to various out of distribution tasks and outperforms SCL for few-shot training scenarios (Section 5.2). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We demystify the robustness of unsupervised representations by investigating the feature similarity, measured by centered kernel alignment (CKA) (Kornblith et al., 2019) between two independent UCL and SCL models and between UCL and SCL models. We notice that two unsupervised model representations have a relatively high feature similarity compared to two supervised representations. Furthermore, in all cases, two models have high similarity in lower layers indicating that they learn similar low-level features. Further, we measure the $\\ell _ { 2 }$ distance between model parameters (Neyshabur et al., 2020) and visually compare the feature representations learned by different SCL and UCL strategies. We observe that UCL obtains human perceptual feature patterns for previous tasks, demonstrating their effectiveness to alleviate catastrophic forgetting (Section 5.3). We conjecture that this is due to their characteristic ability to learn general-purpose features (Doersch et al., 2020), which makes them transfer better and comparatively more robust to catastrophic forgetting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To gain further insights, we visualize the loss landscape (Li et al., 2018) of the UCL and SCL models and observe that UCL obtains a flatter and smoother loss landscape than SCL. Additionally, we propose a simple yet effective technique coined Lifelong Unsupervised Mixup (LUMP), which utilizes mixup (Zhang et al., 2018) for unlabelled training instances. In particular, LUMP interpolates between the current task examples and examples from previous instances to minimize catastrophic forgetting. We emphasize that LUMP is easy to implement, does not require additional hyperparameters, and simply trains on the interpolated instances. To this end, LUMP requires little, or no modification to existing rehearsal-based methods effectively minimizes catastrophic forgetting even with uniformly selecting the examples from replay buffer. We show that LUMP with UCL outperforms the state-ofthe-art supervised continual learning methods across multiple experimental settings with significantly lower catastrophic forgetting. In summary, our contributions are as follows: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• We attempt to bridge the gap between continual learning and representation learning and tackle the two crucial problems of continual learning with unlabelled data and representation learning on a sequence of tasks. • Systematic quantitative analysis shows that UCL achieves better performance over SCL with significantly lower catastrophic forgetting on Sequential CIFAR-10, CIFAR-100, and Tiny-ImageNet. Additionally, we evaluate out-of-distribution tasks and few-shot training demonstrating the expressive power of unsupervised representations. • We provide visualization of the representations and loss landscapes, which show that UCL learns discriminative, human perceptual patterns and achieves a flatter and smoother loss landscape. Furthermore, we propose Lifelong Unsupervised Mixup (LUMP) for UCL, which effectively alleviates catastrophic forgetting and provides better qualitative interpretations. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Continual learning. We can partition the existing continual learning methods into three categories. The regularization approaches (Li & Hoiem, 2016; Zenke et al., 2017; Schwarz et al., 2018; Ahn et al., 2019) impose a regularization constraint to the objective that mitigates catastrophic forgetting. The architectural approaches (Rusu et al., 2016; Yoon et al., 2018; Li et al., 2019) avoid this problem by including task-specific parameters and allowing the expansion of the network during continual learning. The rehearsal approaches (Rebuffi et al., 2017; Rolnick et al., 2019; Aljundi et al., 2019) allocate a small memory buffer to store and replay the examples from the previous task. However, all these methods are confined to supervised learning, which limits their application in real-life problems. Rao et al. (2019); Smith et al. (2021) tackled the problem of continual unsupervised representation learning; however, their methods are restricted to simple low-resolution tasks and not scalable to large-scale continual learning datasets. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Representational learning. A large number of works have addressed the unsupervised learning problem in the standard machine learning framework. Specifically, contrastive learning frameworks (He et al., 2020; Chen et al., 2020a; Grill et al., 2020; Chen et al., 2020b;c) that learn the representations by measuring the similarities of positive and negative pairs have gained a lot of attention in the community. However, all these methods require large batches and negative sample pairs, which restrict the scalability of these networks. Chen & He (2021) tackled these limitations and proposed SimSiam, that use standard Siamese networks (Bromley et al., 1994) with the stop-gradient operation to prevent the collapsing of Siamese networks to a constant. Recently, Zbontar et al. (2021) formulated an objective that pushes the cross-correlation matrix between the embeddings of distorted versions of a sample closer to the identity matrix. However, all these methods assume the presence of large datasets for pre-training, which is impractical in real-world applications. In contrast, we tackle the problem of incremental representational learning and learn the representations sequentially while maximizing task adaptation and minimizing catastrophic forgetting. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.1 PROBLEM SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We consider the continual learning setting, where we learn on a continuum of data consisting of $T$ $\\mathcal { T } _ { 1 : T } = ( \\mathcal { T } _ { 1 } \\dots \\mathcal { T } _ { T } )$ In superonding , where taskwithdistri nsists a task descriptorexamples. Each input-ion. Let us consider a $\\tau \\in \\{ 1 \\ldots T \\}$ $\\mathcal { D } _ { \\tau } = \\{ ( { \\boldsymbol { x } } _ { i , \\tau } , y _ { i , \\tau } ) _ { i = 1 } ^ { n _ { \\tau } } \\}$ $n _ { \\tau }$ $( { \\bf x } _ { i , \\tau } , \\bar { y _ { i , \\tau } } ) \\in \\mathcal { X } _ { \\tau } \\times \\mathcal { Y } _ { \\tau }$ $( \\mathcal { X } _ { \\tau } , \\mathcal { Y } _ { \\tau } )$ network $f _ { \\Theta } : \\mathcal { X } _ { \\tau } \\to \\mathbb { R } ^ { D }$ parametrized by $\\mathbf { \\Theta } \\Theta = \\{ \\pmb { w } _ { l } \\} _ { l = 1 } ^ { l = L }$ , where $\\mathbb { R } ^ { D }$ and $L$ denote $D$ -dimensional embedding space and number of layers respectively. The classifier is denoted by $h _ { \\psi } : \\mathbb { R } ^ { D }  \\mathcal { V } _ { \\tau }$ . The network error using cross entropy loss (CE) for SCL with finetuning can be formally defined as: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fabd1e0d3449e0127af1eeb333e509257da899fbb26d781ad1e5ae7b97d2267c.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { S C L } } ^ { \\mathrm { F I N E T U N E } } = \\mathrm { C E } \\left( h _ { \\psi } \\left( f _ { \\Theta } \\left( \\pmb { x } _ { i , \\tau } \\right) , \\tau \\right) , y _ { i , \\tau } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this work, we assume the absence of label supervision during training and focus on unsupervised continual learning. In particular, each task consists of $\\mathcal { U } _ { \\tau } \\bar { = } \\{ ( { \\pmb x } _ { i , \\tau } ) _ { i = 1 } ^ { n _ { \\tau } } \\}$ , $\\mathbf { \\boldsymbol { x } } _ { i , \\tau } ~ \\in ~ \\mathcal { X } _ { \\tau }$ with $n _ { \\tau }$ examples. Our aim is to learn the representations $f _ { \\Theta } : \\mathcal { X } _ { \\tau } \\to \\mathbb { R } ^ { D }$ on a sequence of tasks while preserving the knowledge of the previous tasks. We introduce the representation learning framework and propose LUMP in Section 4 to learn unsupervised representations while effectively mitigating catastrophic forgetting. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 LEARNING PROTOCOL AND EVALUATION METRICS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Currently, the traditional continual learning strategies follow the standard training protocol, where we learn the network representations $f _ { \\Theta } : \\mathcal { X } _ { \\tau } \\to \\mathcal { Y } _ { \\tau }$ on a sequence of tasks. In contrast, our goal is to learn the feature representations $f _ { \\Theta } : \\mathcal { X } _ { \\tau } \\to \\mathbb { R } ^ { D }$ , so we follow a two-step learning protocol to obtain the model predictions. First, we pre-train the representations on a sequence of tasks $T _ { 1 : T } = ( \\tau _ { \\cdot \\cdot \\cdot } \\mathcal { T } _ { T } )$ to obtain the representations. Next, we evaluate the quality of our pre-trained representations using a $\\mathbf { K }$ -nearest neighbor (KNN) classifier (Wu et al., 2018) following the setup in Chen et al. (2020a); Chen & He (2021); Zbontar et al. (2021). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To validate knowledge transfer of the learned representations, we adopt the metrics from the SCL literature (Chaudhry et al., 2019b; Mirzadeh et al., 2020). Let $\\boldsymbol { a } _ { \\tau , i }$ denote the test accuracy of task $i$ after learning task $\\mathcal { T } _ { \\tau }$ using a KNN on frozen pre-trained representations on task $\\mathcal { T } _ { \\tau }$ . More formally, we can define the metrics to evaluate the continually learned representations as follow: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "1. Average accuracy is the average test accuracy of all the tasks completed until the continual learning of task τ : Aτ = 1τ Pτi=1 aτ,i 2. Average Forgetting is the average performance decrease of each task between its maximum accuracy and accuracy at the completion of training: $\\begin{array} { r } { F = \\frac { 1 } { T - 1 } \\sum _ { i = 1 } ^ { T - 1 } \\operatorname* { m a x } _ { \\tau \\in \\{ 1 , \\dots , T \\} } \\left( a _ { \\tau , i } - a _ { T , i } \\right) } \\end{array}$ ",
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+    {
+        "type": "text",
+        "text": "4 UNSUPERVISED CONTINUAL LEARNING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 CONTINUOUS REPRESENTATION LEARNING WITH SEQUENTIAL TASKS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To learn feature representations, contrastive learning (Chen et al., 2020a;b; He et al., 2020) maximizes the similarity of representations between the images of the same views (positive pairs) and minimizes the similarity between images of different views (negative pairs). However, these methods require large batches, negative sample pairs (Chen et al., 2020a;b), or architectural modifications (He et al., 2020; Chen et al., 2020c), or non-differentiable operators (Caron et al., 2020), which makes their application difficult for continual learning scenarios. In this work, we focus on SimSiam (Chen & He, 2021) and BarlowTwins (Zbontar et al., 2021), which tackle these limitations and achieve state-of-the-art performance on standard representation learning benchmarks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "SimSiam (Chen & He, 2021) uses a variant of Siamese networks (Bromley et al., 1994) for learning input data representations. It consists of an encoder network $f _ { \\Theta }$ , which is composed of a backbone network, and is shared across a projection MLP and prediction MLP head $h ( \\cdot )$ . Specifically, SimSiam minimizes the cosine-similarity between the output vectors of the projector and the predictor MLP across two different augmentations for an image. Initially, we consider FINETUNE, which is a a naive CL baseline that minimizes the cosine-similarity between the projector output $( z _ { i , \\tau } ^ { 1 } = f _ { \\Theta } ( x _ { i , \\tau } ^ { 1 } ) )$ and the predictor output $( p _ { i , \\tau } ^ { 2 } = h ( f _ { \\Theta } ( x _ { i , \\tau } ^ { 2 } ) )$ on a sequence of tasks as follows: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/4672cdddefec03800b8f0e0ce4166a6f589f17b6ccb864eac30429c8915cc150.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\mathrm { U C L } } ^ { \\mathrm { F I N E T U N E } } = \\displaystyle \\frac { 1 } { 2 } D ( p _ { i , \\tau } ^ { 1 } , \\mathrm { s t o p g r a d } ( z _ { i , \\tau } ^ { 2 } ) ) + \\frac { 1 } { 2 } D ( p _ { i , \\tau } ^ { 2 } , \\mathrm { s t o p g r a d } ( z _ { i , \\tau } ^ { 1 } ) ) , } \\\\ { \\mathrm { w h e r e } D ( p _ { i , \\tau } ^ { 1 } , z _ { i , \\tau } ^ { 2 } ) = - \\frac { p _ { i , \\tau } ^ { 1 } } { \\| p _ { i , \\tau } ^ { 2 } \\| _ { 2 } } \\cdot \\frac { z _ { i , \\tau } ^ { 2 } } { \\| z _ { i , \\tau } ^ { 2 } \\| _ { 2 } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$x _ { i , \\tau } ^ { 1 }$ nd -n $x _ { i , \\tau } ^ { 2 }$ are two randomly augmented views of an input example  Note that, the stopgrad is a crucial component in Si $x _ { i , \\tau } \\in \\mathcal { T } _ { \\tau }$ and preve $\\lVert \\cdot \\rVert _ { 2 }$ denotese trivial $\\ell _ { 2 }$ solutions obtained by Siamese networks. Due to its simplicity and effectiveness, we chose Simsiam to analyze the performance of unsupervised representations for continual learning. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "BarlowTwins (Zbontar et al., 2021) minimizes the redundancy between the embedding vector components of the distorted versions of an instance while conserving the maximum information inspired from Barlow (1961). In particular, the objective function eliminates the SimSiam stopgrad component and instead makes the cross-correlation matrix computed between the outputs of two identical networks closer to the identity matrix. Let $\\mathcal { C }$ be the cross-correlation matrix between the outputs of two Siamese branches along the batch dimension and $Z _ { 1 }$ and $Z _ { 2 }$ denote the batch embeddings of the distorted views for all images of a batch from the current task $( x _ { \\tau } \\in \\mathcal { U } _ { \\tau }$ ). The objective function for UCL with finetuning and BarlowTwins can then be defined as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a31e8caafdcb8d5a288a1474ba72cffcb80ca7026a9764f66d9b2fda6c02e67.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { U C L } } ^ { \\mathrm { F I N E T U N E } } = \\sum _ { i } ( 1 - \\mathcal { C } _ { i i } ) ^ { 2 } + \\ \\lambda \\cdot \\sum _ { i } \\sum _ { j \\neq i } \\mathcal { C } _ { i j } ^ { 2 } , \\mathrm { w h e r e } \\ \\mathcal { C } _ { i j } = \\frac { \\sum _ { B } z _ { B , i } ^ { 1 } z _ { B , j } ^ { 2 } } { \\sqrt { \\sum _ { B } { ( z _ { B , i } ^ { 1 } ) } ^ { 2 } } \\sqrt { \\sum _ { B } { ( z _ { B , j } ^ { 2 } ) } ^ { 2 } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\lambda$ is a positive constant trading off the importance of the invariance and redundancy reduction terms of the loss, $i$ and $j$ denote the network’s output vector dimensions. Similar to SimSiam, BarlowTwins is simple, easy to implement, and can be applied to existing continual learning strategies with little or no modification. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Learning feature representations from labelled instances on a sequence of tasks has been long studied in continual learning. However, the majority of these learning strategies are not directly applicable to UCL. To compare with the regularization-based strategies, we extend Synaptic Intelligence (SI) (Zenke et al., 2017) to UCL and consider the online per-synapse consolidation during the entire training trajectory of the unsupervised representations. For architectural-based strategies, we investigate Progressive Neural Networks (PNN) (Rusu et al., 2016) and learn the feature representations progressively using the representations learning frameworks proposed in Section 4.1. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We also formulate Dark Experience Replay (DER) (Buzzega et al., 2020) for UCL. DER for SCL alleviates catastrophic forgetting by matching the network logits across a sequence of tasks during the optimization trajectory. Notably, the loss for SCL-DER can be defined as follow: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3e24618f51d26f742545a9d0b23849eb513a16952b68a40a92aaa51dff6092d.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { S C L } } ^ { \\mathrm { D E R } } = \\mathcal { L } _ { \\mathrm { S C L } } ^ { \\mathrm { F I N E T U N E } } + ~ \\alpha \\cdot \\mathbb { E } _ { ( \\boldsymbol { x } , \\boldsymbol { p } ) \\sim \\mathcal { M } } \\big [ \\| \\mathrm { s o f t m a x } ( \\boldsymbol { p } ) - \\mathrm { s o f t m a x } ( h _ { \\psi } ( \\boldsymbol { x } _ { i , \\tau } ) ) \\| _ { 2 } ^ { 2 } \\big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            300
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where p = hψτ (x), LFINETSCL $\\mathcal { L } _ { \\mathrm { S C L } } ^ { \\mathrm { F I N E T U N E } }$ denotes the cross-entropy loss on the current task (see Equation (1)) and random examples are selected using reservoir sampling from the replay-buffer $\\mathcal { M }$ . Since, we do not have access to the labels for UCL, we cannot minimize the aforementioned objective. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Instead, we utilize the output of the projected output by the backbone network to preserve the knowledge of the past tasks over the entire training trajectory. In particular, DER for UCL consists of a combination of two terms. The first term learns the representations using SimSiam from Equation (2) or BarlowTwins from Equation (3) and the second term minimizes the Euclidean distance between the projected outputs to minimize catastrophic forgetting. More formally, UCL-DER minimizes the following loss: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ebf045e3c6d6cbf7df52bb95dff49a5c74f519bc7cc4eed3bbe678587ad75797.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { U C L } } ^ { \\mathrm { D E R } } = \\mathcal { L } _ { \\mathrm { U C L } } ^ { \\mathrm { F I N E T U N E } } + \\ \\alpha \\cdot \\mathbb { E } _ { ( x ) \\sim \\mathcal { M } } \\big [ \\| f _ { \\Theta _ { \\tau } } ( x ) - f _ { \\Theta } ( x _ { i , \\tau } ) \\| _ { 2 } ^ { 2 } \\big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "However, the performance of the rehearsal-based methods is sensitive to the choice of $\\alpha$ and often requires supervised training setup, task identities, and boundaries. To tackle this issue, we propose Lifelong Unsupervised Mixup in the subsequent subsection, which interpolates between the current and past task instances to mitigate catastrophic forgetting effectively. ",
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+    {
+        "type": "text",
+        "text": "4.3 LIFELONG UNSUPERVISED MIXUP",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The standard Mixup (Zhang et al., 2018) training constructs virtual training examples based on the principle of Vicinal Risk Minimization . In particular, let $( x _ { i } , y _ { i } )$ and $( x _ { j } , y _ { j } )$ denote two random feature-target pairs sampled from the training data distribution and let $( \\tilde { x } , \\tilde { y } )$ denote the interpolated feature-target pair in the vicinity of these examples; mixup then minimizes the following objective: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c6b620b09d7a8eb72d9c20c6cb8b6d7551f3efb0f3e8264a2d29e6b67749649e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } ^ { \\mathrm { M x U P } } ( \\tilde { x } , \\tilde { y } ) = \\mathrm { C E } \\left( h _ { \\psi } \\left( f _ { \\Theta } \\left( \\tilde { x } \\right) \\right) , \\tilde { y } \\right) , } \\\\ & { \\quad \\quad \\mathrm { w h e r e } \\tilde { x } = \\lambda \\cdot x _ { i } + \\left( 1 - \\lambda \\right) \\cdot x _ { j } \\mathrm { a n d } \\tilde { y } = \\lambda \\cdot y _ { i } + \\left( 1 - \\lambda \\right) \\cdot y _ { j } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "$\\lambda \\sim \\operatorname { B e t a } ( \\alpha , \\alpha )$ , for $\\alpha \\in ( 0 , \\infty )$ . In this work, we focus on lifelong self-supervised learning and propose Lifelong Unsupervised Mixup (LUMP) that utilizes mixup for UCL by incorporating the instances stored in the replay-buffer from the previous tasks into the vicinal distribution. In particular, LUMP interpolates between the examples of the current task $( x _ { i , \\tau } ) \\in \\mathcal { U } _ { \\tau }$ and random examples selected using uniform sampling from the replay buffer, which encourages the model to behave linearly across a sequence of tasks. More formally, LUMP minimizes the objective in Equation (2) and Equation (3) on the following interpolated instances $\\tilde { x } _ { i , \\tau }$ for the current task $\\tau$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/86930e96d5faa951e6075f5b8db24e758afc3c15852af48ce8092312fae7842a.jpg",
+        "text": "$$\n\\boldsymbol { \\tilde { x } } _ { i , \\tau } = \\lambda \\cdot \\boldsymbol { x } _ { i , \\tau } + ( 1 - \\lambda ) \\cdot \\boldsymbol { x } _ { j , \\boldsymbol { M } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $x _ { j , \\mathcal { M } } \\sim \\mathcal { M }$ denotes the example selected using uniform sampling from replay buffer $\\mathcal { M }$ . The interpolated examples not only augments the past tasks’ instances in the replay buffer but also approximates a regularized loss minimization (Zhang et al., 2021). During UCL, LUMP enhances the robustness of learned representation by revisiting the attributes of the past task that are similar to the current task. Recently, Kim et al. (2020); Lee et al. (2021); Verma et al. (2021); Shen et al. (2022) also employed mixup for contrastive learning. Our work is different from these existing works in that our objective is different, and we focus on unsupervised continual learning. To this end, LUMP successively mitigates catastrophic forgetting and learns discriminative & human-perceptual features over the current state-of-the-art SCL strategies (see Table 1 and Figure 4). ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Baselines. We compare with multiple supervised and unsupervised continual learning baselines across different categories of continual learning methods. ",
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+    {
+        "type": "text",
+        "text": "1. Supervised continual learning. FINETUNE is a vanilla supervised learning method trained on a sequence of tasks without regularization or episodic memory and MULTITASK optimizes the model on complete data. For regularization-based CL methods, we compare against SI (Zenke et al., 2017) and AGEM (Chaudhry et al., 2019a). We include PNN (Rusu et al., 2016) for architecture-based methods. Lastly, we consider GSS (Aljundi et al., 2019) that populates the replay-buffer using solid-angle minimization and DER (Buzzega et al., 2020) matches the network logits sampled through the optimization trajectory for rehearsal during continual learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2. Unsupervised continual learning. We consider the unsupervised variants of various SCL baselines to show the utility of the unsupervised representations for sequential learning. Specifically, we use SIMSIAM (Chen & He, 2021) and BARLOWTWINS (Zbontar et al., 2021), which are the state-of-the-art representational learning techniques for learning the unsupervised continual representations. We compare with FINETUNE and MULTITASK following the supervised learning baselines, and SI (Zenke et al., 2017), PNN (Rusu et al., 2016) for unsupervised regularization and architecture CL methods respectively. For rehearsal-based method, we compare with the UCL variant of DER (Buzzega et al., 2020) described in Section 4.2 ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Datasets. We compare the performance of SCL and UCL on various continual learning benchmarks using single-head ResNet-18 (He et al., 2016) architecture. Split CIFAR-10 (Krizhevsky, 2012) consists of two random classes out of the ten classes for each task. Split CIFAR-100 (Krizhevsky, 2012) consists of five random classes out of the 100 classes for each task. Split Tiny-ImageNet is a variant of the ImageNet dataset (Deng et al., 2009) containing five random classes out of the 100 classes for each task with the images sized $6 4 \\times 6 4$ pixels. ",
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+    {
+        "type": "text",
+        "text": "Training and evaluation setup. We follow the hyperparameter setup of Buzzega et al. (2020) for all the SCL strategies and tune them for the UCL representation learning strategies. All the learned representations are evaluated with KNN classifier (Wu et al., 2018) across three independent runs. Further, we use the hyper-parameters obtained by SimSiam for training UCL strategies with BarlowTwins to analyze the sensitivity of UCL to hyper-parameters and for a fair comparison between different methods. We train all the UCL methods for 200 epochs and evaluate with the KNN classifier (Wu et al., 2018). We provide the hyper-parameters in detail in Table A.5. ",
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+    {
+        "type": "text",
+        "text": "5.2 QUANTITATIVE RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Evaluation on SimSiam. Table 1 shows the evaluation results for supervised and unsupervised representations learnt by SimSiam (Chen & He, 2021) across various continual learning strategies. In all cases, continual learning with unsupervised representations achieves significantly better performance than supervised representations with substantially lower forgetting. For instance, SI with UCL obtains better performance and $6 8 \\%$ , $5 4 \\%$ , and $4 4 \\%$ lower forgetting relative to the best-performing SCL strategy on Split CIFAR-10, Split CIFAR-100, and Split Tiny-ImageNet, respectively. Surprisingly, FINETUNE with UCL achieves higher performance and significantly lower forgetting in comparison to all SCL strategies except DER. Furthermore, LUMP improves upon the UCL strategies: $2 . 8 \\%$ and $5 . 9 \\%$ relative increase in accuracy and $1 5 \\%$ and $5 7 . 1 \\%$ relative decrease in forgetting on Split CIFAR-100 and Split Tiny-ImageNet, respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Evaluation on BarlowTwins. To verify that unsupervised representations are indeed more robust to catastrophic forgetting, we train BarlowTwins (Zbontar et al., 2021) on a sequence of tasks. We notice that the representations learned with BarlowTwins substantially improve the accuracy and forgetting over SCL: $7 1 . 4 \\%$ , $6 9 . 7 \\%$ and $7 3 . 2 \\%$ decrease in forgetting with FINETUNE on Split CIFAR-10, Split CIFAR-100 and Split Tiny-ImageNet respectively. Similarly, we observe that SI, and DER are more robust to catastrophic forgetting; however, PNN underperforms on complicated tasks since feature accumulation using adaptor modules is insufficient to construct useful representations for current task adaptation. Interestingly, representations learnt with BarlowTwins achieve lower forgetting for FINETUNE, DER and LUMP than SimSiam with comparable accuracy across all the datasets. ",
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+    {
+        "type": "table",
+        "img_path": "images/404b290d782b25d10227baaefec667204c6d0999f3d801114b15919ba7db1a3d.jpg",
+        "table_caption": [
