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+# META-GRAPH: FEW SHOT LINK PREDICTION VIA META LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We consider the task of few shot link prediction, where the goal is to predict missing edges across multiple graphs using only a small sample of known edges. We show that current link prediction methods are generally ill-equipped to handle this task—as they cannot effectively transfer knowledge between graphs in a multigraph setting and are unable to effectively learn from very sparse data. To address this challenge, we introduce a new gradient-based meta learning framework, Meta-Graph, that leverages higher-order gradients along with a learned graph signature function that conditionally generates a graph neural network initialization. Using a novel set of few shot link prediction benchmarks, we show that MetaGraph enables not only fast adaptation but also better final convergence and can effectively learn using only a small sample of true edges.
+
+# 1 INTRODUCTION
+
+Given a graph representing known relationships between a set of nodes, the goal of link prediction is to learn from the graph and infer novel or previously unknown relationships (Liben-Nowell & Kleinberg, 2003). For instance, in a social network we may use link prediction to power a friendship recommendation system (Aiello et al., 2012), or in the case of biological network data we might use link prediction to infer possible relationships between drugs, proteins, and diseases (Zitnik & Leskovec, 2017). However, despite its popularity, previous work on link prediction generally focuses only on one particular problem setting: it generally assumes that link prediction is to be performed on a single large graph and that this graph is relatively complete, i.e., that at least $50 \%$ of the true edges are observed during training (e.g., see Grover & Leskovec, 2016; Kipf & Welling, 2016b; Liben-Nowell & Kleinberg, 2003; Lu & Zhou, 2011). ¨
+
+In this work, we consider the more challenging setting of few shot link prediction, where the goal is to perform link prediction on multiple graphs that contain only a small fraction of their true, underlying edges. This task is inspired by applications where we have access to multiple graphs from a single domain but where each of these individual graphs contains only a small fraction of the true, underlying edges. For example, in the biological setting, high-throughput interactomics offers the possibility to estimate thousands of biological interaction networks from different tissues, cell types, and organisms (Barrios-Rodiles et al., 2005); however, these estimated relationships can be noisy and sparse, and we need learning algorithms that can leverage information across these multiple graphs in order to overcome this sparsity. Similarly, in the e-commerce and social network settings, link prediction can often have a large impact in cases where we must quickly make predictions on sparsely-estimated graphs, such as when a service has been recently deployed to a new locale. That is to say to link prediction for a new sparse graph can benefit from transferring knowledge from other, possibly more dense, graphs assuming there is exploitable shared structure.
+
+We term this problem of link prediction from sparsely-estimated multi-graph data as few shot link prediction analogous to the popular few shot classification setting (Miller et al., 2000; Lake et al., 2011; Koch et al., 2015). The goal of few shot link prediction is to observe many examples of graphs from a particular domain and leverage this experience to enable fast adaptation and higher accuracy when predicting edges on a new, sparsely-estimated graph from the same domain—a task that can can also be viewed as a form of meta learning, or learning to learn (Bengio et al., 1990; 1992; Thrun & Pratt, 2012; Schmidhuber, 1987) in the context of link prediction. This few shot link prediction setting is particularly challenging as current link prediction methods are generally ill-equipped to transfer knowledge between graphs in a multi-graph setting and are also unable to effectively learn from very sparse data.
+
+Present work. We introduce a new framework called Meta-Graph for few shot link prediction and also introduce a series of benchmarks for this task. We adapt the classical gradient-based metalearning formulation for few shot classification (Miller et al., 2000; Lake et al., 2011; Koch et al., 2015) to the graph domain. Specifically, we consider a distribution over graphs as the distribution over tasks from which a global set of parameters are learnt, and we deploy this strategy to train graph neural networks (GNNs) that are capable of few-shot link prediction. To further bootstrap fast adaptation to new graphs we also introduce a graph signature function, which learns how to map the structure of an input graph to an effective initialization point for a GNN link prediction model. We experimentally validate our approach on three link prediction benchmarks. We find that our MetaGraph approach not only achieves fast adaptation but also converges to a better overall solution in many experimental settings, with an average improvement of $5 . { \bar { 3 } } \%$ in AUC at convergence over non-meta learning baselines.
+
+![](images/3d27283a5af3fc62b05e5f3045f1d1a1e6778a8917e954e47649f583cf08cdb7.jpg)  
+Figure 1: Left: Graphical model for Meta-Graph vs. MAML. Right: Meta-Graph architecture.
+
+# 2 PRELIMINARIES AND PROBLEM DEFINITION
+
+The basic set-up for few shot link prediction is as follows: We assume that we have a distribution $p ( \mathcal G )$ over graphs, from which we can sample training graphs $\mathcal { G } _ { i } ~ \sim ~ p ( \mathcal { G } )$ , where each $\mathcal { G } _ { i } = ( \mathcal { V } _ { i } , \mathcal { E } _ { i } , X _ { i } )$ is defined by a set of nodes $\nu _ { i }$ , edges $\mathcal { E } _ { i }$ , and matrix of real-valued node attributes $X \in \mathbb { R } ^ { | \mathcal { V } _ { i } | \times d }$ . When convenient, we will also equivalently represent a graph as $\mathcal { G } _ { i } = ( \mathcal { V } _ { i } , A _ { i } , X _ { i } )$ , where $A _ { i } \in \mathbb { Z } ^ { | \mathcal { V } _ { i } | \times | \mathcal { V } _ { i } | }$ is an adjacency matrix representation of the edges in $\mathcal { E } _ { i }$ . We assume that each of these sampled graphs, $\mathcal { G } _ { i }$ , is a simple graph (i.e., contain a single type of relation and no self loops) and that every node $v \in \mathcal V _ { i }$ in the graph is associated with a real valued attribute vector $\mathbf { x } _ { v } \in \bar { \mathbb { R } ^ { d } }$ from a common vector space. We further assume that for each graph $\mathcal { G } _ { i }$ we have access to only a sparse subset of the true edges $\mathcal { E } _ { i } ^ { \mathrm { t r a i n } } \subset \mathcal { E } _ { i }$ (with $| \mathcal { E } _ { i } ^ { \mathrm { t r a i n } } | < < | \mathcal { E } _ { i } | )$ during training. In terms of distributional assumptions we assume that this $p ( \mathcal G )$ is defined over a set of related graphs (e.g., graphs drawn from a common domain or application setting).
+
+Our goal is to learn a global or meta link prediction model from a set of sampled training graphs $\mathcal { G } _ { i } \sim p ( \mathcal { G } ) , i = 1 . . . n$ , such that we can use this meta model to quickly learn an effective link prediction model on a newly sampled graph $\mathcal { G } _ { * } \sim p ( \mathcal { G } )$ . More specifically, we wish to optimize a global set of parameters $\theta$ , as well as a graph signature function $\psi ( \mathcal { G } _ { i } )$ , which can be used together to generate an effective parameter initialization, $\phi _ { i }$ , for a local link prediction model on graph $\mathcal { G } _ { i }$ .
+
+Relationship to standard link prediction. Few shot link prediction differs from standard link prediction in three important ways:
+
+1. Rather than learning from a single graph $\mathcal { G }$ , we are learning from multiple graphs $\{ { \mathcal { G } } _ { 1 } , . . . , { \mathcal { G } } _ { n } \}$ sampled from a common distribution or domain.
+
+2. We presume access to only a very sparse sample of true edges. Concretely, we focus on settings where at most $30 \%$ of the edges in $\mathcal { E } _ { i }$ are observed during training, i.e., where $\frac { | \mathcal { E } ^ { \mathrm { t r a i n } } | } { | \mathcal { E } | } \leq 0 . 3$ . 1
+
+3. We distinguish between the global parameters, which are used to encode knowledge about the underlying distribution of graphs, and the local parameters $\phi _ { i }$ , which are optimized to perform link prediction on a specific graph $\mathcal { G } _ { i }$ . This distinction allows us to consider leveraging information from multiple graphs, while still allowing for individually-tuned link prediction models on each specific graph.
+
+Relationship to traditional meta learning. Traditional meta learning for few-shot classification, generally assumes a distribution $p ( \mathcal { T } )$ over classification tasks, with the goal of learning global parameters that can facilitate fast adaptation to a newly sampled task $\mathcal { T } _ { i } \sim p ( \mathcal { T } )$ with few examples. We instead consider a distribution $p ( \mathcal G )$ over graphs with the goal of performing link prediction on a newly sampled graph. An important complication of this graph setting is that the individual predictions for each graph (i.e., the training edges) are not i.i.d.. Furthermore, for few shot link prediction we require training samples as a sparse subset of true edges that represents a small percentage of all edges in a graph. Note that for very small percentages we effectively break all graph structure and recover the supervised setting for few shot classification and thus simplifying the problem.
+
+# 3 PROPOSED APPROACH
+
+We now outline our proposed approach, Meta-Graph, to the few shot link prediction problem. We first describe how we define the local link prediction models, which are used to perform link prediction on each specific graph $\mathcal { G } _ { i }$ . Next, we discuss our novel gradient-based meta learning approach to define a global model that can learn from multiple graphs to generate effective parameter initializations for the local models. The key idea behind Meta-Graph is that we use gradient-based meta learning to optimize a shared parameter initialization $\theta$ for the local models, while also learning a parametric encoding of each graph $\mathcal { G } _ { i }$ that can be used to modulate this parameter initialization in a graph-specific way (Figure 1).
+
+# 3.1 LOCAL LINK PREDICTION MODEL
+
+In principle, our framework can be combined with a wide variety of GNN-based link prediction approaches, but here we focus on variational graph autoencoders (VGAEs) (Kipf & Welling, 2016b) as our base link prediction framework. Formally, given a graph $\mathcal { G } = ( \nu , A , X )$ , the VGAE learns an inference model, $q _ { \phi }$ , that defines a distribution over node embeddings $q _ { \phi } ( Z | A , X )$ , where each row $z _ { v } \in \mathbb { R } ^ { d }$ of $Z \in \mathbb { R } ^ { | \nu | \times d }$ is a node embedding that can be used to score the likelihood of an edge existing between pairs of nodes. The parameters of the inference model are shared across all the nodes in $\mathcal { G }$ , to define the approximate posterior $q _ { \phi } ( z _ { v } | A , X ) = \mathcal { N } ( z _ { v } | \mu _ { v } , \mathrm { d i a g } ( \sigma _ { v } ^ { 2 } ) )$ , where the parameters of the normal distribution are learned via GNNs:
+
+$$
+\mu = { \bf G } { \bf N } { \bf N } _ { \mu } ( A , X ) , \qquad \mathrm { a n d } \qquad \log ( \sigma ) = { \bf G } { \bf N } { \bf N } _ { \sigma } ( A , X ) .
+$$
+
+The generative component of the VGAE is then defined as
+
+$$
+p ( A | Z ) = \prod _ { i = 1 } ^ { N } \prod _ { j = 1 } ^ { N } p ( A _ { u , v } | z _ { u } , z _ { v } ) , \qquad \mathrm { w i t h } \qquad p ( A _ { u , v } | z _ { u } , z _ { v } ) = \sigma ( z _ { u } ^ { \top } z _ { v } ) ,
+$$
+
+i.e., the likelihood of an edge existing between two nodes, $u$ and $v$ , is proportional to the dot product of their node embeddings. Given the above components, the inference GNNs can be trained to minimize the variational lower bound on the training data:
+
+$$
+\mathcal { L } _ { G } = \mathbb { E } _ { q _ { \phi } } [ \log p ( A ^ { \operatorname { t r a i n } } | Z ) ] - K L [ q _ { \phi } ( Z | X , A ^ { \operatorname { t r a i n } } ) | | p ( z ) ] ,
+$$
+
+where a Gaussian prior is used for $p ( z )$ .
+
+We build upon VGAEs due to their strong performance on standard link prediction benchmarks (Kipf & Welling, 2016b), as well as the fact that they have a well-defined probabilistic interpretation that generalizes many embedding-based approaches to link prediction (e.g., node2vec (Grover & Leskovec, 2016)). We describe the specific GNN implementations we deploy for the inference model in Section 3.3.
+
+# 3.2 OVERVIEW OF META-GRAPH
+
+The key idea behind Meta-Graph is that we use gradient-based meta learning to optimize a shared parameter initialization $\theta$ for the inference models of a VGAE, while also learning a parametric encoding $\psi ( \mathcal { G } _ { i } )$ that modulates this parameter initialization in a graph-specific way. Specifically, given a sampled training graph $\mathcal { G } _ { i }$ , we initialize the inference model $q _ { \phi _ { i } }$ for a VGAE link prediction model using a combination of two learned components:
+
+• A global initialization, $\theta$ , that is used to initialize all the parameters of the GNNs in the inference model. The global parameters $\theta$ are optimized via second-order gradient descent to provide an effective initialization point for any graph sampled from the distribution $p ( \mathcal G )$ . • A graph signature $s _ { \mathcal { G } _ { i } } ~ = ~ \psi ( \mathcal { G } _ { i } )$ that is used to modulate the parameters of inference model $\phi _ { i }$ based on the history of observed training graphs. In particular, we assume that the inference model $q _ { \phi _ { i } }$ for each graph $\mathcal { G } _ { i }$ can be conditioned on the graph signature. That is, we augment the inference model to $g _ { \phi _ { i } } ( Z | A , X , s _ { \mathcal { G } _ { i } } )$ , where we also include the graph signature $s _ { \mathcal { G } _ { i } }$ as a conditioning input. We use a $\mathbf { k }$ -layer graph convolutional network (GCN) (Kipf & Welling, 2016a), with sum pooling to compute the signature:
+
+$$
+s _ { \mathcal { G } } = \psi ( \mathcal { G } ) = \mathbf { M } \mathbf { L } \mathbf { P } ( \sum _ { v \in \mathcal { V } } z _ { v } ) \qquad \mathrm { w i t h } \qquad Z = \mathbf { G } \mathbf { C } \mathbf { N } ( A , X ) ,
+$$
+
+where GCN denotes a k-layer GCN (as defined in (Kipf & Welling, 2016a)), MLP denotes a densely-connected neural network, and we are summing over the node embeddings $z _ { v }$ output from the GCN. As with the global parameters $\theta$ , the graph signature model $\psi$ is optimized via second-order gradient descent.
+
+The overall Meta-Graph architecture is detailed in Figure 1 and the core learning algorithm is summarized in the algorithm block below.
+
+# Algorithm 1: Meta-Graph for Few Shot Link Prediction
+
+Result: Global parameters $\theta$ , Graph signature function $\psi$   
+Initialize learning rates: $\alpha , \epsilon$   
+Sample a mini-batch of graphs, $\mathcal { G } _ { b a t c h }$ from $p ( \mathcal G )$ ;   
+for each $\mathcal { G } \in \mathcal { G } _ { b a t c h }$ do ${ \mathcal { E } } = { \mathcal { E } } ^ { \mathrm { t r a i n } } \cup { \mathcal { E } } ^ { \mathrm { v a l } } \cup { \mathcal { E } } ^ { \mathrm { t e s t } } / /$ / Split edges into train, val, and test $s _ { \mathcal { G } } = \psi ( \mathcal { G } , \mathcal { E } ^ { \mathrm { t r a i n } } )$ // Compute graph signature Initialize: $\phi ^ { ( 0 ) }  \theta / /$ Initialize local parameters via global parameters for $k$ in $[ 1 : K ]$ do $s _ { \mathcal { G } } = \mathrm { s t o p g r a d } ( s _ { \mathcal { G } } )$ // Stop Gradients to Graph Signature $\mathcal { L } _ { t r a i n } = \mathbb { E } _ { q } [ \log p ( A ^ { \mathrm { { t r a i n } } } | Z ) ] - K L [ q _ { \phi } ( Z | \mathcal { E } ^ { \mathrm { t r a i n } } , s _ { \mathcal { G } } ) | | p ( z ) ]$ Update ${ \phi } ^ { ( k ) } \gets { \phi } ^ { ( k - 1 ) } - \alpha \nabla _ { \phi } \mathcal { L } _ { t r a i n }$ end Initialize: $\theta  \phi _ { K }$ $s _ { \mathcal { G } } = \psi ( \mathcal { G } , \mathcal { E } ^ { \mathrm { v a l } } \cup \mathcal { E } ^ { \mathrm { t r a i n } } )$ // Compute graph signature with validation edges $\begin{array} { r } { \dot { \mathcal { L } } _ { v a l } = \mathbb { E } _ { q } [ \log p ( A ^ { \mathrm { v a l } } | \dot { Z } ) ] - K \dot { L } [ q ( \bar { Z } | \dot { \mathcal { E } } ^ { \mathrm { v a l } } \cup \dot { \mathcal { E } } ^ { \mathrm { t r a i n } } , s _ { \mathcal { G } } ) | | p ( z ) ] } \end{array}$ Update $\theta  \theta - \epsilon \nabla _ { \theta } \mathcal { L } _ { v a l }$ Update $\psi  \psi - \epsilon \nabla _ { \psi } \mathcal { L } _ { v a l }$   
+end
+
+The basic idea behind the algorithm is that we (i) sample a batch of training graphs, (ii) initialize VGAE link prediction models for these training graphs using our global parameters and signature function, (iii) run $K$ steps of gradient descent to optimize each of these VGAE models, and (iv) use second order gradient descent to update the global parameters and signature function based on a held-out validation set of edges. As depicted in Fig 1, this corresponds to updating the GCN based encoder for the local link prediction parameters $\phi _ { j }$ and global parameters $\theta$ along with the graph signature function $\psi$ using second order gradients. Note that since we are running $K$ steps of gradient descent within the inner loop of Algorithm 1, we are also “meta” optimizing for fast adaptation, as $\theta$ and $\psi$ are being trained via second-order gradient descent to optimize the local model performance after $K$ gradient updates, where generally $K \in \{ 0 , 1 , \ldots , 5 \}$ .
+
+# 3.3 VARIANTS OF META-GRAPH
+
+We consider several concrete instantiations of the Meta-Graph framework, which differ in terms of how the output of the graph signature function is used to modulate the parameters of the VGAE inference models. For all the Meta-Graph variants, we build upon the standard GCN propagation rule (Kipf & Welling, 2016a) to construct the VGAE inference models. In particular, we assume that all the inference GNNs (Equation 1) are defined by stacking $K$ neural message passing layers of the form:
+
+$$
+h _ { v } ^ { ( k ) } = \mathrm { R e L U } \left( \sum _ { u \in \mathcal { N } ( v ) \cup \{ v \} } \frac { m _ { s _ { \mathcal { G } } } \left( W ^ { ( k ) } h _ { u } ^ { ( k - 1 ) } \right) } { \sqrt { | \mathcal { N } ( v ) | | \mathcal { N } ( u ) | } } \right) ,
+$$
+
+where $h _ { v } \in \mathbb { R } ^ { d }$ denotes the embedding of node $v$ at layer $k$ of the model, $\mathcal { N } ( v ) = \{ u \in \mathcal { V } : e _ { u , v } \in$ $\mathcal { E } \}$ denotes the nodes in the graph neighborhood of $v$ , and $W ^ { ( k ) } \in \mathbb { R } ^ { d \times d }$ is a trainable weight matrix for layer $k$ . The key difference between Equation 5 and the standard GCN propagation rule is that we add the modulation function $m _ { s _ { \mathcal G } }$ , which is used to modulate the message passing based on the graph signature $s _ { \mathcal { G } } = \psi ( \mathcal { G } )$ .
+
+We describe different variations of this modulation below. In all cases, the intuition behind this modulation is that we want to compute a structural signature from the input graphs that can be used to condition the initialization of the local link prediction models. Intuitively, we expect this graph signature to encode structural properties of sampled graphs $\mathcal { G } _ { i } \sim p ( \mathcal { G } )$ in order to modulate the parameters of the local VGAE link prediction models and adapt it to the current graph.
+
+GS-Modulation. Inspired by Brockschmidt (2019), we experiment with basic feature-wise linear modulation (Strub et al., 2018) to define the modulation function $m _ { s _ { \mathcal G } }$ :
+
+$$
+\begin{array} { c } { \beta _ { k } , \gamma _ { k } , = \psi ( \mathcal { G } ) } \\ { m _ { \beta _ { k } , \gamma _ { k } } \left( W ^ { ( k ) } h _ { u } ^ { ( k - 1 ) } \right) = \gamma _ { k } \odot W h ^ { ( k - 1 ) } + \beta _ { k } . } \end{array}
+$$
+
+Here, we restrict the modulation terms $\beta _ { k }$ and $\gamma _ { k }$ output by the signature function to be in $[ - 1 , 1 ]$ by applying a tanh non-linearity after Equation 4.
+
+GS-Gating. Feature-wise linear modulation of the GCN parameters (Equation 6) is an intuitive and simple choice that provides flexible modulation while still being relatively constrained. However, one drawback of the basic linear modulation is that it is “always on”, and there may be instances where the modulation could actually be counter-productive to learning. To allow the model to adaptively learn when to apply modulation, we extend the feature-wise linear modulation using a sigmoid gating term, $\rho _ { k }$ (with $[ 0 , 1 ]$ entries), that gates in the influence of $\gamma$ and $\beta$ :
+
+$$
+\begin{array} { r } { \begin{array} { c } { \beta _ { k } , \gamma _ { k } , \rho _ { k } = \psi ( \mathcal { G } ) } \\ { \beta _ { k } = \rho _ { k } \odot \beta _ { k } + \left( \mathbb { 1 } - \rho _ { k } \right) \odot \mathbb { 1 } } \\ { \gamma _ { k } = \rho _ { k } \odot \gamma _ { k } + \left( \mathbb { 1 } - \rho _ { k } \right) \odot \mathbb { 1 } } \\ { m _ { \beta _ { k } , \gamma _ { k } } \left( W ^ { ( k ) } h _ { u } ^ { ( k - 1 ) } \right) = \gamma _ { k } \odot W h ^ { ( k - 1 ) } + \beta _ { k } . } \end{array} } \end{array}
+$$
+
+GS-Weights. In the final variant of Meta-Graph, we extend the gating and modulation idea by separately aggregating graph neighborhood information with and without modulation and then merging these two signals via a convex combination:
+
+$$
+\begin{array} { r l } & { \beta _ { k } , \gamma _ { k } , \rho _ { k } = \psi ( \mathcal { G } ) } \\ & { \quad h _ { v } ^ { ( k ) , 1 } = \mathrm { R e L U } \left( \displaystyle \sum _ { u \in N ( v ) \cup \{ v \} } \frac { W ^ { ( k ) } h _ { u } ^ { ( k - 1 ) } } { \sqrt { | \mathcal { N } ( v ) | | \mathcal { N } ( u ) | } } \right) } \\ & { \quad h _ { v } ^ { ( k ) , 2 } = \mathrm { R e L U } \left( \displaystyle \sum _ { u \in N ( v ) \cup \{ v \} } \frac { m _ { s _ { \beta _ { k } , \gamma _ { k } } } \left( W ^ { ( k ) } h _ { u } ^ { ( k - 1 ) } \right) } { \sqrt { | \mathcal { N } ( v ) | | \mathcal { N } ( u ) | } } \right) } \\ & { \quad h _ { \eta } ^ { ( k ) } = \rho _ { k } \odot h _ { v } ^ { ( k ) , 1 } + ( 1 - \rho _ { k } ) \odot h _ { \eta } ^ { ( k ) , 2 } , } \end{array}
+$$
+
+where we use the basic linear modulation (Equation 6) to define $m _ { s _ { \beta _ { k } } , \gamma _ { k } }$
+
+# 3.4 MAML FOR LINK PREDICTION AS A SPECIAL CASE
+
+Note that a simplification of Meta-Graph, where the graph signature function is removed, can be viewed as an adaptation of model agnostic meta learning (MAML) (Finn et al., 2017) to the few shot link prediction setting. As discussed in Section 2, there are important differences in the setup for few shot link prediction, compared to traditional few shot classification. Nonetheless, the core idea of leveraging an inner and outer loop of training in Algorithm 1—as well as using second order gradients to optimize the global parameters—can be viewed as an adaptation of MAML to the graph setting, and we provide comparisons to this simplified MAML approach in the experiments below. We formalize the key differences by depicting the graphical model of MAML as first depicted in (Grant et al., 2018) and contrasting it with the graphical model for Meta-Graph, in Figure 1. MAML when reinterpreted for a distribution over graphs, maximizes the likelihood over all edges in the distribution. On the other hand, Meta-Graph when recast in a hierarchical Bayesian framework adds a graph signature function that influences $\tilde { \phi _ { j } }$ to produce the modulated parameters $\phi _ { j }$ from $N$ sampled edges. This explicit influence of $\psi$ is captured by the term $p ( \tilde { \phi _ { j } } | \psi , \phi _ { j } )$ in Equation 7 below:
+
+$$
+p ( \mathcal { E } | \theta ) = \prod _ { j } ^ { J } \left( \int \int p ( \mathcal { E } _ { j } | \phi _ { j } ) p ( \phi _ { j } | \psi , \tilde { \phi } _ { j } ) p ( \tilde { \phi _ { j } } | \theta ) d \phi _ { j } d \tilde { \phi _ { j } } \right)
+$$
+
+For computational tractability we take the likelihood of the modulated parameters as a point estimate —i.e., $p \bar { ( \phi _ { j } | \psi , \tilde { \phi _ { j } } ) } = \delta ( \psi \cdot \tilde { \tilde { \phi _ { j } } } )$ .
+
+# 4 EXPERIMENTS
+
+We design three novel benchmarks for the few-shot link prediction task. All of these benchmarks contain a set of graphs drawn from a common domain. In all settings, we use $80 \%$ of these graphs for training and $10 \%$ as validation graphs, where these training and validation graphs are used to optimize the global model parameters (for Meta-Graph) or pre-train weights (for various baseline approaches). We then provide the remaining $10 \%$ of the graphs as test graphs, and our goal is to fine-tune or train a model on these test graphs to achieve high link prediction accuracy. Note that in this few shot link prediction setting, there are train/val/test splits at both the level of graphs and edges: for every individual graph, we are optimizing a model using the training edges to predict the likelihood of the test edges, but we are also training on multiple graphs with the goal of facilitating fast adaptation to new graphs via the global model parameters.
+
+Our goal is to use our benchmarks to investigate four key empirical questions:
+
+Q1 How does the overall performance of Meta-Graph compare to various baselines, including (i) a simple adaptation of MAML (Finn et al., 2017) (i.e., an ablation of Meta-Graph where the graph signature function is removed), (ii), standard pre-training approaches where we pre-train the VGAE model on the training graphs before fine-tuning on the test graphs, and (iii) naive baselines that do not leverage multi-graph information (i.e., a basic VGAE without pre-training, the Adamic-Adar heuristic (Adamic & Adar, 2003), and DeepWalk (Perozzi et al., 2014))?   
+Q2 How well does Meta-Graph perform in terms of fast adaption? Is Meta-Graph able to achieve strong performance after only a small number of gradient steps on the test graphs?   
+Q3 How necessary is the graph signature function for strong performance, and how do the different variants of the Meta-Graph signature function compare across the various benchmark settings?   
+Q4 What is learned by the graph signature function? For example, do the learned graph signatures correlate with the structural properties of the input graphs, or are they more sensitive to node feature information?
+
+Datasets. Two of our benchmarks are derived from standard multi-graph datasets from proteinprotein interaction (PPI) networks (Zitnik & Leskovec, 2017) and 3D point cloud data (FirstMMDB) (Neumann et al., 2013). These benchmarks are traditionally used for node and graph classification, respectively, but we adapt them for link prediction. We also create a novel multi-graph dataset based upon the AMINER citation data (Tang et al., 2008), where each node corresponds to a paper and links represent citations. We construct individual graphs from AMINER data by sampling ego networks around nodes and create node features using embeddings of the paper abstracts (see Appendix for details). We preprocess all graphs in each domain such that each graph contains a minimum of 100 nodes and up to a maximum of 20000 nodes. For all datasets, we perform link prediction by training on a small subset (i.e., a percentage) of the edges and then attempting to predict the unseen edges (with $2 0 \%$ of the held-out edges used for validation). Key dataset statistics are summarized in Table 1.
+
+Table 1: Statistics for the three datasets used to test Meta-Graph.   
+
+<table><tr><td colspan="2">DATASET</td><td colspan="2">#GRAPHS</td><td colspan="2">AVG.NODES</td><td colspan="2">AVG.EDGES</td><td colspan="2">#NODE FEATS</td><td></td></tr><tr><td colspan="2">PPI</td><td colspan="2">24</td><td colspan="2">2,331</td><td colspan="2">64,596</td><td colspan="3">50</td></tr><tr><td colspan="2">FIRSTMMDB</td><td colspan="2">41</td><td colspan="2">1,377</td><td colspan="2">6,147</td><td colspan="3">5</td></tr><tr><td colspan="2">EGO-AMINER</td><td colspan="2">72</td><td colspan="2">462</td><td colspan="2">2245</td><td colspan="3">300</td></tr><tr><td colspan="9">PPI</td><td rowspan="2">Ego-AMINER</td><td colspan="2"></td></tr><tr><td colspan="2">Edges</td><td>10%</td><td>20%</td><td>30%</td><td>10%</td><td>20%</td><td>FirstMMDB 30%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td colspan="2">Meta-Graph</td><td>0.795</td><td>0.833</td><td>0.845</td><td>0.782</td><td>0.786</td><td>0.783</td><td>0.626</td><td></td><td></td><td>0.786</td></tr><tr><td colspan="2">MAML</td><td>0.770</td><td>0.815</td><td>0.828</td><td>0.776</td><td>0.782</td><td></td><td>0.793</td><td>0.561</td><td>0.738 0.662</td><td>0.667</td></tr><tr><td colspan="2">Random</td><td>0.578</td><td>0.651</td><td>0.697</td><td>0.742</td><td>0.732</td><td></td><td>0.720</td><td>0.500</td><td>0.500</td><td>0.500</td></tr><tr><td colspan="2">No Fintune</td><td>0.738</td><td>0.786</td><td>0.801</td><td>0.740</td><td>0.710</td><td></td><td>0.734</td><td>0.548</td><td>0.621</td><td>0.673</td></tr><tr><td colspan="2">Finetune</td><td>0.752</td><td>0.801</td><td>0.821</td><td>0.752</td><td>0.735</td><td></td><td>0.723</td><td>0.623</td><td>0.691</td><td>0.723</td></tr><tr><td colspan="2">Adamic</td><td>0.540</td><td>0.623</td><td>0.697</td><td>0.504</td><td>0.519</td><td></td><td>0.544</td><td>0.515</td><td>0.549</td><td>0.597</td></tr><tr><td colspan="2">Deepwalk</td><td>0.664</td><td>0.673</td><td>0.694</td><td>0.487</td><td>0.473</td><td></td><td>0.510</td><td>0.602</td><td>0.638</td><td>0.672</td></tr><tr><td colspan="2"></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 2: Convergence AUC results for different training edge splits.
+
+Baseline details. Several baselines correspond to modifications or ablations of Meta-Graph, including the straightforward adaptation of MAML (which we term MAML in the results), a finetune baseline where we pre-train a VGAE on the training graphs observed in a sequential order and finetune on the test graphs (termed Finetune). We also consider a VGAE trained individually on each test graph (termed No Finetune). For Meta-Graph and all of these baselines we employ Bayesian optimization with Thompson sampling (Kandasamy et al., 2018) to perform hyperparameter selection using the validation sets. We use the recommended default hyperparameters for DeepWalk and Adamic-Adar baseline is hyperparameter-free. 2
+
+# 4.1 RESULTS
+
+Q1: Overall Performance. Table 2 shows the link prediction AUC for Meta-Graph and the baseline models when trained to convergence using $10 \%$ , $20 \%$ or $30 \%$ of the graph edges. In this setting, we adapt the link prediction models on the test graphs until learning converges, as determined by performance on the validation set of edges, and we report the average link prediction AUC over the test edges of the test graphs. Overall, we find that Meta-Graph achieves the highest average AUC in all but one setting, with an average relative improvement of $4 . 8 \%$ in AUC compared to the MAML approach and an improvement of $5 . 3 \%$ compared to the Finetune baseline. Notably, MetaGraph is able to maintain especially strong performance when using only $1 0 \%$ of the graph edges for training, highlighting how our framework can learn from very sparse samples of edges. Interestingly, in the Ego-AMINER dataset, unlike PPI and FIRSTMM DB, we observe the relative difference in performance between Meta-Graph and MAML to increase with density of the training set. We hypothesize that this is due to fickle nature of optimization with higher order gradients in MAML (Antoniou et al., 2018) which is somewhat alleviated in GS-gating due to the gating mechanism. With respect to computational complexity we observe a slight overhead when comparing MetaGraph to MAML which can be reconciled by realizing that the graph signature function is not updated in the inner loop update but only in outer loop. In the Appendix, we provide additional results when using larger sets of training edges, and, as expected, we find that the relative gains of Meta-Graph decrease as more and more training edges are available.
+
+Q2: Fast Adaptation. Table 3 highlights the average AUCs achieved by Meta-Graph and the baselines after performing only 5 gradient updates on the batch of training edges. Note that in this setting we only compare to the MAML, Finetune, and No Finetune baselines, as fast adaption in this setting is not well defined for the DeepWalk and Adamic-Adar baselines. In terms of fast adaptation, we again find that Meta-Graph is able to outperform all the baselines in all but one setting, with an average relative improvement of $9 . 4 \%$ compared to MAML and $8 . 0 \%$ compared to the Finetune baseline—highlighting that Meta-Graph can not only learn from sparse samples of edges but is also able to quickly learn on new data using only a small number of gradient steps. Also, we observe poor performance for MAML in the Ego-AMINER dataset dataset which we hypothesize is due to extremely low learning rates —i.e. $1 e - 7$ needed for any learning, the addition of a graph signature alleviates this problem. Figure 2 shows the learning curves for the various models on the PPI and FirstMM DB datasets, where we can see that Meta-Graph learns very quickly but can also begin to overfit after only a small number of gradient updates, making early stopping essential.
+
+<table><tr><td></td><td colspan="3">PPI</td><td colspan="3">FirstMM DB</td><td colspan="3">Eg0-AMINER</td></tr><tr><td>Edges</td><td>10%</td><td>20%</td><td>30%</td><td>10%</td><td>20%</td><td>30%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td>Meta-Graph</td><td>0.795</td><td>0.824</td><td>0.847</td><td>0.773</td><td>0.767</td><td>0.737</td><td>0.620</td><td>0.585</td><td>0.732</td></tr><tr><td>MAML</td><td>0.728</td><td>0.809</td><td>0.804</td><td>0.763</td><td>0.750</td><td>0.750</td><td>0.500</td><td>0.504</td><td>0.500</td></tr><tr><td> No Fintune</td><td>0.600</td><td>0.697</td><td>0.717</td><td>0.708</td><td>0.680</td><td>0.709</td><td>0.500</td><td>0.500</td><td>0.500</td></tr><tr><td>Finetune</td><td>0.582</td><td>0.727</td><td>0.774</td><td>0.705</td><td>0.695</td><td>0.704</td><td>0.608</td><td>0.675</td><td>0.713</td></tr></table>
+
+Table 3: 5-gradient update AUC results with various fractions of training edges.
+
+![](images/37b689acaa1f4fc5bc7594ba671ec19c5464a82b14bb67bf1a5fa8e725e3edbc.jpg)
+
+![](images/01fcf45c53d7b493495aa9afbd34ee2891c5b0a2264e5031268f3e66a6b1d2f5.jpg)  
+Figure 2: AUC scores on PPI (Left) and FirstMM DB (Right) graphs with $1 0 \%$ of edges observed.
+
+Q3: Choice of Meta-Graph Architecture. We study the impact of the graph signature function and its variants GS-Gating and GS-Weights by performing an ablation study using the FirstMM DB dataset. Figure 3 shows the performance of the different model variants and baselines considered as the training progresses. In addition to models that utilize different signature functions we report a random baseline where parameters are initialized but never updated allowing us to assess the inherent power of the VGAE model for few-shot link prediction. To better understand the utility of using a GCN based inference network we also report a VGAE model that uses a simple MLP on the node features and is trained analogously to Meta-Graph as a baseline. As shown in Figure 3 many versions of the signature function start at a better initialization point or quickly achieve higher AUC scores in comparison to MAML and the other baselines, but simple modulation and GS-Gating are superior to GS-Weights after a few gradient steps.
+
+Q4: What is learned by the graph signature? To gain further insight into what knowledge is transferable among graphs we use the FirstMM DB and Ego-AMINER datasets to probe and compare the output of the signature function with various graph heuristics. In particular, we treat the output of ${ \bar { s _ { \mathscr { G } } } } = \psi ( { \mathscr { G } } )$ as a vector and compute the cosine similarity between all pairs of graph in the training set (i.e., we compute the pairwise cosine similarites between graph signatures, $s _ { \mathcal { G } }$ ). We similarly compute three pairwise graph statistics—namely, the cosine similarity between average node features in the graphs, the difference in number of nodes, and the difference in number of edges—and we compute the Pearson correlation between the pairwise graph signature similarities and these other pairwise statistics. As shown in Table 4 we find strong positive correlation in terms of Pearson correlation coefficient between node features and the output of the signature function for both datasets, indicating that the graph signature function is highly sensitive to feature information. This observation is not entirely surprising given that we use such sparse samples of edges—meaning that many structural graph properties are likely lost and making the meta-learning heavily reliant on node feature information. We also observe moderate negative correlation with respect to the average difference in nodes and edges between pairs of graphs for FirstMM DB dataset. For Ego-AMINER we observe small positive correlation for difference in nodes and edges.
+
+![](images/b16fab5eab70141cf24bba12e5766df02f86d177d0c73d38b34090c50bb1d346.jpg)  
+Figure 3: Ablation study on PPI (Left) and FirstMM DB (Right) graphs with $1 0 \%$ of edges.
+
+![](images/9e75963c562b5c3a338965c0cea1e509b82d55090b5f5cf547344711f0f0ef9a.jpg)
+
+<table><tr><td></td><td colspan="3">FirstMMDB</td><td colspan="3">Ego-AMINER</td></tr><tr><td>% Edges</td><td>10%</td><td>20%</td><td>30%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td>Node Feats</td><td>0.928</td><td>0.950</td><td>0.761</td><td>0.473</td><td>0.385</td><td>0.448</td></tr><tr><td>Diff Num. Nodes</td><td>-0.093</td><td>-0.196</td><td>-0.286</td><td>0.095</td><td>0.086</td><td>0.085</td></tr><tr><td>Diff Num. Edges</td><td>-0.093</td><td>-0.195</td><td>-0.281</td><td>0.093</td><td>0.072</td><td>0.075</td></tr></table>
+
+Table 4: Pearson scores between graph signature output and other graph statistics.
+
+# 5 RELATED WORK
+
+We now briefly highlight related work on link prediction, meta-learning, few-shot classification, and few-shot learning in knowledge graphs. Link prediction considers the problem of predicting missing edges between two nodes in a graph that are likely to have an edge. (Liben-Nowell & Kleinberg, 2003). Common successful applications of link prediction include friend and content recommendations (Aiello et al., 2012), shopping and movie recommendation (Huang et al., 2005), knowledge graph completion (Nickel et al., 2015) and even important social causes such as identifying criminals based on past activities (Hasan et al., 2006). Historically, link prediction methods have utilized topological graph features such as common neighbors yielding strong baselines like Adamic/Adar measure (Adamic & Adar, 2003), Jaccard Index among others. Other approaches include Matrix Factorization (Menon & Elkan, 2011) and more recently deep learning and graph neural networks based approaches (Grover & Leskovec, 2016; Wang et al., 2015; Zhang & Chen, 2018) have risen to prominence. A commonality among all the above approaches is that the link prediction problem is define over a single dense graph where the objective is to predict unknown/future links within the same graph. Unlike these previous approaches, our approach considers link prediction tasks over multiple sparse graphs which are drawn from distribution over graphs akin to real world scenario such as protein-protein interaction graphs, 3D point cloud data and citation graphs in different communities.
+
+In meta-learning or learning to learn (Bengio et al., 1990; 1992; Thrun & Pratt, 2012; Schmidhuber, 1987), the objective is to learn from prior experiences to form inductive biases for fast adaptation to unseen tasks. Meta-learning has been particularly effective in few-shot learning tasks with a few notable approaches broadly classified into metric based approaches (Vinyals et al., 2016; Snell et al., 2017; Koch et al., 2015), augmented memory (Santoro et al., 2016; Kaiser et al., 2017; Mishra et al., 2017) and optimization based approaches (Finn et al., 2017; Lee & Choi, 2018). Recently, there are several works that lie at the intersection of meta-learning for few-shot classification and graph based learning. In Latent Embedding Optimization, Rusu et al. (2018) learn a graph between tasks in embedding space while Liu et al. (2019) introduce a message propagation rule between prototypes of classes. However, both these methods are restricted to the image domain and do not consider meta-learning over a distribution of graphs as done here.
+
+Another related line of work considers the task of few-shot relation prediction in knowledge graphs. Xiong et al. (2018) developed the first method for this task, which leverages a learned matching metric using both a learned embedding and one-hop graph structures. More recently Chen et al. (2019) introduce Meta Relational Learning framework (MetaR) that seeks to transfer relation-specific meta information to new relation types in the knowledge graph. A key distinction between few-shot relation setting and the one which we consider in this work is that we assume a distribution over graphs while in the knowledge graph setting there is only a single graph and the challenge is generalizing to new types of relations within this graph.
+
+# 6 DISCUSSION AND CONCLUSION
+
+We introduce the problem of few-shot link prediction—where the goal is to learn from multiple graph datasets to perform link prediction using small samples of graph data—and we develop the Meta-Graph framework to address this task. Our framework adapts gradient-based meta learning to optimize a shared parameter initialization for local link prediction models, while also learning a parametric encoding, or signature, of each graph, which can be used to modulate this parameter initialization in a graph-specific way. Empirically, we observed substantial gains using Meta-Graph compared to strong baselines on three distinct few-shot link prediction benchmarks. In terms of limitations and directions for future work, one key limitation is that our graph signature function is limited to modulating the local link prediction model through an encoding of the current graph, which does not explicitly capture the pairwise similarity between graphs in the dataset. Extending Meta-Graph by learning a similarity metric or kernel between graphs—which could then be used to condition meta-learning—is a natural direction for future work. Another interesting direction for future work is extending the Meta-Graph approach to multi-relational data, and exploiting similarities between relation types through a suitable Graph Signature function.
+
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+
+# 7 APPENDIX
+
+# 7.1 A: EGO-AMINER DATASET CONSTRUCTION
+
+To construct the Ego-Aminer dataset we first create citation graphs from different fields of study. We then select the top 100 graphs in terms number of nodes for further pre-processing. Specifically, we take the 5-core of each graph ensuring that each node has a minimum of 5-edges. We then construct ego networks by randomly sampling a node from the 5-core graph and taking its two hop neighborhood. Finally, we remove graphs with fewer than 100 nodes and greater than 20000 nodes which leads to a total of 72 graphs as reported in Table 1.
+
+# 7.2 B: ADDITIONAL RESULTS
+
+We list out complete results when using larger sets of training edges for PPI, FIRSTMM DB and Ego-Aminer datasets. We show the results for two metrics i.e. Average AUC across all test graphs. As expected, we find that the relative gains of Meta-Graph decrease as more and more training edges are available.