+            "Table 1: Accuracy and forgetting of the learnt representations on Split CIFAR-10, Split CIFAR-100 and Split Tiny-ImageNet on Resnet-18 architecture with KNN classifier (Wu et al., 2018). All the values are measured by computing mean and standard deviation across three trials. The best and second-best results are highlighted in bold and underline respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td colspan=\"2\">SPLIT CIFAR-10</td><td colspan=\"2\">SPLIT CIFAR-100</td><td colspan=\"2\">SPLIT TINY-IMAGENET</td></tr><tr><td colspan=\"7\">ACCURACY FORGETTING ACCURACY FORGETTING ACCURACY FORGETTING</td></tr><tr><td colspan=\"7\">SUPERVISED CONTINUAL LEARNING</td></tr><tr><td>FINETUNE</td><td>82.87 (± 0.47)</td><td>14.26 (± 0.52)</td><td>61.08 (± 0.04)</td><td>31.23 (± 0.41)</td><td>53.10 (± 1.37)</td><td>33.15 (± 1.22)</td></tr><tr><td>PNN (Rusu et al., 2016)</td><td>82.74 (± 2.12)</td><td></td><td>66.05 (±0.86)</td><td></td><td>64.38 (± 0.92)</td><td></td></tr><tr><td>SI (Zenke et al., 2017)</td><td>85.18 (± 0.65)</td><td>11.39 (± 0.77)</td><td>63.58 (± 0.37)</td><td>27.98 (± 0.34)</td><td>44.96 (± 2.41)</td><td>26.29 (± 1.40)</td></tr><tr><td>A-GEM (Chaudhry et al., 2019a)</td><td>82.41 (± 1.24)</td><td>13.82 (± 1.27)</td><td>59.81 (± 1.07)</td><td>30.08 (± 0.91)</td><td>60.45 (± 0.24)</td><td>24.94 (± 1.24)</td></tr><tr><td>Gss (Aljundi et al.,2019)</td><td>89.49 (± 1.75)</td><td>7.50 (± 1.52)</td><td>70.78 (± 1.67)</td><td>21.28 (± 1.52)</td><td>70.96 (± 0.72)</td><td>14.76 (± 1.22)</td></tr><tr><td>DER (Buzzega et al., 2020)</td><td>91.35 (± 0.46)</td><td>5.65 (± 0.35)</td><td>79.52 (± 1.88)</td><td>12.80 (± 1.47)</td><td>68.03 (±0.85)</td><td>17.74 (± 0.65)</td></tr><tr><td>MULTITASK</td><td>97.77 (± 0.15)</td><td></td><td>93.89 (±0.78)</td><td></td><td>91.79 (± 0.46)</td><td></td></tr><tr><td colspan=\"7\">UNSUPERVISED CONTINUAL LEARNING</td></tr><tr><td colspan=\"7\">FINETUNE PNN (Rusu et al., 2016)</td></tr><tr><td></td><td>90.11 (±0.12) 90.93 (± 0.22)</td><td>5.42 (±0.08)</td><td>75.42 (± 0.78)</td><td>10.19 (± 0.37)</td><td>71.07 (± 0.20)</td><td>9.48 (±0.56)</td></tr><tr><td>SIISIIN SI (Zenke et al., 2017)</td><td>92.75 (± 0.06)</td><td></td><td>66.58 (± 1.00)</td><td>5.54 (± 1.30)</td><td>62.15 (± 1.35) 72.34 (±0.42)</td><td></td></tr><tr><td>DER (Buzzega et al.,2020)</td><td>91.22 (± 0.30)</td><td>1.81 (± 0.21)</td><td>80.08 (± 1.30)</td><td></td><td>71.90 (± 1.44)</td><td>8.26 (± 0.64)</td></tr><tr><td>LUMP</td><td>91.00 (± 0.40)</td><td>4.63 (±0.26) 2.92 (± 0.53)</td><td>77.27 (± 0.30)</td><td>9.31 (± 0.09)</td><td></td><td>8.36 (± 2.06)</td></tr><tr><td></td><td></td><td></td><td>82.30 (± 1.35)</td><td>4.71 (± 1.52)</td><td>76.66 (± 2.39)</td><td>3.54 (± 1.04)</td></tr><tr><td>MULTITASK</td><td>95.76 (± 0.08)</td><td></td><td>86.31 (±0.38)</td><td></td><td>82.89 (± 0.49)</td><td></td></tr><tr><td colspan=\"7\">FINETUNE</td></tr><tr><td>PNN (Rusu et al., 2016)</td><td>87.72 (± 0.32)</td><td>4.08 (± 0.56)</td><td>71.97 (± 0.54)</td><td>9.45 (± 1.01)</td><td>66.28 (± 1.23)</td><td>8.89 (±0.66)</td></tr><tr><td>SI (Zenke et al., 2017)</td><td>87.52 (± 0.33)</td><td></td><td>57.93 (± 2.98)</td><td></td><td>48.70 (± 2.59)</td><td></td></tr><tr><td>DER (Buzzega et al., 2020)</td><td>90.21 (± 0.08) 88.67 (± 0.24)</td><td>2.03 (±0.22)</td><td>75.04 (± 0.63)</td><td>7.43 (± 0.67)</td><td>56.96 (± 1.48)</td><td>17.04 (± 0.89)</td></tr><tr><td>LUMP</td><td></td><td>2.41 (± 0.26)</td><td>73.48 (± 0.53)</td><td>7.98 (± 0.29)</td><td>68.56 (± 1.47)</td><td>7.87 (± 0.44)</td></tr><tr><td>PPITSIITTIS</td><td>90.31 (± 0.30)</td><td>1.13 (± 0.18)</td><td>80.24 (± 1.04)</td><td>3.53 (± 0.83)</td><td>72.17 (± 0.89)</td><td>2.43 (± 1.00)</td></tr><tr><td>MULTITASK</td><td>95.48 (± 0.14)</td><td></td><td>87.16 (± 0.52)</td><td></td><td>82.42 (± 0.74)</td><td></td></tr><tr><td colspan=\"7\">accuracy overdata size (h hrieteigreeere forgetting overdata size SCL-FT UCL-FT SCL-DER SCL-SI UCL-SI UCL-FT SCL-DER LUMP 10 UCL-SI LUMP 0.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "",
+        "image_caption": [
+            "Figure 2: Evaluation on Few-shot training for Split CIFAR-100 across different number of training instances per task. The results are measured across three independent trials. ",
+            "Figure 3: CKA Feature similarity between two independent UCL models (red), two independent SCL models (blue), and UCL and SCL model (green) for different strategies on Split CIFAR-100 test distribution. "
+        ],
+        "image_footnote": [],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Evaluation on Few-shot training. Figure 2 compares the effect of few-shot training on UCL and SCL, where each task has a limited number of training instances. Specifically, we conduct the experimental evaluation using 100, 200, 500, and 2500 training instances for each task in split CIFAR-100 dataset. Surprisingly, we observe that the gap in average accuracy between SCL and UCL methods widens with a decrease in the number of training instances. Note that UCL decreases the accuracy by $1 5 . 7 8 \\% p$ on average with lower forgetting when the number of training instances decreases from 2500 to 100; whereas, SCL obtains a severe $3 2 . 2 1 \\% p$ deterioration in accuracy. We conjecture that this is an outcome of the discriminative feature embeddings learned by UCL, which discriminates all the images in the dataset and captures more than class-specific information as also observed in Doersch et al. (2020). Furthermore, LUMP improves the performance over all the baselines with a significant margin across all few-shot experiments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Evaluation on OOD datasets. We evaluate the learnt representations on various out-of-distribution (OOD) datasets in Table 2 to measure their generalization to unseen data distributions. In particular, we conduct the OOD evaluation on MNIST (LeCun, 1998), Fashion-MNIST (FMNIST) (Xiao et al., 2017), SVHN (Netzer et al., 2011), CIFAR-10 and CIFAR-100 (Krizhevsky, 2012) using a KNN classifier (Wu et al., 2018). We observe that unsupervised representations outperform the supervised representations in all cases across all the datasets. In particular, the UCL representations learned with Simsiam, and SI on Split-CIFAR-10 improves the absolute performance over the best-performing SCL strategy by $4 . 5 8 \\%$ , $6 . 0 9 \\%$ , $1 5 . 2 6 \\%$ , and $1 7 . 0 7 \\%$ on MNIST, FMNIST, SVHN, and CIFAR-100 respectively. Further, LUMP trained on Split-CIFAR-100 outperforms SI across all datasets and obtains comparable performance with Split CIFAR-10 dataset. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/f106866471529ae507ba6b60bc0e62d5237da93ebb88b497607ef5933fa24e33.jpg",
+        "table_caption": [
+            "Table 2: Comparison of accuracy on out of distribution datasets using a KNN classifier (Wu et al., 2018) on pretrained SCL and UCL representations. We consider MNIST (LeCun, 1998), Fashion-MNIST (FMNIST) (Xiao et al., 2017), SVHN (Netzer et al., 2011) as out of distribution for Split CIFAR-100 and Split CIFAR-10. All the values are measured by computing mean and standard deviation across three trials. The best and second-best results are highlighted in bold and underline respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>IN-CLASS</td><td colspan=\"4\">SPLIT CIFAR-10</td><td colspan=\"4\">SPLIT CIFAR-100</td></tr><tr><td>OUT-OF-CLASS</td><td>MNIST</td><td>FMNIST</td><td>SVHN</td><td>CIFAR-100</td><td>MNIST</td><td>FMNIST</td><td>SVHN</td><td>CIFAR-10</td></tr><tr><td colspan=\"9\">SUPERVISED CONTINUAL LEARNING</td></tr><tr><td>FINETUNE</td><td>86.42 (± 1.11)</td><td>74.47 (±0.84)4</td><td>41.00 (±0.85)</td><td>17.42 (± 0.96)</td><td>75.02 (±3.97)</td><td>62.37 (± 3.20)</td><td>38.05 (±0.73)</td><td>39.18 (± 0.83)</td></tr><tr><td>SI (Zenke et al., 2017)</td><td>87.08 (± 0.79)</td><td>76.41 (± 0.81)</td><td>42.62 (± 1.31)</td><td>19.14 (± 0.91)</td><td>79.96 (± 2.63)</td><td>63.71 (± 1.36)</td><td>40.92 (± 1.64)</td><td>40.41 (± 1.71)</td></tr><tr><td>A-GEM (Chaudhry et al., 2019a)</td><td>86.07 (± 1.94)</td><td>74.74(± 3.21)</td><td>37.77 (± 3.49)</td><td>16.11 (± 0.38)</td><td>77.56 (± 3.21)</td><td>64.16 (± 2.29)</td><td>37.48 (± 1.73)</td><td>37.91 (± 1.33)</td></tr><tr><td>Gss (Aljundi et al., 2019)</td><td>70.36 (± 3.54)</td><td>69.20 (± 2.51)</td><td>33.11 (± 2.26)</td><td>18.21 (± 0.39)</td><td>76.54 (± 0.46)</td><td>65.31 (± 1.72)</td><td>35.72 (± 2.37)</td><td>49.41 (± 1.81)</td></tr><tr><td>DER (Buzzega et al., 2020)</td><td>80.32 (± 1.91)</td><td>70.49 (± 1.54)</td><td>41.48 (± 2.76)</td><td>17.72 (± 0.25)</td><td>87.71 (± 2.23)</td><td>75.97 (± 1.29)</td><td>50.26 (± 0.95)</td><td>59.07 (± 1.06)</td></tr><tr><td>MULTITASK</td><td></td><td>88.79 (± 1.13) 79.50 (±0.52)41.26(± 1.95)</td><td></td><td>27.68 (±0.66)</td><td>92.29 (± 3.37)</td><td>86.12 (± 1.87)</td><td>54.94 (± 1.77)</td><td>54.04 (± 3.68)</td></tr><tr><td colspan=\"9\">UNSUPERVISED CONTINUAL LEARNING</td></tr><tr><td>FINETUNE</td><td>89.23 (± 0.99)</td><td>80.05 (±0.34)</td><td>49.66 (±0.81)</td><td>34.52 (± 0.12)</td><td>85.99 (±0.86)</td><td>76.90 (± 0.11)</td><td>50.09 (± 1.41)</td><td>57.15 (± 0.96)</td></tr><tr><td>SI (Zenke et al., 2017)</td><td>93.72 (± 0.58)</td><td>82.50 (± 0.51)</td><td>57.88 (±0.16)</td><td>36.21 (±0.69)</td><td>91.50 (± 1.26)</td><td>80.57 (±0.93)</td><td>54.07 (± 2.73)</td><td>60.55 (± 2.54)</td></tr><tr><td>DER (Buzzega et al., 2020)</td><td>88.35 (±0.82)</td><td>79.33 (± 0.62)</td><td>48.83 (± 0.55))</td><td>30.68 (± 0.36)</td><td>87.96 (± 2.04)</td><td>76.21 (± 0.63)</td><td>47.70 (± 0.94)</td><td>56.26 (± 0.16)</td></tr><tr><td>WAISNIS LUMP</td><td>91.03 (± 0.22)</td><td>80.78 (±0.88)</td><td>45.18 (± 1.57)</td><td>31.17 (± 1.83)</td><td>91.76 (± 1.17)</td><td>81.61 (± 0.45)</td><td>50.13 (±0.71)</td><td>63.00 (±0.53)</td></tr><tr><td>MULTITASK</td><td>90.69 (± 0.13)</td><td>80.65 (±0.42)</td><td>47.67 (± 0.45)</td><td>39.55 (± 0.18)</td><td>90.35 (±0.24)</td><td>81.11 (± 1.86)</td><td>52.20 (± 0.61)</td><td>70.19 (± 0.15)</td></tr><tr><td>FINETUNE</td><td>86.86 (± 1.62)</td><td>78.37 (± 0.74)</td><td>44.64 (± 2.39)</td><td>28.03 (±0.52)</td><td>76.08 (± 2.86)</td><td>76.82 (±0.83)</td><td>42.95 (±0.90)</td><td>53.12 (± 0.13)</td></tr><tr><td>SI (Zenke et al., 2017)</td><td>90.31 (± 0.69)</td><td>80.58 (± 0.68)</td><td>49.18 (± 0.51)</td><td>31.80 (± 0.4)</td><td>85.24 (± 0.99)</td><td>78.82 (± 0.67)</td><td>45.18 (± 1.37)</td><td>53.99 (± 0.56)</td></tr><tr><td>DER (Buzzega et al.,2020)</td><td>85.15 (± 2.19)</td><td>77.96 (± 0.59)</td><td>45.68 (± 0.93)</td><td>27.83 (± 0.86)</td><td>78.08 (± 1.95)</td><td>76.67 (±0.68)</td><td>44.58 (± 1.01)</td><td>53.24 (±0.82)</td></tr><tr><td>LUMP</td><td>88.73 (± 0.54)</td><td>81.69 (± 0.45)</td><td>51.53(± 0.41)</td><td>31.53 (± 0.36)</td><td>90.22 (± 1.39)</td><td>81.28 (± 0.91)</td><td>50.24 (± 0.95)</td><td>60.76 (± 0.87)</td></tr><tr><td>PBPIIIITTIS MULTITASK</td><td>88.63 (± 1.38)</td><td>79.49 (± 0.29)</td><td>49.24 (± 2.44)</td><td>36.33 (±0.29)</td><td>86.98 (± 1.70)</td><td>79.40 (± 1.10)</td><td>50.19 (±0.81)</td><td>49.50 (± 0.38)</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "5.3 QUALITATIVE ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Similarity in feature and parameter space. We analyze the similarity between the representations learnt between (i) Two independent UCL models, (ii) Two independent SCL models (iii) SCL and UCL models using centered kernel alignment (CKA) (Kornblith et al., 2019) in Figure 3, which provides a score between 0 and 1 measuring the similarity between a pair of hidden representations. For two representations Θ1 : X → Rd1 and Θ2 : X → Rd1 , CKA(Θ1, Θ2) = ||Cov(Θ1(x),Θ2(x))||2F||Cov(Θ1(x))||F ·||Cov(Θ2(x))||F , where covariances are with respect to the test distribution. Additionally, we measure the $\\ell _ { 2 }$ distance (Neyshabur et al., 2020) between the parameters of two independent UCL models (see Table 3) and two independent SCL models (see Table 4). First, we observe that the representations learned by two independent UCL methods have a high feature similarity and lower $\\ell _ { 2 }$ distance compared to the two independent SCL methods, demonstrating UCL representations’ robustness. Second, we note that the representations between any two independent models are highly similar in the lower layers indicating that they learn similar high-level features, including edges and shapes; however, the features are dissimilar for the higher modules. Lastly, we see that the representations between a UCL and SCL model are similar in the lower layers but diverge in the higher layers across all CL strategies. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Visualization of feature space. Next, we visualize the learned features to dissect further the representations learned by UCL and SCL strategies. Figure 4 shows the visualization of the latent feature maps for tasks $\\mathcal { T } _ { 0 }$ and $\\mathcal { T } _ { 1 3 }$ after the completion of continual learning. For $\\mathcal { T } _ { 0 }$ , we observe that the SCL methods are prone to catastrophic forgetting, as the features appear noisy and do not have coherent patterns. In contrast, the features learned by UCL strategies are perceptually relevant and robust to catastrophic forgetting, with LUMP learning the most distinctive features. Similar to $\\mathcal { T } _ { 0 }$ , we observe that the UCL features are more relevant and distinguishable than SCL for $\\mathcal { T } _ { 1 3 }$ . Note that we randomly selected the examples and feature maps for all visualizations. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Loss landscape visualization. To gain further insights, we visualize the loss landscape of task $\\mathcal { T } _ { 0 }$ after the completion of training on task $\\mathcal { T } _ { 0 }$ and $\\mathcal { T } _ { 1 9 }$ for various UCL and SCL strategies in Figure 5. We measure the cross-entropy loss for all methods with a randomly initialized linear classifier for a fair evaluation of two different directions. We use the visualization tool from Li et al. (2018) that searches the task loss surface by repeatedly adding random perturbations to model weights. We observe that the loss landscape after $\\mathcal { T } _ { 0 }$ looks quite similar across all the strategies since the forgetting does not exist yet. However, after training $\\mathcal { T } _ { 1 9 }$ , there is a clear difference with the UCL strategies obtaining a flatter and smoother loss landscape because UCL methods are more stable and robust to the forgetting, which hurts the loss landscapes of past tasks for SCL. It is important to observe that LUMP obtains a smoother landscape than other UCL strategies, demonstrating its effectiveness. We defer further analyses for feature and loss landscape visualization to Appendix A.2. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/81cf7ed318e16b19d3a3635127a9daed3c72e13c0eb056f18b1b606b7d49f4a1.jpg",
+        "table_caption": [
+            "Table 3: $\\ell _ { 2 }$ distance between UCL parameters after completion of training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MODEL</td><td>FINETUNE</td><td>S1</td><td>DER</td><td>MULTITASK</td></tr><tr><td>FINETUNE</td><td>60.00 (± 1.70)</td><td></td><td></td><td></td></tr><tr><td>SI</td><td>76.46 (± 0.48)</td><td>92.35 (± 0.61)</td><td></td><td></td></tr><tr><td>DER</td><td>55.60 (± 1.42)</td><td>75.54 (± 0.97)</td><td>48.76 (± 1.54)</td><td></td></tr><tr><td>MULTITASK</td><td>61.32 (± 0.59)</td><td>79.95 (± 0.40)</td><td>57.90 (± 0.86)</td><td>61.42 (± 0.78)</td></tr></table>",
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+        "type": "table",
+        "img_path": "images/04d1fcd00bd2b5cc3cf74596e8df470a57384d77fd485c8204033c1f352eafaf.jpg",
+        "table_caption": [
+            "Table 4: $\\ell _ { 2 }$ distance between SCL paraneters after completion of training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>MODEL</td><td>FINETUNE</td><td>SI</td><td>DER</td><td>MULTITASK</td></tr><tr><td>FINETUNE</td><td>183.31 (± 0.10)</td><td></td><td></td><td></td></tr><tr><td>SI</td><td>206.16 (± 0.28)</td><td>226.05 (± 0.13)</td><td></td><td></td></tr><tr><td>DER</td><td>202.61 (± 0.46)</td><td>224.78 (± 0.75)</td><td>219.06 (± 0.27)</td><td></td></tr><tr><td>MULTITASK</td><td>258.12 (± 0.26)</td><td>277.30 (± 0.69)</td><td>271.48 (± 0.45)</td><td>314.84 (± 0.92)</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/c3ac687f242d718cd3c73bee19640019de4ed74e1f83c4ef3f892aa16f6a4dc8.jpg",
+        "image_caption": [
+            "Figure 4: Visualization of feature maps for the second block representations learnt by SCL and UCL strategies (with Simsiam) for ResNet-18 architecture after the completion of CL for Split CIFAR-100 dataset $( n = 2 0 )$ ). "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/9462ac68652082e8aaec80aa8da988e820a12909990da7aa46015a8992027101.jpg",
+        "image_caption": [
+            "Figure 5: Loss landscape visualization of $\\mathcal { T } _ { 0 }$ after the completion of training on task $\\mathcal { T } _ { 0 }$ (top) and $\\mathcal { T } _ { 1 9 }$ (bottom) for Split CIFAR-100 dataset on ResNet-18 architecture. We use Simsiam for UCL methods. "
+        ],
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This work attempts to bridge the gap between unsupervised representation learning and continual learning. In particular, we establish the following findings for unsupervised continual learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Surpassing supervised continual learning. Our empirical evaluation across various CL strategies and datasets shows that UCL representations are more robust to catastrophic forgetting than SCL representations. Furthermore, we notice that UCL generalizes better to OOD tasks and achieves stronger performance on few-shot learning tasks. We propose Lifelong unsupervised mixup (LUMP), which interpolates the unsupervised instances between the current task and past task and obtains higher performance with lower catastrophic forgetting across a wide range of tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Dissecting the learned representations. We conduct a systematic analysis to understand the differences between the representations learned by UCL and SCL strategies. By investigating the similarity between the representations, we observe that UCL and SCL strategies have high similarities in the lower layers but are dissimilar in the higher layers. We also show that UCL representations learn coherent and discriminative patterns and smoother loss landscape than SCL. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Limitations and future work. In this work, we do not consider the high-resolution tasks for CL. We intend to evaluate the forgetting of the learnt representations on ImageNet (Deng et al., 2009) in future work, since UCL shows lower catastrophic forgetting and representation learning has made significant progress on ImageNet over the past years. In follow-up work, we intend to conduct further analysis to understand the behavior of UCL and develop sophisticated methods to continually learn unsupervised representations under various setups, such as class-incremental or task-agnostic CL. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We thank the anonymous reviewers for their insightful comments and suggestions. This work was supported by Microsoft Research Asia, the Engineering Research Center Program through the National Research Foundation of Korea (NRF) funded by the Korean Government MSIT (NRF2018R1A5A1059921), Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program (KAIST) and 2021-0-01696). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies. ",
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+    {
+        "type": "text",
+        "text": "AUTHOR CONTRIBUTIONS ",
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+    {
+        "type": "text",
+        "text": "Divyam Madaan conceived of the presented idea, developed the experimental framework, carried out OOD evaluation, CKA visualization and took the lead in writing the manuscript. Jaehong Yoon performed the hyperparameter search, carried out the visualization of loss landscape and feature maps and performed the few-shot training analysis. Yuanchun Li, Yunxin Liu, and Sung Ju Hwang supervised the project. ",
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+        "text": "Friedemann Zenke, Ben Poole, and Surya Ganguli. Continual learning through synaptic intelligence. In Proceedings of the International Conference on Machine Learning (ICML), 2017. ",
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+            168,
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+    {
+        "type": "text",
+        "text": "Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. In Proceedings of the International Conference on Learning Representations (ICLR), 2018. ",
+        "bbox": [
+            173,
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+    {
+        "type": "text",
+        "text": "Linjun Zhang, Zhun Deng, Kenji Kawaguchi, Amirata Ghorbani, and James Zou. How does mixup help with robustness and generalization? In Proceedings of the International Conference on Learning Representations (ICLR), 2021. ",
+        "bbox": [
+            174,
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+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A SUPPLEMENTARY MATERIAL ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            102,
+            450,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Organization. In the supplementary material, we provide the implementation details followed by the hyper-parameter configurations in Appendix A.1. Further, we show the other experiments we conducted and additional visualizations and results in Appendix A.2. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            176
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.1 EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            193,
+            395,
+            207
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Implementations. We use the DER (Buzzega et al., 2020) open-source codebase1 for all the experiments. In particular, we reproduce all their experimental results for supervised continual learning and use various models with their set of hyper-parameters as our baselines. We follow the original representations for $\\mathrm { S i m } \\mathrm { S i a m } ^ { 2 }$ and BarlowTwins3 for unsupervised continual learning. We verify our implementation by reproducing the reported results on CIFAR-10 in the original paper, where we train the representations on the complete CIFAR-10 dataset and evaluate on the test-set using KNN classifier (Wu et al., 2018). In particular, (Wu et al., 2018) stores the features for each instance in the task-level training set in a discrete memory bank. The optimal feature-level embeddings are then learned by instance-level discrimination, which maximally scatters the features of the training samples. Following prior works in representation learning, we use the task-level training set without any augmentation in the task-incremental setup for the supervised and unsupervised KNN evaluation. ",
+        "bbox": [
+            174,
+            218,
+            825,
+            372
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameter configurations. We use the tuned hyper-parameters reported by Buzzega et al. (2020) for all the SCL experiments. On the other hand, we tune the hyper-parameters for continual learning strategies for UCL. We provide the hyper-parameters setup for UCL for different datasets in Table A.5. We train all the UCL methods with a batch size of 256 for 200 epochs, while training the SCL methods with a batch size of 32 for 50 epochs following Buzzega et al. (2020). We observed that training the SCL methods further lead to a degredation in performance for all the methods. We use the same set of augmentations for both SCL and UCL except that we use RandomResizedCrop with scale in [0.2, 1.0] for UCL (Wu et al., 2018; Chen & He, 2021) and RandomCrop for SCL. For rehearsal-based methods, we use the buffer size 200 for Split CIFAR-10, Split CIFAR-100 and 256 for Split Tiny-ImageNet dataset. We use a learning rate of 0.03 for SGD optimizer with weight decay 5e-4 and momentum 0.9. ",
+        "bbox": [
+            173,
+            378,
+            825,
+            530
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/0d060a69610ff1a5a26f58a698d75afb49745e1a7e329e34a854aa749b5623b6.jpg",
+        "table_caption": [
+            "Table A.5: Hyperparameter configurations for all the datasets on ResNet-18 architecture. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td>SPLIT CIFAR-10</td><td>SPLIT CIFAR-100</td><td>SEQ.TINY-IMAGENET</td></tr><tr><td>S1</td><td>c : 100 m:1</td><td>c : 0.1 m:1</td><td>c : 0.01 m:1</td></tr><tr><td>PNN</td><td>wd : 64</td><td>wd : 12</td><td>wd : 8</td></tr><tr><td>DER</td><td>α : 0.1</td><td>α : 0.1</td><td>α : 0.01</td></tr><tr><td>LUMP</td><td>入: 0.1</td><td>入: 0.1</td><td>入: 0.4</td></tr></table>",
+        "bbox": [
+            223,
+            560,
+            774,
+            647
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2 ADDITIONAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            671,
+            415,
+            685
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We provide additional loss landscape on Split CIFAR-100 in Figure A.6 and Figure A.7, Figure A.8 show the second and third block feature visualizations on Split CIFAR-100 respectively. Figure A.9 shows the feature visualizations for Split Tiny-ImageNet on ResNet-18 architecture. ",
+        "bbox": [
+            176,
+            696,
+            823,
+            739
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/69dfb7f41d6d34dd4ca483cb846c642db74f91d3e8a8f0a208134f58e0d8b215.jpg",
+        "image_caption": [
+            "Figure A.6: Loss landscape visualization of $\\mathcal { T } _ { 0 }$ after the completion of training on task $\\mathcal { T } _ { 0 } , \\mathcal { T } _ { 1 7 } , \\mathcal { T } _ { 1 8 }$ , and $\\mathcal { T } _ { 1 9 }$ for Split CIFAR-100 dataset on ResNet-18 architecture. We use Simsiam for UCL methods. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            195,
+            849,
+            795
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/59887c9b6bc8aced71be07cae33f26a0069fe68cfc5e774efeb90cee9dbe05bd.jpg",
+        "image_caption": [
+            "Figure A.7: Visualization of feature maps for the second block representations learnt by SCL and UCL strategies (with Simsiam) for Resnet-18 architecture after the completion of continual learning for Split CIFAR-100 dataset $n = 2 0$ ). The accuracy is the mean across three runs for the corresponding task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            127,
+            833,
+            851
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/ea5a16a4f7f0ee8d4c02d67c79a880ba9151dc89a3655d440e628d31abcc528e.jpg",
+        "image_caption": [
+            "Figure A.8: Visualization of feature maps for the third block representations learnt by SCL and UCL strategies (with Simsiam) for Resnet-18 architecture after the completion of continual learning for Split CIFAR-100 dataset $n = 2 0$ ). The accuracy is the mean across three runs for the corresponding task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            128,
+            834,
+            851
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/3b14374517bbb4507f4f16d6e77d6f2933fb6e36c554d2969976955b539a8d1a.jpg",
+        "image_caption": [
+            "Figure A.9: Visualization of feature maps for the second block representations learnt by SCL and UCL strategies (with Simsiam) for Resnet-18 architecture after the completion of continual learning for Split Tiny-ImageNet dataset $n = 2 0$ ). The accuracy is the mean across three runs for the corresponding task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            121,
+            834,
+            849
+        ],
+        "page_idx": 17
+    }
+]

parse/dev/9jsZiUgkCZP/9jsZiUgkCZP.md

@@ -0,0 +1,427 @@
+# UNIFIED VISUAL TRANSFORMER COMPRESSION
+
+Shixing $\mathbf { V } \mathbf { u } ^ { 1 , * }$ , Tianlong $\mathbf { C h e n ^ { 1 , * } }$ , Jiayi Shen2, Huan Yuan3, Jianchao Tan3,   
+Sen Yang3, Ji Liu3, Zhangyang Wang1   
+1University of Texas at Austin, 2Texas A&M University, 3Kwai Inc.   