+
+Table 5: AUC Convergence results for PPI dataset for training edge splits   
+
+<table><tr><td colspan="8">PPI</td></tr><tr><td>Convergence</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.795</td><td>0.831</td><td>0.846</td><td>0.853</td><td>0.848</td><td>0.853</td><td>0.855</td></tr><tr><td>MAML</td><td>0.745</td><td>0.820</td><td>0.840</td><td>0.852</td><td>0.854</td><td>0.856</td><td>0.863</td></tr><tr><td>Random</td><td>0.578</td><td>0.651</td><td>0.697</td><td>0.729</td><td>0.756</td><td>0.778</td><td>0.795</td></tr><tr><td>No Finetune</td><td>0.738</td><td>0.786</td><td>0.801</td><td>0.817</td><td>0.827</td><td>0.837</td><td>0.836</td></tr><tr><td>Finetune</td><td>0.752</td><td>0.8010</td><td>0.821</td><td>0.832</td><td>0.818</td><td>0.856</td><td>0.841</td></tr><tr><td>Adamic</td><td>0.540</td><td>0.623</td><td>0.697</td><td>0.756</td><td>0.796</td><td>0.827</td><td>0.849</td></tr><tr><td>MAML-MLP</td><td>0.603</td><td>0.606</td><td>0.606</td><td>0.606</td><td>0.604</td><td>0.604</td><td>0.605</td></tr><tr><td>Deepwalk</td><td>0.664</td><td>0.673</td><td>0.694</td><td>0.727</td><td>0.731</td><td>0.747</td><td>0.761</td></tr></table>
+
+Table 6: 5-gradient update AUC results for PPI for training edge splits   
+
+<table><tr><td>PPI-5 updates</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.795</td><td>0.829</td><td>0.847</td><td>0.853</td><td>0.848</td><td>0.854</td><td>0.856</td></tr><tr><td>MAML</td><td>0.756</td><td>0.837</td><td>0.840</td><td>0.852</td><td>0.855</td><td>0.855</td><td>0.856</td></tr><tr><td>No Finetune</td><td>0.600</td><td>0.697</td><td>0.717</td><td>0.784</td><td>0.814</td><td>0.779</td><td>0.822</td></tr><tr><td>Finetune</td><td>0.582</td><td>0.727</td><td>0.774</td><td>0.702</td><td>0.804</td><td>0.718</td><td>0.766</td></tr><tr><td>MAML-MLP</td><td>0.603</td><td>0.606</td><td>0.603</td><td>0.604</td><td>0.603</td><td>0.606</td><td>0.605</td></tr></table>
+
+Table 7: AUC Convergence results for FIRSTMM DB dataset for training edge splits   
+
+<table><tr><td colspan="8">FirstMMDB</td></tr><tr><td>Convergence</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.782</td><td>0.786</td><td>0.783</td><td>0.781</td><td>0.760</td><td>0.746</td><td>0.739</td></tr><tr><td>MAML</td><td>0.776</td><td>0.782</td><td>0.793</td><td>0.785</td><td>0.791</td><td>0.663</td><td>0.788</td></tr><tr><td>Random</td><td>0.742</td><td>0.732</td><td>0.720</td><td>0.714</td><td>0.705</td><td>0.698</td><td>0.695</td></tr><tr><td>No Finetune</td><td>0.740</td><td>0.710</td><td>0.734</td><td>0.722</td><td>0.712</td><td>0.710</td><td>0.698</td></tr><tr><td>Finetune</td><td>0.752</td><td>0.735</td><td>0.723</td><td>0.734</td><td>0.749</td><td>0.700</td><td>0.695</td></tr><tr><td>Adamic</td><td>0.504</td><td>0.519</td><td>0.544</td><td>0.573</td><td>0.604</td><td>0.643</td><td>0.678</td></tr><tr><td>Deepwalk</td><td>0.487</td><td>0.473</td><td>0.510</td><td>0.608</td><td>0.722</td><td>0.832</td><td>0.911</td></tr></table>
+
+Table 8: 5-gradient update AUC results for FIRSTMM DB for training edge splits   
+
+<table><tr><td colspan="8">FirstMMDB</td></tr><tr><td>5 updates</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.773</td><td>0.767</td><td>0.743</td><td>0.759</td><td>0.742</td><td>0.732</td><td>0.688</td></tr><tr><td>MAML</td><td>0.763</td><td>0.750</td><td>0.624</td><td>0.776</td><td>0.759</td><td>0.663</td><td>0.738</td></tr><tr><td>No Finetune</td><td>0.708</td><td>0.680</td><td>0.709</td><td>0.701</td><td>0.685</td><td>0.683</td><td>0.653</td></tr><tr><td>Finetune</td><td>0.705</td><td>0.695</td><td>0.704</td><td>0.704</td><td>0.696</td><td>0.658</td><td>0.670</td></tr></table>
+
+Table 9: AUC Convergence results for Ego-Aminer dataset for training edge splits   
+
+<table><tr><td colspan="8">Ego-Aminer</td></tr><tr><td>Convergence</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.626</td><td>0.738</td><td>0.786</td><td>0.791</td><td>0.792</td><td>0.817</td><td>0.786</td></tr><tr><td>MAML</td><td>0.561</td><td>0.662</td><td>0.667</td><td>0.682</td><td>0.720</td><td>0.741</td><td>0.768</td></tr><tr><td>Random</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td></tr><tr><td>No Finetune</td><td>0.548</td><td>0.621</td><td>0.673</td><td>0.702</td><td>0.652</td><td>0.7458</td><td>0.769</td></tr><tr><td>Finetune</td><td>0.623</td><td>0.691</td><td>0.723</td><td>0.764</td><td>0.767</td><td>0.792</td><td>0.781</td></tr><tr><td>Adamic</td><td>0.515</td><td>0.549</td><td>0.597</td><td>0.655</td><td>0.693</td><td>0.744</td><td>0.772</td></tr><tr><td>Deepwalk</td><td>0.602</td><td>0.638</td><td>0.672</td><td>0.686</td><td>0.689</td><td>0.711</td><td>0.731</td></tr></table>
+
+Table 10: 5-gradient update AUC results for Ego-Aminer for training edge splits   
+
+<table><tr><td colspan="8">Ego-Aminer</td></tr><tr><td>5 updates</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.620</td><td>0.5850</td><td>0.732</td><td>0.500</td><td>0.790</td><td>0.733</td><td>0.500</td></tr><tr><td>MAML</td><td>0.500</td><td>0.504</td><td>0.500</td><td>0.500</td><td>0.519</td><td>0.500</td><td>0.500</td></tr><tr><td>No Finetune</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td></tr><tr><td>Finetune</td><td>0.608</td><td>0.675</td><td>0.713</td><td>0.755</td><td>0.744</td><td>0.706</td><td>0.671</td></tr></table>

parse/train/BJepcaEtwB/BJepcaEtwB_content_list.json

@@ -0,0 +1,1553 @@
+[
+    {
+        "type": "text",
+        "text": "META-GRAPH: FEW SHOT LINK PREDICTION VIA META LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We consider the task of few shot link prediction, where the goal is to predict missing edges across multiple graphs using only a small sample of known edges. We show that current link prediction methods are generally ill-equipped to handle this task—as they cannot effectively transfer knowledge between graphs in a multigraph setting and are unable to effectively learn from very sparse data. To address this challenge, we introduce a new gradient-based meta learning framework, Meta-Graph, that leverages higher-order gradients along with a learned graph signature function that conditionally generates a graph neural network initialization. Using a novel set of few shot link prediction benchmarks, we show that MetaGraph enables not only fast adaptation but also better final convergence and can effectively learn using only a small sample of true edges. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            446,
+            336,
+            463
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Given a graph representing known relationships between a set of nodes, the goal of link prediction is to learn from the graph and infer novel or previously unknown relationships (Liben-Nowell & Kleinberg, 2003). For instance, in a social network we may use link prediction to power a friendship recommendation system (Aiello et al., 2012), or in the case of biological network data we might use link prediction to infer possible relationships between drugs, proteins, and diseases (Zitnik & Leskovec, 2017). However, despite its popularity, previous work on link prediction generally focuses only on one particular problem setting: it generally assumes that link prediction is to be performed on a single large graph and that this graph is relatively complete, i.e., that at least $50 \\%$ of the true edges are observed during training (e.g., see Grover & Leskovec, 2016; Kipf & Welling, 2016b; Liben-Nowell & Kleinberg, 2003; Lu & Zhou, 2011). ¨ ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we consider the more challenging setting of few shot link prediction, where the goal is to perform link prediction on multiple graphs that contain only a small fraction of their true, underlying edges. This task is inspired by applications where we have access to multiple graphs from a single domain but where each of these individual graphs contains only a small fraction of the true, underlying edges. For example, in the biological setting, high-throughput interactomics offers the possibility to estimate thousands of biological interaction networks from different tissues, cell types, and organisms (Barrios-Rodiles et al., 2005); however, these estimated relationships can be noisy and sparse, and we need learning algorithms that can leverage information across these multiple graphs in order to overcome this sparsity. Similarly, in the e-commerce and social network settings, link prediction can often have a large impact in cases where we must quickly make predictions on sparsely-estimated graphs, such as when a service has been recently deployed to a new locale. That is to say to link prediction for a new sparse graph can benefit from transferring knowledge from other, possibly more dense, graphs assuming there is exploitable shared structure. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We term this problem of link prediction from sparsely-estimated multi-graph data as few shot link prediction analogous to the popular few shot classification setting (Miller et al., 2000; Lake et al., 2011; Koch et al., 2015). The goal of few shot link prediction is to observe many examples of graphs from a particular domain and leverage this experience to enable fast adaptation and higher accuracy when predicting edges on a new, sparsely-estimated graph from the same domain—a task that can can also be viewed as a form of meta learning, or learning to learn (Bengio et al., 1990; 1992; Thrun & Pratt, 2012; Schmidhuber, 1987) in the context of link prediction. This few shot link prediction setting is particularly challenging as current link prediction methods are generally ill-equipped to transfer knowledge between graphs in a multi-graph setting and are also unable to effectively learn from very sparse data. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Present work. We introduce a new framework called Meta-Graph for few shot link prediction and also introduce a series of benchmarks for this task. We adapt the classical gradient-based metalearning formulation for few shot classification (Miller et al., 2000; Lake et al., 2011; Koch et al., 2015) to the graph domain. Specifically, we consider a distribution over graphs as the distribution over tasks from which a global set of parameters are learnt, and we deploy this strategy to train graph neural networks (GNNs) that are capable of few-shot link prediction. To further bootstrap fast adaptation to new graphs we also introduce a graph signature function, which learns how to map the structure of an input graph to an effective initialization point for a GNN link prediction model. We experimentally validate our approach on three link prediction benchmarks. We find that our MetaGraph approach not only achieves fast adaptation but also converges to a better overall solution in many experimental settings, with an average improvement of $5 . { \\bar { 3 } } \\%$ in AUC at convergence over non-meta learning baselines. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/3d27283a5af3fc62b05e5f3045f1d1a1e6778a8917e954e47649f583cf08cdb7.jpg",
+        "image_caption": [
+            "Figure 1: Left: Graphical model for Meta-Graph vs. MAML. Right: Meta-Graph architecture. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES AND PROBLEM DEFINITION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            527,
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+            544
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The basic set-up for few shot link prediction is as follows: We assume that we have a distribution $p ( \\mathcal G )$ over graphs, from which we can sample training graphs $\\mathcal { G } _ { i } ~ \\sim ~ p ( \\mathcal { G } )$ , where each $\\mathcal { G } _ { i } = ( \\mathcal { V } _ { i } , \\mathcal { E } _ { i } , X _ { i } )$ is defined by a set of nodes $\\nu _ { i }$ , edges $\\mathcal { E } _ { i }$ , and matrix of real-valued node attributes $X \\in \\mathbb { R } ^ { | \\mathcal { V } _ { i } | \\times d }$ . When convenient, we will also equivalently represent a graph as $\\mathcal { G } _ { i } = ( \\mathcal { V } _ { i } , A _ { i } , X _ { i } )$ , where $A _ { i } \\in \\mathbb { Z } ^ { | \\mathcal { V } _ { i } | \\times | \\mathcal { V } _ { i } | }$ is an adjacency matrix representation of the edges in $\\mathcal { E } _ { i }$ . We assume that each of these sampled graphs, $\\mathcal { G } _ { i }$ , is a simple graph (i.e., contain a single type of relation and no self loops) and that every node $v \\in \\mathcal V _ { i }$ in the graph is associated with a real valued attribute vector $\\mathbf { x } _ { v } \\in \\bar { \\mathbb { R } ^ { d } }$ from a common vector space. We further assume that for each graph $\\mathcal { G } _ { i }$ we have access to only a sparse subset of the true edges $\\mathcal { E } _ { i } ^ { \\mathrm { t r a i n } } \\subset \\mathcal { E } _ { i }$ (with $| \\mathcal { E } _ { i } ^ { \\mathrm { t r a i n } } | < < | \\mathcal { E } _ { i } | )$ during training. In terms of distributional assumptions we assume that this $p ( \\mathcal G )$ is defined over a set of related graphs (e.g., graphs drawn from a common domain or application setting). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to learn a global or meta link prediction model from a set of sampled training graphs $\\mathcal { G } _ { i } \\sim p ( \\mathcal { G } ) , i = 1 . . . n$ , such that we can use this meta model to quickly learn an effective link prediction model on a newly sampled graph $\\mathcal { G } _ { * } \\sim p ( \\mathcal { G } )$ . More specifically, we wish to optimize a global set of parameters $\\theta$ , as well as a graph signature function $\\psi ( \\mathcal { G } _ { i } )$ , which can be used together to generate an effective parameter initialization, $\\phi _ { i }$ , for a local link prediction model on graph $\\mathcal { G } _ { i }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Relationship to standard link prediction. Few shot link prediction differs from standard link prediction in three important ways: ",
+        "bbox": [
+            174,
+            797,
+            823,
+            827
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. Rather than learning from a single graph $\\mathcal { G }$ , we are learning from multiple graphs $\\{ { \\mathcal { G } } _ { 1 } , . . . , { \\mathcal { G } } _ { n } \\}$ sampled from a common distribution or domain. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. We presume access to only a very sparse sample of true edges. Concretely, we focus on settings where at most $30 \\%$ of the edges in $\\mathcal { E } _ { i }$ are observed during training, i.e., where $\\frac { | \\mathcal { E } ^ { \\mathrm { t r a i n } } | } { | \\mathcal { E } | } \\leq 0 . 3$ . 1 ",
+        "bbox": [
+            179,
+            864,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. We distinguish between the global parameters, which are used to encode knowledge about the underlying distribution of graphs, and the local parameters $\\phi _ { i }$ , which are optimized to perform link prediction on a specific graph $\\mathcal { G } _ { i }$ . This distinction allows us to consider leveraging information from multiple graphs, while still allowing for individually-tuned link prediction models on each specific graph. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Relationship to traditional meta learning. Traditional meta learning for few-shot classification, generally assumes a distribution $p ( \\mathcal { T } )$ over classification tasks, with the goal of learning global parameters that can facilitate fast adaptation to a newly sampled task $\\mathcal { T } _ { i } \\sim p ( \\mathcal { T } )$ with few examples. We instead consider a distribution $p ( \\mathcal G )$ over graphs with the goal of performing link prediction on a newly sampled graph. An important complication of this graph setting is that the individual predictions for each graph (i.e., the training edges) are not i.i.d.. Furthermore, for few shot link prediction we require training samples as a sparse subset of true edges that represents a small percentage of all edges in a graph. Note that for very small percentages we effectively break all graph structure and recover the supervised setting for few shot classification and thus simplifying the problem. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 PROPOSED APPROACH",
+        "text_level": 1,
+        "bbox": [
+            176,
+            329,
+            392,
+            344
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We now outline our proposed approach, Meta-Graph, to the few shot link prediction problem. We first describe how we define the local link prediction models, which are used to perform link prediction on each specific graph $\\mathcal { G } _ { i }$ . Next, we discuss our novel gradient-based meta learning approach to define a global model that can learn from multiple graphs to generate effective parameter initializations for the local models. The key idea behind Meta-Graph is that we use gradient-based meta learning to optimize a shared parameter initialization $\\theta$ for the local models, while also learning a parametric encoding of each graph $\\mathcal { G } _ { i }$ that can be used to modulate this parameter initialization in a graph-specific way (Figure 1). ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 LOCAL LINK PREDICTION MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            494,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In principle, our framework can be combined with a wide variety of GNN-based link prediction approaches, but here we focus on variational graph autoencoders (VGAEs) (Kipf & Welling, 2016b) as our base link prediction framework. Formally, given a graph $\\mathcal { G } = ( \\nu , A , X )$ , the VGAE learns an inference model, $q _ { \\phi }$ , that defines a distribution over node embeddings $q _ { \\phi } ( Z | A , X )$ , where each row $z _ { v } \\in \\mathbb { R } ^ { d }$ of $Z \\in \\mathbb { R } ^ { | \\nu | \\times d }$ is a node embedding that can be used to score the likelihood of an edge existing between pairs of nodes. The parameters of the inference model are shared across all the nodes in $\\mathcal { G }$ , to define the approximate posterior $q _ { \\phi } ( z _ { v } | A , X ) = \\mathcal { N } ( z _ { v } | \\mu _ { v } , \\mathrm { d i a g } ( \\sigma _ { v } ^ { 2 } ) )$ , where the parameters of the normal distribution are learned via GNNs: ",
+        "bbox": [
+            173,
+            521,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e11eebbccec104bcb6d4586ab6b5a96ae3a8c1d66509bf3c3e2511025bb8402.jpg",
+        "text": "$$\n\\mu = { \\bf G } { \\bf N } { \\bf N } _ { \\mu } ( A , X ) , \\qquad \\mathrm { a n d } \\qquad \\log ( \\sigma ) = { \\bf G } { \\bf N } { \\bf N } _ { \\sigma } ( A , X ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            645,
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+            662
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The generative component of the VGAE is then defined as ",
+        "bbox": [
+            174,
+            672,
+            558,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2d7f1565f4ad87842a7d0fc2af0af9a0345bf3f60ccd164b92fd4076bb164fb3.jpg",
+        "text": "$$\np ( A | Z ) = \\prod _ { i = 1 } ^ { N } \\prod _ { j = 1 } ^ { N } p ( A _ { u , v } | z _ { u } , z _ { v } ) , \\qquad \\mathrm { w i t h } \\qquad p ( A _ { u , v } | z _ { u } , z _ { v } ) = \\sigma ( z _ { u } ^ { \\top } z _ { v } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            695,
+            746,
+            742
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "i.e., the likelihood of an edge existing between two nodes, $u$ and $v$ , is proportional to the dot product of their node embeddings. Given the above components, the inference GNNs can be trained to minimize the variational lower bound on the training data: ",
+        "bbox": [
+            173,
+            751,
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+            792
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/48abd887f4e65f2cca268df098bbdf9914dabf45064e4c17cb77dbc568fe9cc4.jpg",
+        "text": "$$\n\\mathcal { L } _ { G } = \\mathbb { E } _ { q _ { \\phi } } [ \\log p ( A ^ { \\operatorname { t r a i n } } | Z ) ] - K L [ q _ { \\phi } ( Z | X , A ^ { \\operatorname { t r a i n } } ) | | p ( z ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            803,
+            686,
+            823
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where a Gaussian prior is used for $p ( z )$ . ",
+        "bbox": [
+            176,
+            833,
+            436,
+            848
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We build upon VGAEs due to their strong performance on standard link prediction benchmarks (Kipf & Welling, 2016b), as well as the fact that they have a well-defined probabilistic interpretation that generalizes many embedding-based approaches to link prediction (e.g., node2vec (Grover & Leskovec, 2016)). We describe the specific GNN implementations we deploy for the inference model in Section 3.3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.2 OVERVIEW OF META-GRAPH ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "The key idea behind Meta-Graph is that we use gradient-based meta learning to optimize a shared parameter initialization $\\theta$ for the inference models of a VGAE, while also learning a parametric encoding $\\psi ( \\mathcal { G } _ { i } )$ that modulates this parameter initialization in a graph-specific way. Specifically, given a sampled training graph $\\mathcal { G } _ { i }$ , we initialize the inference model $q _ { \\phi _ { i } }$ for a VGAE link prediction model using a combination of two learned components: ",
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+        "type": "text",
+        "text": "• A global initialization, $\\theta$ , that is used to initialize all the parameters of the GNNs in the inference model. The global parameters $\\theta$ are optimized via second-order gradient descent to provide an effective initialization point for any graph sampled from the distribution $p ( \\mathcal G )$ . • A graph signature $s _ { \\mathcal { G } _ { i } } ~ = ~ \\psi ( \\mathcal { G } _ { i } )$ that is used to modulate the parameters of inference model $\\phi _ { i }$ based on the history of observed training graphs. In particular, we assume that the inference model $q _ { \\phi _ { i } }$ for each graph $\\mathcal { G } _ { i }$ can be conditioned on the graph signature. That is, we augment the inference model to $g _ { \\phi _ { i } } ( Z | A , X , s _ { \\mathcal { G } _ { i } } )$ , where we also include the graph signature $s _ { \\mathcal { G } _ { i } }$ as a conditioning input. We use a $\\mathbf { k }$ -layer graph convolutional network (GCN) (Kipf & Welling, 2016a), with sum pooling to compute the signature: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/d26cef0fdc5a9e2400c2d07353a038c843270bd09d009ca40d180c5e60db1243.jpg",
+        "text": "$$\ns _ { \\mathcal { G } } = \\psi ( \\mathcal { G } ) = \\mathbf { M } \\mathbf { L } \\mathbf { P } ( \\sum _ { v \\in \\mathcal { V } } z _ { v } ) \\qquad \\mathrm { w i t h } \\qquad Z = \\mathbf { G } \\mathbf { C } \\mathbf { N } ( A , X ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where GCN denotes a k-layer GCN (as defined in (Kipf & Welling, 2016a)), MLP denotes a densely-connected neural network, and we are summing over the node embeddings $z _ { v }$ output from the GCN. As with the global parameters $\\theta$ , the graph signature model $\\psi$ is optimized via second-order gradient descent. ",
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+    {
+        "type": "text",
+        "text": "The overall Meta-Graph architecture is detailed in Figure 1 and the core learning algorithm is summarized in the algorithm block below. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: Meta-Graph for Few Shot Link Prediction ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Result: Global parameters $\\theta$ , Graph signature function $\\psi$   \nInitialize learning rates: $\\alpha , \\epsilon$   \nSample a mini-batch of graphs, $\\mathcal { G } _ { b a t c h }$ from $p ( \\mathcal G )$ ;   \nfor each $\\mathcal { G } \\in \\mathcal { G } _ { b a t c h }$ do ${ \\mathcal { E } } = { \\mathcal { E } } ^ { \\mathrm { t r a i n } } \\cup { \\mathcal { E } } ^ { \\mathrm { v a l } } \\cup { \\mathcal { E } } ^ { \\mathrm { t e s t } } / /$ / Split edges into train, val, and test $s _ { \\mathcal { G } } = \\psi ( \\mathcal { G } , \\mathcal { E } ^ { \\mathrm { t r a i n } } )$ // Compute graph signature Initialize: $\\phi ^ { ( 0 ) }  \\theta / /$ Initialize local parameters via global parameters for $k$ in $[ 1 : K ]$ do $s _ { \\mathcal { G } } = \\mathrm { s t o p g r a d } ( s _ { \\mathcal { G } } )$ // Stop Gradients to Graph Signature $\\mathcal { L } _ { t r a i n } = \\mathbb { E } _ { q } [ \\log p ( A ^ { \\mathrm { { t r a i n } } } | Z ) ] - K L [ q _ { \\phi } ( Z | \\mathcal { E } ^ { \\mathrm { t r a i n } } , s _ { \\mathcal { G } } ) | | p ( z ) ]$ Update ${ \\phi } ^ { ( k ) } \\gets { \\phi } ^ { ( k - 1 ) } - \\alpha \\nabla _ { \\phi } \\mathcal { L } _ { t r a i n }$ end Initialize: $\\theta  \\phi _ { K }$ $s _ { \\mathcal { G } } = \\psi ( \\mathcal { G } , \\mathcal { E } ^ { \\mathrm { v a l } } \\cup \\mathcal { E } ^ { \\mathrm { t r a i n } } )$ // Compute graph signature with validation edges $\\begin{array} { r } { \\dot { \\mathcal { L } } _ { v a l } = \\mathbb { E } _ { q } [ \\log p ( A ^ { \\mathrm { v a l } } | \\dot { Z } ) ] - K \\dot { L } [ q ( \\bar { Z } | \\dot { \\mathcal { E } } ^ { \\mathrm { v a l } } \\cup \\dot { \\mathcal { E } } ^ { \\mathrm { t r a i n } } , s _ { \\mathcal { G } } ) | | p ( z ) ] } \\end{array}$ Update $\\theta  \\theta - \\epsilon \\nabla _ { \\theta } \\mathcal { L } _ { v a l }$ Update $\\psi  \\psi - \\epsilon \\nabla _ { \\psi } \\mathcal { L } _ { v a l }$   \nend ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The basic idea behind the algorithm is that we (i) sample a batch of training graphs, (ii) initialize VGAE link prediction models for these training graphs using our global parameters and signature function, (iii) run $K$ steps of gradient descent to optimize each of these VGAE models, and (iv) use second order gradient descent to update the global parameters and signature function based on a held-out validation set of edges. As depicted in Fig 1, this corresponds to updating the GCN based encoder for the local link prediction parameters $\\phi _ { j }$ and global parameters $\\theta$ along with the graph signature function $\\psi$ using second order gradients. Note that since we are running $K$ steps of gradient descent within the inner loop of Algorithm 1, we are also “meta” optimizing for fast adaptation, as $\\theta$ and $\\psi$ are being trained via second-order gradient descent to optimize the local model performance after $K$ gradient updates, where generally $K \\in \\{ 0 , 1 , \\ldots , 5 \\}$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 VARIANTS OF META-GRAPH ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We consider several concrete instantiations of the Meta-Graph framework, which differ in terms of how the output of the graph signature function is used to modulate the parameters of the VGAE inference models. For all the Meta-Graph variants, we build upon the standard GCN propagation rule (Kipf & Welling, 2016a) to construct the VGAE inference models. In particular, we assume that all the inference GNNs (Equation 1) are defined by stacking $K$ neural message passing layers of the form: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/18c9f267e26b29b59ce7371d5085a2b07ea09040beb16867a8154345c22db21b.jpg",