+{shixingyu, tianlong.chen, atlaswang}@utexas.edu, asjyjya-617@tamu.edu, {yuanhuan9412, senyang.nlpr, ji.liu.uwisc}@gmail.com, jianchaotan@kuaishou.com
+
+# ABSTRACT
+
+Vision transformers (ViTs) have gained popularity recently. Even without customized image operators such as convolutions, ViTs can yield competitive performance when properly trained on massive data. However, the computational overhead of ViTs remains prohibitive, due to stacking multi-head self-attention modules and else. Compared to the vast literature and prevailing success in compressing convolutional neural networks, the study of Vision Transformer compression has also just emerged, and existing works focused on one or two aspects of compression. This paper proposes a unified ViT compression framework that seamlessly assembles three effective techniques: pruning, layer skipping, and knowledge distillation. We formulate a budget-constrained, end-to-end optimization framework, targeting jointly learning model weights, layer-wise pruning ratios/masks, and skip configurations, under a distillation loss. The optimization problem is then solved using the primal-dual algorithm. Experiments are conducted with several ViT variants, e.g. DeiT and T2T-ViT backbones on the ImageNet dataset, and our approach consistently outperforms recent competitors. For example, DeiT-Tiny can be trimmed down to $50 \%$ of the original FLOPs almost without losing accuracy. Codes are available online: https://github.com/VITA-Group/UVC.
+
+# 1 INTRODUCTION
+
+Convolution neural networks (CNNs) (LeCun et al., 1989; Krizhevsky et al., 2012; He et al., 2016) have been the de facto architecture choice for computer vision tasks in the past decade. Their training and inference cost significant and ever-increasing computational resources. Recently, drawn by the scaling success of attention-based models (Vaswani et al., 2017) in natural language processing (NLP) such as BERT (Devlin et al., 2018), various works seek to leverage the Transformer architecture to computer vision (Parmar et al., 2018; Child et al., 2019; Chen et al., 2020a). The Vision Transformer (ViT) architecture (Dosovitskiy et al., 2020), and its variants, have been demonstrated to achieve comparable or superior results on a series of image understanding tasks compared to the state of the art CNNs, especially when pretrained on datasets with sufficient model capacity (Han et al., 2021).
+
+Despite the emerging power of ViTs, such architecture is shown to be even more resource-intensive than CNNs, making its deployment impractical under resource-limited scenarios. That is due to the absence of customized image operators such as convolution, the stack of self-attention modules that suffer from quadratic complexity with regard to the input size, among other factors. Owing to the substantial architecture differences between CNNs and ViTs, although there is a large wealth of successful CNN compression techniques (Liu et al., 2017; Li et al., 2016; He et al., 2017; 2019), it is not immediately clear whether they are as effective as for ViTs. One further open question is how to best integrate their power for ViT compression, as one often needs to jointly exploit multiple compression means for CNNs (Mishra & Marr, 2018; Yang et al., 2020b; Zhao et al., 2020b).
+
+On the other hand, the NLP literature has widely explored the compression of BERT (Ganesh et al., 2020), ranging from unstructured pruning (Gordon et al., 2020; Guo et al., 2019), attention head pruning (Michel et al., 2019) and encoder unit pruning (Fan et al., 2019); to knowledge distillation (Sanh et al., 2019), layer factorization (Lan et al., 2019), quantization (Zhang et al., 2020; Bai et al.,
+
+2020) and dynamic width/depth inference (Hou et al., 2020). Lately, earlier works on compressing ViTs have also drawn ideas from those similar aspects: examples include weight/attention pruning (Zhu et al., 2021; Chen et al., 2021b; Pan et al., 2021a), input feature (token) selection (Tang et al., 2021; Pan et al., 2021a), and knowledge distillation (Touvron et al., 2020; Jia et al., 2021). Yet up to our best knowledge, there has been no systematic study that strives to either compare or compose (even naively cascade) multiple individual compression techniques for ViTs – not to mention any joint optimization like (Mishra & Marr, 2018; Yang et al., 2020b; Zhao et al., 2020b) did for CNNs. We conjecture that may potentially impede more performance gains from ViT compression.
+
+This paper aims to establish the first all-in-one compression framework that organically integrates three different compression strategies: (structured) pruning, block skipping, and knowledge distillation. Rather than ad-hoc composition, we propose a unified vision transformer compression (UVC) framework, which seamlessly integrates the three effective compression techniques and jointly optimizes towards the task utility goal under the budget constraints. UVC is mathematically formulated as a constrained optimization problem and solved using the primal-dual algorithm from end to end. Our main contributions are outlined as follows:
+
+• We present UVC that unleashes the potential of ViT compression, by jointly leveraging multiple ViT compression means for the first time. UVC only requires to specify a global resource budget, and can automatically optimize the composition of different techniques.   
+• We formulate and solve UVC as a unified constrained optimization problem. It simultaneously learns model weights, layer-wise pruning ratios/masks, and skip configurations, under a distillation loss and an overall budget constraint.   
+• Extensive experiments are conducted with popular variants of ViT, including several DeiT backbones and T2T-ViT on ImageNet, and our proposal consistently performs better than or comparably with existing methods. For example, UVC on DeiT-Tiny (with/without distillation tokens) yields around $50 \%$ FLOPs reduction, with little performance degradation (only $0 . 3 \% / 0 . 9 \%$ loss compared to the uncompressed baseline).
+
+# 2 RELATED WORK
+
+# 2.1 VISION TRANSFORMER
+
+Transformer (Vaswani et al., 2017) architecture stems from natural language processing (NLP) applications first, with the renowned technique utilizing Self-Attention to exploit information from sequential data. Though intuitively the transformer model seems inept to the special inductive bias of space correlation for images-oriented tasks, it has proved itself of capability on vision tasks just as good as CNNs (Dosovitskiy et al., 2020). The main point of Vision Transformer is that they encode the images by partitioning them into sequences of patches, projecting them into token embeddings, and feeding them to transformer encoders (Dosovitskiy et al., 2020). ViT outperforms convolutional nets if given sufficient training data on various image classification benchmarks.
+
+Since then, ViT has been developed to various different variants first on data efficiency towards training, like DeiT (Touvron et al., 2020) and T2T-ViT (Yuan et al., 2021) are proposed to enhance ViT’s training data efficiency, by leveraging teacher-student and better-crafted architectures respectively. Then modifications are made to the general structure of ViT to tackle other popular downstream computer vision tasks, including object detection (Zheng et al., 2020; Carion et al., 2020; Dai et al., 2021; Zhu et al., 2020), semantic segmentation (Wang et al., 2021a;b), image enhancement (Chen et al., 2021a; Yang et al., 2020a), image generation (Jiang et al., 2021), video understanding (Bertasius et al., 2021), and 3D point cloud processing (Zhao et al., 2020a).
+
+# 2.2 MODEL COMPRESSION
+
+Pruning. Pruning methods can be broadly categorized into: unstructured pruning (Dong et al., 2017; Lee et al., 2018; Xiao et al., 2019) by removing insignificant weight via certain criteria; and structured pruning (Luo et al., 2017; He et al., 2017; 2018; Yu et al., 2018; Lin et al., 2018; Guo et al., 2021; Yu et al., 2021; Chen et al., 2021b; Shen et al., 2021) by zero out parameters in a structured group manner. Unstructured pruning can be magnitude-based (Han et al., 2015a;b), hessian-based (LeCun et al., 1990; Dong et al., 2017), and so on. They result in irregular sparsity, causing sparse matrix operations that are hard to accelerate on hardware (Buluc & Gilbert, 2008; Gale et al., 2019). This can be addressed with structured pruning where algorithms usually calculate an importance score for some group of parameters (e.g., convolutional channels, or matrix rows). Liu et al. (2017) uses the scaling factor of the batch normalization layer as the sensitivity metric. Li et al. (2016) proposes channel-wise summation over weights as the metric. Lin et al. (2020) proposes to use channel rank as sensitivity metric while (He et al., 2019) uses the geometric median of the convolutional filters as pruning criteria. Particularly for transformer-based models, the basic structures that many consider pruning with include blocks, attention heads, and/or fully-connected matrix rows (Chen et al., 2020b; 2021b). For example, (Michel et al., 2019) canvasses the behavior of multiple attention heads and proposes an iterative algorithm to prune redundant heads. (Fan et al., 2019) prunes entire layers to extract shallow models at inference time.
+
+Knowledge Distillation. Knowledge distillation (KD) is a special technique that does not explicitly compress the model from any dimension of the network. KD lets a student model leverage ··soft” labels coming from a teacher network (Hinton et al., 2015) to boost the performance of a student model. This can be regarded as a form of compression from the teacher model into a smaller student. The soft labels from the teacher are well known to be more informative than hard labels and leads to better student training (Yuan et al., 2020; Wei et al., 2020)
+
+Skip Configuration. Skip connection plays a crucial role in transformers (Raghu et al., 2021), by tackling the vanishing gradient problem (Vaswani et al., 2017), or by preventing their outputs from degenerating exponentially quickly with respect to the network depth (Dong et al., 2021).
+
+Meanwhile, transformer has an inborn advantage of a uniform block structure. A basic transformer block contains a Self-Attention module and a Multi-layer Perceptron module, and its output size matches the input size. That implies the possibility to manipulate the transformer depth by directly skipping certain layers or blocks. (Xu et al., 2020) proposes to randomly replace the original modules with their designed compact substitutes to train the compact modules to mimic the behavior of the original modules. (Zhang & He, 2020) designs a Switchable-Transformer Blocks to progressively drop layers from the architecture. To flexibly adjust the size and latency of transformers by selecting adaptive width and depth, DynaBERT (Hou et al., 2020) first trains a width-adaptive BERT and then allows for both adaptive width and depth. LayerDrop (Fan et al., 2019) randomly drops layers at training time; while at test time, it allows for sub-network selection to any desired depth.
+
+# 3 METHODOLOGY
+
+# 3.1 PRELIMINARY
+
+Vision Transformer (ViT) Architecture. To unfold the unified algorithm in the following sections, here we first introduce the notations. There are totally $L$ transformer blocks. In each block $l$ of the ViT, there are two constituents, namely the Multi-head Self-Attention (MSA) module and the MLP module. Uniformly, each MSA for transformer block $l$ has $H$ attention heads originally.
+
+In the Multi-head Self-Attention module, $W _ { Q } ^ { ( l ) } , W _ { K } ^ { ( l ) } , W _ { V } ^ { ( l ) }$ are the weights of the three linear projection matrix in block $l$ that uses the block input $X ^ { l }$ to calculate attention matrices: $Q ^ { l } , K ^ { l }$ , ${ \bf \widehat V } ^ { l }$ . The weights of the projection module that follows self-attention calculation are denoted as $W ^ { ( l , 1 ) }$ , represent the first linear projection module in block l. The MLP module consists of two linear projection modules $W ^ { ( l , 2 ) }$ and $\bar { W } ^ { ( l , 3 ) }$ .
+
+Compression Targets The main parameters that can be potentially compressed in a ViT block are W (l)Q , $\bar { W } _ { Q } ^ { ( l ) } , W _ { K } ^ { ( l ) } , W _ { V } ^ { \bar { ( l ) } }$ and $W ^ { ( l , 1 ) }$ , $W ^ { ( l , 2 ) }$ , $W ^ { ( l , 3 ) }$ . Our goal is to prune the head number and head dimensions simultaneously inside each layer, associated with the layer level skipping, solved in a unified framework. Currently, we do not extend the scope to reduce other dimensions such as input patch number or token size. However, our framework can also pack these parts together easily.
+
+For head number and head dimensions pruning, instead of going into details of QKV computation, we innovate to use $\{ W ^ { ( l , 1 ) } \} _ { 1 \le l \le L }$ to be the proxy pruning targets. Pruning on these linear layers is equivalent to the pruning of head number and head dimension. We also add $\{ W ^ { ( l , 3 ) } \} _ { 1 \leq l \leq L }$ as our pruning targets, since these linear layers do not have dimension alignment issues with other parts, they can be freely pruned, while the output of $\{ W ^ { ( l , 2 ) } \}$ should match with the dimension of block input. We do not prune $W _ { Q } ^ { ( l ) } , W _ { K } ^ { ( l ) } , W _ { V } ^ { ( \bar { l } ) }$ inside each head, since $Q ^ { l } , K ^ { l } ,$ , $V ^ { l }$ should be of the same shape for computing self-attention.
+
+![](images/21e421ed1101839888cf838d9284c553a300e465036c071b49d53e37f58f8890.jpg)  
+Figure 1: The overall framework of UVC, that integrates three compression strategies: (1) Pruning within a block: In a transformer block, we targeting on pruning Self-Attention head numbers $( s ^ { ( l , 1 ) } )$ , neuron numbers within a Self-Attention head $( r ^ { l , i } )$ and the hidden size of MLP module $( s ^ { ( l , 3 ) } )$ as well. (2) Skipping manipulation across blocks: When $\mathbf { \chi } _ { g t } ^ { ( l , 0 ) }$ dominates, directly skip block $l$ and send $X ^ { l }$ into block $l + 1$ ; Otherwise, pass $X ^ { l }$ into block $l$ without skip connection. (3) Knowledge distillation: The original uncompressed model is used to provide soft labels for knowledge distillation.
+
+Besides, skip connection is recognized as another important component to enhance the performance of ViTs (Raghu et al., 2021). The dimension between $X ^ { l }$ and the output of the linear projection module should be aligned. For this sake, $W ^ { ( l , 2 ) }$ is excluded from our compression. Eventually, the weights to be compressed in our subsequent framework are $\{ W ^ { ( l , 1 ) } , W ^ { ( l , 3 ) } \} _ { 1 \le l \le L }$ .
+
+# 3.2 RESOURCE-CONSTRAINED END-TO-END VIT COMPRESSION
+
+We target a principled constrained optimization framework jointly optimizing all weights and compression hyperparameters. We consider two strategies to compress ViT: (1) structural pruning of linear projection modules in a ViT block; and (2) adjusting the skipping patterns across different ViT blocks, including skipping/dropping an entire block. The second point, to our best knowledge, is the first time to be ever considered in ViT compression. The full framework is illustrated in Figure 1.
+
+Alternatively, the two strategies could be considered as enforcing mixed-level group sparsity: the head dimension level, the head number level, and the block number level. The rationale is: when enforced with the same pruning ratios, models that are under finer-grained sparsity (i.e., pruning in smaller groups) are unfriendly to latency, while models that are under coarser-grained sparsity (i.e., pruning in larger groups) is unfriendly to accuracy. The mixed-level group sparsity can hence more flexibly trade-off between latency and accuracy.
+
+The knowledge distillation is further incorporated into the objective to guide the end-to-end process.   
+Below we walk through each component one-by-one, before presenting the unified optimization.
+
+Pruning within a Block To compress each linear projection module on width, we first denote $s ^ { ( l , 3 ) }$ as the number of columns to be pruned for weights $\bar { W } ^ { ( l , 3 ) }$ . Compression for weights $W ^ { ( l , 1 ) }$ is more complicated as it is decided by two degrees of freedom. As the input tensor to be multiplied with $W ^ { ( l , 1 ) }$ is the direct output of attention heads. Hence, within each layer $l$ , we denote $s ^ { ( l , 1 \bar { ) } }$ to be the attention head numbers that need to be pruned, and $r ^ { ( l , i ) }$ the number of output neurons to be pruned for each attention head $i$ . Figure 2 illustrates the two sparsity levels: the head dimension level as controlled by $r ^ { ( l , i ) }$ , and the head number level as controlled by $s ^ { ( l , 1 ) }$ . They give more flexibility to transformer compression by selecting the optimal architecture in a multi-grained way. We emphasize again that those variables above are not manually picked, but rather optimized globally.
+
+Skipping Manipulation across Blocks The next component to jointly optimize is the skip connectivity pattern, across different ViT blocks. For vanilla ViTs, skip connection plays a crucial role in ensuring gradient flow and avoiding feature collapse (Dong et al., 2021). Raghu et al. (2021) suggests that skip connections in ViT are even more influential than in ResNets, having strong effects on performance and representation similarity. However, few existing studies have systematically studied how to strategically adjust skip connections provided a ViT architecture, in order to optimize the accuracy and efficiency trade-off.
+
+In general, it is known that more skip connections help CNNs’ accuracy without increasing the parameter volume (Huang et al., 2017); and ViT seems to favor the same trend (Raghu et al., 2021). Moreover, adding a new skip connection over an existing block would allow us to “skip" it during inference, or “drop" it. This represents more aggressive “layer-wise" pruning, compared to the aforementioned element-wise or conventional structural pruning
+
+While directly dropping blocks might look aggressive at the first glance, there are two unique enabling factors in ViTs that facilitate so. Firstly, unlike in CNNs, ViTs have uniform internal structures: by default, the input/output sizes of Self-Attention (SA) module, MLP module and thus the entire block, are all identical. That allows to painlessly drop any of those components and directly reconcatenating the others, without causing any feature dimension incompatibility. Secondly, Phang et al. (2021) observed that the top few blocks of fine-tuned transformers can be discarded without hurting performance, even with no further tuning. That is because deeper layer features tend to have high cross-layer similarities (while being dissimilar to early layer features), making additional blocks add little to the discriminative power. Zhou et al. (2021) also reported that in deep ViTs, the attention maps gradually become similar and even nearly identical after certain layers. Such cross-block “feature collapse" justifies our motivation to drop blocks.