+        "text": "$$\nh _ { v } ^ { ( k ) } = \\mathrm { R e L U } \\left( \\sum _ { u \\in \\mathcal { N } ( v ) \\cup \\{ v \\} } \\frac { m _ { s _ { \\mathcal { G } } } \\left( W ^ { ( k ) } h _ { u } ^ { ( k - 1 ) } \\right) } { \\sqrt { | \\mathcal { N } ( v ) | | \\mathcal { N } ( u ) | } } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            208,
+            669,
+            260
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $h _ { v } \\in \\mathbb { R } ^ { d }$ denotes the embedding of node $v$ at layer $k$ of the model, $\\mathcal { N } ( v ) = \\{ u \\in \\mathcal { V } : e _ { u , v } \\in$ $\\mathcal { E } \\}$ denotes the nodes in the graph neighborhood of $v$ , and $W ^ { ( k ) } \\in \\mathbb { R } ^ { d \\times d }$ is a trainable weight matrix for layer $k$ . The key difference between Equation 5 and the standard GCN propagation rule is that we add the modulation function $m _ { s _ { \\mathcal G } }$ , which is used to modulate the message passing based on the graph signature $s _ { \\mathcal { G } } = \\psi ( \\mathcal { G } )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We describe different variations of this modulation below. In all cases, the intuition behind this modulation is that we want to compute a structural signature from the input graphs that can be used to condition the initialization of the local link prediction models. Intuitively, we expect this graph signature to encode structural properties of sampled graphs $\\mathcal { G } _ { i } \\sim p ( \\mathcal { G } )$ in order to modulate the parameters of the local VGAE link prediction models and adapt it to the current graph. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "GS-Modulation. Inspired by Brockschmidt (2019), we experiment with basic feature-wise linear modulation (Strub et al., 2018) to define the modulation function $m _ { s _ { \\mathcal G } }$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a4bc38eb6af86e4cca9ad33ef529b4801264d1a3399836fa30bce83e0831b1d.jpg",
+        "text": "$$\n\\begin{array} { c } { \\beta _ { k } , \\gamma _ { k } , = \\psi ( \\mathcal { G } ) } \\\\ { m _ { \\beta _ { k } , \\gamma _ { k } } \\left( W ^ { ( k ) } h _ { u } ^ { ( k - 1 ) } \\right) = \\gamma _ { k } \\odot W h ^ { ( k - 1 ) } + \\beta _ { k } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here, we restrict the modulation terms $\\beta _ { k }$ and $\\gamma _ { k }$ output by the signature function to be in $[ - 1 , 1 ]$ by applying a tanh non-linearity after Equation 4. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "GS-Gating. Feature-wise linear modulation of the GCN parameters (Equation 6) is an intuitive and simple choice that provides flexible modulation while still being relatively constrained. However, one drawback of the basic linear modulation is that it is “always on”, and there may be instances where the modulation could actually be counter-productive to learning. To allow the model to adaptively learn when to apply modulation, we extend the feature-wise linear modulation using a sigmoid gating term, $\\rho _ { k }$ (with $[ 0 , 1 ]$ entries), that gates in the influence of $\\gamma$ and $\\beta$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/01baf7cb4e672152c496689a3998baa763c489d8fbb69531b6f8bed547690904.jpg",
+        "text": "$$\n\\begin{array} { r } { \\begin{array} { c } { \\beta _ { k } , \\gamma _ { k } , \\rho _ { k } = \\psi ( \\mathcal { G } ) } \\\\ { \\beta _ { k } = \\rho _ { k } \\odot \\beta _ { k } + \\left( \\mathbb { 1 } - \\rho _ { k } \\right) \\odot \\mathbb { 1 } } \\\\ { \\gamma _ { k } = \\rho _ { k } \\odot \\gamma _ { k } + \\left( \\mathbb { 1 } - \\rho _ { k } \\right) \\odot \\mathbb { 1 } } \\\\ { m _ { \\beta _ { k } , \\gamma _ { k } } \\left( W ^ { ( k ) } h _ { u } ^ { ( k - 1 ) } \\right) = \\gamma _ { k } \\odot W h ^ { ( k - 1 ) } + \\beta _ { k } . } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "GS-Weights. In the final variant of Meta-Graph, we extend the gating and modulation idea by separately aggregating graph neighborhood information with and without modulation and then merging these two signals via a convex combination: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e3dcbe1f3ee5128b914219266fc3197a35770a4df0a3a3d3a1af8bb36eceb1be.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\beta _ { k } , \\gamma _ { k } , \\rho _ { k } = \\psi ( \\mathcal { G } ) } \\\\ & { \\quad h _ { v } ^ { ( k ) , 1 } = \\mathrm { R e L U } \\left( \\displaystyle \\sum _ { u \\in N ( v ) \\cup \\{ v \\} } \\frac { W ^ { ( k ) } h _ { u } ^ { ( k - 1 ) } } { \\sqrt { | \\mathcal { N } ( v ) | | \\mathcal { N } ( u ) | } } \\right) } \\\\ & { \\quad h _ { v } ^ { ( k ) , 2 } = \\mathrm { R e L U } \\left( \\displaystyle \\sum _ { u \\in N ( v ) \\cup \\{ v \\} } \\frac { m _ { s _ { \\beta _ { k } , \\gamma _ { k } } } \\left( W ^ { ( k ) } h _ { u } ^ { ( k - 1 ) } \\right) } { \\sqrt { | \\mathcal { N } ( v ) | | \\mathcal { N } ( u ) | } } \\right) } \\\\ & { \\quad h _ { \\eta } ^ { ( k ) } = \\rho _ { k } \\odot h _ { v } ^ { ( k ) , 1 } + ( 1 - \\rho _ { k } ) \\odot h _ { \\eta } ^ { ( k ) , 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where we use the basic linear modulation (Equation 6) to define $m _ { s _ { \\beta _ { k } } , \\gamma _ { k } }$ ",
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+    {
+        "type": "text",
+        "text": "3.4 MAML FOR LINK PREDICTION AS A SPECIAL CASE",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that a simplification of Meta-Graph, where the graph signature function is removed, can be viewed as an adaptation of model agnostic meta learning (MAML) (Finn et al., 2017) to the few shot link prediction setting. As discussed in Section 2, there are important differences in the setup for few shot link prediction, compared to traditional few shot classification. Nonetheless, the core idea of leveraging an inner and outer loop of training in Algorithm 1—as well as using second order gradients to optimize the global parameters—can be viewed as an adaptation of MAML to the graph setting, and we provide comparisons to this simplified MAML approach in the experiments below. We formalize the key differences by depicting the graphical model of MAML as first depicted in (Grant et al., 2018) and contrasting it with the graphical model for Meta-Graph, in Figure 1. MAML when reinterpreted for a distribution over graphs, maximizes the likelihood over all edges in the distribution. On the other hand, Meta-Graph when recast in a hierarchical Bayesian framework adds a graph signature function that influences $\\tilde { \\phi _ { j } }$ to produce the modulated parameters $\\phi _ { j }$ from $N$ sampled edges. This explicit influence of $\\psi$ is captured by the term $p ( \\tilde { \\phi _ { j } } | \\psi , \\phi _ { j } )$ in Equation 7 below: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ab17b95b3c009d46179d03a6953062362d1ad968d0a7400c795d55d81c5ff9b.jpg",
+        "text": "$$\np ( \\mathcal { E } | \\theta ) = \\prod _ { j } ^ { J } \\left( \\int \\int p ( \\mathcal { E } _ { j } | \\phi _ { j } ) p ( \\phi _ { j } | \\psi , \\tilde { \\phi } _ { j } ) p ( \\tilde { \\phi _ { j } } | \\theta ) d \\phi _ { j } d \\tilde { \\phi _ { j } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For computational tractability we take the likelihood of the modulated parameters as a point estimate —i.e., $p \\bar { ( \\phi _ { j } | \\psi , \\tilde { \\phi _ { j } } ) } = \\delta ( \\psi \\cdot \\tilde { \\tilde { \\phi _ { j } } } )$ . ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We design three novel benchmarks for the few-shot link prediction task. All of these benchmarks contain a set of graphs drawn from a common domain. In all settings, we use $80 \\%$ of these graphs for training and $10 \\%$ as validation graphs, where these training and validation graphs are used to optimize the global model parameters (for Meta-Graph) or pre-train weights (for various baseline approaches). We then provide the remaining $10 \\%$ of the graphs as test graphs, and our goal is to fine-tune or train a model on these test graphs to achieve high link prediction accuracy. Note that in this few shot link prediction setting, there are train/val/test splits at both the level of graphs and edges: for every individual graph, we are optimizing a model using the training edges to predict the likelihood of the test edges, but we are also training on multiple graphs with the goal of facilitating fast adaptation to new graphs via the global model parameters. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our goal is to use our benchmarks to investigate four key empirical questions: ",
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+        "type": "text",
+        "text": "Q1 How does the overall performance of Meta-Graph compare to various baselines, including (i) a simple adaptation of MAML (Finn et al., 2017) (i.e., an ablation of Meta-Graph where the graph signature function is removed), (ii), standard pre-training approaches where we pre-train the VGAE model on the training graphs before fine-tuning on the test graphs, and (iii) naive baselines that do not leverage multi-graph information (i.e., a basic VGAE without pre-training, the Adamic-Adar heuristic (Adamic & Adar, 2003), and DeepWalk (Perozzi et al., 2014))?   \nQ2 How well does Meta-Graph perform in terms of fast adaption? Is Meta-Graph able to achieve strong performance after only a small number of gradient steps on the test graphs?   \nQ3 How necessary is the graph signature function for strong performance, and how do the different variants of the Meta-Graph signature function compare across the various benchmark settings?   \nQ4 What is learned by the graph signature function? For example, do the learned graph signatures correlate with the structural properties of the input graphs, or are they more sensitive to node feature information? ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Datasets. Two of our benchmarks are derived from standard multi-graph datasets from proteinprotein interaction (PPI) networks (Zitnik & Leskovec, 2017) and 3D point cloud data (FirstMMDB) (Neumann et al., 2013). These benchmarks are traditionally used for node and graph classification, respectively, but we adapt them for link prediction. We also create a novel multi-graph dataset based upon the AMINER citation data (Tang et al., 2008), where each node corresponds to a paper and links represent citations. We construct individual graphs from AMINER data by sampling ego networks around nodes and create node features using embeddings of the paper abstracts (see Appendix for details). We preprocess all graphs in each domain such that each graph contains a minimum of 100 nodes and up to a maximum of 20000 nodes. For all datasets, we perform link prediction by training on a small subset (i.e., a percentage) of the edges and then attempting to predict the unseen edges (with $2 0 \\%$ of the held-out edges used for validation). Key dataset statistics are summarized in Table 1. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/34ff9bc6ac081fd3daf13fecf4b1df5e8aa1f2b7ac754d38f9ca61f19f7b9436.jpg",
+        "table_caption": [
+            "Table 1: Statistics for the three datasets used to test Meta-Graph. "
+        ],
+        "table_footnote": [
+            "Table 2: Convergence AUC results for different training edge splits. "
+        ],
+        "table_body": "<table><tr><td colspan=\"2\">DATASET</td><td colspan=\"2\">#GRAPHS</td><td colspan=\"2\">AVG.NODES</td><td colspan=\"2\">AVG.EDGES</td><td colspan=\"2\">#NODE FEATS</td><td></td></tr><tr><td colspan=\"2\">PPI</td><td colspan=\"2\">24</td><td colspan=\"2\">2,331</td><td colspan=\"2\">64,596</td><td colspan=\"3\">50</td></tr><tr><td colspan=\"2\">FIRSTMMDB</td><td colspan=\"2\">41</td><td colspan=\"2\">1,377</td><td colspan=\"2\">6,147</td><td colspan=\"3\">5</td></tr><tr><td colspan=\"2\">EGO-AMINER</td><td colspan=\"2\">72</td><td colspan=\"2\">462</td><td colspan=\"2\">2245</td><td colspan=\"3\">300</td></tr><tr><td colspan=\"9\">PPI</td><td rowspan=\"2\">Ego-AMINER</td><td colspan=\"2\"></td></tr><tr><td colspan=\"2\">Edges</td><td>10%</td><td>20%</td><td>30%</td><td>10%</td><td>20%</td><td>FirstMMDB 30%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td colspan=\"2\">Meta-Graph</td><td>0.795</td><td>0.833</td><td>0.845</td><td>0.782</td><td>0.786</td><td>0.783</td><td>0.626</td><td></td><td></td><td>0.786</td></tr><tr><td colspan=\"2\">MAML</td><td>0.770</td><td>0.815</td><td>0.828</td><td>0.776</td><td>0.782</td><td></td><td>0.793</td><td>0.561</td><td>0.738 0.662</td><td>0.667</td></tr><tr><td colspan=\"2\">Random</td><td>0.578</td><td>0.651</td><td>0.697</td><td>0.742</td><td>0.732</td><td></td><td>0.720</td><td>0.500</td><td>0.500</td><td>0.500</td></tr><tr><td colspan=\"2\">No Fintune</td><td>0.738</td><td>0.786</td><td>0.801</td><td>0.740</td><td>0.710</td><td></td><td>0.734</td><td>0.548</td><td>0.621</td><td>0.673</td></tr><tr><td colspan=\"2\">Finetune</td><td>0.752</td><td>0.801</td><td>0.821</td><td>0.752</td><td>0.735</td><td></td><td>0.723</td><td>0.623</td><td>0.691</td><td>0.723</td></tr><tr><td colspan=\"2\">Adamic</td><td>0.540</td><td>0.623</td><td>0.697</td><td>0.504</td><td>0.519</td><td></td><td>0.544</td><td>0.515</td><td>0.549</td><td>0.597</td></tr><tr><td colspan=\"2\">Deepwalk</td><td>0.664</td><td>0.673</td><td>0.694</td><td>0.487</td><td>0.473</td><td></td><td>0.510</td><td>0.602</td><td>0.638</td><td>0.672</td></tr><tr><td colspan=\"2\"></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+        "text": "Baseline details. Several baselines correspond to modifications or ablations of Meta-Graph, including the straightforward adaptation of MAML (which we term MAML in the results), a finetune baseline where we pre-train a VGAE on the training graphs observed in a sequential order and finetune on the test graphs (termed Finetune). We also consider a VGAE trained individually on each test graph (termed No Finetune). For Meta-Graph and all of these baselines we employ Bayesian optimization with Thompson sampling (Kandasamy et al., 2018) to perform hyperparameter selection using the validation sets. We use the recommended default hyperparameters for DeepWalk and Adamic-Adar baseline is hyperparameter-free. 2 ",
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+    {
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+        "text": "4.1 RESULTS ",
+        "text_level": 1,
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+        "text": "Q1: Overall Performance. Table 2 shows the link prediction AUC for Meta-Graph and the baseline models when trained to convergence using $10 \\%$ , $20 \\%$ or $30 \\%$ of the graph edges. In this setting, we adapt the link prediction models on the test graphs until learning converges, as determined by performance on the validation set of edges, and we report the average link prediction AUC over the test edges of the test graphs. Overall, we find that Meta-Graph achieves the highest average AUC in all but one setting, with an average relative improvement of $4 . 8 \\%$ in AUC compared to the MAML approach and an improvement of $5 . 3 \\%$ compared to the Finetune baseline. Notably, MetaGraph is able to maintain especially strong performance when using only $1 0 \\%$ of the graph edges for training, highlighting how our framework can learn from very sparse samples of edges. Interestingly, in the Ego-AMINER dataset, unlike PPI and FIRSTMM DB, we observe the relative difference in performance between Meta-Graph and MAML to increase with density of the training set. We hypothesize that this is due to fickle nature of optimization with higher order gradients in MAML (Antoniou et al., 2018) which is somewhat alleviated in GS-gating due to the gating mechanism. With respect to computational complexity we observe a slight overhead when comparing MetaGraph to MAML which can be reconciled by realizing that the graph signature function is not updated in the inner loop update but only in outer loop. In the Appendix, we provide additional results when using larger sets of training edges, and, as expected, we find that the relative gains of Meta-Graph decrease as more and more training edges are available. ",
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+        "text": "Q2: Fast Adaptation. Table 3 highlights the average AUCs achieved by Meta-Graph and the baselines after performing only 5 gradient updates on the batch of training edges. Note that in this setting we only compare to the MAML, Finetune, and No Finetune baselines, as fast adaption in this setting is not well defined for the DeepWalk and Adamic-Adar baselines. In terms of fast adaptation, we again find that Meta-Graph is able to outperform all the baselines in all but one setting, with an average relative improvement of $9 . 4 \\%$ compared to MAML and $8 . 0 \\%$ compared to the Finetune baseline—highlighting that Meta-Graph can not only learn from sparse samples of edges but is also able to quickly learn on new data using only a small number of gradient steps. Also, we observe poor performance for MAML in the Ego-AMINER dataset dataset which we hypothesize is due to extremely low learning rates —i.e. $1 e - 7$ needed for any learning, the addition of a graph signature alleviates this problem. Figure 2 shows the learning curves for the various models on the PPI and FirstMM DB datasets, where we can see that Meta-Graph learns very quickly but can also begin to overfit after only a small number of gradient updates, making early stopping essential. ",
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+            "Table 3: 5-gradient update AUC results with various fractions of training edges. "
+        ],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">PPI</td><td colspan=\"3\">FirstMM DB</td><td colspan=\"3\">Eg0-AMINER</td></tr><tr><td>Edges</td><td>10%</td><td>20%</td><td>30%</td><td>10%</td><td>20%</td><td>30%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td>Meta-Graph</td><td>0.795</td><td>0.824</td><td>0.847</td><td>0.773</td><td>0.767</td><td>0.737</td><td>0.620</td><td>0.585</td><td>0.732</td></tr><tr><td>MAML</td><td>0.728</td><td>0.809</td><td>0.804</td><td>0.763</td><td>0.750</td><td>0.750</td><td>0.500</td><td>0.504</td><td>0.500</td></tr><tr><td> No Fintune</td><td>0.600</td><td>0.697</td><td>0.717</td><td>0.708</td><td>0.680</td><td>0.709</td><td>0.500</td><td>0.500</td><td>0.500</td></tr><tr><td>Finetune</td><td>0.582</td><td>0.727</td><td>0.774</td><td>0.705</td><td>0.695</td><td>0.704</td><td>0.608</td><td>0.675</td><td>0.713</td></tr></table>",
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+            "Figure 2: AUC scores on PPI (Left) and FirstMM DB (Right) graphs with $1 0 \\%$ of edges observed. "
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+        "text": "Q3: Choice of Meta-Graph Architecture. We study the impact of the graph signature function and its variants GS-Gating and GS-Weights by performing an ablation study using the FirstMM DB dataset. Figure 3 shows the performance of the different model variants and baselines considered as the training progresses. In addition to models that utilize different signature functions we report a random baseline where parameters are initialized but never updated allowing us to assess the inherent power of the VGAE model for few-shot link prediction. To better understand the utility of using a GCN based inference network we also report a VGAE model that uses a simple MLP on the node features and is trained analogously to Meta-Graph as a baseline. As shown in Figure 3 many versions of the signature function start at a better initialization point or quickly achieve higher AUC scores in comparison to MAML and the other baselines, but simple modulation and GS-Gating are superior to GS-Weights after a few gradient steps. ",
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+    {
+        "type": "text",
+        "text": "Q4: What is learned by the graph signature? To gain further insight into what knowledge is transferable among graphs we use the FirstMM DB and Ego-AMINER datasets to probe and compare the output of the signature function with various graph heuristics. In particular, we treat the output of ${ \\bar { s _ { \\mathscr { G } } } } = \\psi ( { \\mathscr { G } } )$ as a vector and compute the cosine similarity between all pairs of graph in the training set (i.e., we compute the pairwise cosine similarites between graph signatures, $s _ { \\mathcal { G } }$ ). We similarly compute three pairwise graph statistics—namely, the cosine similarity between average node features in the graphs, the difference in number of nodes, and the difference in number of edges—and we compute the Pearson correlation between the pairwise graph signature similarities and these other pairwise statistics. As shown in Table 4 we find strong positive correlation in terms of Pearson correlation coefficient between node features and the output of the signature function for both datasets, indicating that the graph signature function is highly sensitive to feature information. This observation is not entirely surprising given that we use such sparse samples of edges—meaning that many structural graph properties are likely lost and making the meta-learning heavily reliant on node feature information. We also observe moderate negative correlation with respect to the average difference in nodes and edges between pairs of graphs for FirstMM DB dataset. For Ego-AMINER we observe small positive correlation for difference in nodes and edges. ",
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+        "image_caption": [
+            "Figure 3: Ablation study on PPI (Left) and FirstMM DB (Right) graphs with $1 0 \\%$ of edges. "
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+        "table_footnote": [
+            "Table 4: Pearson scores between graph signature output and other graph statistics. "
+        ],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">FirstMMDB</td><td colspan=\"3\">Ego-AMINER</td></tr><tr><td>% Edges</td><td>10%</td><td>20%</td><td>30%</td><td>10%</td><td>20%</td><td>30%</td></tr><tr><td>Node Feats</td><td>0.928</td><td>0.950</td><td>0.761</td><td>0.473</td><td>0.385</td><td>0.448</td></tr><tr><td>Diff Num. Nodes</td><td>-0.093</td><td>-0.196</td><td>-0.286</td><td>0.095</td><td>0.086</td><td>0.085</td></tr><tr><td>Diff Num. Edges</td><td>-0.093</td><td>-0.195</td><td>-0.281</td><td>0.093</td><td>0.072</td><td>0.075</td></tr></table>",
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+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+        "text": "We now briefly highlight related work on link prediction, meta-learning, few-shot classification, and few-shot learning in knowledge graphs. Link prediction considers the problem of predicting missing edges between two nodes in a graph that are likely to have an edge. (Liben-Nowell & Kleinberg, 2003). Common successful applications of link prediction include friend and content recommendations (Aiello et al., 2012), shopping and movie recommendation (Huang et al., 2005), knowledge graph completion (Nickel et al., 2015) and even important social causes such as identifying criminals based on past activities (Hasan et al., 2006). Historically, link prediction methods have utilized topological graph features such as common neighbors yielding strong baselines like Adamic/Adar measure (Adamic & Adar, 2003), Jaccard Index among others. Other approaches include Matrix Factorization (Menon & Elkan, 2011) and more recently deep learning and graph neural networks based approaches (Grover & Leskovec, 2016; Wang et al., 2015; Zhang & Chen, 2018) have risen to prominence. A commonality among all the above approaches is that the link prediction problem is define over a single dense graph where the objective is to predict unknown/future links within the same graph. Unlike these previous approaches, our approach considers link prediction tasks over multiple sparse graphs which are drawn from distribution over graphs akin to real world scenario such as protein-protein interaction graphs, 3D point cloud data and citation graphs in different communities. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In meta-learning or learning to learn (Bengio et al., 1990; 1992; Thrun & Pratt, 2012; Schmidhuber, 1987), the objective is to learn from prior experiences to form inductive biases for fast adaptation to unseen tasks. Meta-learning has been particularly effective in few-shot learning tasks with a few notable approaches broadly classified into metric based approaches (Vinyals et al., 2016; Snell et al., 2017; Koch et al., 2015), augmented memory (Santoro et al., 2016; Kaiser et al., 2017; Mishra et al., 2017) and optimization based approaches (Finn et al., 2017; Lee & Choi, 2018). Recently, there are several works that lie at the intersection of meta-learning for few-shot classification and graph based learning. In Latent Embedding Optimization, Rusu et al. (2018) learn a graph between tasks in embedding space while Liu et al. (2019) introduce a message propagation rule between prototypes of classes. However, both these methods are restricted to the image domain and do not consider meta-learning over a distribution of graphs as done here. ",
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+    {
+        "type": "text",
+        "text": "Another related line of work considers the task of few-shot relation prediction in knowledge graphs. Xiong et al. (2018) developed the first method for this task, which leverages a learned matching metric using both a learned embedding and one-hop graph structures. More recently Chen et al. (2019) introduce Meta Relational Learning framework (MetaR) that seeks to transfer relation-specific meta information to new relation types in the knowledge graph. A key distinction between few-shot relation setting and the one which we consider in this work is that we assume a distribution over graphs while in the knowledge graph setting there is only a single graph and the challenge is generalizing to new types of relations within this graph. ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND CONCLUSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We introduce the problem of few-shot link prediction—where the goal is to learn from multiple graph datasets to perform link prediction using small samples of graph data—and we develop the Meta-Graph framework to address this task. Our framework adapts gradient-based meta learning to optimize a shared parameter initialization for local link prediction models, while also learning a parametric encoding, or signature, of each graph, which can be used to modulate this parameter initialization in a graph-specific way. Empirically, we observed substantial gains using Meta-Graph compared to strong baselines on three distinct few-shot link prediction benchmarks. In terms of limitations and directions for future work, one key limitation is that our graph signature function is limited to modulating the local link prediction model through an encoding of the current graph, which does not explicitly capture the pairwise similarity between graphs in the dataset. Extending Meta-Graph by learning a similarity metric or kernel between graphs—which could then be used to condition meta-learning—is a natural direction for future work. Another interesting direction for future work is extending the Meta-Graph approach to multi-relational data, and exploiting similarities between relation types through a suitable Graph Signature function. ",
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+    {
+        "type": "text",
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+        "text": "Sebastian Thrun and Lorien Pratt. Learning to learn. Springer Science & Business Media, 2012.   \nOriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In Advances in neural information processing systems, pp. 3630–3638, 2016.   \nPeng Wang, BaoWen Xu, YuRong Wu, and XiaoYu Zhou. Link prediction in social networks: the state-of-the-art. Science China Information Sciences, 58(1):1–38, 2015.   \nWenhan Xiong, Mo Yu, Shiyu Chang, Xiaoxiao Guo, and William Yang Wang. One-shot relational learning for knowledge graphs. arXiv preprint arXiv:1808.09040, 2018.   \nMuhan Zhang and Yixin Chen. Link prediction based on graph neural networks. In Advances in Neural Information Processing Systems, pp. 5165–5175, 2018.   \nMarinka Zitnik and Jure Leskovec. Predicting multicellular function through multi-layer tissue networks. Bioinformatics, 33(14):i190–i198, 2017. ",
+        "bbox": [
+            169,
+            99,
+            828,
+            308
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "7 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            294,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "7.1 A: EGO-AMINER DATASET CONSTRUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            516,
+            148
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To construct the Ego-Aminer dataset we first create citation graphs from different fields of study. We then select the top 100 graphs in terms number of nodes for further pre-processing. Specifically, we take the 5-core of each graph ensuring that each node has a minimum of 5-edges. We then construct ego networks by randomly sampling a node from the 5-core graph and taking its two hop neighborhood. Finally, we remove graphs with fewer than 100 nodes and greater than 20000 nodes which leads to a total of 72 graphs as reported in Table 1. ",
+        "bbox": [
+            174,
+            160,
+            825,
+            243
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "7.2 B: ADDITIONAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            260,
+            395,
+            275
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We list out complete results when using larger sets of training edges for PPI, FIRSTMM DB and Ego-Aminer datasets. We show the results for two metrics i.e. Average AUC across all test graphs. As expected, we find that the relative gains of Meta-Graph decrease as more and more training edges are available. ",
+        "bbox": [
+            174,
+            285,
+            825,
+            342
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/9c1e6bf9afa2709cec5321870e0f9dc0d240dd57eab9e28901c7c0949c928d33.jpg",
+        "table_caption": [
+            "Table 5: AUC Convergence results for PPI dataset for training edge splits "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"8\">PPI</td></tr><tr><td>Convergence</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.795</td><td>0.831</td><td>0.846</td><td>0.853</td><td>0.848</td><td>0.853</td><td>0.855</td></tr><tr><td>MAML</td><td>0.745</td><td>0.820</td><td>0.840</td><td>0.852</td><td>0.854</td><td>0.856</td><td>0.863</td></tr><tr><td>Random</td><td>0.578</td><td>0.651</td><td>0.697</td><td>0.729</td><td>0.756</td><td>0.778</td><td>0.795</td></tr><tr><td>No Finetune</td><td>0.738</td><td>0.786</td><td>0.801</td><td>0.817</td><td>0.827</td><td>0.837</td><td>0.836</td></tr><tr><td>Finetune</td><td>0.752</td><td>0.8010</td><td>0.821</td><td>0.832</td><td>0.818</td><td>0.856</td><td>0.841</td></tr><tr><td>Adamic</td><td>0.540</td><td>0.623</td><td>0.697</td><td>0.756</td><td>0.796</td><td>0.827</td><td>0.849</td></tr><tr><td>MAML-MLP</td><td>0.603</td><td>0.606</td><td>0.606</td><td>0.606</td><td>0.604</td><td>0.604</td><td>0.605</td></tr><tr><td>Deepwalk</td><td>0.664</td><td>0.673</td><td>0.694</td><td>0.727</td><td>0.731</td><td>0.747</td><td>0.761</td></tr></table>",
+        "bbox": [
+            258,
+            353,
+            738,
+            494
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/742522d573b90b29a99b3f828c5950c6bebb53eeba9c4379c3c10989298c01d2.jpg",
+        "table_caption": [
+            "Table 6: 5-gradient update AUC results for PPI for training edge splits "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>PPI-5 updates</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.795</td><td>0.829</td><td>0.847</td><td>0.853</td><td>0.848</td><td>0.854</td><td>0.856</td></tr><tr><td>MAML</td><td>0.756</td><td>0.837</td><td>0.840</td><td>0.852</td><td>0.855</td><td>0.855</td><td>0.856</td></tr><tr><td>No Finetune</td><td>0.600</td><td>0.697</td><td>0.717</td><td>0.784</td><td>0.814</td><td>0.779</td><td>0.822</td></tr><tr><td>Finetune</td><td>0.582</td><td>0.727</td><td>0.774</td><td>0.702</td><td>0.804</td><td>0.718</td><td>0.766</td></tr><tr><td>MAML-MLP</td><td>0.603</td><td>0.606</td><td>0.603</td><td>0.604</td><td>0.603</td><td>0.606</td><td>0.605</td></tr></table>",
+        "bbox": [
+            261,
+            537,
+            736,
+            628
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/a60e45cd6042af821f7e23f6b37b2ed7872a78934006f30fe03fb9445c17a190.jpg",
+        "table_caption": [
+            "Table 7: AUC Convergence results for FIRSTMM DB dataset for training edge splits "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"8\">FirstMMDB</td></tr><tr><td>Convergence</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.782</td><td>0.786</td><td>0.783</td><td>0.781</td><td>0.760</td><td>0.746</td><td>0.739</td></tr><tr><td>MAML</td><td>0.776</td><td>0.782</td><td>0.793</td><td>0.785</td><td>0.791</td><td>0.663</td><td>0.788</td></tr><tr><td>Random</td><td>0.742</td><td>0.732</td><td>0.720</td><td>0.714</td><td>0.705</td><td>0.698</td><td>0.695</td></tr><tr><td>No Finetune</td><td>0.740</td><td>0.710</td><td>0.734</td><td>0.722</td><td>0.712</td><td>0.710</td><td>0.698</td></tr><tr><td>Finetune</td><td>0.752</td><td>0.735</td><td>0.723</td><td>0.734</td><td>0.749</td><td>0.700</td><td>0.695</td></tr><tr><td>Adamic</td><td>0.504</td><td>0.519</td><td>0.544</td><td>0.573</td><td>0.604</td><td>0.643</td><td>0.678</td></tr><tr><td>Deepwalk</td><td>0.487</td><td>0.473</td><td>0.510</td><td>0.608</td><td>0.722</td><td>0.832</td><td>0.911</td></tr></table>",
+        "bbox": [
+            266,
+            671,
+            732,
+            801
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/fa3c0fa7dd735e2f393202063e625dc2e0183375fb9ebbae2aed54ce6aab963a.jpg",
+        "table_caption": [
+            "Table 8: 5-gradient update AUC results for FIRSTMM DB for training edge splits "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"8\">FirstMMDB</td></tr><tr><td>5 updates</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.773</td><td>0.767</td><td>0.743</td><td>0.759</td><td>0.742</td><td>0.732</td><td>0.688</td></tr><tr><td>MAML</td><td>0.763</td><td>0.750</td><td>0.624</td><td>0.776</td><td>0.759</td><td>0.663</td><td>0.738</td></tr><tr><td>No Finetune</td><td>0.708</td><td>0.680</td><td>0.709</td><td>0.701</td><td>0.685</td><td>0.683</td><td>0.653</td></tr><tr><td>Finetune</td><td>0.705</td><td>0.695</td><td>0.704</td><td>0.704</td><td>0.696</td><td>0.658</td><td>0.670</td></tr></table>",
+        "bbox": [
+            264,
+            172,
+            733,
+            263
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/e8a116633e9812c491ec76f4394dbfa47815bb8393cab4ab94196a76fc15dc22.jpg",
+        "table_caption": [
+            "Table 9: AUC Convergence results for Ego-Aminer dataset for training edge splits "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"8\">Ego-Aminer</td></tr><tr><td>Convergence</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.626</td><td>0.738</td><td>0.786</td><td>0.791</td><td>0.792</td><td>0.817</td><td>0.786</td></tr><tr><td>MAML</td><td>0.561</td><td>0.662</td><td>0.667</td><td>0.682</td><td>0.720</td><td>0.741</td><td>0.768</td></tr><tr><td>Random</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td></tr><tr><td>No Finetune</td><td>0.548</td><td>0.621</td><td>0.673</td><td>0.702</td><td>0.652</td><td>0.7458</td><td>0.769</td></tr><tr><td>Finetune</td><td>0.623</td><td>0.691</td><td>0.723</td><td>0.764</td><td>0.767</td><td>0.792</td><td>0.781</td></tr><tr><td>Adamic</td><td>0.515</td><td>0.549</td><td>0.597</td><td>0.655</td><td>0.693</td><td>0.744</td><td>0.772</td></tr><tr><td>Deepwalk</td><td>0.602</td><td>0.638</td><td>0.672</td><td>0.686</td><td>0.689</td><td>0.711</td><td>0.731</td></tr></table>",
+        "bbox": [
+            263,
+            436,
+            736,
+            564
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/c96849f5117ecfc2499323a1d4b5a368b45cb979df36a361f7a58b579605fc0b.jpg",
+        "table_caption": [
+            "Table 10: 5-gradient update AUC results for Ego-Aminer for training edge splits "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"8\">Ego-Aminer</td></tr><tr><td>5 updates</td><td>10%</td><td>20%</td><td>30%</td><td>40%</td><td>50%</td><td>60%</td><td>70%</td></tr><tr><td>Meta-Graph</td><td>0.620</td><td>0.5850</td><td>0.732</td><td>0.500</td><td>0.790</td><td>0.733</td><td>0.500</td></tr><tr><td>MAML</td><td>0.500</td><td>0.504</td><td>0.500</td><td>0.500</td><td>0.519</td><td>0.500</td><td>0.500</td></tr><tr><td>No Finetune</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td><td>0.500</td></tr><tr><td>Finetune</td><td>0.608</td><td>0.675</td><td>0.713</td><td>0.755</td><td>0.744</td><td>0.706</td><td>0.671</td></tr></table>",
+        "bbox": [
+            263,
+            738,
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+            828
+        ],
+        "page_idx": 14
+    }
+]

parse/train/KIfbqntFnOc/KIfbqntFnOc_content_list.json

@@ -0,0 +1,1344 @@
+[
+    {
+        "type": "text",
+        "text": "ROBUST ENSEMBLES OF NEURAL NETWORKS USING ITOˆ PROCESSES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "Residual neural networks (ResNets) can be modeled as dynamical systems where the evolution of dynamical systems represents the inference in ResNets. We exploit this connection and the theory of stochastic dynamical systems to construct a novel ensemble of Ito processes as a new deep learning representation ˆ that is more robust than classical residual networks. An Ito process obtained by ˆ solving a suitably-formulated stochastic differential equation derived from a residual network has a probability density function that is not readily perturbed by small changes in the neural network’s inputs. Our robust stochastic Itoˆ ensemble of neural networks achieve an accuracy of $7 3 . 9 1 \\%$ on the CIFAR-10 dataset against the PGD attack with $\\epsilon \\ : = \\ : 2 . 0$ under the $L _ { 2 }$ norm, while the accuracy of Madry’s robustness toolbox on the same attack is $1 8 . 5 9 \\%$ . Similarly, our stochastic Ito ensemble of neural networks achieves an accuracy of ˆ $7 9 . 6 6 \\%$ on PGD attack with $\\epsilon = 1 6 / 2 5 5$ under the $L _ { \\infty }$ norm, while the accuracy of Madry’s robustness toolbox on the same attack is $1 8 . 1 3 \\%$ . The Ito ensemble ˆ trained on ImageNet achieves an accuracy of $2 8 . 5 3 \\%$ against PGD attacks under the $L _ { \\infty }$ norm with $\\epsilon = 1 6 / 2 5 5$ and accuracy of $6 5 . 7 4 \\%$ under the $L _ { 2 }$ norm with $\\epsilon = 3 . 0$ , respectively. This significantly improves state-of-the-art accuracy of $5 \\%$ and $3 5 . 1 6 \\%$ for Madry’s robustness tool against the same PGD attacks under the $L _ { \\infty }$ and $L _ { 2 }$ norms, respectively. Further, our approach achieves these high robustness values without any explicit adversarial training or a significant loss of accuracy on benign inputs. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Deep neural networks (DNNs) have emerged as a very effective learning representation achieving near human-level performance in many domains such as computer vision (Gkioxari et al., 2015), natural language processing (Majumder et al., 2017), and speech recognition (Hannun et al., 2014). Despite this success, the use of deep learning models in high-assurance systems with safety and security requirements such as autonomous vehicles (Bojarski et al., 2016) and medical diagnoses (De Fauw et al., 2018) faces a trust deficit. The lack of robustness of these models and their susceptibility to adversarial attacks (Kurakin et al., 2016; Szegedy et al., 2013) that can change the prediction of a deep neural network via small imperceptible perturbations make deep learning models less trustworthy. This limitation is further aggravated by deep neural networks generally exhibiting very high confidence on incorrect predictions (Guo et al., 2017a; Hendrycks & Gimpel, 2016). Consequently, this lack of robustness hinders their deployment in safety-critical applications. There is a pressing need for a principled approach to learning robust deep learning models that are resilient to adversarial attacks and can abstain from making decisions on inputs for which they are likely to make a wrong prediction. ",
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+        "text": "A number of approaches have been recently proposed to increase the robustness of deep learning models. Adversarial training (Tramer et al., 2017; Engstrom et al., 2020) uses adversarial samples \\` in the training phase to make the models more robust. Another set of alternative approaches use the projection of inputs to data manifold (Lamb et al., 2018; Ilyas et al., 2017; Jang et al., 2020) or other preprocessing methods (Xie et al., 2019; Guo et al., 2017b). These approaches are robust to existing attack methods but their use of adversarial samples or predefined transformations (often achieved via another deep neural network such as autonecoders) makes these approaches susceptible to newer attack strategies. Certifiable-defense approaches (Wong et al., 2018; Dvijotham et al., 2018; ",
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+        "text": "Raghunathan et al., 2018; Dutta et al., 2018) have also been recently proposed to make deep learning models robust against worst-case input over a defined range of perturbations. These theoretical guarantees on worst-case inputs hold only for small perturbations; consequently, their use is limited in practice and their performance is typically inferior to approaches based on adversarial training, particularly for high-dimensional inputs. ",
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+        "text": "In this paper, we address this challenge of robust and trustworthy deep learning using a new representation that exploits the connection between dynamical systems and residual neural networks (ResNets), and uses the theory of stochastic dynamical systems. Dynamical systems can model ResNets where the inference in the network is represented by the evolution of the dynamical system (Chen et al., 2015; Chang et al., 2017; Sonoda & Murata, 2017; Chen et al., 2018; Lu et al., 2018). We construct a novel deep ensemble using a special class of stochastic dynamical systems, namely the Ito drift-diffusion process with suitably bounded diffusion term. It ˆ o process is the sum ˆ of the integral of a process over time and of another process over a Brownian motion. The drift over time models the typical inference in a ResNet and the diffusion Brownian motion models the added stochastic noise that makes the model robust to adversarial perturbations. We form an ensemble of these Ito processes by considering multiple such models and multiple inferences over the same model. If a majority of the ensemble agrees on a particular prediction, Ito ensemble ˆ makes that prediction; otherwise, it abstains from making a decision. ",
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+        "type": "table",
+        "img_path": "images/1862a43a9ba25a46861aaa5f8879ced6decd94e92b926ee5dd59737972b7d827.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Robustness Approach</td><td>Accuracy (%)</td><td rowspan=\"2\">Benchmark</td><td rowspan=\"2\">Norm</td><td colspan=\"2\">Accuracy (%)</td></tr><tr><td>Ito Ensemble</td><td>84.60</td><td>Ito Ensemble</td><td>Madry toolbox</td></tr><tr><td>Engstrom et al. (2020)</td><td>53.49</td><td>CIFAR-10</td><td>L2</td><td>73.91</td><td>18.59</td></tr><tr><td>Balunovic &amp; Vechev (2020)</td><td>46.2</td><td>ImageNet</td><td>L2</td><td>69.51</td><td>43.04</td></tr><tr><td>Zhang et al. (2019)</td><td>40.5</td><td>CIFAR-10</td><td>L</td><td>79.66</td><td>18.13</td></tr><tr><td>Pang et al. (2019) (∈=0.01)</td><td>48.4</td><td>ImageNet</td><td>L8</td><td>28.53</td><td>5.00</td></tr></table>",
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+        "text": "Table 1: (left) Our Ito ensemble approach outperforms SOTA defenses for the PGD attack on ˆ CIFAR-10 with $\\epsilon = 8 / 2 5 5$ unless specified otherwise. (right) Ito ensemble outperforms Madry ˆ toolbox (Engstrom et al., 2020) under PGD attack with $L _ { 2 }$ norm, $\\epsilon = 2 . 0$ , and $L _ { \\infty }$ norm $\\epsilon = 1 6 / 2 5 5$ ",
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+        "text": "We highlight a few results demonstrating the robustness of Ito process ensembles in Table 1. Our It ˆ oˆ ensemble approach has higher accuracy compared to several state-of-the-art robustness approaches. The accuracy of our approach is $8 4 . 6 0 \\%$ on CIFAR-10 against the PGD attack in $L _ { \\infty }$ norm with $\\epsilon = 8 / 2 5 5$ and the next best approach is Engstrom et al. (2020) (Madry toolbox) with an accuracy of $5 3 . 4 9 \\%$ . On CIFAR-10 and ImageNet benchmarks, our Ito ensemble approach is significantly more ˆ robust than Engstrom et al. (2020) (Madry toolbox) against PGD attacks in both $L _ { 2 }$ and $L _ { \\infty }$ norms for different values of attack strength $\\epsilon$ . Thus, our Ito ensembles exhibit remarkable robustness ˆ against adversarial attacks without any explicit adversarial training. ",
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+    {
+        "type": "image",
+        "img_path": "images/ee651e75a7ef35f0c3b36ce537787418b8143175e1fba6412c1b4db6a90edb24.jpg",
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+            "Figure 1: Examples of benign images on which our approach using Ito processes abstains from ˆ making a decision while the original ResNet model makes a decision despite high uncertainty. The first image is found by Ito process ensemble to be confusing between ˆ binoculars and cannon, the second between a radiator and a projector, the third between a trench-coat and bicycle, and the last one between stove and coffee-pot. This uncertainty in Ito ensemble resembles human judgement. ˆ "
+        ],
+        "image_footnote": [],
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+        "text": "Our approach using Ito ensembles can abstain from making decisions on a confusing input. In ˆ Section 4, we demonstrate that abstentions further improve the robustness of the Ito ensembles ˆ to adversarial examples compared to the state-of-the-art approaches. Further, we notice that Itoˆ ensembles abstain even on benign data inputs where manual inspection demonstrates high aleatoric or epistemic uncertainty as shown in Figure 1. Our experiments show that this new approach of using Ito ensembles achieves high robustness without significant loss in accuracy on benign inputs. ˆ ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Residual neural networks are a common neural network architecture that learn only the residuals not learned by the previous layers. ResNets (He et al., 2016) are residual neural networks where residual learning is adopted for every few stacked neural network layers and such building blocks are used to design the complete residual neural network. The dynamics of ResNets and other similar neural networks can be described using ordinary and partial differential equations (Chen et al., 2015; Chang et al., 2017; Sonoda & Murata, 2017; Weinan, 2017; Chen et al., 2018; Lu et al., 2018). One timestep of the dynamics models each building block of the ResNets. Such a dynamical model enables memory efficiency in training and adaptive inference. In contrast, we use stochastic differential equations (Ito processes) and demonstrate their robustness to adversarial attacks. ˆ ",
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+        "type": "text",
+        "text": "A number of adversarial attacks on deep neural networks have been proposed in literature and shown to be effective across different architectures. Attacks such as the fast gradient sign method (FGSM) (Szegedy et al., 2013), the projected gradient decent (PGD) (Madry et al., 2017) and other approaches (Nicolae et al., 2018) have demonstrated the fragility of deep neural networks to small perturbations in their inputs. The most effective state-of-art defenses use adversarial training (Tramer et al., 2017; Engstrom et al., 2020) or some projection or transformation of \\` inputs (Lamb et al., 2018; Ilyas et al., 2017; Jang et al., 2020; Xie et al., 2019; Guo et al., 2017b). The use of adversarial examples or predefined transformations makes these approaches vulnerable to new attacks. In contrast, our approach using Ito process does not need adversarial examples. ˆ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our use of stochastic dynamical systems is inspired by their presence in biological systems (Kitano, 2004; Bressloff, 2014; Allen, 2010) where they impart robustness to external perturbations. As an example, (Arkin et al., 1998) study gene expression using Gillespie’s stochastic formulation of chemical kinetics and show that protein numbers can vary markedly from one cell to another with important consequences for biological robustness (Gonze et al., 2002). ",
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+    {
+        "type": "text",
+        "text": "3 ROBUST LEARNING USING ITOˆ ENSEMBLES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Residual networks (ResNets) can be modeled as dynamical systems where the evolution of the dynamical system represents the inference in ResNets (Chen et al., 2015; Chang et al., 2017; Sonoda & Murata, 2017; Chen et al., 2018; Lu et al., 2018). We connect this view to the theory of stochastic differential equations and construct an ensemble using a class of Ito processes with suitably bounded ˆ diffusion term. This Ito process ensemble exhibits remarkable robustness against adversarial attacks ˆ without any explicit adversarial training. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "FROM RESNETS TO STOCHASTIC ITOˆ PROCESSES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A building block of a residual neural network (He et al., 2016) with the residual mapping $\\mathcal { F } ( \\mathbf { x } ( i ) , \\bar { \\mathbf { W } } ( i ) )$ can be described using the following equation: $\\mathbf { x } ( i + 1 ) = \\mathcal { F } ( \\mathbf { x } ( i ) , \\mathbf { W } ( i ) ) + \\bar { \\mathbf { x } } ( i )$ . Here, ${ \\bf x } ( i )$ is the input to the $i ^ { t h }$ residual network building block and $\\mathbf { x } ( i + 1 )$ is the corresponding output that serves as an input to the next building block. The weights of the neural network layers in this ResNet building block are denoted by $\\mathbf { W } ( i )$ . ",
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+        "type": "text",
+        "text": "After taking suitable limits, the evolution of the ResNet can be described by the ResNet ordinary differential equation (ODE): $\\begin{array} { r } { \\frac { d \\mathbf { x } ( t ) } { d t } = \\mathcal { G } ( \\mathbf { x } ( t ) , \\mathbf { W } ( t ) ) } \\end{array}$ . Here, $\\begin{array} { r } { \\mathcal { G } ( \\mathbf { x } ( t ) , \\mathbf { W } ( t ) ) = \\operatorname* { l i m } _ { \\delta t  0 } \\frac { \\mathcal { F } ( \\mathbf { x } ( t ) , \\mathbf { W } ( t ) ) } { \\delta t } } \\end{array}$ and ${ \\bf x } ( 0 )$ is the input to the neural network. The ResNet ODE can be naturally generalized into an Ito process by using a Brownian motion term with diffusion coefficient ˆ $\\Sigma ( t ) \\dot { = } \\overline { { ( \\sigma _ { i j } ( t ) ) } }$ : $d { \\bf x } ( t ) =$ $\\mathcal { G } ( \\bar { \\mathbf { x } } ( t ) , \\mathbf { W } ( t ) ) ~ d t + \\bar { \\Sigma } ( t ) ~ d B ( t )$ . There are two competing objectives here: ",