+
+In view of those, we introduce skip manipulation as a unique compression means to ViT for the first time. Specifically, for each transformer block with the skip connection, we denote $g t ^ { ( l , 0 ) }$ and $g t ^ { ( l , 1 ) }$ as two binary gating variables (summing up to 1), to decide whether to skip this block or not.
+
+The Constraints: We next formulate weight sparsity constraints for the purpose of pruning. As discussed in Sec 3.1, the target of the proposed method is to prune the head number and head dimension simultaneously, which can actually be modeled as a two-level group sparsity problem when choosing $\{ W ^ { ( l , 1 ) } \} _ { 1 \leq l \leq L }$ as proxy compression targets. Specifically, the input dimension of $\{ W ^ { ( l , 1 ) } \} _ { 1 \le l \le L }$ is equivalent to the sum of the dimensions of all heads. Then we can put a two-level group sparsity regularization on $\{ W ^ { ( l , 1 ) } \} _ { 1 \le l \le L }$ input dimension to compress head number and head dimension at the same time, as shown in 1a and 1b. $r ^ { ( l , i ) }$ corresponding to the pruned size of $i _ { t h }$ head. $s ^ { ( l , 1 ) }$ means the pruned number of heads. For $\{ W ^ { ( l , 3 ) } \} _ { 1 \le l \le L }$ , it is our compression target, we just perform standard one level group sparsity regularization on its input dimension, as shown in 1c.
+
+$$
+\begin{array} { l } { { \displaystyle \sum _ { j } \mathbb { I } \left( \left\| W _ { g _ { i j \cdot } } ^ { ( l , 1 ) } \right\| _ { 2 } ^ { 2 } = 0 \right) \geq r ^ { ( l , i ) } } , } \\ { { \displaystyle \sum _ { i } \mathbb { I } \left( \left\| W _ { g _ { i \cdot } } ^ { ( l , 1 ) } \right\| _ { 2 } ^ { 2 } = 0 \right) \geq s ^ { ( l , 1 ) } } , } \\ { { \displaystyle \sum _ { i } \mathbb { I } \left( \left\| W _ { i , \cdot } ^ { ( l , 3 ) } \right\| _ { 2 } ^ { 2 } = 0 \right) \geq s ^ { ( l , 3 ) } } , } \\ { { \displaystyle \forall l = 1 , 2 , . . . , L , \forall i = 1 , 2 , . . . , H } } \end{array}
+$$
+
+where $W _ { i , \mathbf { \alpha } }$ ,. denotes the $i$ -th column of $W ; W _ { g _ { i } }$ ,. denotes the $i$ -th grouped column matrix of $W$ , i.e. the $i$ -th head; and $W _ { g _ { i j } }$ ,. the $j$ -th column of $W _ { g _ { i } , . }$ , which is the $j$ -th column of the $i$ -th head. In other words, in Eqn. 1b, column matrices are grouped by attention heads. Hence among the original $H$ heads in total at block $l$ , at least $s ^ { ( l , 1 ) }$ heads should be discarded. Similarly, Eqn. 1c demands that at least $s ^ { ( l , 3 ) }$ input neurons should be pruned at this linear projection module. Furthermore, Eqn. 1a requests that in the $i$ -th attention head at block $l$ , $r ( l , i )$ of the output units should be set to zeros.
+
+Our method is formulated as a resource constrained compression framework, given a target resource budget, it will compress the model until the budget is reached. Given a backbone architecture, the FLOPs is the function of $s , r$ and $_ { g t }$ , denoted as $\mathcal { R } _ { \mathrm { F l o p s } } ( s , r , g t )$ . We have the constraint as:
+
+$$
+\mathcal { R } _ { \mathrm { F l o p s } } ( s , r , g t ) \leq \mathcal { R } _ { \mathrm { b u d g e t } } ,
+$$
+
+![](images/90b4da141c9976aa17f972933e0610d05ed073b16981112d3cd54d76bba016db.jpg)  
+Figure 2: The two sparsity levels for pruning within a block: the head dimension level as controlled by $r ^ { ( l , i ) }$ , and the head number level as controlled by $s ^ { ( l , 1 ) }$ . When reaching the same pruning ratio, neuron-level sparsity will not remove any head, which is usually unfriendly to latency; while head level sparsity will only remove attention heads, which is usually unfriendly to accuracy.
+
+where $\pmb { s } = \{ s ^ { ( l , 1 ) } , s ^ { ( l , 3 ) } \} _ { 1 \leq l \leq L }$ and $\pmb { r } = \{ r ^ { ( l , i ) } \} _ { 1 \leq l \leq L , 1 \leq i \leq H }$ . $\mathcal { R } _ { \mathrm { b u d g e t } }$ is the resource budget. We present detailed flops computation equations in terms of $\pmb { s }$ , $\mathbfit { \Delta } \mathbf { r }$ and $_ { g t }$ in Appendix.
+
+Those inequalities can be further rewritten into equation forms to facilitate optimization. As an example, we follow (Tono et al., 2017) to reformulate Eqn. 1b as:
+
+$$
+\sum _ { i } \mathbb { I } \left( \left. W _ { \cdot , g _ { i } } \right. _ { 2 } ^ { 2 } = 0 \right) \geq s \Leftrightarrow \left. W _ { \cdot , g } \right. _ { s , 2 } ^ { 2 } = 0 .
+$$
+
+where $\| \mathbf { W } _ { \cdot , g } \| _ { s , 2 } ^ { 2 }$ denotes the Frobenius norm of the sub-matrix of $W$ consisting of $s$ groups of $W$ with smallest group norms. Eqn. 1a and Eqn. 1c have same conversions.
+
+The Objective: The target objective could be written as ( $\lambda$ is a coefficient):
+
+$$
+\operatorname* { m i n } _ { W , g t } \ \mathcal { L } ( W , g t ) = \ell ( W , g t ) + \lambda \ell _ { d i s t i l l } ( W , W _ { t } ) ,
+$$
+
+where $W _ { t }$ denotes the weights from teacher model, i.e., the uncompressed transformer model, and $\ell _ { d i s t i l l } ( * , * )$ is defined as the knowledge distillation loss. We choose the simper $\ell _ { 2 }$ norm as its implementation, as its performance was found to be comparable to the more traditional ${ \mathrm { K } } { \mathrm { - } } { \mathrm { L } }$ divergence (Kim et al., 2021).
+
+The Final Unified Formulation: Summarizing all above, we arrive at our unified optimization as a mini-max formulation, by leveraging the primal-dual method (Buchbinder & Naor, 2009):
+
+$$
+\operatorname* { m i n } _ { W , s , r , g t p , y , z \geq 0 } \operatorname* { m a x } _ { y , s , r , g t \in p , y , z \geq 0 } \operatorname* { m a x } _ { W , s , r , g t p , y , z \geq 0 } \mathcal { L } ( W , g t ) + \underbrace { z \left( R _ { \mathrm { F l o p s } } ( s , r , g t ) - R _ { \mathrm { b u d g e t } } \right) } _ { \mathrm { ~ } } + \underbrace { \sum _ { i = 1 } ^ { N } \sum _ { p , r , g t \geq 0 } \mathcal { L } ( W , g t ) } _ { \mathrm { ~ } } = \mathrm { ~ c ~ o ~ n ~ s ~ t ~ . ~ }
+$$
+
+$$
+\sum _ { l = 1 } ^ { L } \left( y ^ { ( l , 1 ) } \left. W _ { \cdot , g } ^ { ( l , 1 ) } \right. _ { \lceil s ^ { ( l , 1 ) } \rceil , 2 } ^ { 2 } + y ^ { ( l , 3 ) } \left. W _ { \cdot , \cdot } ^ { ( l , 3 ) } \right. _ { \lceil s ^ { ( l , 3 ) } \rceil , 2 } ^ { 2 } \right) + \sum _ { l = 1 } ^ { L } \sum _ { i = 1 } ^ { H } p ^ { ( l , i ) } \left. W _ { \cdot , g _ { i } . } ^ { ( l , 1 ) } \right. _ { \lceil r ^ { ( l , i ) } \rceil , 2 } ^ { 2 }
+$$
+
+Eventually, we use the solution of the above mini-max problem, i.e. $\pmb { s }$ and $\pmb { r }$ to determine the compression ratio of each layer, and $_ { g t }$ to determine the selection of skip configuration. Here, we select the pruned groups of the parameters by directly ranking columns by their norms, and remove those with the smallest magnitudes.
+
+The general updating policy follows the idea of primal-dual algorithm. The full algorithm is outlined in Algorithm 1 in Appendix. We introduce the solutions to each sub-problem in Appendix as well.
+
+# 4 EXPERIMENTAL RESULTS
+
+Datasets and Benchmarks We conduct experiments for image classification on ImageNet (Krizhevsky et al., 2012). We implement UVC on DeiT (Touvron et al., 2020), which has basically the identical architecture compared with ViT (Dosovitskiy et al., 2020) except for an extra distillation token; and a variant of ViT – T2T-ViT (Yuan et al., 2021), with a special token embedding layer that aggregates neighboring tokens into one token and a backbone with a deep-narrow structure. Experiment has been conducted on DeiT-Tiny/Small/Base and T2T-ViT-14 models. For DeiT-Tiny, we also try it without/with the distillation token. We measure all resource consumptions (including the UVC resource constraints) in terms of inference FLOPs.
+
+Training Settings The whole process of our method consists of two steps.
+
+• Step 1: UVC training. We firstly conduct the primal-dual algorithm to the pretrained DeiT model to produce the compressed model under the given resource budget. • Step 2: Post training. When we have the sparse model, we finetune it for another round of training to regain its accuracy loss during compression.
+
+In the two steps mentioned above, we mainly follow the training settings of DeiT (Touvron et al., 2020) except for a relatively smaller learning rate which benefits finetuning of converged models.
+
+For distillation, we select the pretrained uncompressed model to teach the pruned model.
+
+Numerically, the learning rate for parameter $z$ is always changing during the primal-dual algorithm process. Thurs, we propose to use a dynamic learning rate for the parameter $z$ that controls the budget constraint. We use a four-step schedule of $\{ 1 , 5 , 9 , \bar { 1 3 } , 1 7 \}$ in practice.
+
+Baseline methods We adopt several latest compression methods which fall under two categories:
+
+• Category 1: Input Patch Reduction, including $( i )$ PoWER (Goyal et al., 2020) : which accelerates language model BERT inference by eliminating word-vector in a progressive way. $( i i )$ HVT (Pan et al., 2021b): which reduce the token sequence dimension by using max-pooling hierarchically. (iii) PatchSlimming (Tang et al., 2021): which identifies the effective patches in the last layer and then use them to guide the patch selection process of previous layers; and $( i v ) \mathbf { I A - R E D ^ { 2 } }$ (Pan et al., 2021a): which dynamically and hierarchical drops visual tokens at different levels using a learned policy network.
+
+• Category 2: Model Weight Pruning, including $( v )$ SCOP (Tang et al., 2020): which is originally a channel pruning method used on CNN, we follow (Tang et al., 2021) and implement it on ViT. $( v i )$ Vision Transformer Pruning (VTP) (Zhu et al., 2021): which trains transformers with sparsity regularization to let important dimensions emerge. (vii) SViTE (Chen et al., 2021b): which jointly optimizes model parameters and explores sparse connectivity throughout training, ending up with one final sparse network. SViTE belongs to the most competitive accuracy-efficiency trade-off achieved so far, for ViT pruning. We choose its structured variant to be fair with UVC.
+
+# 4.1 MAIN RESULTS
+
+The main results are listed in Tab. 1. Firstly, we notice that most existing methods cannot save beyond $50 \%$ FLOPs without sacrificing too much accuracy. In comparison, UVC can easily go with larger compression rates (up to $\geq 6 0 \%$ FLOPS saving) without compromising as much. For example, when compressing DeiT-Tiny (with distillation token), UVC can trim the model down to a compelling $2 5 0 \%$ of the original FLOPs while losing only $0 . 3 \%$ accuracy (Tab. 3). Especially, compared with the latest SViTE (Chen et al., 2021b) that can only save up to around $30 \%$ FLOPs, we observe UVC to significantly outperform ig at DeiT-Tiny/Small, at less accuracy drops (0.9/0.4, versus 2.1/0.6) with much more aggressive FLOPs savings $( 5 0 . 7 \% / 4 2 . 4 \%$ , versus $2 3 . 7 \% / 3 1 . 6 3 )$ ). For the larger DeiT-Base, while UVC can save $45 \%$ of its FLOPs, we fin that SViTE cannot be stably trained at such high sparsity. UVC also yield comparable accuracy with the other ViT model pruning approach, VTP (Zhu et al., 2021), with more than $10 \%$ FLOPs savings w.r.t. the original backbone.
+
+Secondly, UVC obtains strong results compared with Patch Reduction based methods. UVC performs clearly better than IA-RED2 (Pan et al., 2021a) at DeiT-Base, and outperforms HVT (Pan et al., 2021b) on both DeiT-Tiny/Small models, which are latest and strong competitors. Compared to Patch Slimming Tang et al. (2021), UVC at DeiT-Small reaches $7 9 . 4 4 \%$ top-1 accuracy, while the compression ratio measured by FLOPs is comparable. On other models, we observe UVC to generally save more FLOPs, yet also sacrificing more accuracies. Moreover, as we explained in
+
+Table 1: Comparison of the vision transformers compressed by UVC with different benchmarks on ImageNet. FLOPs remained denotes the remained ratio of FLOPs to the full-model FLOPs.   
+
+<table><tr><td>Model</td><td>Method</td><td>Top-1 Acc. (%)</td><td>FLOPs(G)</td><td>FLOPs remained(%)</td></tr><tr><td rowspan="7">DeiT-Tiny</td><td>Baseline</td><td>72.2</td><td>1.3</td><td>100</td></tr><tr><td>SViTE</td><td>70.12 (-2.08)</td><td>0.99</td><td>76.31</td></tr><tr><td>PatchSlimming</td><td>72.0 (-0.2)</td><td>0.7</td><td>53.8</td></tr><tr><td>UVC</td><td>71.8 (-0.4)</td><td>0.69</td><td>53.1</td></tr><tr><td>HVT</td><td>69.7 (-2.5)</td><td>0.64</td><td>49.23</td></tr><tr><td>UVC</td><td>71.3 (-0.9)</td><td>0.64</td><td>49.23</td></tr><tr><td>UVC</td><td>70.6 (-1.6)</td><td>0.51</td><td>39.12</td></tr><tr><td rowspan="8">DeiT-Small</td><td>Baseline</td><td>79.8</td><td>4.6</td><td>100</td></tr><tr><td>SViTE</td><td>79.22 (-0.58)</td><td>3.14</td><td>68.36</td></tr><tr><td>PoWER</td><td>78.3 (-1.5)</td><td>2.7</td><td>58.7</td></tr><tr><td>UVC</td><td>79.44 (-0.36)</td><td>2.65</td><td>57.61</td></tr><tr><td>SCOP</td><td>77.5 (-2.3)</td><td>2.6</td><td>56.4</td></tr><tr><td>PatchSlimming</td><td>79.4 (-0.4)</td><td>2.6</td><td>56.5</td></tr><tr><td>HVT</td><td>78.0 (-1.8)</td><td>2.40</td><td>52.2</td></tr><tr><td>UVC</td><td>78.82 (-0.98)</td><td>2.32</td><td>50.41</td></tr><tr><td rowspan="6">DeiT-Base</td><td>Baseline</td><td>81.8</td><td>17.6</td><td>100</td></tr><tr><td>IA-RED²</td><td>80.3 (-1.5)</td><td>11.8</td><td>67.04</td></tr><tr><td>SViTE</td><td>82.22 (+0.42)</td><td>11.87</td><td>66.87</td></tr><tr><td>VTP</td><td>80.7 (-1.1)</td><td>10.0</td><td>56.8</td></tr><tr><td>PatchSlimming</td><td>81.5 (-0.3)</td><td>9.8</td><td>55.7</td></tr><tr><td>UVC</td><td>80.57 (-1.23)</td><td>8.0</td><td>45.50</td></tr><tr><td rowspan="5">T2T-ViT-14</td><td>Baseline</td><td>81.5</td><td>4.8</td><td>100</td></tr><tr><td>PoWER</td><td>79.9 (-1.6)</td><td>3.5</td><td>72.9</td></tr><tr><td>UVC</td><td>80.4 (-1.1)</td><td>2.90</td><td>60.4</td></tr><tr><td>UVC</td><td>79.6 (-1.9)</td><td>2.47</td><td>51.5</td></tr><tr><td>UVC</td><td>78.9 (-2.6)</td><td>2.11</td><td>44.0</td></tr></table>
+
+Table 2: Ablation study on the modules implemented in UVC. The first part present single technique ablation. The second part presents the result to sequentially apply Skip configuration and Pruning.   
+
+<table><tr><td>Method</td><td colspan="2">DeiT-Tiny Acc. (%) FLOPs remained(%)</td></tr><tr><td>Uncompressed baseline</td><td>72.2</td><td>100</td></tr><tr><td>Only skip manipulation</td><td>68.72</td><td>51.85</td></tr><tr><td>Only pruning within a block</td><td>70.52</td><td>50.69</td></tr><tr><td>Without Knowledge Distillation</td><td>69.34</td><td>51.23</td></tr><tr><td>Skip-&gt;Prune 71.49%-&gt;49.39%</td><td>66.84</td><td>49.39</td></tr><tr><td>Prune-→Skip 69.96%-&gt;54.88%</td><td>69.68</td><td>54.88</td></tr><tr><td>Prune-→Skip 63.25%-&gt;50.73%</td><td>70.02</td><td>50.73</td></tr><tr><td>UvC</td><td>71.3</td><td>49.23</td></tr></table>
+
+Section 3.1, those input token reduction methods represent an orthogonal direction to the model weight sparsification way that UVC is pursuing. UVC can also be seamlessly extended to include token reduction into the joint optimization - a future work that we would pursue.
+
+Thirdly, we test UVC on compressing T2T-ViT (Yuan et al., 2021). In the last row of Tab. 1, UVC with $\overline { { 4 4 } } \%$ FLOPs achieves only a $2 . 6 \%$ accuracy drop. Meanwhile, UVC achieves a $1 . 9 \%$ accuracy drop with $5 1 . 5 \%$ of the original FLOPs. With $6 0 . 4 \%$ FLOPs, UVC only suffers from a $1 . 1 \%$ accuracy drop, already outperforming PoWER with $7 2 . 9 \%$ FLOPs.
+
+As UVC highlights the integration of three different compression techniques into one joint optimization framework, it would be natural to question whether each moving part is necessary for the pipeline, and how they each contribute to the final result. We conduct this ablation study, and the results are presented in Tab. 2.
+
+Skip Manipulation only. We first present the result when we only conduct skip manipulation and knowledge distillation in our framework, but not pruning within a block. All other protocols remain unchanged. This two-in-a-way method is actually reduced into LayerDropping (Lin et al., 2018). We have the following findings: Firstly, implementing only skip connection manipulation will incur high instability. During optimization, the objective value fluctuates heavily due to the large architecture changes (adding or removing one whole block) and barely converges to a stable solution. Secondly, only applying skip manipulation will be damaging the accuracy remarkably, e.g., by nearly $4 \%$ drop on DeiT-Tiny. That is expected, as only manipulating the model architecture with such a coarse granularity (keeping or skipping a whole transformer block) is very inflexible and prone to collapse.
+
+Pruning within a Block only. We then conduct experiments without using the gating control for skip connections. That leads to only integrating neuron-level and attention-head-level pruning, with knowledge distillation. It turns out to deliver much better accuracy than only skip manipulation, presumably owing to its finer-grained operation. However, it still has a margin behind our joint UVC method, as the latter also benefits from removing block-level redundancy a priori, which is recently observed to widely exist in fine-tuned transformers Phang et al. (2021). Overall, the ablation results endorse the effectiveness of jointly optimizing those components altogether.
+
+Applying Individual Compression Methods Sequentially Our method is a joint optimization process for skip configuration manipulation and pruning, but is this joint optimization necessary?
+
+To address this curiosity, we mark Prune Skip as the process of pruning first, manipulating skip configuration follows, while Skip Prune vice versa. Note that Skip Prune $7 1 . 4 9 \%  4 9 . 3 9 \%$ denotes the pipeline that use skip configuration first to enforce the model FLOPs to be $7 1 . 4 9 \%$ of the original FLOPs, then apply pruning to further compress the model to $4 9 . 3 9 \%$ . Without loss of generality, we approximate both the pruning procedure and skipping procedure to compress $70 \%$ of the previous model, which will approximately result in a model with $50 \%$ of the original FLOPs. Based on this, we implement: $( i )$ Skip Prune $7 1 . 4 9 \% \to 4 9 . 3 9 \%$ It first compresses the model to $71 \%$ by skipping then applies pruning on the original model and results in the worse performance at $6 6 . 8 4 \%$ . (ii)Prune Skip $6 9 . 9 6 \%  5 4 . 8 8 \%$ It applies pruning to $70 \%$ FLOPs then skipping follows. We observe that applying pruning first can stabilize the choice of skip configuration and result in a top-1 accuracy of $6 9 . 6 8 \%$ . Both results endorse the superior trade-off found by UVC.
+
+Furthermore, we design a better setting (Prune Skip $6 3 . 2 5 \%  5 0 . 7 3 \%$ ) to simulate the compression ratio found by UVC. Details can be found in Appendix. The obtained result is much better than previous ones, but still at a $1 \%$ performance gap from the result provided by UVC. We assign the extra credit to jointly optimizing the architecture and model weights.
+
+# 5 CONCLUSION
+
+In this paper, we propose UVC, a unified ViT compression framework that seamlessly assembles pruning, layer skipping, and knowledge distillation as one. A budget-constrained optimization framework is formulated for joint learning. Experiments demonstrate that UVC can aggressively trim down the prohibit computational costs in an end-to-end way. Our future work will extend UVC to incorporating weight quantization as part of the end-to-end optimization as well.
+
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+
+# A APPENDIX
+
+# A.1 UPDATING POLICY
+
+Updating Weights $W$ Different from other pruning methods in CNN that use fixed pre-trained weights to select the pruned channels/groups with certain pre-defined metrics (Lin et al., 2020; He et al., 2019), here our subproblem could be considered as following dynamic pruning criteria that will be updated along. Specifically, we solve the following subproblem:
+
+$$
+\mathrm { P r o x } _ { \eta _ { 1 } S ( y , s , p , r , W ) } ( \bar { W } ) = \mathrm { a r g m i n } _ { W } \frac { 1 } { 2 } \left. W - \bar { W } \right. ^ { 2 } + \eta _ { 1 } S \left( y , s , p , r , W \right) ,
+$$
+
+where $\bar { W } = W ^ { t } - \eta _ { 1 } \hat { \nabla } _ { W } \ell \left( W ^ { t } \right)$ . The solution admits a bi-level projection (Yang et al., 2016):
+
+$$
+\begin{array}{c} W _ { \cdot , g _ { i j } } ^ { ( l , 1 ) * } = \left\{ \frac { \bar { W } _ { \cdot , g _ { i j } } ^ { ( l , 1 ) } , } { 1 + 2 \eta _ { 1 } p ^ { ( l , i ) } } , \quad \mathrm { ~ i f ~ } \Big \| \bar { W } _ { \cdot , g _ { i j } } ^ { ( l , 1 ) } \Big \| _ { 2 } ^ { 2 } \geq \Big \| W _ { \cdot , g _ { i \mathrm { l e a s t } \cdot \lceil r ^ { ( l , i ) } \rceil } } ^ { ( l , 1 ) } \Big \| _ { 2 } ^ { 2 } ,  \end{array} \right.