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+        "type": "text",
+        "text": "• Very large values of the diffusion term $\\Sigma ( t )$ can completely overshadow the drift term $\\mathcal { G } ( \\mathbf { x } ( t ) , \\mathbf { W } ( t ) )$ leading to a poor accuracy even on benign inputs. When the diffusion term is very large, the paths of the Ito process can completely diverge from the solution of the ˆ original ResNet from which the Ito process was obtained. ˆ   \nVery small values of $\\Sigma ( t )$ make the model closer to the original ResNet and equally non-robust. $\\Sigma ( t ) = 0$ reproduces the original non-stochastic ResNet with no additional robustness. As we increase the diffusion term, the robustness of the neural network increases; this is experimentally demonstrated in Section 4. ",
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+        "type": "text",
+        "text": "So, a natural question to ask is: How do we select the diffusion term $\\Sigma ( t )$ such that the Ito process ˆ satisfies these two competing objectives of accuracy and robustness? ",
+        "bbox": [
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+        "type": "text",
+        "text": "At one hand, the generated Ito process must retain similar accuracy on benign models as the original ˆ ResNet, that is, its solutions are determined mainly by the term $\\mathcal { G } ( \\mathbf { \\dot { x } } ( t ) , \\mathbf { W } ( \\bar { t } ) )$ and Brownian motion noise does not make it diverge significantly. On the other hand, the choice of added diffusion term $\\Sigma ( t )$ must make the model robust enough to be resilient to adversarial perturbations on the inputs. ",
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+    {
+        "type": "text",
+        "text": "ROBUSTNESS OF STOCHASTIC ITOˆ RESNET ENSEMBLES ",
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+    {
+        "type": "text",
+        "text": "Given the Ito process ˆ ${ \\bf x } ( t )$ satisfying the stochastic differential equation $d \\mathbf { x } ( t ) = { \\mathcal { G } } ( \\mathbf { x } ( t ) , \\mathbf { W } ( t ) ) d t + \\Sigma ( t ) d { \\dot { B } } ( t )$ , it is known (Oksendal, 1992) that the probability density $\\boldsymbol { p } ( \\mathbf { x } , t )$ can be mathematically characterized by the following equation: ",
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+        "type": "equation",
+        "img_path": "images/7a815f90d3338a008f4ba42b9a8e1c828a5b38dfb2d65e68a011ecb999710aa8.jpg",
+        "text": "$$\n\\frac { \\mathrm { \\partial } p ( \\mathbf { x } , t ) } { \\partial t } + \\mathcal { G } ( \\mathbf { x } , \\mathbf { W } ) \\nabla p ( \\mathbf { x } , t ) = - p ( \\mathbf { x } , t ) \\sum _ { i } \\frac { \\partial \\mathcal { G } } { \\partial \\mathbf { x } _ { i } } + \\frac { 1 } { 2 } \\sum _ { i } \\sum _ { j } \\frac { \\partial ^ { 2 } } { \\partial \\mathbf { x } _ { i } \\partial \\mathbf { x } _ { j } } \\left( ( \\sum _ { k } \\sigma _ { i k } ( t ) \\sigma _ { j k } ( t ) ) p ( \\mathbf { x } , t ) \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "map itself, that is For robust networks, the rate of change of the residual learning map is much smaller than the residual $\\begin{array} { r } { \\sum _ { i } \\frac { \\partial \\mathcal { G } } { \\partial \\mathbf { x } _ { i } } < \\eta _ { 1 } \\mathcal { G } ( \\mathbf { \\bar { x } } , \\mathbf { W } ) \\frac { \\nabla p ( \\mathbf { x } , t ) } { p ( \\mathbf { x } , t ) } } \\end{array}$ for some small $\\eta _ { 1 }  0$ . Hence, the probability density $\\boldsymbol { p } ( \\mathbf { x } , t )$ can be simplified to the following equation: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24121998b58c7b10cfd05fdab991d0d4044b1f964a36a7a56be128e48ef41c66.jpg",
+        "text": "$$\n\\frac { \\partial p ( { \\bf x } , t ) } { \\partial t } + ( 1 + \\eta _ { 1 } ) \\mathcal { G } ( { \\bf x } , { \\bf W } ) \\nabla p ( { \\bf x } , t ) = \\frac { 1 } { 2 } \\sum _ { i } \\sum _ { j } \\frac { \\partial ^ { 2 } } { \\partial { \\bf x } _ { i } \\partial { \\bf x } _ { j } } \\left( \\left( \\sum _ { k } \\sigma _ { i k } ( t ) \\sigma _ { j k } ( t ) \\right) p ( { \\bf x } , t ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Using the fact that the double derivative of the probability density for robust neural networks is much smaller thfor some small an the re tself, that is 12 Pi Pj ∂2∂xi∂xj sidual map i, we choose $\\begin{array} { r } { \\frac { 1 } { 2 } \\sum _ { i } \\sum _ { j } \\frac { \\partial ^ { 2 } } { \\partial \\mathbf { x } _ { i } \\partial \\mathbf { x } _ { j } } \\left( p ( \\mathbf { x } , t ) \\right) < \\eta _ { 2 } \\mathcal { G } ( \\mathbf { x } , \\mathbf { W } ) \\frac { \\nabla p ( \\mathbf { x } , t ) } { p ( \\mathbf { x } , t ) } } \\end{array}$ $\\eta _ { 2 }  0$ $\\textstyle \\sigma _ { i j } ( t ) \\leq { \\frac { \\omega } { 1 + t } }$ $\\omega$ probability density function of a ResNet with $n$ -dimensional inputs can be further simplified as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/882654c00f7a9ec86aaf33dbc91dfcfa5ff996ad1ee98efd84711cde062e5fd8.jpg",
+        "text": "$$\n\\frac { \\partial p ( { \\bf x } , t ) } { \\partial t } + ( 1 + \\eta _ { 1 } - n \\omega ^ { 2 } \\eta _ { 2 } ) \\mathcal { G } ( { \\bf x } , { \\bf W } ) \\nabla p ( { \\bf x } , t ) = 0\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As $\\eta _ { 1 }  0$ and $\\eta _ { 2 }  0$ for robust neural networks, our choice of $\\textstyle \\sigma _ { i j } ( t ) \\leq { \\frac { \\omega } { 1 + t } }$ for a constant $\\omega$ reduces the equation describing the probability density function to ∂p(x,t)∂t + G(x, W)∇p(x, t) = 0. Interpreting p(x, t) as a function that is constant along the trajectories of a ordinary differential equation i.e. dp(x,t)dt = 0, p(x, t) corresponds to the following differential equation: $\\begin{array} { r } { \\frac { d \\mathbf { x } ( t ) } { d t } \\ = \\ \\mathcal { G } ( \\mathbf { x } ( t ) , \\mathbf { W } ( t ) ) } \\end{array}$ . Hence, under our choice of $\\textstyle \\sigma _ { i j } ( t ) \\leq { \\frac { \\omega } { 1 + t } }$ for a constant $\\omega$ , the solution to the stochastic differential equation agrees with the ResNet ODE for robust neural networks. Our implementation of the stochastic robust Ito ensemble of residual ˆ neural network is formed by discretizing the stochastic differential equation dx(t) = G(x(t), W(t)) dt + Σ(t) dB(t) with the constraint that σij (t) = ω1+t . ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We train stochastic Ito ensembles of residual neural networks using both CIFAR-10 (Krizhevsky ˆ et al., 2014) and ImageNet (Deng et al., 2009) benchmarks. We evaluate the robustness of our stochastic Ito ensembles against two popular adversarial attacks: the fast gradient sign method ˆ (FGSM) (Szegedy et al., 2013) and the projected gradient descent (PGD) (Kurakin et al., 2016) under both $L _ { 2 }$ and $L _ { \\infty }$ norms. We use the conformance in prediction of our Ito ensemble to exploit ˆ their robustness. If a majority of residual network models in our ensemble predict the same label for a given data item, the ensemble makes a prediction as this majority label. Otherwise, the stochastic Ito ensemble assigns no label and abstains from making any decision on the given input ˆ data. Our experiments indicate that this capability of the Ito ensemble to abstain from making ˆ decisions on non-conforming inputs not only helps defend the model against adversarial attacks, it also decreases incorrect predictions on benign data. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "CIFAR-10 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our experiments are performed on a 40-core 256GB RAM server with 4 NVIDIA V100 GPUs ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Question 1: Does the Ito ensemble achieve competitive accuracy on benign inputs? ˆ ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We train the standard ResNet models on benign data and compare their accuracy with the accuracy of our stochastic Ito ensembles on benign data. We obtain the stochastic ResNet models in the ˆ stochastic Ito ensemble by starting with the weights of a standard ResNet model and training them ˆ for 40 epochs with a learning rate of 0.0001 using the Adam optimizer. Table 2 compares the accuracy of the standard model with Ito ensembles. ˆ ",
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/1902d05d0e2cf8cdf3bfa525beb434e81750cfeb1d6cb7ba5402185bbdbb1b46.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Architecture</td><td colspan=\"2\">Accuracy (%)</td><td colspan=\"2\">Incorrect Prediction (%)</td><td rowspan=\"2\">Correct + Abstention (%) Ito Ensemble (%)</td></tr><tr><td></td><td>Original</td><td>Ito Ensemble</td><td>Original</td><td>Itó Ensemble</td></tr><tr><td>ResNet-18</td><td>93.33</td><td>91.51</td><td>6.67</td><td>5.72</td><td>94.28</td></tr><tr><td>ResNet-34</td><td>92.92</td><td>91.33</td><td>7.08</td><td>5.80</td><td>94.20</td></tr><tr><td>ResNet-50</td><td>93.86</td><td>91.59</td><td>6.14</td><td>4.64</td><td>95.29</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Our stochastic Ito ensembles and the standard ResNet neural network architectures haveˆ similar accuracy on CIFAR-10 test data. Because of its ability to abstain from assigning a label when majority of predictions do not conform, the fraction of data where the Ito ensemble predicts ˆ an incorrect label is lower that the fraction of data where the original ResNet model is incorrect. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our stochastic Ito ensembles ofˆ 20 models in Table 2 are trained using a diffusion term corresponding to $\\omega ~ = ~ 0 . 2$ . 20 inferences are obtained from each stochastic model in our Itoˆ ensemble. The accuracy of the stochastic Ito ensembles on CIFAR-10 test data is comparable to ˆ that of the standard models on three ResNet architectures: ResNet-18, ResNet-34, and ResNet-50. Our stochastic Ito ensemble abstains when a majority of the ensemble models do not agree on a ˆ single prediction. This lowers the incorrect predictions of Ito ensemble compared to the original ˆ model. As shown in Figure 1, some of the correct predictions by original ResNet are on images with high aleatoric uncertainty on which Ito ensemble correctly abstains. ˆ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Question 2: Is the stochastic Ito ensemble robust against adversarial attacks? ˆ ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We train stochastic Ito ensembles of 20 ResNet-50 models on CIFAR-10 data with diffusion terms ˆ corresponding to $\\omega = 0 . 2$ and $\\omega = 0 . 4$ . 20 independent inferences are drawn from each stochastic model in the Ito ensemble. We evaluate the robustness of our stochastic It ˆ o ensemble against FGSM ˆ and PGD under both $L _ { 2 }$ and $L _ { \\infty }$ norms. We compare the accuracy of predictions from our stochastic Ito ensembles with that of Madry’s robustness toolbox (Engstrom et al., 2020). ˆ ",
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/6b7166f029d7b6b9cdee17d4c6fb0b3683dd2cf82e9d980228ea7088c6f275af.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>E</td><td>Accuracy for PGD (%)</td><td>Correct + Abstention for PGD (%)</td><td>Accuracy for FGSM (%)</td><td>Correct + Abstention for FGSM (%)</td></tr><tr><td>0.2</td><td>91.34</td><td>94.83</td><td>91.41</td><td>94.93</td></tr><tr><td>0.5</td><td>90.37</td><td>94.53</td><td>90.78</td><td>94.87</td></tr><tr><td>1.0</td><td>86.39</td><td>91.41</td><td>89.22</td><td>93.82</td></tr><tr><td>2.0</td><td>73.91</td><td>79.80</td><td>83.31</td><td>89.41</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 3: The accuracy of our stochastic Ito ensemble with diffusion term corresponding to ˆ $\\omega = 0 . 2$ on the PGD attack for different values of $\\epsilon$ under the $L _ { 2 }$ norm. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Robustness under the $L _ { 2 }$ norm. The accuracy of our stochastic Ito ensembles on the fast gradient ˆ sign method (FGSM) (Szegedy et al., 2013) and the projected gradient descent (PGD) (Kurakin et al., 2016) under the $L _ { 2 }$ norm is shown in Table 3. Our stochastic Ito ensembles use the ResNet-50 ˆ architecture with the diffusion term corresponding to $\\omega = 0 . 2$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our stochastic Ito ensemble approach with a diffusion terms corresponding to ˆ $\\omega = 0 . 2$ shows an accuracy of $7 3 . 9 1 \\%$ against the PGD attack with $\\epsilon = 2 . 0$ under the $L _ { 2 }$ norm. This compare favorably with the $1 8 . 5 9 \\%$ accuracy of Madry’s robustness toolbox on the same PGD attack. The accuracy of our stochastic Ito ensemble approach improves to ˆ $7 9 . 0 7 \\%$ when the diffusion term corresponds to $\\omega = 0 . 4$ . Further, the sum of correct labels and abstentions from our Ito ensemble approach is ˆ $8 8 . 4 3 \\%$ against the PGD attack with $\\epsilon = 2 . 0$ under the $L _ { 2 }$ norm. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/73db714b5b8a9637033c38ef79b706340e1d993d9ff32a9950e1ff1232084cfa.jpg",
+        "image_caption": [
+            "Figure 2: The accuracy of our stochastic Ito ensemble with a diffusion term corresponding to ˆ $\\omega =$ 0.2 (left) and $\\omega = 0 . 4$ (right) on CIFAR-10 compares favorably with Madry’s Robustness Toolbox using the $L _ { 2 }$ norm for different values of $\\epsilon$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 shows the accuracy of our Ito ensemble approach and compares it with the accuracy of ˆ Madry’s robustness toolbox (Engstrom et al., 2020) on CIFAR-10 test data. Both our stochastic Ito ensembles with ˆ $\\omega = 0 . 2$ and $\\omega = 0 . 4$ have higher accuracy on benign data and their accuracy remains higher than Madry’s robustness toolbox for all values of $\\epsilon$ under the $L _ { 2 }$ norm. The accuracy of the Ito ensemble degrades more gracefully as the value of ˆ $\\epsilon$ increases under the $L _ { 2 }$ norm. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Robustness under the $L _ { \\infty }$ norm. We investigate the accuracy of our stochastic Ito ensemble ˆ approach under the $L _ { \\infty }$ norm and compare it to the accuracy of Madry’s robustness toolbox. Table 4 shows the accuracy of our Ito ensemble with diffusion term corresponding to ˆ $\\omega = 0 . 2$ . ",
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+    {
+        "type": "table",
+        "img_path": "images/acf56422f0f94c5315ad2d5267b0e506d8c92ed70e1aefb4f88896e48d3f7513.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>E</td><td>Accuracy for PGD (%)</td><td>Correct + Abstention for PGD (%)</td><td>Accuracy for FGSM (%)</td><td>Correct + Abstention for FGSM (%)</td></tr><tr><td>4 255</td><td>85.82</td><td>92.91</td><td>86.02</td><td>93.00</td></tr><tr><td>8 255</td><td>84.60</td><td>92.48</td><td>85.24</td><td>92.84</td></tr><tr><td>16 255</td><td>79.66</td><td>89.06</td><td>82.74</td><td>91.20</td></tr><tr><td>32 255</td><td>62.97</td><td>74.05</td><td>72.49</td><td>84.08</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 4: The accuracy of our stochastic Ito ensemble with diffusion term corresponding to ˆ $\\omega = 0 . 2$ on the PGD attack for different values of $\\epsilon$ under the $L _ { \\infty }$ norm. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We study the performance of our stochastic Ito ensemble on PGD and FGSM and attacks of varying ˆ magnitudes under the $L _ { \\infty }$ norm, and determine that our stochastic Ito ensemble is robust against ˆ adversarial noise. Figure 3 shows the accuracy of Madry’s robustness toolbox and our Ito ensemble ˆ on adversarial images under the PGD attack. The accuracy of our stochastic Ito ensemble with ˆ diffusion term corresponding to $\\omega = 0 . 4$ is higher than that of the model from Madry’s toolbox for both the original unperturbed images and PGD adversarial images with $\\epsilon = 8 / 2 5 5$ and $\\epsilon = 1 6 / 2 5 5$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Question 3: How does the robustness of the stochastic Ito ensemble approach change with the ˆ number of models in the ensemble and the number of inferences? ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The Ito ensemble has two sources of diversity - different stochastic models and multiple inferences ˆ on the same model. We investigate the accuracy of our Ito ensemble with different number of ˆ stochastic models and different number of inferences from each stochastic model. Figure 4 (left) illustrates the results of our investigations. A significant increase of $5 . 1 \\%$ is observed for the sum of correct outcomes and abstentions by increasing the number of stochastic models from 3 to 20 and the number of sampled independent inferences for each stochastic model from 3 to 20. Increasing the number of stochastic models in the ensemble has more significant influence on the performance of the Ito ensemble than increasing the number of independent inferences from each stochastic model. ˆ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/8e6fb227444f94e631b31f923b7f994419a2350bbea61dae1d1b4df91709e6e0.jpg",
+        "image_caption": [
+            "Figure 3: The accuracy of our stochastic Ito ensemble on CIFAR-10 compares favorably withˆ Madry’s Robustness Toolbox using the $L _ { \\infty }$ norm. (left) $\\omega = 0 . 2$ (right) $\\omega = 0 . 4$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/069a158f9f8721be1248155639fd870d1b9c30fb5fa3bdb99692a524a67895e6.jpg",
+        "image_caption": [
+            "Figure 4: (left) The impact of the number of models and inferences on the sum of correct outcomes and abstentions for our stochastic Ito ensemble with ˆ $\\omega = 0 . 4$ . (right) Diffusion with different values of $\\omega$ in stochastic Ito ensemble vs. PGD accuracy under the ˆ $L _ { \\infty }$ norm. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Question 4: How can we control the diffusion term $\\omega$ to trade-off robustness and accuracy? ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 4 (right) shows the effect of varying diffusion terms with different $\\omega$ on the accuracy of the stochastic Ito ensemble under the ˆ $L _ { \\infty }$ norm for PGD attacks with $\\epsilon = 8 / 2 5 5 , 1 6 / 2 5 5$ and 32/255. The accuracy of the stochastic Ito ensemble first increases as the value ofˆ $\\omega$ increases and then starts decreasing for any given value of $\\omega$ . This shows a tradeoff between the accuracy of the neural network on benign data and its ability to be robust to large adversarial perturbations. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "IMAGENET RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+            739,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "These experiments are performed on a 92-core 480GB RAM server with 8 NVIDIA V100 GPUs. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Question 1: Does the Ito ensemble achieve competitive accuracy on benign inputs? ˆ ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We study the accuracy of our Ito ensemble of 5 ResNet-50 models with 20 independent inferences ˆ per model for different values of $\\omega$ and associated diffusion terms. As shown in Table 5, small values of $\\omega$ do not significantly reduce the accuracy of the Ito ensemble on benign data. ˆ ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Question 2: Is the Ito ensemble approach robust against adversarial attacks? ˆ ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 shows the accuracy of our stochastic Ito ensemble approach on ImageNet against the PGD ˆ attack with various values of $\\epsilon$ under $L _ { 2 }$ as well as $L _ { \\infty }$ norms. The accuracy of our Ito ensemble ˆ compares favorably with the results from Madry’s robustness toolbox Engstrom et al. (2020). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b9183063dbae89bb5fc865691a6fd37113678f35efc87622fa957b91e4f9253c.jpg",
+        "table_caption": [
+            "Table 5: Accuracy of Ito ensembles on benign data with diffusion corresponding to different ˆ $\\omega$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Diffusion Term w</td><td>Original Accuracy</td><td>Itó Ensemble Accuracy</td><td>% Decrease in Accuracy</td><td>Correct + Abstentions (%)</td></tr><tr><td>0.1</td><td>76.13</td><td>76.04</td><td>0.09</td><td>77.87</td></tr><tr><td>0.2</td><td>76.13</td><td>73.59</td><td>2.54</td><td>78.29</td></tr><tr><td>0.3</td><td>76.13</td><td>67.79</td><td>8.34</td><td>76.40</td></tr><tr><td>0.4</td><td>76.13</td><td>61.66</td><td>14.47</td><td>76.42</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/21162b88fb4ee53ccc2c070dd16b40f76e3af9b8a9bd3276f4cb2b4c88ffdb94.jpg",
+        "image_caption": [
+            "Figure 5: The accuracy of our stochastic Ito ensemble with ˆ $\\omega { = } 0 . 2$ on ImageNet compares favorably with Robustness Toolbox using the $L _ { 2 }$ norm (left) and the $L _ { \\infty }$ norm (right) for different values of $\\epsilon$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            238,
+            802,
+            425
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Question 3: How can we control the diffusion term $\\omega$ to trade-off robustness and accuracy? ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            492,
+            808,
+            507
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/e043f0493fd99409dd8437c4d40cda05de8656c110f0bf1a0dc10d5bf22fdab9.jpg",
+        "table_caption": [
+            "Table 5 and Table 6 show that the diffusion parameter $\\omega$ can be used to establish a desired trade-off between robustness and benign accuracy for the ImageNet data set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Diffusion Term w</td><td>Ito Ensemble Benign Accuracy (%)</td><td>Ito Ensemble PGD Accuracy (%)</td><td>Correct + Abstentions (%)</td></tr><tr><td>0.1</td><td>76.04</td><td>26.23</td><td>27.89</td></tr><tr><td>0.2</td><td>73.59</td><td>53.42</td><td>61.32</td></tr><tr><td>0.3</td><td>67.79</td><td>60.11</td><td>70.01</td></tr><tr><td>0.4</td><td>61.66</td><td>58.57</td><td>73.55</td></tr></table>",
+        "bbox": [
+            243,
+            554,
+            754,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Accuracy of our Ito ensemble approach on PGD attack ˆ $( \\epsilon = 8 / 2 5 5 )$ with different diffusion parameters $\\omega$ . Higher values of $\\omega$ lead to more robust models. ",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            722,
+            318,
+            738
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We have shown that ensembles of neural networks corresponding to a class of Ito processes are ˆ more robust than classical residual networks. An Ito process obtained by solving a ˆ suitably-formulated stochastic differential equation derived from a residual network has a probability density function that is robust to adversarial input perturbations. Further, the achieved robustness does not require any explicit adversarial training; hence, it is likely to generalize to unforeseen attacks. We empirically evaluated the robustness of our Ito ensembles and demonstrated ˆ that they achieve higher accuracy under FGSM/PGD attacks over the $L _ { 2 } / L _ { \\infty }$ norm compared to state-of-the-art methods. This robustness is attained without significantly sacrificing accuracy on benign data. Further, Ito ensemble abstains on benign inputs with high uncertainty reflecting ˆ uncertainty-aware learning. Our paper is a step towards the use of Ito processes and stochastic ˆ differential equation models to build robust ensembles in deep learning. This will aid the adoption of deep learning in safety-critical applications. ",
+        "bbox": [
+            174,
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+            920
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            287,
+            118
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Linda JS Allen. An introduction to stochastic processes with applications to biology. CRC Press, 2010. ",
+        "bbox": [
+            173,
+            126,
+            825,
+            155
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Adam Arkin, John Ross, and Harley H McAdams. Stochastic kinetic analysis of developmental pathway bifurcation in phage $\\lambda$ -infected escherichia coli cells. Genetics, 149(4):1633–1648, 1998. ",
+        "bbox": [
+            176,
+            165,
+            823,
+            207
+        ],
+        "page_idx": 8
+    },
+    {
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parse/train/MDMV2SxCboX/MDMV2SxCboX.md

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+# Why Do Pretrained Language Models Help in Downstream Tasks? An Analysis of Head and Prompt Tuning
+
+Colin Wei Sang Michael Xie Tengyu Ma
+
+Department of Computer Science Stanford University
+
+{colinwei,xie,tengyuma}@cs.stanford.edu
+
+# Abstract
+
+Pretrained language models have achieved state-of-the-art performance when adapted to a downstream NLP task. However, theoretical analysis of these models is scarce and challenging since the pretraining and downstream tasks can be very different. We propose an analysis framework that links the pretraining and downstream tasks with an underlying latent variable generative model of text — the downstream classifier must recover a function of the posterior distribution over the latent variables. We analyze head tuning (learning a classifier on top of the frozen pretrained model) and prompt tuning in this setting. The generative model in our analysis is either a Hidden Markov Model (HMM) or an HMM augmented with a latent memory component, motivated by long-term dependencies in natural language. We show that 1) under certain non-degeneracy conditions on the HMM, simple classification heads can solve the downstream task, 2) prompt tuning obtains downstream guarantees with weaker non-degeneracy conditions, and 3) our recovery guarantees for the memory-augmented HMM are stronger than for the vanilla HMM because task-relevant information is easier to recover from the long-term memory. Experiments on synthetically generated data from HMMs back our theoretical findings.