+$$
+
+$$
+W _ { \cdot , g _ { i } } ^ { ( l , 1 ) * } = \left\{ \frac { \bar { W } _ { \cdot , g _ { i } } ^ { ( l , 1 ) } , } { 1 + 2 \eta _ { 1 } y ^ { ( l , 1 ) } } \right. \quad \mathrm { i f } \left\| \bar { W } _ { \cdot , g _ { i } } ^ { ( l , 1 ) } \right\| _ { 2 } ^ { 2 } \geq \left\| W _ { \cdot , g _ { \mathrm { l e a s i - } \lceil s ^ { ( l , 1 ) } \rceil } } ^ { ( l , 1 ) } \right\| _ { 2 } ^ { 2 } ,
+$$
+
+$$
+W _ { \cdot , i } ^ { ( l , 3 ) * } = \{ \frac { \bar { W } _ { \cdot , i } ^ { ( l , 3 ) } , } { 1 + 2 \eta _ { 1 } y ^ { ( l , 3 ) } } , \quad \mathrm { i f } \| \bar { W } _ { \cdot , i } ^ { ( l , 3 ) } \| _ { 2 } ^ { 2 } \geq \| W _ { \cdot , \mathrm { l e a s t - } \lceil s ^ { ( l , 3 ) } \rceil } ^ { ( l , 3 ) } \| _ { 2 } ^ { 2 } ,
+$$
+
+where least- $j$ denotes the index of the (group) columns of $W$ that have $j$ -th least (group) norm.
+
+Updating gt $\pmb { g t } ^ { l } = ( \pmb { g t } ^ { ( l , 0 ) } , \pmb { g t } ^ { ( l , 1 ) } )$ are used to generate a binomial categorical distribution to decide whether pass through block $l$ or directly skip it. As the two variables are discrete, we apply the renowned Gumbel-Softmax (GSM) trick (Jang et al., 2016) to obtain differentiable and polarized sampling. For $\pmb { g t } ^ { l } = ( \pmb { g t } ^ { ( l , 0 ) } , \pmb { g t } ^ { ( l , 1 ) } )$ , given i.i.d Gumbel noise $g$ drawn from $G u m b e l ( 0 , 1 )$ distribution, a soft categorical sample can be drawn by
+
+$$
+G ^ { l } = G S M ( g t ^ { l } ) = S o f t m a x ( ( l o g ( \pmb { g } t ^ { l } ) + g ) / \tau ) \in \mathbb { R } ^ { 2 } ,
+$$
+
+where $G ^ { ( l , 1 ) }$ refers to the continuous possibility to preserve current block $l$ while $G ^ { ( l , 0 ) }$ to drop it.   
+Directly applying the chain rule on $\scriptstyle { \mathcal { L } } ( W , g t )$ w.r.t $_ { g t }$ can now calculate $\tilde { \nabla } _ { g t } \mathcal { L } ( W ^ { t } , g t ^ { t } )$ .
+
+Also, $_ { g t }$ participates in the calculation of FLOPs, by deciding whether to pass certain blocks. Since passing or skipping a block is a dynamic choice during training, we estimate the FLOPs of the $l$ -th block $\mathcal { R } _ { \mathrm { F l o p s } } ( s , r , g t )$ with skip gating by using its expectation. To be specific,
+
+$$
+\begin{array} { r l } & { \mathcal { R } _ { \mathrm { F l o p s } _ { l } } ( s _ { l } , r _ { l } , g t _ { l } ) = \mathbb { E } [ \mathcal { R } _ { \mathrm { F l o p s } _ { l } } ( s _ { l } , r _ { l } , g t _ { l } ) | s _ { l } , r _ { l } ] } \\ & { \quad \quad \quad = G ^ { ( l , 0 ) } \mathcal { R } _ { \mathrm { F l o p s } _ { l } } ( I d e n t i t y ) + G ^ { ( l , 1 ) } \mathcal { R } _ { \mathrm { F l o p s } _ { l } } ( s _ { l } , r _ { l } ) } \\ & { \quad \quad \quad = G ^ { ( l , 1 ) } \mathcal { R } _ { \mathrm { F l o p s } _ { l } } ( s _ { l } , r _ { l } ) } \end{array}
+$$
+
+Hence, updating policy for $_ { g t }$ is formulated as:
+
+$$
+\begin{array} { r l } & { g t ^ { t + 1 } = g t ^ { t } - \eta _ { 4 } \left( \tilde { \nabla } _ { g t } \mathcal { L } \left( W , g t ^ { t } \right) + \tilde { \nabla } _ { g t } z \left( R _ { \mathrm { F l o p s } } \left( s , r , g t ^ { t } \right) - R _ { \mathrm { b u d g e t } } \right) \right) } \\ & { \qquad = g t ^ { t } - \eta _ { 4 } \left( \tilde { \nabla } _ { g t } \mathcal { L } \left( W , g t ^ { t } \right) + \tilde { \nabla } _ { g t } z R _ { \mathrm { F l o p s } } ( s , r ) G S M ( g t ^ { t } ) \right) } \end{array}
+$$
+
+Updating $\pmb { s }$ and $\mathbfit { \Delta } \mathbf { r }$ Similar to the updating policy of $_ { g t }$ , one gradient term w.r.t. $\pmb { s }$ and $\pmb { r }$ are $\tilde { \nabla } _ { s } z \left( R _ { \mathrm { F l o p s } } \left( s , r , g t \right) - R _ { \mathrm { b u d g e t } } \right)$ , $\tilde { \nabla } _ { \boldsymbol { r } } z \left( R _ { \mathrm { F l o p s } } \left( s , \boldsymbol { r } , \boldsymbol { g } t \right) - R _ { \mathrm { b u d g e t } } \right)$ respectively.
+
+The other gradient term is calculated on the unified formulation Eqn. 5. Refer to that, $\pmb { s }$ and $\mathbfit { \Delta } \mathbf { r }$ are floating-point numbers during the optimization process. In practise, ceiling functions are operated on them to determine the integer number that should be pruned for each layer. However, the ceiling function ⌈.⌉ is non-differentiable. To solve this problem, we implement Straight-through estimator(STE) (Bengio et al., 2013) to provide a proxy of the gradient when performing the backward pass. We set $\frac { \tilde { \partial } \lceil s \rceil } { \tilde { \partial } s } = 1$
+
+As for $\| \mathbf { W } _ { \cdot , g } \| _ { s , 2 } ^ { 2 }$ term in the sparsity loss, we use $\lVert \boldsymbol { W } _ { \cdot , g } \rVert _ { s + 1 , 2 } ^ { 2 } - \lVert \boldsymbol { W } _ { \cdot , g } \rVert _ { s , 2 } ^ { 2 }$ as the proxy of partial derivative of $\| \mathbf { W } _ { \cdot , g } \| _ { s , 2 } ^ { 2 }$ with respect to $s$ :
+
+$$
+\frac { \tilde { \partial } \left\| W _ { \cdot , g } \right\| _ { s , 2 } ^ { 2 } } { \tilde { \partial } s } = \left\| W _ { \cdot , g _ { \mathrm { l e a s t } \cdot \mathrm { m i n } \left\{ \mathrm { D i m } ( W ) , s + 1 \right\} } } \right\| _ { 2 } ^ { 2 } ,
+$$
+
+where $\operatorname { D i m } ( W )$ is the number of column groups of $W$ . The other two terms in the sparsity loss can be processed similarly.
+
+# A.2 MAIN ALGORITHM
+
+The general updating policy follows the idea of primal-dual algorithm. The full algorithm is outlined in Algorithm 1.
+
+<table><tr><td></td><td colspan="2">Algorithm1: Gradient-based algorithm to solve problem (5) for Unified ViT Compression.</td></tr><tr><td colspan="2">Input: Resource budget Rbudget,learning rates n1, N2, N3, N4, N5, 76, number of total iterations T. Result: Transformer pruned weights W*.</td></tr><tr><td colspan="2">Initialize t=1,W1;</td></tr><tr><td colspan="2">// random or a pre-trained dense model 2 fort←1tordo</td></tr><tr><td colspan="2">Wt+1= ProxmS(yt,st,pt,rt,wt) (Wt -n1VwL(Wt,gt)); 3</td></tr><tr><td colspan="2">// Proximal-SGD gt+1=gt-n2(sS(y+,st,pt,rt,gtt,Wt+1)+zt(RFlops(st,rt,gt）-Rbudget)); 4</td></tr><tr><td colspan="2">// Gradient (STE） Descent</td></tr><tr><td colspan="2">rt+1=rt-n3(S(y,st+1,pt,rt,gtt,Wt+1)+r2+(RFlops(st+1,rt,gt）-Rbudget)); 5</td></tr><tr><td colspan="2">// Gradient (STE） Descent</td></tr><tr><td colspan="2">gt+1=gt-nA （(gtC(Wt+1,gt²)+gtzt (RFlps(8+1,rt+1,gt）-Rbudgt); 6</td></tr><tr><td colspan="2">// Gradient Descent 2t+1=2t+n7(RFlops(st+1,rt+1,gt+1)-Rbudget); 7</td></tr><tr><td colspan="2">// Gradient Ascent 8 // Gradient Ascent</td></tr><tr><td colspan="2">=y(l,1）+s y(l,1)t+1 [[s(l,1)t+1] ； ,2</td></tr><tr><td colspan="2" rowspan="2">yl,3)+ns y(,3)t+1 W(,3)t+1|12</td></tr><tr><td colspan="2">,∀l=1,,L ，</td></tr><tr><td>p(,i)t+1</td><td>[s(l,3)t+1], ,2 W(1)t+1|2 ∀i=1,,H,∀l=1,,L</td></tr></table>
+
+10 W ∗ = W
+
+# A.3 VISUALIZATION
+
+To reveal more intuitions how UVC optimize the architecture, we present two groups of visualizations:
+
+• Sparse connectivity patterns: We first visualize the sparse connectivity patterns in Fig. 3 (a), i.e, how many attention heads are preserved, and how they (sparsely) distribute over all blocks. For the smaller DeiT-Tiny model, UVC tends to more evenly drop attention heads across layers; meanwhile, for the larger DeiT-Base, UVC clearly prefers to drop more attention heads in the early layers.
+
+![](images/930a06420313014f0eaf8b93497ce0147eb00e80a3e3296bbc3be25537362dec.jpg)  
+Figure 3: Visualizations of: (a) sparse attention head patterns for DeiT-Tiny (left) and DeiT-Base (right); (b) skip connection patterns for DeiT-Tiny (top) and DeiT-Base (bottom).
+
+Table 3: Result with distillation token of DeiT   
+
+<table><tr><td>Model</td><td>Method</td><td>Top-1 Acc. (%)</td><td>FLOPs(G)</td><td>FLOPs remained(%)</td></tr><tr><td>DeiT-Tiny</td><td>Baseline</td><td>74.4</td><td>1.3</td><td>100</td></tr><tr><td>(dist token)</td><td>UvC</td><td>74.1(-0.3)</td><td>0.66</td><td>50.58</td></tr></table>
+
+• Skip connection patterns: We next draw the result of skip connection patterns Fig. 3 (b), showing what layers are learned to be directly skipped under unified optimization. We observe UVC’s obvious tendency to drop later layers, which coincides nicely with the observations by (Phang et al., 2021; Zhou et al., 2021).
+
+As is observed in Fig. 3, DeiT-Tiny discarded 2 blocks in total, which results in $84 \%$ of compression ratio. Thus, in Sec. 4.2, to design an optimal architecture for sequentially applying compression methods, we first prune the DeiT-Tiny to $63 \%$ and apply skipping to indulge a $50 \%$ FLOPs’ model to mimick the behavior of UVC.
+
+# A.4 EXTRA RESULTS
+
+In this section, we present the result of DeiT with distillation token to extract the best performance vision transformer can reach under UVC framework. Results are shown in 3.

parse/dev/Bct2f8fRd8S/Bct2f8fRd8S.md

@@ -0,0 +1,331 @@
+# The Unreliability of Explanations in Few-shot Prompting for Textual Reasoning
+
+Xi Ye Greg Durrett Department of Computer Science The University of Texas at Austin {xiye,gdurrett}@cs.utexas.edu
+
+# Abstract
+
+Does prompting a large language model (LLM) like GPT-3 with explanations improve in-context learning? We study this question on two NLP tasks that involve reasoning over text, namely question answering and natural language inference. We test the performance of four LLMs on three textual reasoning datasets using prompts that include explanations in multiple different styles. For these tasks, we find that including explanations in the prompts for OPT, GPT-3 (davinci), and InstructGPT (text-davinci-001) only yields small to moderate accuracy improvements over standard few-show learning. However, text-davinci-002 is able to benefit more substantially.
+
+We further show that explanations generated by the LLMs may not entail the models’ predictions nor be factually grounded in the input, even on simple tasks with extractive explanations. However, these flawed explanations can still be useful as a way to verify LLMs’ predictions post-hoc. Through analysis in our three settings, we show that explanations judged by humans to be good—logically consistent with the input and the prediction—more likely cooccur with accurate predictions. Following these observations, we train calibrators using automatically extracted scores that assess the reliability of explanations, allowing us to improve performance post-hoc across all of our datasets.1
+
+![](images/7e3d6e2fb5650e1d21e0b886189e9e8693241f6aef1af0a1c67f6cca21457fa5.jpg)  
+Figure 1: Prompting GPT-3 with explanations. By including explanations in the in-context examples, we can cause GPT-3 to generate an explanation for the test example as well. In this case, the generated explanation is nonfactual, despite the simple reasoning involved here. However, we show this nonfactuality actually provides a signal that can help calibrate the model.
+
+# 1 Introduction
+
+Recent scaling of pre-training has empowered large language models (LLMs) to learn NLP tasks from just a few training examples “in context,” without updating the model’s parameters (Brown et al., 2020). However, this learning process is still poorly understood: models are biased by the order of in-context examples (Zhao et al., 2021) and may not leverage the instructions or even the labels of the examples in the ways one expects (Min et al., 2022; Webson and Pavlick, 2022). Existing tools for interpreting model predictions have high computational cost (Ribeiro et al., 2016) or require access to gradients (Simonyan et al., 2014; Sundararajan et al., 2017), making them unsuitable for investigating in-context learning or explaining the predictions of prompted models.
+
+One appealing way to gain more insight into predictions obtained through in-context learning is to let the language model “explain itself” (Nye et al., 2021; Wei et al., 2022; Chowdhery et al., 2022; Marasovic et al., 2022; Lampinen et al., 2022). In addition to input-label training pairs in context, one ´ can prompt the language model with an explanation for each pair and trigger the model to generate an explanation for its prediction (Figure 1). Prompting with explanations introduces much richer information compared to using labels alone, which might guide the inference process and allow the model to learn more information from the examples.
+
+In this work, we investigate the nature of the explanations that LLMs generate and whether they can improve few-shot in-context learning for textual reasoning tasks, specifically QA and NLI. Recent prior work that finds success with this approach largely targets symbolic reasoning tasks with a very different structure, such as math word problem solving (Nye et al., 2021; Wei et al., 2022). We experiment on three different datasets spanning QA and NLI with four LLMs: OPT, GPT-3 (davinci), InstructGPT (text-davinci-001), and text-davinci-002. The results suggest that explanations only substantially improve accuracy for text-davinci-002, but give a smaller improvement or even hurt the performance with the other LLMs.
+
+Surprisingly, we find that the explanations generated by LLMs can be unreliable, even for a very simple synthetic dataset. We evaluate the explanations along two axes: factuality, whether the explanation is correctly grounded in the input, and consistency, whether the explanation entails the final prediction. LLMs tend to generate consistent explanations that account for the predictions, but the explanations may not be factual, as as shown in Figure 1. Furthermore, our analysis suggests an unreliable explanation more likely indicates a wrong prediction compared to a reliable explanation.
+
+Despite LLMs’ failures here, we can still benefit from model-generated explanations by using them for calibration. If we are able to automatically assess the reliability of an explanation, we can allow an LLM to return a null answer when its explanation is unreliable, since the prediction in this case is less likely to be correct. Unfortunately, there is no automated way to perfectly assess the reliability, but we can extract features that approximately reflect it. We use these features to calibrate InstructGPT’s2 predictions, and successfully improve the in-context learning performance across all the datasets.
+
+In summary, our main findings are: (1) Simply plugging explanations into the prompt does not always substantially boost the in-context learning performance for textual reasoning. (2) LLMs generate explanations consistent with their predictions, but these explanations might not be factually grounded in the inputs. (3) The factuality of an explanation can serve as an indicator for the correctness of the corresponding prediction. (4) Using features that can approximate the factuality of explanations, we successfully use explanations to improve the in-context learning performance across all tasks.
+
+# 2 Does Prompting with Explanations Improve In-Context Learning?
+
+In this paper, we specifically focus on tasks involving reasoning over natural language. These are tasks where explanations have been traditionally studied (Camburu et al., 2018; Rajani et al., 2019), but which are more complex than tasks like sentiment analysis which are well explained by extractive rationales (Zaidan et al., 2007; DeYoung et al., 2020). We experiment on two tasks, reading comprehension question answering (QA) and natural language inference (NLI), on three English-language datasets. For each dataset, we create a test set with 250 examples.
+
+Figure 2: A SYNTH example and an E-SNLI example. See Figure 3 for ADVHOTPOT examples.   
+
+<table><tr><td>HINXS</td><td>Context: Question: Answer:</td><td>Christopher agrees with Kevin.Tifany agrees with Mathew.Mary hangs out with Danielle.James hangs out with Thomas.Kevin is a student.Matthew is a plumber.Danielle is a student. Thomas is a plumber. Who hangs out with a student? Explanation:Danielle is a student and Mary hangs out with Danielle.</td></tr><tr><td></td><td>Mary Premise:</td><td>A toddler in a green jersey is being folowed bya wheelchair bound woman in ared sweater past awooden bench.</td></tr><tr><td>IINS-3</td><td>Hypothesis: Label:</td><td>A toddler is walking near his wheelchair bound grandmother. Neither Explanation: the woman may not be his grandmother.</td></tr></table>
+
+# 2.1 Datasets
+
+Synthetic Multi-hop QA (SYNTH) In order to have a controlled setting where we can easily understand whether explanations are factual and consistent with the answer, we create a synthetic multi-hop QA dataset. Shown in Figure 2, each example in this dataset asks a bridge question (using the terminology of Yang et al. (2018)) over a context consisting of supporting facts paired with controlled distractors. This dataset is carefully designed to avoid spurious correlations, giving us full understanding over the correct reasoning process and the explanation for every example, which naturally consists of the two supporting sentences. See Appendix B for full details of this dataset.3
+
+Adversarial HotpotQA (ADVHOTPOT) We also test on the English-language Adversarial HotpotQA dataset (Yang et al., 2018; Jiang and Bansal, 2019). We use the adversarially augmented version since InstructGPT achieves high performance on the distractor setting of the original dataset. We make a challenging set of examples by balancing sets of questions on which InstructGPT makes correct and incorrect predictions. The context of each question includes two ground truth supporting paragraphs and two adversarial paragraphs. Full details of preprocessing the ADVHOTPOT dataset can be found in Appendix C.
+
+For ADVHOTPOT, we manually annotated explanations for the training examples. Figure 1 shows an example of such an explanation, highlighted in orange. We could use the supporting sentences as the explanations, but we found they are usually too verbose and not sufficient, e.g., with anaphors that resolve outside of the supporting sentences. Therefore, we manually annotate a set of explanations which clearly describe the reasoning path for each question.
+
+E-SNLI E-SNLI (Camburu et al., 2018) is an English-language classification dataset commonly used to study explanations, released under the MIT license. Shown in Figure 2, each example consists of a premise and a hypothesis, and the task is to classify the hypothesis as entailed by, contradicted by, or neutral with respect to the premise. As a notable contrast to the other datasets, the explanations here are more abstract natural language written by human annotators, as opposed to mostly constructed from extracted snippets of context.
+
+# 2.2 Baselines
+
+We study the effectiveness of plugging in explanations by comparing the in-context learning performance of prompting with or without explanations. Prompting without explanations resembles the standard few-shot in-context learning approach (Few-Shot). To incorporate explanations into the prompt, we consider the following two most commonly used paradigms:
+
+Explain-then-Predict (E-P) prepends an explanation before the label (Figure 1). The language model is expected to generate an explanation first followed by the prediction. The prompting style of past work involving computational traces can be categorized into this paradigm, including Nye et al. (2021) and Wei et al. (2022). This approach is also called a pipeline model in other literature on training models using explanations (Jacovi and Goldberg, 2021; Wiegreffe et al., 2021).
+
+Table 1: Results of prompting with explanations on four large language models. Using explanations leads to small to moderate improves performance on OPT, GPT-3, and InstructGPT, and has more prominent effects on text-davinci-002.   
+
+<table><tr><td colspan="2"></td><td>SYNTH</td><td>ADVHOTPOT</td><td>E-SNLI</td></tr><tr><td rowspan="3">OPT (175B)</td><td>FEW-SHOT</td><td>40.52.8</td><td>49.72.6</td><td>44.03.8</td></tr><tr><td>E-P</td><td>29.60.5</td><td>52.66.5</td><td>39.37.8</td></tr><tr><td>P-E</td><td>40.22.6</td><td>43.34.5</td><td>43.41.6</td></tr><tr><td rowspan="3">GPT-3</td><td>FEW-SHOT</td><td>49.50.6</td><td>49.16.2</td><td>43.35.7</td></tr><tr><td>E-P</td><td>47.12.8</td><td>54.14.1</td><td>40.44.5</td></tr><tr><td>P-E</td><td>51.31.8</td><td>48.74.6</td><td>48.72.4</td></tr><tr><td rowspan="3">InstructGPT</td><td>FEW-SHOT</td><td>54.83.1</td><td>53.22.3</td><td>56.82.0</td></tr><tr><td>E-P</td><td>58.52.1</td><td>58.24.1</td><td>41.82.5</td></tr><tr><td>P-E</td><td>53.61.0</td><td>51.52.4</td><td>59.41.0</td></tr><tr><td rowspan="3">text-davinci-002</td><td>FEW-SHOT</td><td>72.01.4</td><td>77.73.2</td><td>69.12.0</td></tr><tr><td>E-P</td><td>86.93.8</td><td>82.45.1</td><td>75.67.6</td></tr><tr><td>P-E</td><td>81.12.8</td><td>77.24.8</td><td>69.45.0</td></tr></table>
+
+Predict-then-Explain (P-E) generates the explanation after the prediction. Unlike E-P, the predicted explanation does not influence the predicted label, since we use greedy inference and the explanation comes afterwards. However, the explanations in the prompt still impact the predictions.
+
+# 2.3 Setup
+
+For few-shot learning, we use roughly the maximum allowed shots in the prompt that can fit the length limit of OPT (Zhang et al., 2022) and GPT-3 (Brown et al., 2020), which is 16 for SYNTH, 6 for ADVHOTPOT, and 32 for E-SNLI, respectively.4 We experiment with four LLMs, including OPT (175B), GPT-3 (davinci), InstructGPT (text-davinci-001), and text-davinci-002. OPT and GPT-3 are trained using the standard causal language modeling objective, whereas InstructGPT and text-davinci-002 are trained with special instruction data and human annotations. We generate outputs with greedy decoding (temperature set to be 0). Our prompt formats follow those in Brown et al. (2020). The explanations are inserted before/after the prediction with conjunction words like because. Please refer to Appendix A for full prompts. Because the results of in-context learning vary with the examples presented in the input prompt, for each dataset, we randomly sample multiple groups of training shots, and report the mean and standard deviation of the results (subscript). We use 5 groups for InstructGPT, the primary LM we are using throughout our paper, and 3 groups for the rest.
+
+# 2.4 Results
+
+As shown in Table 1, OPT, GPT-3, and InstructGPT show small to moderate improvements from using explanations for textual reasoning tasks. On the two QA tasks, SYNTH and ADVHOTPOT, E-P improves the performance of InstructGPT, the best among these three LMs, from 54.8 to 58.5 and 56.8 to 59.4, respectively.5 On E-SNLI, P-E outperforms FEW-SHOT by 2.6, whereas E-P substantially lags FEW-SHOT. Comparing E-P against P-E on SYNTH and E-SNLI, E-P typically degrades performance (except on SYNTH for InstructGPT) and P-E is inconsistent across the different models, whereas E-P consistently leads to performance improvements on ADVHOTPOT. There is no single winner between the two paradigms of using explanations; choosing the most effective way is task-specific. Overall, vanilla LLMs (OPT and GPT-3) see limited benefit from producing explanations, and even the Instruct-series InstructGPT does not see substantial improvements.
+
+The only exception is text-davinci-002. text-davinci-002 greatly benefits from explanations in the prompt across all three tasks, and E-P is consistently more effective than P-E. However, it is unclear what contributes to this difference. As far as we are aware, the differences between text-davinci-002 and InstructGPT are not described in any publication or blog post.6 Comparing GPT-3 and InstructGPT, we see the move to Instruct series models is not sufficient to explain the difference. Given the lack of transparency with this model, we hesitate to make scientific claims about the results it yields.
+
+Figure 3: Explanations generated for ADVHOTPOT. InstructGPT may generate nonfactual explanations containing hallucination (red) or inconsistent explanations contradicting the answer (red).   
+
+<table><tr><td>naetatr</td><td>Pedro Rubens! The individual chapters were published into 64 &quot;tankobon&quot;by Kodansha. Yoko Shoji(born4June195o,in Mobara,Chiba)isaJapanese mangaartist.She is bestknown for writing &quot;Seito Shokun! Mulder Scully!The individual chapters were published into 14 &quot;tankobon&quot; by Kodansha. Seito Shokun! The individual chapters were published into 24 &quot;tankobon&quot; by Kodansha between. Q:How many chapters does Yoko Shoji&#x27;s most famous manga have?</td></tr><tr><td>Trrnsrtetetr</td><td>A: First,Yoko Shoji&#x27;s mostfamous manga is &quot;Seito Shokun!&quot;.Second,&quot;Seito Shokun!&quot;has 64chapters.The answer is 64. Tim Minchin (December 29,1808 July 31,1875) was the President of the United States. Andrew Johnson (December 29,1808 July 31,1875) was the President of the United States. George Andrew Atzerodt (June 12,1835-July 7,1865)was a conspirator, with John Wilkes Booth. Jesse Andrew Williams (June 12,1835-July7,1865) was a conspirator, with John Wilkes Booth. Q: Who was older, George Atzerodt or Andrew Johnson? A: First,George Atzerodt was bornonJune12,1835.Second,Andrew Johnson was bornon December 29,1808.The answer is George Atzerodt.</td></tr></table>
+
+Table 2: Left: factuality (Fac) and consistency (Con) of the generated explanations. Right: the $\%$ of the examples whose explanation factuality/consistency is congruent with the prediction accuracy. In general, LLMs tend to generate consistent but less likely factual explanations.   
+
+<table><tr><td colspan="2"></td><td>Acc</td><td>Fac</td><td>Con</td><td>Acc=Fac</td><td>Acc=Con</td></tr><tr><td rowspan="7">InstructGPT</td><td colspan="6">reliability of explanations generated by InstructGPT</td></tr><tr><td>SYNTH (E-P)</td><td>58.4</td><td>72.8</td><td>64.8</td><td>66.5</td><td>68.8</td></tr><tr><td>SYNTH (P-E)</td><td>54.8</td><td>51.6</td><td>95.2</td><td>89.6</td><td>57.2</td></tr><tr><td>ADVHP (E-P)</td><td>62.0</td><td>79.6</td><td>91.2</td><td>80.0</td><td>68.4</td></tr><tr><td>ADVHP (P-E)</td><td>54.0</td><td>69.2</td><td>82.0</td><td>77.6</td><td>67.2</td></tr><tr><td>E-SNLI(P-E)</td><td>62.0</td><td>1</td><td>98.8</td><td>1</td><td>62.0</td></tr><tr><td colspan="6">reliability of explanations generated by other LLMs on SYNTH</td></tr><tr><td rowspan="2">OPT (175B)</td><td>SYNTH (E-P)</td><td>30.0</td><td>77.2</td><td>47.2</td><td>45.6</td><td>58.8</td></tr><tr><td>SYNTH (P-E)</td><td>39.6</td><td>64.0</td><td>81.2</td><td>69.2</td><td>49.6</td></tr><tr><td rowspan="2">GPT-3</td><td>SYNTH (E-P)</td><td>46.8</td><td>59.2</td><td>64.8</td><td>66.8</td><td>61.2</td></tr><tr><td>SYNTH (P-E)</td><td>52.4</td><td>52.4</td><td>83.2</td><td>78.4</td><td>58.0</td></tr><tr><td rowspan="2">text-davinci-002</td><td>SYNTH (E-P)</td><td>86.0</td><td>91.6</td><td>85.2</td><td>91.2</td><td>84.8</td></tr><tr><td>SYNTH (P-E)</td><td>81.6</td><td>83.2</td><td>96.4</td><td>95.8</td><td>82.8</td></tr></table>
+
+Our results do not suggest immediate strong improvements from incorporating explanations across all LLMs, even for our synthetic dataset, contradicting recent prior work. This can be attributed to the difference between the tasks we study. The tasks that receive significant benefits from using explanations in Nye et al. (2021) and Wei et al. (2022) are all program-like (e.g., integer addition and program execution), whereas the tasks in this work emphasize textual reasoning grounded in provided inputs. In fact, in Wei et al. (2022) and Chowdhery et al. (2022), explanations only show mild benefit on open-domain QA tasks like StrategyQA (Geva et al., 2021) that are closer to our setting.
+
+# 3 Can LLMs Generate Factual and Consistent Explanations?
+
+Prompting LLMs with explanations and having models generate them may not guarantee higher performance on our tasks. But what about the quality of the model-generated explanations themselves? We assess the reliability of the explanations for the three datasets, measured in terms of two aspects.
+
+Factuality refers to whether a generated explanation is faithfully grounded in the corresponding input context (context for QA and premise/hypothesis pair for NLI). A factual explanation should not contain hallucinations that contradict the context. See Figure 3 for a nonfactual explanation.
+
+Consistency measures if the explanation entails the prediction. Our concept of consistency resembles plausibility as described in Jacovi and Goldberg (2021), in that we assess whether the prediction follows from the explanation as perceived by a human. See Figure 3 for an inconsistent explanation.
+
+For SYNTH, we use rules to automatically judge whether an explanation is factual and consistent on all four LLMs. For ADVHOTPOT and E-SNLI, the authors manually inspected the explanations generated by InstructGPT and annotated them for these two characteristics (more details in Appendix D). Note for each setting, the results are based on the explanations and predictions obtained with a single set of training shots. We only show the results of P-E on E-SNLI, as E-P is substantially worse here.
+
+Results We summarize the results in Table 2. We only report consistency on E-SNLI, as the explanations for ESNLI often require some external commonsense knowledge which cannot be easily grounded in the inputs or judged as true or false (examples in Appendix F). The results suggest a disconnect between the model predictions and the “reasoning” in explanations. On InstructGPT, though using explanations improves its performance across three tasks, the generated explanations are unreliable (upper section), even for the straightforward synthetic setting. Comparing the factuality of explanations for SYNTH generated by GPT-3, InstructGPT, and text-davinci-002, we see that instruction tuning improves the factuality, but even the most powerful text-davinci-002 still fails to generate explanations that are perfectly grounded in the input context. Overall, LLMs tend to generate consistent explanations $( > 8 0 \%$ for all three datasets with the right prompt structure), but the explanations are less likely to be factual, which is concerning as they can deceive a user of the system into believing the model’s answer.
+
+![](images/1587e11713a5abeaab67eca8ef5e40f84e75c8296eec9142aad635647d9a3db3.jpg)  
+of Correct/Incorrect Predictions by Factuality/Consistency   
+Figure 4: Explanations are more likely to be nonfactual than to be inconsistent, and a nonfactual explanation usually indicates an incorrect prediction.
+
+# 3.1 Reliability of Explanations and Prediction Accuracy
+
+LLMs may hallucinate problematic explanations, but this could actually be advantageous if it gives us a way of spotting when the model’s “reasoning” has failed. We investigate the connection between the reliability of an explanation and the accuracy of a prediction and ask whether a reliable explanation indicates an accurate prediction. (This resembles the linguistic calibration of Mielke et al. (2022), but using a different signal for calibration.)
+
+As shown in Table 2 (right), accuracy and factuality/consistency are typically correlated, especially factuality. By knowing whether an explanation is factual, we can guess the model’s accuracy a high fraction of the time (Accuracy $=$ Factuality). A nonfactual explanation very likely means an incorrect prediction on the SYNTH dataset across all four LLMs. On ADVHOTPOT, factuality and InstructGPT’s prediction correspond $8 0 . 0 \%$ of the time, substantially surpassing the prediction accuracy itself. We show fractions of correct and incorrect predictions when the explanations are factual/nonfactual and consistent/inconsistent in Figure 4 for two of our settings. Factual explanations are much more likely paired with correct predictions compared to nonfactual explanations. Consistency is also connected to accuracy but is an inferior indicator compared to factuality in general (Table 2).
+
+# 4 Calibrating In-Context Learning using Explanations
+
+From Section 3.1, we see that a human oracle assessment of the factuality of an explanation could be of substantial use for calibrating the corresponding prediction. Can we automate this process?
+
+We first show how to achieve this goal on the perfectly controlled SYNTH dataset (Section 4.1). On our other two datasets, we use surface lexical matching to approximate semantic matching and give real-valued scores approximately reflecting factuality. Following past work on supervised calibration (Kamath et al., 2020; Chen et al., 2021; Ye and Durrett, 2022), we can learn a calibrator that tunes the probabilities of a prediction based on the score of its explanation (Section 4.2). We show such a calibrator can be trained with a handful of examples beyond those used for in-context learning and successfully improve the in-context learning performance on realistic datasets.7 We note that, as mentioned before, the experiments in this section are conducted on InstructGPT.
+
+# 4.1 Motivating Example: Improving SYNTH Dataset
+
+We first show how post-hoc calibration functions in the controlled SYNTH setting, where we can simply check the factuality of an explanation. Since the generated explanation always follows the format “B is [profession] and A [verb] B.” (example in Figure 2), we can split the explanation into two sentences. The explanation is factual if and only if each of the two sentences exactly matches one of the sentences in the context.
+
+We use the assessment to improve the performance of P-E for SYNTH, where a nonfactual explanation typically indicates an incorrect prediction. This gives us a way to reject presumably incorrect answers. Specifically, we iterate through the top 5 candidate answers (restricted by the API) given by InstructGPT and reject any answer-explanation pair if the explanation is nonfactual until we find a factual one. This procedure dramatically improves the accuracy from $5 2 . 4 \%$ to $7 4 . 8 \%$ . Note that this SYNTH dataset is a challenging task given its lack of reasoning shortcuts: for reference, neither ROBERTA (Liu et al., 2019) nor DEBERTA (He et al., 2021) finetuned with 16 examples can achieve an accuracy surpassing $50 \%$ . With the help of the explanations and the checking procedure, we can use InstructGPT to achieve strong results using few-shot learning.
+
+# 4.2 Learning-based Calibration Framework
+
+Framework We now introduce the framework that can leverage the factuality assessment of an explanation to calibrate a prediction. Let $\pmb { p }$ be the vector of predicted probabilities associated with each class label in NLI (or the probability score of predicted answer in QA). Let $v$ be a scalar value extracted from the explanation to describe the factuality. Then, we can adjust the probabilities accordingly using a linear model: $\pmb { \hat { p } } = \mathrm { s o f t m a x } ( W [ \pmb { p } ; \pmb { v } ] + b )$ , where $\hat { p }$ is the tuned probabilities.
+
+Our calibration framework is extended from classical calibration methods (Platt, 1999; Guo et al., 2017; Zhao et al., 2021), which apply an affine transformation on the probabilities alone: $\hat { p } =$ softmax $( W p + b )$ . In contrast, we use an additional factor $v$ in calibration to incorporate the factuality assessment of the explanation.
+
+There are a small number of parameters ( $W$ and $b$ ) that need to be trained in such a calibration framework. We will rely on a few more examples in addition to the shots we use in the prompt to train the calibrator. Specifically, we use the prompt examples to generate the predictions and explanations for these extra examples, and extract predicted probabilities, factors, and target probabilities triples to construct training data points used to train the calibrator. Note this procedure requires no explanation annotations for the extra examples.
+
+Approximating Factuality We approximate the factuality using lexical overlap between the explanations and the inputs, which we found to work fairly well for our tasks.
+
+ADVHOTPOT: We use an explanation consisting of two sentences (examples in Figure 3) as an illustration. Let $\mathcal { E } = ( E ^ { ( 1 ) } , E ^ { ( 2 ) } )$ be the generated explanation, where $E ^ { ( 1 ) }$ and $E ^ { ( 2 ) }$ are the two sentences, and the $E ^ { ( i ) } = ( e _ { 1 } , e _ { 2 } , \cdot \cdot \cdot )$ contain tokens $e _ { 1 } , e _ { 2 } , \cdots$ . Similarly, let $\mathcal { P } =$ $( P ^ { ( 1 ) } , P ^ { ( 2 ) } , P ^ { ( 3 ) } , P ^ { ( 4 ) } )$ be the context paragraphs, and $P ^ { ( i ) } = ( p _ { 1 } , p _ { 2 } , \cdot \cdot \cdot )$ be the tokens. The factuality estimation of one explanation sentence $E ^ { ( i ) }$ is defined as: $\begin{array} { r } { \mathcal { V } ( E ^ { ( i ) } ) = \operatorname* { m a x } _ { P \in \mathcal { P } } \frac { | E ^ { ( i ) } \cap P | } { | E ^ { ( i ) } | } } \end{array}$ .
+
+Intuitively, the factuality score for a sentence $E$ is defined as the maximum number of overlapping tokens over all paragraphs $P$ , normalized by the number of tokens in $E$ . We then define the factuality score for the whole explanation as $\begin{array} { r } { \mathcal { V } ( \mathcal { E } ) = \mathrm { \bar { \ m i n } } _ { E \in \mathcal { E } } \mathcal { V } ( E ) } \end{array}$ , as it requires all sentences to be factual in order to make the entire explanation factual.8
+
+E-SNLI: The explanations of E-SNLI do not really involve a concept of factuality. Nevertheless, we use an analogous score following the same principle by viewing the premise as the context. Let $\boldsymbol { E } = ( e _ { 1 } , e _ { 2 } , \cdots )$ be the explanation and $P = ( p _ { 1 } , p _ { 2 } , \cdots )$ be the premise. We simply score the explanation by $\begin{array} { r } { \mathcal { V } ( E ) = \frac { | E | \cap | P | } { | E | } } \end{array}$ . The more an explanation overlaps with the premise, the more factual we judge it to be.
+
+# 4.3 Calibrating E-SNLI
+
+Setup For E-SNLI, we use calibration methods to postprocess the final probabilities. Unlike classical temperature scaling (Platt, 1999), note that the methods we use here can actually change the prediction; we will therefore evaluate on accuracy of the calibrated model.
+
+We study the effectiveness of our explanationbased calibrator under different training data sizes varying from 32 to 128. Recall that we only require explanation annotations for 32 data points, and only need the labels for the rest to train the calibrator. For E-SNLI, we calibrate P-E, which is shown to be more effective than E-P in this setting (Section 2.4).
+
+Baselines We provide the performance of finetuned ROBERTA (Liu et al., 2019) model as a reference, finding this to work better than DeBERTa (He et al., 2021). To isolate the effec
+
+Table 3: Accuracy $\mathrm { ( m e a n _ { s t d d e v } ) }$ of various methods on E-SNLI under different data conditions. L denotes number of labels (as well as the total number of examples); E denotes the number of explanations. Calibrating using explanations successfully improves the performance of in-context learning.   
+
+<table><tr><td>w/o Explanation</td><td>32L</td><td>64L</td><td>96L</td><td>128L</td></tr><tr><td>RoBERTa</td><td>40.14.7</td><td>43.05.1</td><td>49.05.2</td><td>54.94.8</td></tr><tr><td>FEW-SHOT</td><td>56.82.0</td><td>1</td><td>1</td><td>1</td></tr><tr><td>FEW-SHOT(NN)</td><td>1</td><td>1</td><td>1</td><td>58.91.0</td></tr><tr><td>FEW-SHOT+PROBCAL</td><td>61.93.8</td><td>62.42.6</td><td>63.22.9</td><td>63.91.2</td></tr><tr><td>w/ Explanation</td><td>32L+32E</td><td>64L+32E</td><td>96L+32E</td><td>128L+32E</td></tr><tr><td>P-E</td><td>59.42.0</td><td>一</td><td></td><td>1</td></tr><tr><td>P-E+PROBCAL</td><td>64.41.8</td><td>65.41.2</td><td>65.41.6</td><td>65.41.9</td></tr><tr><td>P-E+EXPLCAL</td><td>64.22.6</td><td>65.81.3</td><td>67.61.6</td><td>68.51.2</td></tr><tr><td>P-E+ZHANG</td><td>63.03.2</td><td>65.22.2</td><td>65.41.5</td><td>65.92.5</td></tr></table>
+
+tiveness of using explanations for calibration, we introduce three additional baselines using nonexplanation-based calibrators. We apply the probability-based calibrator as described in Section 4.2 on the results obtained on few-shot learning (FEW-SHOT $^ { \cdot + }$ PROBCAL) and predict-then-explain pipeline $( { \mathrm { P } } { \cdot } { \mathrm { E } } { + } { \mathrm { P R O B C A L } }$ ). We note that the parameters of these calibrators are trained using the additional data points, as opposed to being heuristically determined as in Zhao et al. (2021). Furthermore, we experiment with a recently proposed supervised calibrator from Zhang et al. (2021), which uses the CLS representations from an additional language model as features in the calibrator. The probabilities are tuned using $\pmb { \hat { p } } = \mathrm { s o f t m a x } ( W [ \pmb { p } ; \pmb { h } ] + \bar { \boldsymbol { b } } )$ , where $^ { h }$ is the CLS representation. Since we do not have access to the embeddings obtained by GPT-3, we use ROBERTA to extract the vectors instead. We use such a calibrator on top of our best-performing base model, P-E, resulting $\mathrm { P } \mathrm { - } \mathrm { E } +$ ZHANG ET AL. (2021).
+
+Limited by the maximum prompt length, in-context learning is not able to take as input the additional data used for training the calibrator. For a fair comparison, we can allow the in-context model to use this data by varying the prompts across test examples, dynamically choosing the prompt examples to maximize performance. Choosing closer data points for prompting is a common and effective way of scaling up the training data size for in-context learning (Shin et al., 2021; Liu et al., 2021). Following Liu et al. (2021), we test the performance of choosing nearest neighbors for the prompt based on CLS embedding produced by a ROBERTA model (Liu et al., 2019), referred as FEW-SHOT(NN). It is worth clarifying that the FEW-SHOT and FEW-SHOT $^ +$ PROBCAL approaches use the same set of 32 training shots in the prompt for every test example, whereas the shot sets vary from example to example in FEW-SHOT(NN).
+
+Results We show the results in Table 3. We use 5 different groups of training examples and report the mean and standard deviation across the groups. For FEW-SHOT(NN), we only report the results obtained using 128 examples; results using a smaller number of examples will be worse than this.
+
+Under 128 training examples, applying a trained calibrator on top of prompting with explanation (i.e., P- $\Xi +$ EXPLCAL) achieves the best accuracy of $6 8 . 5 \%$ , which is $12 \%$ higher than the performance of the vanilla uncalibrated few-shot in-context learning (FEW-SHOT). P-E+EXPLCAL also outperforms FEW-SHOT $^ +$ PROBCAL and P- $\Xi +$ PROBCAL by $5 \%$ and $3 \%$ , respectively. Using explanations is more effective than using probabilities alone. In addition, $\mathrm { P } \mathrm { - } \mathrm { E } +$ EXPLCAL also outperforms P- $\mathsf { E } +$ ZHANG ET AL. (2021), whose performance is on par with P-E+PROBCAL. This suggests the additional CLS information is not very helpful in this setting.
+
+As the data size increases from 32 to 128, the performance of the explanation-based calibrator keeps improving notably, whereas the performance of probability-based calibrators nearly saturates at a data size of 96. The performance of FEW-SHOT(NN) with 128 training instances only improves the performance by $3 . 3 \%$ , compared to FEW-SHOT with 32 training instances. Choosing nearest neighbors as the shots, while being effective when having access to a large amount of data, is not helpful in the extreme data-scarce regime. Calibrating using explanations is an effective way of using a few extra data points that cannot fit in the prompt, which is a pitfall of standard in-context learning.
+
+Table 4: AUC scores $\left( \mathrm { { m e a n } _ { \mathrm { { s t d } \ d e v } } } \right)$ on ADVHOTPOT under different data conditions. $\mathbf { L }$ and $\mathbf { E }$ denotes the number of label annotations and explanation annotations, respectively. Explanationbased calibration successfully improves the performance on top of prompting with explanations.   
+
+<table><tr><td>w/o Explanation</td><td>6L</td><td>32L</td><td>64L</td></tr><tr><td>FEW-SHOT FEW-SHOT(NN)</td><td>59.62.4</td><td>1</td><td>1</td></tr><tr><td>w/Explanation</td><td>1 6L+6E</td><td>1 32L+6E</td><td>61.30.9 64L+6E</td></tr><tr><td>E-P</td><td>64.42.9</td><td></td><td></td></tr><tr><td>E-P+EXPLCAL</td><td></td><td>66.03.9</td><td>68.83.0</td></tr><tr><td>E-P+ZHANG</td><td>1 1</td><td>65.63.9</td><td>66.13.2</td></tr></table>
+
+![](images/ac62f2c8bc287230803bb7f2b84c7493d086a7bca98eb99419c3217e5589a8e9.jpg)  
+Figure 5: Coverage-Acc curves of various methods on ADVHOTPOT. E- $\mathrm { P } +$ EXPLCAL is better calibrated compared to uncalbrated E-P as well as the other approaches.
+
+Finally, ROBERTA finetuned using 128 shots only achieves an accuracy of $5 4 . 9 \%$ , lagging the performance of GPT-3 based models. The limited training data size is insufficient for finetuning smaller language models like ROBERTA, but is sufficient for $\mathrm { P } \mathrm { - } \mathrm { E } +$ EXPLCAL to be effective.
+
+# 4.4 Calibrating ADVHOTPOT
+
+Setup For the ADVHOTPOT dataset, our calibration takes the form of tuning the confidence scores of the predicted answers to better align them with the correctness of predictions. These confidence scores can be used in a “selective QA” setting (Kamath et al., 2020), where the model can abstain on a certain fraction of questions where it assigns low confidence to its answers. We use the area under coverage-accuracy curve (AUC) to evaluate how well a model is calibrated as in past literature (Kamath et al., 2020; Chen et al., 2021; Zhang et al., 2021; Garg and Moschitti, 2021; Ye and Durrett, 2022). The curve plots the average accuracy with varying fractions (coverage) of questions being answered (examples in Figure 5). For any given coverage, a better calibrated model should be able to identify questions that it performs best on, hence resulting a higher AUC.
+
+We experiment with training data set sizes of 6, 32, and 64. We report the results averaged from 5 trials using different training sets. For ADVHOTPOT, we calibrate E-P, which is shown to be more effective than P-E in this setting (Section 2.4). Our approach is also effective for calibrating P-E; please refer to Appendix E for details.
+
+Results We show the AUC scores in Table 4. By leveraging explanations, $\mathrm { E } { \mathrm { - } } \mathrm { P } +$ EXPLCAL successfully achieves an AUC of 68.8, surpassing both FEW-SHOT by 7 points and E-P by 4 points. We note this is a substantial improvement, given that the upperbound of AUC is constrained by the accuracy of the answers and cannot reach 100. Figure 5 shows the coverage-accuracy curves of various methods averaged across the 5 training runs. E- $\mathrm { P } +$ EXPLCAL always achieves a higher accuracy than its uncalibrated counterpart, E-P, under a certain coverage, and the gap is especially large in the most confident intervals (coverage $< 5 0 \%$ ). E- $\mathrm { P } +$ ZHANG ET AL. (2021) is able to calibrate the predictions on this dataset, but still lags our explanation-based calibrator, E-P+EXPLCAL.