+
+# 1 Introduction
+
+Natural language processing (NLP) has been revolutionized by large-scale pretrained language models such as BERT [4] and GPT [25], which are adapted to a variety of downstream NLP tasks. Although a large body of empirical work seeks to understand the effectiveness of pretrained models [7, 5, 12, 35, 34, 11, 27, 15], theoretical understanding is scarce. Theoretically analyzing the relationship between the pretraining and downstream tasks is challenging because pretraining and downstream settings can greatly differ.
+
+The key starting point for our analysis is to link the pretraining and downstream settings through an underlying generative model of the data. We model the data distribution as a latent variable model and the downstream task as a function of the latent variables. Assuming that pretraining on a large corpus allows us to learn the generative model, the conditional token probabilities predicted by the pretrained model carry information about the hidden variables. In downstream adaptation, we aim to recover this information to solve the downstream task.
+
+Though full finetuning is the de facto empirical standard, analyzing it is challenging because it requires characterizing the weights of the pretrained model. In this paper, we focus on head tuning and prompt tuning, which both freeze all pretrained parameters and allow us to treat the pretrained model as a black box. Head tuning [22] trains task-specific heads on top of the pretrained model outputs. Prompt tuning [31, 19, 9, 21] optimizes a task-specific “prompt” that is concatenated to the model input. Studying prompt tuning is particularly interesting since it can match the performance of full finetuning with less computation time [19, 9, 21].
+
+Our work contrasts with prior theoretical work [28], which assumes that downstream labels are recoverable via a linear head applied to the conditional token probabilities, and analyze how errors in pretraining or model misspecification propagate downstream. We consider specific generative distributions for which we can prove these assumptions, showing that head and prompt tuning can recover the downstream labels.
+
+Our analysis considers two data-generating distributions with increasing realism. First, we consider data generated from a Hidden Markov Model (HMM), where the downstream task is to learn a linear classifier on the posterior distribution over the hidden states (Section 3). We prove that, under strong non-degeneracy conditions on token emission probabilities, a linear head applied to a pretrained model $G$ which outputs exact conditional token probabilities $( G _ { i } ( x ) = P [ X _ { i } \bar { | } \bar { x } _ { - i } ] )$ can recover the downstream label (Theorem 3.3). Furthermore, we can prove better recovery guarantees with relaxed non-degeneracy assumptions (Assumption 3.1) by using continuous prompt tuning (Theorem 3.6), reflecting the strong empirical performance of prompt tuning [19, 9, 21]. Intuitively, prompt tuning conditions the latent variables so that nonessential information for the downstream task can be ignored during the tuning phase, making task-essential information easier to recover.
+
+Second, we also strengthen our analysis by leveraging additional structure in the data. Motivated by long-range dependences in natural language, we analyze HMM variants with additional latent “memory” variables that can store long-term information more easily than vanilla HMMs (Section 4). Here, the downstream task is to learn a linear classifier on the posterior distribution of the memory variables. We show that, under weaker non-degeneracy conditions than the first setting, an attentionbased classification head can recover ground-truth downstream labels from pretrained model outputs (Theorem 4.3). Intuitively, our recovery guarantees improve because the classification head can focus on the persistent, task-essential information in the memory while ignoring other transient and nonessential aspects of the latent variables. As with the vanilla HMM, we analyze prompt tuning for relaxing the non-degeneracy conditions even further (Theorem 4.6).
+
+In summary, we relate the pretraining and downstream tasks by assuming that the downstream task is to learn a classifier on the posterior distributions of the latent variables defined by an underlying generative model of text. Our theoretical contributions are: 1) in this setting we analyze an HMM generative model show that simple classification heads can recover the true downstream labels under certain non-degeneracy assumptions, 2) we prove that soft prompt tuning can relax the non-degeneracy assumptions needed for downstream recovery making it easier to extract task-specific information, and 3) our recovery guarantees are stronger for memory-augmented HMMs in comparison to the vanilla HMM when tuning an attention-based classfication head.
+
+We empirically evaluate our theoretical results with language models pretrained on synthetically generated data from HMMs. We find that prompt tuning obtains good downstream performance when our non-degeneracy conditions are relaxed, whereas head tuning performs poorly. Furthermore, we show that head tuning obtains better downstream performance when data is generated from a memory-augmented HMM, compared to a vanilla HMM, as is predicted by our theory.
+
+# 1.1 Related works
+
+The black box nature of BERT and related models has inspired a variety of empirical works which seek to understand them. Probing papers study whether a pretrained model computes various types of structured information (e.g., syntactic [35, 11]) by evaluating the performance of simple classifiers, or probes, on the representations [7, 12, 34, 27, 15]. Other papers ablate various aspects of pretraining, such as changing the masking scheme [14, 20, 40] or permuting the word order [32].
+
+In comparison, theoretical analysis of pretrained language models is limited. Besides [28], which we discussed in Section 1, Zhang and Hashimoto [40] analyze using a linear classifier to approximately recover the latent variable in a Gaussian graphical model with sparse dependencies between observed variables. However, their analysis and setting are focused towards understanding syntactic dependencies between tokens, whereas we directly model and analyze downstream performance.
+
+Prompt-based tuning [31, 19, 9, 21, 13, 6, 41, 2, 23], which has improved empirical downstream performance for lightweight adaptation methods beyond head tuning to approach full finetuning, is an important focus of our theoretical analysis. Shin et al. [31] employ task-specific prompts that are optimized over the discrete token space. Schick and Schütze [29, 30] reformulate natural language tasks as cloze-style phrases to enable few-shot learning. Subsequent methods [19, 9, 21] optimize “soft” prompts, or continuous embedding vectors. Lester et al. [19] employ soft prompts on pretrained large-scale T5 [26] models and show that as the model size increases, prompt tuning performance can eventually match finetuning. Hambardzumyan et al. [9] applies a variant of soft prompt tuning to MLM models. Li and Liang [21] propose prefix tuning, which prepends a trainable prefix embedding sequence to all layers of the transformer.
+
+More broadly, Lee et al. [18] analyze reconstruction-based self-supervised learning methods in a general setting and show that under certain conditional independence assumptions, predicting one observed variable from another allows recovery of the latent with a linear head. Other theoretical works analyzing self-supervised or constrastive learning include [1, 10, 36, 38, 37], but they do not directly relate to our particular setting.
+
+# 2 Formulations and notations
+
+We analyze models pretrained on masked language modeling (MLM) objectives. Let $\mathcal { X }$ denote a finite vocabulary of input tokens, $\mathcal { X } ^ { \ast }$ the set of variable-length sequences of tokens, and $X =$ $( X _ { 1 } , \dots , X _ { T } ) \in { \mathcal { X } } ^ { * }$ a random sequence of $T$ tokens. Let $\Delta ^ { | \bar { x } | }$ denote the space of probability distributions over tokens.
+
+Pretraining and downstream task. Let $G ( x ) = ( G _ { 1 } ( x ) , G _ { 2 } ( x ) , . . . )$ denote the masked language model which predicts a probability vector for each timestep in the input $x$ . Our theoretical abstraction is that $G _ { i }$ perfectly computes the distribution of $X _ { i }$ , the $i$ -th token, conditioned on all other tokens: $G _ { i } ( x ) = P [ X _ { i } | X _ { - i } = x _ { - i } ]$ . Here $P [ X _ { i } \mid X _ { - i } = x _ { - i } ] \in \Delta ^ { | \mathcal { X } | }$ is a probability vector. In particular, $G _ { i } ( x )$ does not depend on $x _ { i }$ . The downstream task involves labeled examples $( x , F ^ { \star } ( x ) ) \in \mathcal { X } ^ { * } \times \mathcal { Y }$ , where $F ^ { \star } : \mathcal { X } ^ { * }  \mathcal { Y }$ provides ground-truth downstream labels and $\mathcal { V }$ is a discrete set of labels for classification.
+
+Head and prompt tuning. Head tuning trains a classification head $f$ on top of fixed model outputs, resulting in the classifier $\mathsf { \bar { F } } ( x ) = \mathbb { 1 } ( f ( G ( x ) ) \geqslant 0 )$ . We expect $f$ to be a simple function such as a linear or one layer attention model. We also analyze variants where $f$ also takes the tokens $x$ or embeddings of $x$ as input, which provides additional information. Soft prompt tuning requires viewing the pretrained model $G$ as a function of the token embeddings; we refer to this model by $\overline { { G } }$ . Letting $e ( \boldsymbol { x } ) = e ( x _ { 1 } ) , \ldots , e ( x _ { t } )$ denote the token embeddings, we have $\overline { { G } } ( e ( x ) ) = G ( x )$ . Soft prompt tuning concatenates a trainable prompt $u$ so that the model output is $\overline { { G } } ( ( u , e ( x ) )$ . We consider simultaneously training the prompt parameter $u$ and a classification head to fit the downstream task.
+
+Notations. Let $\Delta ^ { d }$ denote the space of $d$ -dimensional probability vectors. We work with discrete random variables $V$ taking values in a finite set $\nu$ . We use $P [ V ] \in \Delta ^ { | \nu | }$ to denote the distribution of $V$ and $P [ U | V = v ] \in \mathbb { R } ^ { | \mathcal { U } | }$ the conditional distribution of $U$ given $V = v$ . $\operatorname* { P r } ( V = v ) \in [ 0 , 1 ]$ will denote the probability that $V$ takes values $v$ . We also let $P [ U = u \vert V ] \in \mathbb { R } ^ { \vert \nu \vert }$ denote the vector with entries $\operatorname* { P r } ( U = u \vert V = v )$ . $P [ U | V ] \in \mathbb { R } ^ { | \mathcal { U } | \times | \mathcal { V } | }$ will describe the matrix with entries $P [ U | V ] _ { u , v } = \operatorname* { P r } ( U \dot { = } u | V \dot { = } v )$ .
+
+For a sequence $v = ( v _ { 1 } , \ldots , v _ { t } )$ , we use the notation $v _ { i : j }$ for $i \leqslant j$ to denote $( v _ { i } , \ldots , v _ { j } )$ , and $v _ { - i }$ to denote $( v _ { 1 : i - 1 } , v _ { i + 1 : t } )$ . We let 1 denote the indicator function. For set $\nu$ , we let $\mathcal { V } ^ { * } = \mathcal { \bar { V } } ^ { 1 } \cup \mathcal { V } ^ { 2 } \cup \cdots$ denote variable-length sequences of elements of $\nu$ . Let $\odot$ denote elementwise product. Let $\mathbf { 1 } _ { d } , \mathbf { 0 } _ { d }$ denote the $d$ -dimensional all-1’s and all-0’s vector. We omit the subscript if the dimension is clear from context. For two vectors $a , b \in \mathbb { R } ^ { d }$ , we let $a / b$ denote their element-wise division. We use $\operatorname { s u p p } ( a )$ to denote the set of indices where vector $a$ is non-zero.
+
+# 3 Analysis for Hidden Markov Models
+
+Defining a relation between pretraining and downstream tasks is the foremost challenge for analysis. We propose to link the two via latent variable generative assumptions on the input distribution. We model the downstream task as a function of the posterior distribution of the latent variables. Towards a first result, this section studies the case where inputs are generated by HMMs (see Figure 1 (left)), which have been well-studied in the context of language and speech processing (see e.g. [24, 17, 3]).
+
+![](images/565f61d89a958502b132a1b5d80068eabd4d909632a3aadb21ffc7dd9929aad0.jpg)  
+Figure 1: Left: Illustration of HMM graphical model. Right: Overview of the formulation and analysis setting for prompt (and head) tuning. To abstractify soft prompt tuning, we note that every token has a natural embedding, the corresponding row of the emission probability matrix. We view prompt tuning as adding a fake token $\widetilde { z }$ to the vocabulary, assigning it a row $u$ in the emission matrix, and prepending it to the input embedding sequence. More details are provided in Section 3.1.
+
+Data distribution. Let $\mathcal { H }$ denote the hidden state space of the HMM. We use $\begin{array} { r l } { H } & { { } = } \end{array}$ $( H _ { 0 } , H _ { 1 } , \ldots , H _ { T } ) \in \mathcal { H } ^ { * }$ to denote the sequence of hidden states. For all timesteps $i > 0$ , the transition probabilities are time-invariant, i.e. $P [ H _ { i } | H _ { i - 1 } ] = A$ for $A \in \mathbb { R } ^ { | \mathcal { H } | \times | \mathcal { H } | }$ . For each timestep $i \geqslant 1$ , tokens $X _ { i }$ are emitted following some time-invariant probability: $P [ X _ { i } \mid H _ { i } ] = W$ for $W \in \mathbb { R } ^ { | \mathcal { X } | \times | \mathcal { H } | }$ . The joint probability of $X , H$ is
+
+$$
+\operatorname* { P r } ( X , H = x , h \mid T = t ) = \operatorname* { P r } ( H _ { 0 } = h _ { 0 } ) \prod _ { i = 1 } ^ { t } \operatorname* { P r } ( H _ { i } = h _ { i } \mid H _ { i - 1 } = h _ { i - 1 } ) \operatorname* { P r } ( X _ { i } = x _ { i } \mid H _ { i } = h _ { i } ) .
+$$
+
+Downstream tasks. We assume that $H _ { 0 }$ has the meaningful information for the downstream task, which is a binary classification task where the ground-truth labeling $F ^ { \star }$ is assumed to be a linear classifier on the posterior $P [ H _ { 0 } | X _ { 1 : T } = x ]$ :
+
+$$
+F ^ { \star } ( x ) = \mathbb { 1 } ( \mu ^ { \top } P [ H _ { 0 } | X _ { 1 : T } = x ] \geqslant 0 )
+$$
+
+for $\mu \in \mathbb { R } ^ { | \mathcal { H } | }$ . Our results are easily extended to the multiclass setting. We consider tuning a linear head for the downstream classifier, which formally computes $\mathbb { 1 } ( b ^ { \top } G _ { 1 } ( x ) \geqslant 0 )$ for $b \in \mathbb { R } ^ { | \mathcal { X } | }$ . The following non-degeneracy condition is crucial for our recovery result in this setting.
+
+Assumption 3.1 (Non-degeneracy, vanilla HMM). The token emission probability matrix $W$ has linearly independent columns.
+
+We also require the following regularity conditions on $H _ { 0 }$ and the state transitions.
+
+Assumption 3.2 (Regularity). The Markov chain $H _ { 0 } , H _ { 1 } , \ldots$ is ergodic, and $P [ H _ { 0 } ]$ has full support
+
+We show that if $W$ has linearly independent columns, a linear head fits downstream labels.
+
+Theorem 3.3. Assume that non-degeneracy (Assumption 3.1) and regularity (Assumption 3.2) hold. Then any downstream task $F ^ { \star } ( x )$ of the form (3.1) can be computed by a linear head on $G$ applied to $a$ shifted sequence. That is, there exists linear head weights $b \in \mathbb { R } ^ { | \mathcal { X } | }$ such that for all $x \in \operatorname { s u p p } ( P [ X ] )$ ,
+
+$$
+F ^ { \star } ( x ) = \mathbb { 1 } ( b ^ { \top } G _ { 1 } ( x ^ { \prime } ) \geqslant 0 )
+$$
+
+where $\boldsymbol { x } ^ { \prime } = \left( \boldsymbol { \mathcal { O } } , x _ { 1 : t } \right)$ is the concatenation of a special token $\mathcal { D }$ with $x$ .1
+
+The key for the proof is to leverage the following general statement about random variables $U , V , Z$ such that $U \perp V | Z$ , which decomposes the expression for $P [ U | V ]$ .
+
+Proposition 3.4. Let $U , V , Z$ be random variables such that $U \perp V | Z$ . Then for any $v$ , $P [ U | V =$ $v ] = P [ U | Z ] \cdot P [ Z | V = v ]$ . Thus, i $f P [ U | Z ]$ has a left inverse $( P [ U | Z ] ) ^ { \dagger }$ , then $P [ Z | V =$ $v ] = ( P [ U \mid Z ] ) ^ { \dagger } P [ U \mid V = v ]$ .
+
+By the conditional independence structure of the HMM, Proposition 3.4 immediately implies
+
+$$
+G _ { 1 } ( x ^ { \prime } ) = W P [ H _ { 1 } | X _ { 2 : T + 1 } = x ] \implies P [ H _ { 1 } | X _ { 2 : T + 1 } = x ] = W ^ { \dagger } G _ { 1 } ( x ^ { \prime } )
+$$
+
+where $W ^ { \dagger }$ is the left inverse for $W$ , guaranteed to exist by Assumption 3.1. This lets us recover $P [ H _ { 1 } | X _ { 2 : T + 1 } = x ]$ by applying a linear function to $G _ { 1 } ( \bar { x } ^ { \prime } )$ . Additional linear functions will be sufficient to obtain $\mathrm { \bar { \mu } } ^ { \top } P [ H _ { 0 } | X _ { 1 : T } = x ]$ from $P [ H _ { 1 } | X _ { 2 : T + 1 } = x ]$ . We provide the full proof in Section B.
+
+Proposition 3.4 is reminiscent of the arguments of [18], which leverages the independence structure in the same way. Subsequent sections will require more complicated analyses and recovery procedures.
+
+A drawback of Theorem 3.3 is that it relies heavily on assuming $W$ has full column rank, which implies the necessary condition that $| { \mathcal { H } } | \leqslant | { \mathcal { X } } |$ . Without this assumption, it is unclear how to recover $P [ \tilde { H } _ { 0 } | X _ { 1 : T } = x ]$ from $G ( x )$ alone. However, in realistic settings we would expect $| { \mathcal { H } } | > | { \mathcal { X } } |$ , as increasing the size of the hidden state space improves language modeling capabilities of HMMs [3].
+
+# 3.1 Relaxed non-degeneracy assumptions via prompt tuning
+
+In this section, we study applying soft, or continuous, prompt tuning [19, 9] to the setting above. We show that by using soft prompt tuning, we can recover $F ^ { \star }$ using a linear head on $G$ for HMMs where the non-degeneracy assumptions on $W$ are relaxed. Our analysis provides insight into the empirical successes of prompt-tuning: intuitively, prompt tuning enables better recovery of the downstream task by conditioning the output of $G$ to only contain task-specific information.
+
+Soft prompt tuning trains task-specific embedding vectors, but analyzing how the model processes embedding vectors is challenging because it requires opening up the black box of the pretrained model. Thus, we require additional abstractions about how the pretrained model processes the embedding vectors. We will extend the mask language model $G$ to a model $\overline { { G } }$ that maps a sequence of embeddings $e _ { 1 } , \ldots , e _ { t }$ to conditional probabilities $G _ { 1 } ( x ) , \dots , G _ { t } ( x )$ as follows. We observe that each token $z$ in the vocabulary $\mathcal { X }$ naturally corresponds to a $| \mathcal { H } |$ -dimensional vector: the $z$ -th row of the emission probability matrix $W$ , or equivalently, $P [ X _ { i } = z \mid H _ { i } ]$ . We denote this embedding by $e ( z )$ and call the family of embeddings $\bar { \{ e ( z ) : z \in \mathcal { X } \} }$ proper embeddings. A fundamental property of HMMs is that the conditional probability $P [ X _ { i } \mid X _ { - i } = x _ { - i } ]$ only depends on $x _ { 1 } , \ldots , x _ { t }$ through their embeddings $e ( x ) = ( e ( x _ { 1 } ) , \ldots , e ( x _ { t } ) )$ . In other words, there exists a function $\overline { { G } } _ { i }$ such that
+
+$$
+G _ { i } ( x _ { 1 } , \dots , x _ { t } ) = { \overline { { G } } } _ { i } ( e ( x _ { 1 } ) , \dots , e ( x _ { t } ) )
+$$
+
+In particular, we let $\overline { { G } } _ { i }$ compute the standard message passing algorithm [16] that computes the conditional probability of HMMs. This ensures that $\overline { { G } } _ { i }$ is well defined on all sequences of nonnegative vectors in $[ 0 , 1 ] ^ { | \mathcal { H } | }$ , beyond sequences of proper embeddings.We assume that pretraining produces this $\overline { { G } } _ { i }$ , which we treat as a blackbox for prompt tuning.
+
+In particular, for prompt tuning we can consider the case where we pass an arbitrary nonnegative vector $u \in [ 0 , 1 ] ^ { | \mathcal { H } | }$ to $\overline { G }$ in the first argument and proper embeddings at positions $i > 1$ . We can interpret $u$ as the embedding of a fake token $\widetilde { z }$ . Concretely, consider adding a new token $\widetilde { z }$ to the vocabulary $\mathcal { X }$ , and changing the emission probability at position 1 to satisfy $P [ X _ { 1 } = \tilde { z } | H _ { 1 } ] = u$ and for all $z \neq \widetilde { z }$ , $P [ X _ { 1 } = z | H _ { 1 } ] { \propto } ( 1 - u ) \odot e ( z )$ . Then $\overline { { G } } _ { i } ( u , e ( x _ { 1 } ) , \ldots , e ( x _ { t } ) )$ precisely computes the conditional probability $P [ X _ { i } | X _ { - i } = ( { \tilde { z } } , x _ { 1 } , \dots , x _ { t } ) _ { - i } ]$ under the modified HMM. We refer the readers to Section C for the formal definition of $\overline { { G } } _ { i }$ and formal proofs of the interpretation above.
+
+We consider a downstream training algorithm which trains the prompt tuning parameter $u$ described above and a linear classification head. Letting $u$ denote the trainable prompt parameter and $b \in \mathbb { R } ^ { | \mathcal { X } | }$ the trainable linear head weights, the model uses the embedding sequence
+
+$$
+{ \widehat { e } } ( x ) \triangleq ( u , e ( \emptyset ) , e ( x _ { 1 } ) , \dots , e ( x _ { t } ) )
+$$
+
+and outputs the prediction $F ( x ) = \mathbb { 1 } ( b ^ { \top } G _ { 2 } ( { \widehat { e } } ( x ) ) \geqslant 0 )$ . We can provide recovery guarantees for this model if the ground-truth classifier weights $\mu$ (defined in (3.1)) and columns of the HMM transition matrix $A$ satisfy the following relaxation of the requirement in Theorem 3.3 that $W$ is nondegenerate.