+
+In addition, the explanation-based calibrator can be effective with as few as 32 examples. This is because there are only two parameters (the probability of predicted answer and the explanationbased factor) in the calibrator, which can be easily learned in this few-shot setting. Comparing E $\mathbf { \partial } _ { - } \mathbf { P } { + } \mathbf { E }$ XPLCAL against FEW-SHOT(NN), using nearest neighbors in the prompt is also able to improve the performance compared to using a fixed set of shots (FEW-SHOT), yet our lightweight calibrator can better utilize such a small amount of data, and learn to distinguish more accurate predictions based on the explanations.
+
+# 5 Related Work
+
+Our investigation is centered around in-context learning (Brown et al., 2020), which has garnered increasing interest since the breakthrough of various large pretrained language models. Recent work has been devoted to studying different aspects of in-context learning, including its wayward behaviors (Min et al., 2022; Webson and Pavlick, 2022) and approaches to overcome them (Zhao et al., 2021), whereas our exploration focuses on using explanations.
+
+The utility of explanations for few-shot in-context learning has also been discussed concurrently (Nye et al., 2021; Wei et al., 2022; Marasovic et al., 2022; Chowdhery et al., 2022; Lampinen et al., ´ 2022; Wiegreffe et al., 2022), especially in symbolic reasoning tasks. We differ in that we study more free-form explanations in tasks (QA and NLI, specifically) focusing on textual reasoning over provided contexts. Furthermore, our work focuses on the nature of the explanations generated by LLMs, which are found to be unreliable. Regarding our use of calibration, similar ideas of explanation-based performance estimation have been applied to other tasks (Rajani and Mooney, 2018; Ye et al., 2021; Ye and Durrett, 2022), but we rely on the free-text explanations generated by the model instead of interpretations obtained through post-hoc interpretation techniques.
+
+More broadly, how to use explanations in various forms (textual explanation, highlights, etc.) to train better models is a longstanding problem (Zaidan et al., 2007). Past work has built a series of pipeline models that first generate the explanations and then make predictions purely based on the generated explanations (Wiegreffe et al., 2021; Zhou and Tan, 2021; Chen et al., 2022). Prior research has also explored using explanations as additional supervision to train joint models (Hancock et al., 2018; Dua et al., 2020; Lamm et al., 2021; Stacey et al., 2022). Another line of work seeks to align the reasoning process of a trained model with the explanations, which is typically done by interpreting a prediction post-hoc through explanation techniques and optimizing the distance between the obtained explanation and ground truth explanation (Liu and Avci, 2019; Rieger et al., 2020; Plumb et al., 2020; Erion et al., 2021; Yao et al., 2021). These aforementioned methods all update the model parameters and typically require a considerable amount of explanation annotations to be effective. By contrast, our setting treats language models as pure black boxes and only requires few-shot explanations.
+
+# 6 Discussion & Conclusion
+
+Caveats and Risks of Explanations from Large Language Models Our analysis suggests that LLMs’ internal “reasoning” does not always align with explanations that it generates, as shown by our consistency results. More concerning, the explanations might not be factually grounded in the provided prompt. This shortcoming should caution against any deployment of this technology in practice: because the explanations are grammatical English and look very convincing, they may deceive users into believing the system’s responses even when those responses are incorrect. Section 6 of Bender et al. (2021) discusses these risks in additional detail. The fact that language models can hallucinate explanations is also found in other work (Zhou and Tan, 2021). This result is unsurprising in some sense: without sufficient supervision or grounding, language models do not learn meaning as distinct from form (Bender and Koller, 2020), so we should not expect their explanations to be strongly grounded.
+
+We have shown that even explanations which don’t lead to accuracy gains can still be useful for calibration. However, the lexical overlap feature we use here is a weak signal of explanation correctness (see the example in Figure 1). Strong enough entailment models should theoretically be able to perform this task and work across a range of tasks without fine-tuning. This explanation assessment model can even be a language model itself trained for this particular propose to approach the verification tasks for a given domain by in-context learning.
+
+Conclusion We have explored the capabilities of LLMs in using explanations in in-context learning for textual reasoning. Through our experiments with four LLMs and on two QA datasets and an NLI dataset, we find that simply including explanations in the prompt does not always improve the performance of in-context learning. Our manual analysis demonstrates that LLMs tend to generate nonfactual explanations when making wrong predictions, which can be a useful leverage to assess the correctness of the predictions. Lastly, we showcase how to use explanations to build lightweight calibrators, which successfully improve InstructGPT’s in-context learning performance across all three datasets.
+
+# Acknowledgments
+
+We would like to thank Eunsol Choi, Ruiqi Zhong, Jocelyn Chen, Zayne Sprague, and Jiacheng Xu for their helpful feedback on drafts of this work, as well as the anonymous reviewers for their thoughtful reviews. This work was partially supported by NSF Grant IIS-1814522, NSF CAREER Award IIS-2145280, a grant from Open Philanthropy, a gift from Salesforce Inc., and a gift from Adobe.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We use the GPT-3 Instruct-series API (text-davinci-001).
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See reference (Jiang and Bansal, 2019) and (Camburu et al., 2018).   
+(b) Did you mention the license of the assets? [Yes] See Section 2.1.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We included the Synthetic dataset in the supplementary material.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/Bct2f8fRd8S/Bct2f8fRd8S_content_list.json

@@ -0,0 +1,1781 @@
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+    {
+        "type": "text",
+        "text": "The Unreliability of Explanations in Few-shot Prompting for Textual Reasoning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Xi Ye Greg Durrett Department of Computer Science The University of Texas at Austin {xiye,gdurrett}@cs.utexas.edu ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Does prompting a large language model (LLM) like GPT-3 with explanations improve in-context learning? We study this question on two NLP tasks that involve reasoning over text, namely question answering and natural language inference. We test the performance of four LLMs on three textual reasoning datasets using prompts that include explanations in multiple different styles. For these tasks, we find that including explanations in the prompts for OPT, GPT-3 (davinci), and InstructGPT (text-davinci-001) only yields small to moderate accuracy improvements over standard few-show learning. However, text-davinci-002 is able to benefit more substantially. ",
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+    {
+        "type": "text",
+        "text": "We further show that explanations generated by the LLMs may not entail the models’ predictions nor be factually grounded in the input, even on simple tasks with extractive explanations. However, these flawed explanations can still be useful as a way to verify LLMs’ predictions post-hoc. Through analysis in our three settings, we show that explanations judged by humans to be good—logically consistent with the input and the prediction—more likely cooccur with accurate predictions. Following these observations, we train calibrators using automatically extracted scores that assess the reliability of explanations, allowing us to improve performance post-hoc across all of our datasets.1 ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/7e3d6e2fb5650e1d21e0b886189e9e8693241f6aef1af0a1c67f6cca21457fa5.jpg",
+        "image_caption": [
+            "Figure 1: Prompting GPT-3 with explanations. By including explanations in the in-context examples, we can cause GPT-3 to generate an explanation for the test example as well. In this case, the generated explanation is nonfactual, despite the simple reasoning involved here. However, we show this nonfactuality actually provides a signal that can help calibrate the model. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recent scaling of pre-training has empowered large language models (LLMs) to learn NLP tasks from just a few training examples “in context,” without updating the model’s parameters (Brown et al., 2020). However, this learning process is still poorly understood: models are biased by the order of in-context examples (Zhao et al., 2021) and may not leverage the instructions or even the labels of the examples in the ways one expects (Min et al., 2022; Webson and Pavlick, 2022). Existing tools for interpreting model predictions have high computational cost (Ribeiro et al., 2016) or require access to gradients (Simonyan et al., 2014; Sundararajan et al., 2017), making them unsuitable for investigating in-context learning or explaining the predictions of prompted models. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "One appealing way to gain more insight into predictions obtained through in-context learning is to let the language model “explain itself” (Nye et al., 2021; Wei et al., 2022; Chowdhery et al., 2022; Marasovic et al., 2022; Lampinen et al., 2022). In addition to input-label training pairs in context, one ´ can prompt the language model with an explanation for each pair and trigger the model to generate an explanation for its prediction (Figure 1). Prompting with explanations introduces much richer information compared to using labels alone, which might guide the inference process and allow the model to learn more information from the examples. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, we investigate the nature of the explanations that LLMs generate and whether they can improve few-shot in-context learning for textual reasoning tasks, specifically QA and NLI. Recent prior work that finds success with this approach largely targets symbolic reasoning tasks with a very different structure, such as math word problem solving (Nye et al., 2021; Wei et al., 2022). We experiment on three different datasets spanning QA and NLI with four LLMs: OPT, GPT-3 (davinci), InstructGPT (text-davinci-001), and text-davinci-002. The results suggest that explanations only substantially improve accuracy for text-davinci-002, but give a smaller improvement or even hurt the performance with the other LLMs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Surprisingly, we find that the explanations generated by LLMs can be unreliable, even for a very simple synthetic dataset. We evaluate the explanations along two axes: factuality, whether the explanation is correctly grounded in the input, and consistency, whether the explanation entails the final prediction. LLMs tend to generate consistent explanations that account for the predictions, but the explanations may not be factual, as as shown in Figure 1. Furthermore, our analysis suggests an unreliable explanation more likely indicates a wrong prediction compared to a reliable explanation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Despite LLMs’ failures here, we can still benefit from model-generated explanations by using them for calibration. If we are able to automatically assess the reliability of an explanation, we can allow an LLM to return a null answer when its explanation is unreliable, since the prediction in this case is less likely to be correct. Unfortunately, there is no automated way to perfectly assess the reliability, but we can extract features that approximately reflect it. We use these features to calibrate InstructGPT’s2 predictions, and successfully improve the in-context learning performance across all the datasets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In summary, our main findings are: (1) Simply plugging explanations into the prompt does not always substantially boost the in-context learning performance for textual reasoning. (2) LLMs generate explanations consistent with their predictions, but these explanations might not be factually grounded in the inputs. (3) The factuality of an explanation can serve as an indicator for the correctness of the corresponding prediction. (4) Using features that can approximate the factuality of explanations, we successfully use explanations to improve the in-context learning performance across all tasks. ",
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+    {
+        "type": "text",
+        "text": "2 Does Prompting with Explanations Improve In-Context Learning? ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    {
+        "type": "text",
+        "text": "In this paper, we specifically focus on tasks involving reasoning over natural language. These are tasks where explanations have been traditionally studied (Camburu et al., 2018; Rajani et al., 2019), but which are more complex than tasks like sentiment analysis which are well explained by extractive rationales (Zaidan et al., 2007; DeYoung et al., 2020). We experiment on two tasks, reading comprehension question answering (QA) and natural language inference (NLI), on three English-language datasets. For each dataset, we create a test set with 250 examples. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "table",
+        "img_path": "images/31aa1bfd00617012cf6eebeb05c6c5fc4ea72b4ebaf822c31bb1cef4eec61e45.jpg",
+        "table_caption": [
+            "Figure 2: A SYNTH example and an E-SNLI example. See Figure 3 for ADVHOTPOT examples. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>HINXS</td><td>Context: Question: Answer:</td><td>Christopher agrees with Kevin.Tifany agrees with Mathew.Mary hangs out with Danielle.James hangs out with Thomas.Kevin is a student.Matthew is a plumber.Danielle is a student. Thomas is a plumber. Who hangs out with a student? Explanation:Danielle is a student and Mary hangs out with Danielle.</td></tr><tr><td></td><td>Mary Premise:</td><td>A toddler in a green jersey is being folowed bya wheelchair bound woman in ared sweater past awooden bench.</td></tr><tr><td>IINS-3</td><td>Hypothesis: Label:</td><td>A toddler is walking near his wheelchair bound grandmother. Neither Explanation: the woman may not be his grandmother.</td></tr></table>",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.1 Datasets ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Synthetic Multi-hop QA (SYNTH) In order to have a controlled setting where we can easily understand whether explanations are factual and consistent with the answer, we create a synthetic multi-hop QA dataset. Shown in Figure 2, each example in this dataset asks a bridge question (using the terminology of Yang et al. (2018)) over a context consisting of supporting facts paired with controlled distractors. This dataset is carefully designed to avoid spurious correlations, giving us full understanding over the correct reasoning process and the explanation for every example, which naturally consists of the two supporting sentences. See Appendix B for full details of this dataset.3 ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Adversarial HotpotQA (ADVHOTPOT) We also test on the English-language Adversarial HotpotQA dataset (Yang et al., 2018; Jiang and Bansal, 2019). We use the adversarially augmented version since InstructGPT achieves high performance on the distractor setting of the original dataset. We make a challenging set of examples by balancing sets of questions on which InstructGPT makes correct and incorrect predictions. The context of each question includes two ground truth supporting paragraphs and two adversarial paragraphs. Full details of preprocessing the ADVHOTPOT dataset can be found in Appendix C. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For ADVHOTPOT, we manually annotated explanations for the training examples. Figure 1 shows an example of such an explanation, highlighted in orange. We could use the supporting sentences as the explanations, but we found they are usually too verbose and not sufficient, e.g., with anaphors that resolve outside of the supporting sentences. Therefore, we manually annotate a set of explanations which clearly describe the reasoning path for each question. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "E-SNLI E-SNLI (Camburu et al., 2018) is an English-language classification dataset commonly used to study explanations, released under the MIT license. Shown in Figure 2, each example consists of a premise and a hypothesis, and the task is to classify the hypothesis as entailed by, contradicted by, or neutral with respect to the premise. As a notable contrast to the other datasets, the explanations here are more abstract natural language written by human annotators, as opposed to mostly constructed from extracted snippets of context. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Baselines ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We study the effectiveness of plugging in explanations by comparing the in-context learning performance of prompting with or without explanations. Prompting without explanations resembles the standard few-shot in-context learning approach (Few-Shot). To incorporate explanations into the prompt, we consider the following two most commonly used paradigms: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Explain-then-Predict (E-P) prepends an explanation before the label (Figure 1). The language model is expected to generate an explanation first followed by the prediction. The prompting style of past work involving computational traces can be categorized into this paradigm, including Nye et al. (2021) and Wei et al. (2022). This approach is also called a pipeline model in other literature on training models using explanations (Jacovi and Goldberg, 2021; Wiegreffe et al., 2021). ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/db4018bdabc9988846380c844d9d2c381eb5d1fcd9ba25c57a4d122030785075.jpg",
+        "table_caption": [
+            "Table 1: Results of prompting with explanations on four large language models. Using explanations leads to small to moderate improves performance on OPT, GPT-3, and InstructGPT, and has more prominent effects on text-davinci-002. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td>SYNTH</td><td>ADVHOTPOT</td><td>E-SNLI</td></tr><tr><td rowspan=\"3\">OPT (175B)</td><td>FEW-SHOT</td><td>40.52.8</td><td>49.72.6</td><td>44.03.8</td></tr><tr><td>E-P</td><td>29.60.5</td><td>52.66.5</td><td>39.37.8</td></tr><tr><td>P-E</td><td>40.22.6</td><td>43.34.5</td><td>43.41.6</td></tr><tr><td rowspan=\"3\">GPT-3</td><td>FEW-SHOT</td><td>49.50.6</td><td>49.16.2</td><td>43.35.7</td></tr><tr><td>E-P</td><td>47.12.8</td><td>54.14.1</td><td>40.44.5</td></tr><tr><td>P-E</td><td>51.31.8</td><td>48.74.6</td><td>48.72.4</td></tr><tr><td rowspan=\"3\">InstructGPT</td><td>FEW-SHOT</td><td>54.83.1</td><td>53.22.3</td><td>56.82.0</td></tr><tr><td>E-P</td><td>58.52.1</td><td>58.24.1</td><td>41.82.5</td></tr><tr><td>P-E</td><td>53.61.0</td><td>51.52.4</td><td>59.41.0</td></tr><tr><td rowspan=\"3\">text-davinci-002</td><td>FEW-SHOT</td><td>72.01.4</td><td>77.73.2</td><td>69.12.0</td></tr><tr><td>E-P</td><td>86.93.8</td><td>82.45.1</td><td>75.67.6</td></tr><tr><td>P-E</td><td>81.12.8</td><td>77.24.8</td><td>69.45.0</td></tr></table>",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Predict-then-Explain (P-E) generates the explanation after the prediction. Unlike E-P, the predicted explanation does not influence the predicted label, since we use greedy inference and the explanation comes afterwards. However, the explanations in the prompt still impact the predictions. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 Setup",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For few-shot learning, we use roughly the maximum allowed shots in the prompt that can fit the length limit of OPT (Zhang et al., 2022) and GPT-3 (Brown et al., 2020), which is 16 for SYNTH, 6 for ADVHOTPOT, and 32 for E-SNLI, respectively.4 We experiment with four LLMs, including OPT (175B), GPT-3 (davinci), InstructGPT (text-davinci-001), and text-davinci-002. OPT and GPT-3 are trained using the standard causal language modeling objective, whereas InstructGPT and text-davinci-002 are trained with special instruction data and human annotations. We generate outputs with greedy decoding (temperature set to be 0). Our prompt formats follow those in Brown et al. (2020). The explanations are inserted before/after the prediction with conjunction words like because. Please refer to Appendix A for full prompts. Because the results of in-context learning vary with the examples presented in the input prompt, for each dataset, we randomly sample multiple groups of training shots, and report the mean and standard deviation of the results (subscript). We use 5 groups for InstructGPT, the primary LM we are using throughout our paper, and 3 groups for the rest. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 Results ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As shown in Table 1, OPT, GPT-3, and InstructGPT show small to moderate improvements from using explanations for textual reasoning tasks. On the two QA tasks, SYNTH and ADVHOTPOT, E-P improves the performance of InstructGPT, the best among these three LMs, from 54.8 to 58.5 and 56.8 to 59.4, respectively.5 On E-SNLI, P-E outperforms FEW-SHOT by 2.6, whereas E-P substantially lags FEW-SHOT. Comparing E-P against P-E on SYNTH and E-SNLI, E-P typically degrades performance (except on SYNTH for InstructGPT) and P-E is inconsistent across the different models, whereas E-P consistently leads to performance improvements on ADVHOTPOT. There is no single winner between the two paradigms of using explanations; choosing the most effective way is task-specific. Overall, vanilla LLMs (OPT and GPT-3) see limited benefit from producing explanations, and even the Instruct-series InstructGPT does not see substantial improvements. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The only exception is text-davinci-002. text-davinci-002 greatly benefits from explanations in the prompt across all three tasks, and E-P is consistently more effective than P-E. However, it is unclear what contributes to this difference. As far as we are aware, the differences between text-davinci-002 and InstructGPT are not described in any publication or blog post.6 Comparing GPT-3 and InstructGPT, we see the move to Instruct series models is not sufficient to explain the difference. Given the lack of transparency with this model, we hesitate to make scientific claims about the results it yields. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/985ace058484256295f3e9d27023d1a9de798cd084698cdaafa9b990e002398a.jpg",
+        "table_caption": [
+            "Figure 3: Explanations generated for ADVHOTPOT. InstructGPT may generate nonfactual explanations containing hallucination (red) or inconsistent explanations contradicting the answer (red). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>naetatr</td><td>Pedro Rubens! The individual chapters were published into 64 &quot;tankobon&quot;by Kodansha. Yoko Shoji(born4June195o,in Mobara,Chiba)isaJapanese mangaartist.She is bestknown for writing &quot;Seito Shokun! Mulder Scully!The individual chapters were published into 14 &quot;tankobon&quot; by Kodansha. Seito Shokun! The individual chapters were published into 24 &quot;tankobon&quot; by Kodansha between. Q:How many chapters does Yoko Shoji&#x27;s most famous manga have?</td></tr><tr><td>Trrnsrtetetr</td><td>A: First,Yoko Shoji&#x27;s mostfamous manga is &quot;Seito Shokun!&quot;.Second,&quot;Seito Shokun!&quot;has 64chapters.The answer is 64. Tim Minchin (December 29,1808 July 31,1875) was the President of the United States. Andrew Johnson (December 29,1808 July 31,1875) was the President of the United States. George Andrew Atzerodt (June 12,1835-July 7,1865)was a conspirator, with John Wilkes Booth. Jesse Andrew Williams (June 12,1835-July7,1865) was a conspirator, with John Wilkes Booth. Q: Who was older, George Atzerodt or Andrew Johnson? A: First,George Atzerodt was bornonJune12,1835.Second,Andrew Johnson was bornon December 29,1808.The answer is George Atzerodt.</td></tr></table>",
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+    },
+    {
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+        "table_caption": [
+            "Table 2: Left: factuality (Fac) and consistency (Con) of the generated explanations. Right: the $\\%$ of the examples whose explanation factuality/consistency is congruent with the prediction accuracy. In general, LLMs tend to generate consistent but less likely factual explanations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td>Acc</td><td>Fac</td><td>Con</td><td>Acc=Fac</td><td>Acc=Con</td></tr><tr><td rowspan=\"7\">InstructGPT</td><td colspan=\"6\">reliability of explanations generated by InstructGPT</td></tr><tr><td>SYNTH (E-P)</td><td>58.4</td><td>72.8</td><td>64.8</td><td>66.5</td><td>68.8</td></tr><tr><td>SYNTH (P-E)</td><td>54.8</td><td>51.6</td><td>95.2</td><td>89.6</td><td>57.2</td></tr><tr><td>ADVHP (E-P)</td><td>62.0</td><td>79.6</td><td>91.2</td><td>80.0</td><td>68.4</td></tr><tr><td>ADVHP (P-E)</td><td>54.0</td><td>69.2</td><td>82.0</td><td>77.6</td><td>67.2</td></tr><tr><td>E-SNLI(P-E)</td><td>62.0</td><td>1</td><td>98.8</td><td>1</td><td>62.0</td></tr><tr><td colspan=\"6\">reliability of explanations generated by other LLMs on SYNTH</td></tr><tr><td rowspan=\"2\">OPT (175B)</td><td>SYNTH (E-P)</td><td>30.0</td><td>77.2</td><td>47.2</td><td>45.6</td><td>58.8</td></tr><tr><td>SYNTH (P-E)</td><td>39.6</td><td>64.0</td><td>81.2</td><td>69.2</td><td>49.6</td></tr><tr><td rowspan=\"2\">GPT-3</td><td>SYNTH (E-P)</td><td>46.8</td><td>59.2</td><td>64.8</td><td>66.8</td><td>61.2</td></tr><tr><td>SYNTH (P-E)</td><td>52.4</td><td>52.4</td><td>83.2</td><td>78.4</td><td>58.0</td></tr><tr><td rowspan=\"2\">text-davinci-002</td><td>SYNTH (E-P)</td><td>86.0</td><td>91.6</td><td>85.2</td><td>91.2</td><td>84.8</td></tr><tr><td>SYNTH (P-E)</td><td>81.6</td><td>83.2</td><td>96.4</td><td>95.8</td><td>82.8</td></tr></table>",
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+        "type": "text",
+        "text": "Our results do not suggest immediate strong improvements from incorporating explanations across all LLMs, even for our synthetic dataset, contradicting recent prior work. This can be attributed to the difference between the tasks we study. The tasks that receive significant benefits from using explanations in Nye et al. (2021) and Wei et al. (2022) are all program-like (e.g., integer addition and program execution), whereas the tasks in this work emphasize textual reasoning grounded in provided inputs. In fact, in Wei et al. (2022) and Chowdhery et al. (2022), explanations only show mild benefit on open-domain QA tasks like StrategyQA (Geva et al., 2021) that are closer to our setting. ",
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+        "type": "text",
+        "text": "3 Can LLMs Generate Factual and Consistent Explanations? ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Prompting LLMs with explanations and having models generate them may not guarantee higher performance on our tasks. But what about the quality of the model-generated explanations themselves? We assess the reliability of the explanations for the three datasets, measured in terms of two aspects. ",
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+        "text": "Factuality refers to whether a generated explanation is faithfully grounded in the corresponding input context (context for QA and premise/hypothesis pair for NLI). A factual explanation should not contain hallucinations that contradict the context. See Figure 3 for a nonfactual explanation. ",
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+        "type": "text",
+        "text": "Consistency measures if the explanation entails the prediction. Our concept of consistency resembles plausibility as described in Jacovi and Goldberg (2021), in that we assess whether the prediction follows from the explanation as perceived by a human. See Figure 3 for an inconsistent explanation. ",
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+        "type": "text",
+        "text": "For SYNTH, we use rules to automatically judge whether an explanation is factual and consistent on all four LLMs. For ADVHOTPOT and E-SNLI, the authors manually inspected the explanations generated by InstructGPT and annotated them for these two characteristics (more details in Appendix D). Note for each setting, the results are based on the explanations and predictions obtained with a single set of training shots. We only show the results of P-E on E-SNLI, as E-P is substantially worse here. ",
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+        "text": "Results We summarize the results in Table 2. We only report consistency on E-SNLI, as the explanations for ESNLI often require some external commonsense knowledge which cannot be easily grounded in the inputs or judged as true or false (examples in Appendix F). The results suggest a disconnect between the model predictions and the “reasoning” in explanations. On InstructGPT, though using explanations improves its performance across three tasks, the generated explanations are unreliable (upper section), even for the straightforward synthetic setting. Comparing the factuality of explanations for SYNTH generated by GPT-3, InstructGPT, and text-davinci-002, we see that instruction tuning improves the factuality, but even the most powerful text-davinci-002 still fails to generate explanations that are perfectly grounded in the input context. Overall, LLMs tend to generate consistent explanations $( > 8 0 \\%$ for all three datasets with the right prompt structure), but the explanations are less likely to be factual, which is concerning as they can deceive a user of the system into believing the model’s answer. ",
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+            "of Correct/Incorrect Predictions by Factuality/Consistency ",
+            "Figure 4: Explanations are more likely to be nonfactual than to be inconsistent, and a nonfactual explanation usually indicates an incorrect prediction. "
+        ],
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+        "type": "text",
+        "text": "3.1 Reliability of Explanations and Prediction Accuracy ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "LLMs may hallucinate problematic explanations, but this could actually be advantageous if it gives us a way of spotting when the model’s “reasoning” has failed. We investigate the connection between the reliability of an explanation and the accuracy of a prediction and ask whether a reliable explanation indicates an accurate prediction. (This resembles the linguistic calibration of Mielke et al. (2022), but using a different signal for calibration.) ",
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+        "text": "As shown in Table 2 (right), accuracy and factuality/consistency are typically correlated, especially factuality. By knowing whether an explanation is factual, we can guess the model’s accuracy a high fraction of the time (Accuracy $=$ Factuality). A nonfactual explanation very likely means an incorrect prediction on the SYNTH dataset across all four LLMs. On ADVHOTPOT, factuality and InstructGPT’s prediction correspond $8 0 . 0 \\%$ of the time, substantially surpassing the prediction accuracy itself. We show fractions of correct and incorrect predictions when the explanations are factual/nonfactual and consistent/inconsistent in Figure 4 for two of our settings. Factual explanations are much more likely paired with correct predictions compared to nonfactual explanations. Consistency is also connected to accuracy but is an inferior indicator compared to factuality in general (Table 2). ",
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+        "type": "text",
+        "text": "4 Calibrating In-Context Learning using Explanations ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "From Section 3.1, we see that a human oracle assessment of the factuality of an explanation could be of substantial use for calibrating the corresponding prediction. Can we automate this process? ",
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+        "type": "text",
+        "text": "We first show how to achieve this goal on the perfectly controlled SYNTH dataset (Section 4.1). On our other two datasets, we use surface lexical matching to approximate semantic matching and give real-valued scores approximately reflecting factuality. Following past work on supervised calibration (Kamath et al., 2020; Chen et al., 2021; Ye and Durrett, 2022), we can learn a calibrator that tunes the probabilities of a prediction based on the score of its explanation (Section 4.2). We show such a calibrator can be trained with a handful of examples beyond those used for in-context learning and successfully improve the in-context learning performance on realistic datasets.7 We note that, as mentioned before, the experiments in this section are conducted on InstructGPT. ",
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+        "type": "text",
+        "text": "4.1 Motivating Example: Improving SYNTH Dataset ",
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+        "type": "text",
+        "text": "We first show how post-hoc calibration functions in the controlled SYNTH setting, where we can simply check the factuality of an explanation. Since the generated explanation always follows the format “B is [profession] and A [verb] B.” (example in Figure 2), we can split the explanation into two sentences. The explanation is factual if and only if each of the two sentences exactly matches one of the sentences in the context. ",
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+        "text": "We use the assessment to improve the performance of P-E for SYNTH, where a nonfactual explanation typically indicates an incorrect prediction. This gives us a way to reject presumably incorrect answers. Specifically, we iterate through the top 5 candidate answers (restricted by the API) given by InstructGPT and reject any answer-explanation pair if the explanation is nonfactual until we find a factual one. This procedure dramatically improves the accuracy from $5 2 . 4 \\%$ to $7 4 . 8 \\%$ . Note that this SYNTH dataset is a challenging task given its lack of reasoning shortcuts: for reference, neither ROBERTA (Liu et al., 2019) nor DEBERTA (He et al., 2021) finetuned with 16 examples can achieve an accuracy surpassing $50 \\%$ . With the help of the explanations and the checking procedure, we can use InstructGPT to achieve strong results using few-shot learning. ",
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+        "type": "text",
+        "text": "4.2 Learning-based Calibration Framework ",
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+        "type": "text",
+        "text": "Framework We now introduce the framework that can leverage the factuality assessment of an explanation to calibrate a prediction. Let $\\pmb { p }$ be the vector of predicted probabilities associated with each class label in NLI (or the probability score of predicted answer in QA). Let $v$ be a scalar value extracted from the explanation to describe the factuality. Then, we can adjust the probabilities accordingly using a linear model: $\\pmb { \\hat { p } } = \\mathrm { s o f t m a x } ( W [ \\pmb { p } ; \\pmb { v } ] + b )$ , where $\\hat { p }$ is the tuned probabilities. ",
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+        "text": "Our calibration framework is extended from classical calibration methods (Platt, 1999; Guo et al., 2017; Zhao et al., 2021), which apply an affine transformation on the probabilities alone: $\\hat { p } =$ softmax $( W p + b )$ . In contrast, we use an additional factor $v$ in calibration to incorporate the factuality assessment of the explanation. ",
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+        "text": "There are a small number of parameters ( $W$ and $b$ ) that need to be trained in such a calibration framework. We will rely on a few more examples in addition to the shots we use in the prompt to train the calibrator. Specifically, we use the prompt examples to generate the predictions and explanations for these extra examples, and extract predicted probabilities, factors, and target probabilities triples to construct training data points used to train the calibrator. Note this procedure requires no explanation annotations for the extra examples. ",
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+        "text": "Approximating Factuality We approximate the factuality using lexical overlap between the explanations and the inputs, which we found to work fairly well for our tasks. ",
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+        "type": "text",
+        "text": "ADVHOTPOT: We use an explanation consisting of two sentences (examples in Figure 3) as an illustration. Let $\\mathcal { E } = ( E ^ { ( 1 ) } , E ^ { ( 2 ) } )$ be the generated explanation, where $E ^ { ( 1 ) }$ and $E ^ { ( 2 ) }$ are the two sentences, and the $E ^ { ( i ) } = ( e _ { 1 } , e _ { 2 } , \\cdot \\cdot \\cdot )$ contain tokens $e _ { 1 } , e _ { 2 } , \\cdots$ . Similarly, let $\\mathcal { P } =$ $( P ^ { ( 1 ) } , P ^ { ( 2 ) } , P ^ { ( 3 ) } , P ^ { ( 4 ) } )$ be the context paragraphs, and $P ^ { ( i ) } = ( p _ { 1 } , p _ { 2 } , \\cdot \\cdot \\cdot )$ be the tokens. The factuality estimation of one explanation sentence $E ^ { ( i ) }$ is defined as: $\\begin{array} { r } { \\mathcal { V } ( E ^ { ( i ) } ) = \\operatorname* { m a x } _ { P \\in \\mathcal { P } } \\frac { | E ^ { ( i ) } \\cap P | } { | E ^ { ( i ) } | } } \\end{array}$ . ",
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+        "type": "text",
+        "text": "Intuitively, the factuality score for a sentence $E$ is defined as the maximum number of overlapping tokens over all paragraphs $P$ , normalized by the number of tokens in $E$ . We then define the factuality score for the whole explanation as $\\begin{array} { r } { \\mathcal { V } ( \\mathcal { E } ) = \\mathrm { \\bar { \\ m i n } } _ { E \\in \\mathcal { E } } \\mathcal { V } ( E ) } \\end{array}$ , as it requires all sentences to be factual in order to make the entire explanation factual.8 ",
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+        "type": "text",
+        "text": "E-SNLI: The explanations of E-SNLI do not really involve a concept of factuality. Nevertheless, we use an analogous score following the same principle by viewing the premise as the context. Let $\\boldsymbol { E } = ( e _ { 1 } , e _ { 2 } , \\cdots )$ be the explanation and $P = ( p _ { 1 } , p _ { 2 } , \\cdots )$ be the premise. We simply score the explanation by $\\begin{array} { r } { \\mathcal { V } ( E ) = \\frac { | E | \\cap | P | } { | E | } } \\end{array}$ . The more an explanation overlaps with the premise, the more factual we judge it to be. ",
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+    {
+        "type": "text",
+        "text": "4.3 Calibrating E-SNLI ",
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+        "type": "text",
+        "text": "Setup For E-SNLI, we use calibration methods to postprocess the final probabilities. Unlike classical temperature scaling (Platt, 1999), note that the methods we use here can actually change the prediction; we will therefore evaluate on accuracy of the calibrated model. ",
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+        "type": "text",
+        "text": "We study the effectiveness of our explanationbased calibrator under different training data sizes varying from 32 to 128. Recall that we only require explanation annotations for 32 data points, and only need the labels for the rest to train the calibrator. For E-SNLI, we calibrate P-E, which is shown to be more effective than E-P in this setting (Section 2.4). ",
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+        "type": "text",
+        "text": "Baselines We provide the performance of finetuned ROBERTA (Liu et al., 2019) model as a reference, finding this to work better than DeBERTa (He et al., 2021). To isolate the effec",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Accuracy $\\mathrm { ( m e a n _ { s t d d e v } ) }$ of various methods on E-SNLI under different data conditions. L denotes number of labels (as well as the total number of examples); E denotes the number of explanations. Calibrating using explanations successfully improves the performance of in-context learning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>w/o Explanation</td><td>32L</td><td>64L</td><td>96L</td><td>128L</td></tr><tr><td>RoBERTa</td><td>40.14.7</td><td>43.05.1</td><td>49.05.2</td><td>54.94.8</td></tr><tr><td>FEW-SHOT</td><td>56.82.0</td><td>1</td><td>1</td><td>1</td></tr><tr><td>FEW-SHOT(NN)</td><td>1</td><td>1</td><td>1</td><td>58.91.0</td></tr><tr><td>FEW-SHOT+PROBCAL</td><td>61.93.8</td><td>62.42.6</td><td>63.22.9</td><td>63.91.2</td></tr><tr><td>w/ Explanation</td><td>32L+32E</td><td>64L+32E</td><td>96L+32E</td><td>128L+32E</td></tr><tr><td>P-E</td><td>59.42.0</td><td>一</td><td></td><td>1</td></tr><tr><td>P-E+PROBCAL</td><td>64.41.8</td><td>65.41.2</td><td>65.41.6</td><td>65.41.9</td></tr><tr><td>P-E+EXPLCAL</td><td>64.22.6</td><td>65.81.3</td><td>67.61.6</td><td>68.51.2</td></tr><tr><td>P-E+ZHANG</td><td>63.03.2</td><td>65.22.2</td><td>65.41.5</td><td>65.92.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "tiveness of using explanations for calibration, we introduce three additional baselines using nonexplanation-based calibrators. We apply the probability-based calibrator as described in Section 4.2 on the results obtained on few-shot learning (FEW-SHOT $^ { \\cdot + }$ PROBCAL) and predict-then-explain pipeline $( { \\mathrm { P } } { \\cdot } { \\mathrm { E } } { + } { \\mathrm { P R O B C A L } }$ ). We note that the parameters of these calibrators are trained using the additional data points, as opposed to being heuristically determined as in Zhao et al. (2021). Furthermore, we experiment with a recently proposed supervised calibrator from Zhang et al. (2021), which uses the CLS representations from an additional language model as features in the calibrator. The probabilities are tuned using $\\pmb { \\hat { p } } = \\mathrm { s o f t m a x } ( W [ \\pmb { p } ; \\pmb { h } ] + \\bar { \\boldsymbol { b } } )$ , where $^ { h }$ is the CLS representation. Since we do not have access to the embeddings obtained by GPT-3, we use ROBERTA to extract the vectors instead. We use such a calibrator on top of our best-performing base model, P-E, resulting $\\mathrm { P } \\mathrm { - } \\mathrm { E } +$ ZHANG ET AL. (2021). ",
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+    {
+        "type": "text",
+        "text": "Limited by the maximum prompt length, in-context learning is not able to take as input the additional data used for training the calibrator. For a fair comparison, we can allow the in-context model to use this data by varying the prompts across test examples, dynamically choosing the prompt examples to maximize performance. Choosing closer data points for prompting is a common and effective way of scaling up the training data size for in-context learning (Shin et al., 2021; Liu et al., 2021). Following Liu et al. (2021), we test the performance of choosing nearest neighbors for the prompt based on CLS embedding produced by a ROBERTA model (Liu et al., 2019), referred as FEW-SHOT(NN). It is worth clarifying that the FEW-SHOT and FEW-SHOT $^ +$ PROBCAL approaches use the same set of 32 training shots in the prompt for every test example, whereas the shot sets vary from example to example in FEW-SHOT(NN). ",
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+    {
+        "type": "text",
+        "text": "Results We show the results in Table 3. We use 5 different groups of training examples and report the mean and standard deviation across the groups. For FEW-SHOT(NN), we only report the results obtained using 128 examples; results using a smaller number of examples will be worse than this. ",
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+        "type": "text",
+        "text": "Under 128 training examples, applying a trained calibrator on top of prompting with explanation (i.e., P- $\\Xi +$ EXPLCAL) achieves the best accuracy of $6 8 . 5 \\%$ , which is $12 \\%$ higher than the performance of the vanilla uncalibrated few-shot in-context learning (FEW-SHOT). P-E+EXPLCAL also outperforms FEW-SHOT $^ +$ PROBCAL and P- $\\Xi +$ PROBCAL by $5 \\%$ and $3 \\%$ , respectively. Using explanations is more effective than using probabilities alone. In addition, $\\mathrm { P } \\mathrm { - } \\mathrm { E } +$ EXPLCAL also outperforms P- $\\mathsf { E } +$ ZHANG ET AL. (2021), whose performance is on par with P-E+PROBCAL. This suggests the additional CLS information is not very helpful in this setting. ",
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+        "type": "text",
+        "text": "As the data size increases from 32 to 128, the performance of the explanation-based calibrator keeps improving notably, whereas the performance of probability-based calibrators nearly saturates at a data size of 96. The performance of FEW-SHOT(NN) with 128 training instances only improves the performance by $3 . 3 \\%$ , compared to FEW-SHOT with 32 training instances. Choosing nearest neighbors as the shots, while being effective when having access to a large amount of data, is not helpful in the extreme data-scarce regime. Calibrating using explanations is an effective way of using a few extra data points that cannot fit in the prompt, which is a pitfall of standard in-context learning. ",
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+        "table_caption": [
+            "Table 4: AUC scores $\\left( \\mathrm { { m e a n } _ { \\mathrm { { s t d } \\ d e v } } } \\right)$ on ADVHOTPOT under different data conditions. $\\mathbf { L }$ and $\\mathbf { E }$ denotes the number of label annotations and explanation annotations, respectively. Explanationbased calibration successfully improves the performance on top of prompting with explanations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>w/o Explanation</td><td>6L</td><td>32L</td><td>64L</td></tr><tr><td>FEW-SHOT FEW-SHOT(NN)</td><td>59.62.4</td><td>1</td><td>1</td></tr><tr><td>w/Explanation</td><td>1 6L+6E</td><td>1 32L+6E</td><td>61.30.9 64L+6E</td></tr><tr><td>E-P</td><td>64.42.9</td><td></td><td></td></tr><tr><td>E-P+EXPLCAL</td><td></td><td>66.03.9</td><td>68.83.0</td></tr><tr><td>E-P+ZHANG</td><td>1 1</td><td>65.63.9</td><td>66.13.2</td></tr></table>",
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+        "image_caption": [
+            "Figure 5: Coverage-Acc curves of various methods on ADVHOTPOT. E- $\\mathrm { P } +$ EXPLCAL is better calibrated compared to uncalbrated E-P as well as the other approaches. "
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+        "type": "text",
+        "text": "Finally, ROBERTA finetuned using 128 shots only achieves an accuracy of $5 4 . 9 \\%$ , lagging the performance of GPT-3 based models. The limited training data size is insufficient for finetuning smaller language models like ROBERTA, but is sufficient for $\\mathrm { P } \\mathrm { - } \\mathrm { E } +$ EXPLCAL to be effective. ",
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+        "type": "text",
+        "text": "4.4 Calibrating ADVHOTPOT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Setup For the ADVHOTPOT dataset, our calibration takes the form of tuning the confidence scores of the predicted answers to better align them with the correctness of predictions. These confidence scores can be used in a “selective QA” setting (Kamath et al., 2020), where the model can abstain on a certain fraction of questions where it assigns low confidence to its answers. We use the area under coverage-accuracy curve (AUC) to evaluate how well a model is calibrated as in past literature (Kamath et al., 2020; Chen et al., 2021; Zhang et al., 2021; Garg and Moschitti, 2021; Ye and Durrett, 2022). The curve plots the average accuracy with varying fractions (coverage) of questions being answered (examples in Figure 5). For any given coverage, a better calibrated model should be able to identify questions that it performs best on, hence resulting a higher AUC. ",
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+        "type": "text",
+        "text": "We experiment with training data set sizes of 6, 32, and 64. We report the results averaged from 5 trials using different training sets. For ADVHOTPOT, we calibrate E-P, which is shown to be more effective than P-E in this setting (Section 2.4). Our approach is also effective for calibrating P-E; please refer to Appendix E for details. ",
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+        "type": "text",
+        "text": "Results We show the AUC scores in Table 4. By leveraging explanations, $\\mathrm { E } { \\mathrm { - } } \\mathrm { P } +$ EXPLCAL successfully achieves an AUC of 68.8, surpassing both FEW-SHOT by 7 points and E-P by 4 points. We note this is a substantial improvement, given that the upperbound of AUC is constrained by the accuracy of the answers and cannot reach 100. Figure 5 shows the coverage-accuracy curves of various methods averaged across the 5 training runs. E- $\\mathrm { P } +$ EXPLCAL always achieves a higher accuracy than its uncalibrated counterpart, E-P, under a certain coverage, and the gap is especially large in the most confident intervals (coverage $< 5 0 \\%$ ). E- $\\mathrm { P } +$ ZHANG ET AL. (2021) is able to calibrate the predictions on this dataset, but still lags our explanation-based calibrator, E-P+EXPLCAL. ",
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+    {
+        "type": "text",
+        "text": "In addition, the explanation-based calibrator can be effective with as few as 32 examples. This is because there are only two parameters (the probability of predicted answer and the explanationbased factor) in the calibrator, which can be easily learned in this few-shot setting. Comparing E $\\mathbf { \\partial } _ { - } \\mathbf { P } { + } \\mathbf { E }$ XPLCAL against FEW-SHOT(NN), using nearest neighbors in the prompt is also able to improve the performance compared to using a fixed set of shots (FEW-SHOT), yet our lightweight calibrator can better utilize such a small amount of data, and learn to distinguish more accurate predictions based on the explanations. ",
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+    {
+        "type": "text",
+        "text": "5 Related Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our investigation is centered around in-context learning (Brown et al., 2020), which has garnered increasing interest since the breakthrough of various large pretrained language models. Recent work has been devoted to studying different aspects of in-context learning, including its wayward behaviors (Min et al., 2022; Webson and Pavlick, 2022) and approaches to overcome them (Zhao et al., 2021), whereas our exploration focuses on using explanations. ",
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+    {
+        "type": "text",
+        "text": "The utility of explanations for few-shot in-context learning has also been discussed concurrently (Nye et al., 2021; Wei et al., 2022; Marasovic et al., 2022; Chowdhery et al., 2022; Lampinen et al., ´ 2022; Wiegreffe et al., 2022), especially in symbolic reasoning tasks. We differ in that we study more free-form explanations in tasks (QA and NLI, specifically) focusing on textual reasoning over provided contexts. Furthermore, our work focuses on the nature of the explanations generated by LLMs, which are found to be unreliable. Regarding our use of calibration, similar ideas of explanation-based performance estimation have been applied to other tasks (Rajani and Mooney, 2018; Ye et al., 2021; Ye and Durrett, 2022), but we rely on the free-text explanations generated by the model instead of interpretations obtained through post-hoc interpretation techniques. ",
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+    {
+        "type": "text",
+        "text": "More broadly, how to use explanations in various forms (textual explanation, highlights, etc.) to train better models is a longstanding problem (Zaidan et al., 2007). Past work has built a series of pipeline models that first generate the explanations and then make predictions purely based on the generated explanations (Wiegreffe et al., 2021; Zhou and Tan, 2021; Chen et al., 2022). Prior research has also explored using explanations as additional supervision to train joint models (Hancock et al., 2018; Dua et al., 2020; Lamm et al., 2021; Stacey et al., 2022). Another line of work seeks to align the reasoning process of a trained model with the explanations, which is typically done by interpreting a prediction post-hoc through explanation techniques and optimizing the distance between the obtained explanation and ground truth explanation (Liu and Avci, 2019; Rieger et al., 2020; Plumb et al., 2020; Erion et al., 2021; Yao et al., 2021). These aforementioned methods all update the model parameters and typically require a considerable amount of explanation annotations to be effective. By contrast, our setting treats language models as pure black boxes and only requires few-shot explanations. ",
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+    {
+        "type": "text",
+        "text": "6 Discussion & Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Caveats and Risks of Explanations from Large Language Models Our analysis suggests that LLMs’ internal “reasoning” does not always align with explanations that it generates, as shown by our consistency results. More concerning, the explanations might not be factually grounded in the provided prompt. This shortcoming should caution against any deployment of this technology in practice: because the explanations are grammatical English and look very convincing, they may deceive users into believing the system’s responses even when those responses are incorrect. Section 6 of Bender et al. (2021) discusses these risks in additional detail. The fact that language models can hallucinate explanations is also found in other work (Zhou and Tan, 2021). This result is unsurprising in some sense: without sufficient supervision or grounding, language models do not learn meaning as distinct from form (Bender and Koller, 2020), so we should not expect their explanations to be strongly grounded. ",
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+    {
+        "type": "text",
+        "text": "We have shown that even explanations which don’t lead to accuracy gains can still be useful for calibration. However, the lexical overlap feature we use here is a weak signal of explanation correctness (see the example in Figure 1). Strong enough entailment models should theoretically be able to perform this task and work across a range of tasks without fine-tuning. This explanation assessment model can even be a language model itself trained for this particular propose to approach the verification tasks for a given domain by in-context learning. ",
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+    {
+        "type": "text",
+        "text": "Conclusion We have explored the capabilities of LLMs in using explanations in in-context learning for textual reasoning. Through our experiments with four LLMs and on two QA datasets and an NLI dataset, we find that simply including explanations in the prompt does not always improve the performance of in-context learning. Our manual analysis demonstrates that LLMs tend to generate nonfactual explanations when making wrong predictions, which can be a useful leverage to assess the correctness of the predictions. Lastly, we showcase how to use explanations to build lightweight calibrators, which successfully improve InstructGPT’s in-context learning performance across all three datasets. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments ",
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+        "type": "text",
+        "text": "We would like to thank Eunsol Choi, Ruiqi Zhong, Jocelyn Chen, Zayne Sprague, and Jiacheng Xu for their helpful feedback on drafts of this work, as well as the anonymous reviewers for their thoughtful reviews. This work was partially supported by NSF Grant IIS-1814522, NSF CAREER Award IIS-2145280, a grant from Open Philanthropy, a gift from Salesforce Inc., and a gift from Adobe. ",
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