+
+Assumption 3.5 (Relaxed non-degeneracy condition). There exists a set of essential hidden states $\mathcal { H } ^ { \star } \subseteq \mathcal { H }$ , so that the columns of $W$ corresponding to $\mathcal { H } ^ { \star }$ , $\{ W _ { : , h } \} _ { h \in \mathcal { H } ^ { \star } }$ , are linearly independent. Furthermore, $\mathcal { H } ^ { \star }$ covers all meaningful information for the downstream tasks: $\operatorname { s u p p } ( \mu ) \subseteq { \mathcal { H } } ^ { \star }$ .
+
+In addition, a last technical requirement on $\mathcal { H } ^ { \star }$ is as follows: there exists a set $B \subseteq { \mathcal { H } }$ such that ${ \mathcal { H } } ^ { \star } = \cup _ { h \in B } \operatorname { s u p p } ( A _ { : , h } )$ . In other words, $\mathcal { H } ^ { \star }$ must be the set of all states reachable by starting from some state in $\boldsymbol { B }$ and transitioning one step in the hidden Markov chain.
+
+Compared to Assumption 3.1, which required that all columns of $W$ are linearly independent, Assumption 3.5 only requires linear independence on a subset $\mathcal { H } ^ { \star }$ of essential states. In the setting where $| { \mathcal { H } } | > | { \mathcal { X } } |$ , the condition for Theorem 3.3 can never hold. On the other hand, Assumption 3.5 could still hold, for example, if $| \operatorname { s u p p } ( \mu ) | < | \chi |$ and the set of columns of $W$ corresponding to hidden states in $\operatorname { s u p p } ( \mu )$ is linearly independent. The last technical requirement in Assumption 3.5 is also required, which could be satisfied if columns of $A$ are sparse. The following theorem shows that when Assumption 3.5 holds, we can recover $F ^ { \star }$ using soft prompt tuning with a linear head.
+
+Theorem 3.6. In the above setting, assume that Assumptions 3.2 and 3.5 hold. Then $F ^ { \star }$ can be computed using soft prompt tuning with a linear head on $\overline { { G } }$ . Concretely, there is a continuous prompt parameter $u \in \mathbb { R } ^ { | \mathcal { H } | }$ and weight vector $b \in \mathbb { R } ^ { | \mathcal { X } | }$ , such that for all $x \in \operatorname { s u p p } ( P [ X ] )$ ,
+
+$$
+F ^ { \star } ( x ) = \mathbb { 1 } ( b ^ { \top } \overline { { G } } _ { 2 } ( \hat { e } ( x ) ) \geqslant 0 )
+$$
+
+where ep prepends u to the input embedding sequence, as defined in (3.2).
+
+Theorem 3.6 provides a stronger recovery result than Theorem 3.3, which only used a linear head. This is also reflected in our synthetic experiments (Section 5), and prior work which shows that variants of prompt tuning can perform much better than only training the last few layers of the model [21]. Our theory suggests that prompt tuning could help by conditioning the hidden variables to remove nonessential information for the task from the output of $G$ . This makes task-essential information easier to recover.
+
+The key proof intuition is that although recovering $P [ H _ { 0 } | X _ { 1 : T } = x ]$ is impossible without strong non-degeneracy conditions (Assumption 3.1), we can aim to recover $\bar { P [ \cal H _ { 0 } | \bar { X } _ { 1 : T } = x ] }$ on the subset of essential states $\mathcal { H } ^ { \star }$ defined in Assumption 3.5, which suffices for computing $\mu ^ { \top } P [ H _ { 0 } | X _ { 1 : T } = x ]$ , since $\mathcal { H } ^ { \star } \supseteq \mathsf { s u p p } ( \mu )$ . To recover $\bar { P [ H _ { 0 } | X _ { 1 : T } \ = \ x ] }$ on $\mathcal { H } ^ { \star }$ , we observe in Lemma C.2 that prepending the prompt $u$ is equivalent to introducing a modified random sequence $\hat { X }$ and fake token $\widetilde { z }$ which influences the posterior of $H _ { 2 }$ as follows:
+
+$$
+\overline { { G } } _ { 2 } ( \widehat { e } ( x ) ) = r _ { x } W D ( P [ H _ { 2 } | \widehat { X } _ { 1 } = \widetilde { z } ] \odot P [ H _ { 0 } | X _ { 1 : T } = x ] )
+$$
+
+for invertible diagonal matrix $D$ and positive scalar $r _ { x }$ . We choose $u$ so $P [ H _ { 2 } | \hat { X } _ { 1 } = \tilde { z } ] \odot$ $P [ H _ { 0 } | X _ { 1 : T } = \bar { x ] }$ is supported only on $\mathcal { H } ^ { \star }$ . As corresponding columns of $W$ are linearly independent (Assumption 3.5), we recover $\mathrm { P r } ( H _ { 0 } = h | X _ { 1 : T } = x )$ for $h \in \mathcal { H } ^ { \star }$ via a linear function of $\overline { { G } } _ { 2 } ( \widehat { e } ( x ) )$ . This suffices for computing $\mu ^ { \top } P [ H _ { 0 } | X _ { 1 : T } = x ]$ . For more details, see Section $\textrm { C }$ .
+
+# 4 Analysis for memory-augmented Hidden Markov Models
+
+We study a memory-augmented HMM which explicitly disentangles the evolution of hidden states from a persistent “memory” variable. Inspired by natural sentences, this model is intended to better capture the distinction between syntax, which constantly evolves, and semantics, which changes less. This additional structure in the generative model allows us to strengthen our results by relaxing the non-degeneracy conditions on $W$ , the token emission probabilities. Thus, both head and prompt tuning are more powerful in this setting compared to Section 3 and can recover the downstream label with weaker non-degeneracy assumptions on $W$ . In Section 4.2, we show that soft prompt tuning also provides an advantage over head tuning alone.
+
+Data distribution. The memory-augmented HMM, depicted in Figure 2, can be viewed as a generative variant of memory networks [39, 33] and is closely related to Hidden Topic Markov Models [8]. There are two sets of latent variables in the memory-augmented HMM: a Markov chain on hidden states $H _ { 0 } , H _ { 1 } , . . . ,$ meant to model the evolution of syntax, and a persistent “memory” $M = ( M _ { 1 } , \dots , M _ { N } )$ with $N$ total cells, where each $M _ { i }$ takes values in a finite set $\mathcal { M }$ . The full joint
+
+![](images/c5f0766cf68bc506c99186f4cb7010d27a8e6453ddbd36505507a0cd3ad19b4e.jpg)  
+Figure 2: Left: Memory-augmented HMM with a single memory cell. The memory $M$ and hidden state $H _ { i }$ determine the emission probabilities for each state $X _ { i }$ . Right: Memory-augmented HMM with multiple memories $M _ { 1 } , \dots , M _ { N }$ . The hidden state $H _ { i }$ consists of a cell index $J _ { i }$ and syntax state $S _ { i }$ . To sample $X _ { i }$ , we first look up the $J _ { i }$ -th memory cell $M _ { J _ { i } }$ . The token emission probability is then determined by the tuple $( M _ { J _ { i } } , J _ { i } , S _ { i } )$ .
+
+probability is as follows:
+
+$$
+\begin{array} { l } { { \displaystyle { \sf P r } ( X , H , M = x , h , m | T = t ) = } \ ~ } \\ { { \displaystyle ~ { \sf P r } ( M = m ) { \bf P r } ( H _ { 0 } = h _ { 0 } ) \prod _ { i = 1 } ^ { t } { \bf P r } ( H _ { i } = h _ { i } | H _ { i - 1 } = h _ { i - 1 } ) { \bf P r } ( X _ { i } = x _ { i } | M = m , H _ { i } = h _ { i } ) } } \end{array}
+$$
+
+The hidden state is modified to explicitly consist of a disentangled cell index $J \in [ N ]$ and syntax state $S \in S$ , such that $H _ { i } = ( J _ { i } , S _ { i } )$ and $\mathbf { \dot { \mathcal { H } } } = [ N ] \times \mathbf { \mathcal { S } }$ . To sample the token at timestep $i$ given the hidden state $H _ { i } = ( J _ { i } , S _ { i } )$ , we first use $J _ { i }$ to index the memory $M$ , obtaining the random variable $M _ { J _ { i } }$ . $X _ { i }$ is then sampled according to some time-invariant probability depending on $M _ { J _ { i } } , J _ { i } , S _ { i }$ :
+
+$$
+P [ X _ { i } \mid M = m , H _ { i } = ( j , s ) ] = P [ X _ { i } \mid M _ { J _ { i } } = m _ { j } , H _ { i } = ( j , s ) ] = W _ { : , ( m _ { j } , j , s ) }
+$$
+
+Here $W \in \mathbb { R } ^ { | \mathcal { X } | \times | \mathcal { M } | | \mathcal { H } | }$ stores the emission probabilities for each choice of memory cell value and hidden state. Note that in particular, the conditional probabilities for $X _ { i }$ only depend on a single memory cell for each timestep. We also note that memory-augmented HMMs can be viewed as vanilla HMMs with structured transitions because $( H _ { 0 } , M ) , \bar { ( } H _ { 1 } , \bar { M } ) , . . .$ can be viewed as a Markov chain where the memory component does not change.
+
+Example 4.1 (Generating natural sentence with memory-augmented HMM). We consider how this model may generate the sentence “The cow in the pasture rolled on the grass’ happily.” $M _ { 1 }$ could store the subject (“cow”), $M _ { 2 }$ the location (“pasture”), $M _ { 3 }$ the sentiment (“happily”), and $S _ { i }$ could determine part-of-speech. For timesteps where “cow” and “rolled” are emitted $J _ { i } = 1$ because we emit information related to the sentence subject. Timesteps for “pasture” and “grass” have $J _ { i } = 2$ .
+
+Downstream tasks. We consider downstream tasks where ground-truth labels are obtained via a linear classifier on the posterior distribution of a particular memory cell $j ^ { \star } \in [ N ]$ : $F ^ { \star } ( x ) =$ $\begin{array} { r } { \mathbb { 1 } ( \mu ^ { \top } P [ M _ { j ^ { \star } } | X _ { 1 : T } = x ] \geqslant 0 ) } \end{array}$ q, where $\mu \in \mathbb { R } ^ { | \mathcal { M } | }$ . Intuitively, this formulation models downstream tasks which depend on a particular aspect of the semantics but not on syntax (e.g. in the setting of Example 4.1, if $j ^ { \star } = 3$ , the task is sentiment analysis).
+
+# 4.1 Tuning attention head for recovering ground-truth downstream labels
+
+To recover the downstream labeling, we require an attention-based classification head, which is a function of both the input embeddings and outputs of $G$ . Formally, let $q \in \mathbb { R } ^ { | \mathcal { H } | + 1 }$ denote a query parameter and $\beta _ { 1 } , \dots , \beta _ { t } \in \mathbb { R } ^ { | \mathcal { H } | + 1 }$ denote trainable position embeddings. Given pretrained model outputs $G _ { i } ( x )$ and trainable token embeddings $e ( x _ { i } )$ , the attention head $\mathrm { A t t n } ( \cdot )$ applies key and value functions $K , V$ to compute the output as follows:
+
+$$
+\begin{array} { c } { \displaystyle \mathcal { Z } \triangleq \arg \underset { i } { \operatorname* { m a x } } \{ q ^ { \top } ( K ( G _ { i } ( x ) ) + \beta _ { i } ) \} } \\ { \displaystyle \mathrm { A t t n } ( ( G _ { i } ( x ) , e ( x _ { i } ) ) _ { i = 1 } ^ { t } ) \triangleq \frac { 1 } { | \mathcal { Z } | } \sum _ { i \in \mathcal { Z } } V ( G _ { i } ( x ) , e ( x _ { i } ) ) } \end{array}
+$$
+
+where arg max refers to the set of indices achieving the maximum in (4.1). We note that standard attention heads in practice rely on the softmax function, but the expression based on arg max above captures the limiting behavior as $\| q \| _ { 2 } \to \infty$ . We consider linear key functions given by ${ \bar { K } } ( G _ { i } ( x ) ) =$ $\Theta ^ { ( K ) } G _ { i } ( x )$ . The value function $V : \mathbb { R } ^ { | \mathcal { X } | } \times \mathbb { R } ^ { | \mathcal { M } | | \mathcal { H } | } \to \mathbb { R }$ uses parameters $\Theta ^ { ( V ) } \in \mathbb { R } ^ { | \mathcal { M } | | \mathcal { H } | \times | \mathcal { X } | }$ and $b \in \mathbb { R } ^ { | \mathcal { M } | | \mathcal { H } | }$ and computes $V ( G _ { i } ( x ) , e ( x _ { i } ) ) = b ^ { \top } ( ( \Theta ^ { ( V ) } G _ { i } ( x ) ) \odot e ( x _ { i } ) )$ .
+
+Because our generative model disentangles $H$ and $M$ , we can relax the non-degeneracy assumption on the token emission probabilities $W$ , compared to Theorem 3.3. The relaxed assumption only requires the columns $\{ \bar { W } _ { : , ( m , h ) } \} _ { m \in \mathcal { M } , h \in \mathcal { H } ^ { \star } }$ to be linearly independent in a subset $\mathcal { H } ^ { \star }$ of “recoverable” hidden states, whereas Assumption 3.1 required all columns to be linearly independent.
+
+Assumption 4.2 (Existence of “recoverable” hidden states). There exists a set of recoverable hidden states ${ \bar { \mathcal { H } } } ^ { \star } = \{ j ^ { \star } \} \times S ^ { \star }$ , such that the collection of token emission probabilities from $\mathcal { M } \times \mathcal { H } ^ { \star }$ $\{ W _ { : , ( m , h ) } \} _ { m \in { \mathcal { M } } , h \in { \mathcal { H } } ^ { \star } }$ , is a linearly independent set of vectors.
+
+Furthermore, the span of these vectors must be disjoint from the span of token emission probabilities from $\mathcal { M } \times ( \mathcal { H } \backslash \mathcal { H } ^ { \star } )$ $\begin{array} { r } { \colon \mathrm { s p a n } \big ( \{ W _ { : , ( m , h ) } \} _ { m \in \mathcal { M } , h \in \mathcal { H } ^ { \star } } \big ) \cap \mathrm { s p a n } \big ( \{ W _ { : , ( m , h ^ { \prime } ) } \} _ { m \in \mathcal { M } , h \in \mathcal { H } \backslash \mathcal { H } ^ { \star } } \big ) = \{ \mathbf { 0 } _ { | \mathcal { X } | } \} \mathrm { . ~ } } \end{array}$ .
+
+Note that the non-degeneracy condition of Theorem 3.3 would require $\{ W _ { : , ( m , h ) } \} _ { m \in { \mathcal { M } } , h \in { \mathcal { H } } }$ to be linearly independent, whereas Assumption 4.2 only requires linear independence for $h \in \mathcal { H } ^ { \star }$ . The second condition states that $\mathcal { H } ^ { \star }$ and $\mathcal { H } \backslash \mathcal { H } ^ { \star }$ are distinguishable by the token emission probabilities.
+
+We explain Assumption 4.2 in the setting of Example 4.1. For natural language, there might be choices of $h = ( j _ { i } , s _ { i } )$ for which the set $\{ W _ { : , ( m , h ) } \} _ { m \in \mathcal { M } }$ of token emission probabilities is fundamentally not very diverse, and therefore not linearly independent. For example, if the syntax $s _ { i }$ indicates “article”, i.e. words such as “a”, “an”, and “the”, the token emission probabilities would carry little information about $M _ { j _ { i } }$ because the choice of article does not depend much on semantics, so columns corresponding to $s _ { i } =$ “article” would not be linearly independent, violating Assumption 3.1. However, Assumption 4.2 allows us to avoid this issue by placing such $h$ in $\mathcal { H } \backslash \mathcal { H } ^ { \star }$ , a set of hidden states which we can ignore, and only including hidden states which carry a lot of information about $M$ in $\mathcal { H } ^ { \star }$ . In Example 4.1, when $J _ { i } = 2$ (location), $S _ { i } = \mathrm { \ " n o u n } ^ { \mathrm { * } }$ , the position $i$ should convey a lot about the location (in this case, “pasture”), so it is more reasonable to assume that $\{ W _ { : , m , h } \} _ { m \in \mathcal { M } }$ is linearly independent for this hidden state.
+
+Thus, our aim is to focus on recovering information for the downstream task from positions $i$ where $H _ { i } \in { \mathcal { H } } ^ { \star }$ . Formally, we define the following set of input sequences containing positions $i$ where the posterior of $H _ { i }$ given $x _ { - i }$ concentrates on $\mathcal { H } ^ { \star }$ :
+
+$$
+{ \mathcal { R } } \triangleq \{ ( x _ { 1 } , \dots , x _ { t } ) \in \operatorname { s u p p } ( P [ X ] ) : \exists i { \mathrm { ~ w i t h ~ } } \operatorname { s u p p } ( P [ H _ { i } \mid X _ { - i } = x _ { - i } ] ) \subseteq { \mathcal { H } } ^ { \star } \}
+$$
+
+The following theorem shows that under Assumption 4.2, we can recover $F ^ { \star }$ using the attention head described above, if $x \in \mathcal { R }$ is nonempty. Note that $\mathcal { R }$ is nonempty if the posterior of $H _ { i }$ concentrates on $\mathcal { H } ^ { \star }$ for some $i$ . For natural language, it is realistic to assume this can occur because syntactic aspects of a sentence are typically low-entropy when the full sentence is observed.
+
+Theorem 4.3. Assume that non-degeneracy (Assumption 4.2) and regularity (Assumption 3.2) hold. Define $\mathcal { R }$ as in (4.3). Then there exist an attention head on $G ( x )$ and token embeddings $e ( x _ { i } )$ such that the following holds for any $x \in \mathcal { R }$ :
+
+$$
+F ^ { \star } ( x ) = \mathbb { 1 } \bigl ( \mathrm { A t t n } ( ( G _ { i } ( x ) , e ( x _ { i } ) ) _ { i = 1 } ^ { t } ) \geqslant 0 \bigr )
+$$
+
+where the function Attn is in the form described in (4.2).
+
+The idea is to use the attention mechanism to attend to positions $i$ where ${ \mathrm { s u p p } } ( P [ H _ { i } | X _ { - i } = x _ { - i } ] ) \subseteq$ $\mathcal { H } ^ { \star }$ . The intuition of Assumption 4.2 is that such positions are more informative for recovering the latent posteriors; indeed, from the outputs $G _ { i } ( x )$ at such $i$ , the value function in the attention will be able to recover $P [ M _ { j ^ { \star } } | X _ { 1 : T } = x ]$ . A full proof is provided in Section D.1.
+
+# 4.2 Guarantees for prompt-tuning
+
+Though the generative modeling assumptions in this section already allowed relaxed non-degeneracy assumptions, applying soft prompt tuning allows us to relax them even further. For simplicity, we consider the setting where there is a single memory cell, so $M \in \mathcal { M }$ , and the downstream task is a linear classifier on the posterior of the memory: $F ^ { \star } ( x ) = \mathbb { 1 } ( \mu ^ { \top } P [ M | X _ { 1 : T } = x ] \geqslant 0 )$ q. This simplified setting doesn’t require the explicit disentanglement between $J _ { i }$ and $S _ { i }$ in $H _ { i }$ . We analyze continuous prompt-tuning in a setting where the pretrained model $\overline { { G } }$ follows the same abstraction as in Section 3.1. We modify the model to take $| \mathcal { M } | | \mathcal { H } |$ -dimensional vectors, so the proper embedding for token $z$ is given by $e ( z ) = P [ X _ { i } = z | M , H _ { i } ] = W _ { z , : } ^ { \top }$ . In Section D.3, we describe the formal construction and interpretation of $\overline { { G } }$ in the more general setting with more memories.
+
+Letting $u \in \mathbb { R } ^ { | \mathcal { M } | | \mathcal { H } | }$ denote the trainable prompt parameter, we define the input embeddings
+
+$$
+{ \widehat { e } } ( x ) \triangleq ( u , e ( x _ { 1 } ) , \dots , e ( x _ { t } ) )
+$$
+
+The downstream model applies an attention head to the output of $\overline { G }$ : $\begin{array} { r l r l } { F ( x ) } & { { } = } & { } \end{array}$ $\mathbb { 1 } \left( \mathrm { A t t n } ( ( \overline { G } _ { i } ( \widehat { e } ( x ) ) , \widehat { e } _ { i } ( x ) ) _ { i = 1 } ^ { t + 1 } ) \ \geqslant \ 0 \right)$ q, where Attn is defined in (4.2). An additional stationarity assumption on $P [ H _ { 0 } ]$ will simplify the recovery procedure (though it can be removed).
+
+Assumption 4.4 (Stationarity). Assumption 3.2 holds on the Markov chain $H _ { 0 } , H _ { 1 } , \dots$ . Furthermore, $P [ H _ { 0 } ]$ is the stationary distribution: $P [ H _ { 0 } ] = A P [ H _ { 0 } ]$ , where $A$ is the transition matrix.
+
+As before, we assume sparsity of $\mu$ and some non-degeneracy of $W$ , though the assumption is more relaxed and easier to state compared to the vanilla HMM setting.
+
+Assumption 4.5 (Relaxed version of Assumption 4.2). Let $\mathcal { M } ^ { \star } \triangleq \operatorname { s u p p } ( \mu )$ denote the set of non-zero coordinates in $\mu$ . There exists a set of recoverable hidden states $\mathcal { H } ^ { \star }$ , such that the collection of token emission probabilities from $\mathcal { M } ^ { \star } \times \mathcal { H } ^ { \star }$ , $\{ W _ { : , ( m , h ) } \} _ { m \in { \mathcal { M } } ^ { \star } , h \in { \mathcal { H } } ^ { \star } }$ , is linearly independent.
+
+Furthermore, the span of these vectors must be disjoint from the span of token emission probabilities from $\mathcal { M } ^ { \star } \times ( \mathcal { H } \backslash \mathcal { H } ^ { \star } )$ $) \colon \operatorname { s p a n } ( \{ W _ { : , ( m , h ) } \} _ { m \in \mathcal { M } ^ { \star } , h \in \mathcal { H } ^ { \star } } ) \cap \operatorname { s p a n } ( \{ W _ { : , ( m , h ^ { \prime } ) } \} _ { m \in \mathcal { M } ^ { \star } , h \in \mathcal { H } \backslash \mathcal { H } ^ { \star } } ) = \{ \mathbf { 0 } _ { | \mathcal { X } | } \}$ .
+
+We note that Assumption 4.5, and Assumption D.5 for multiple memories, are relaxations of Assumption 4.2, as they only consider memory values in $\operatorname { s u p p } ( \mu )$ , whereas Assumption 4.2 considers all $m \in \mathcal { M }$ . An additional advantage of the memory-augmented HMM is that Assumption 4.2 is simpler than Assumption 3.1 and does not require any conditions on the transition matrix $A$ . We now state our result for recovering $F ^ { \star }$ with soft prompt tuning and an attention head.
+
+Theorem 4.6. In the setting above, suppose that non-degeneracy Assumption 4.5 and stationarity Assumption 4.4 hold. Then there exists a prompt u and attention head on $\overline { { G } } ( \widehat { e } ( x ) )$ and the token embeddings which can compute the ground-truth $F ^ { \star } ( x )$ for any $x \in \mathcal { R }$ , defined in (4.3):
+
+$$
+F ^ { \star } ( x ) = \mathbb { 1 } \left( \mathrm { A t t n } ( ( \overline { G } _ { i } ( \widehat e ( x ) ) , \widehat e _ { i } ( x ) ) _ { i = 1 } ^ { t + 1 } ) \geqslant 0 \right)
+$$
+
+where $\hat { e }$ is the embedding in (4.4) and Attn is defined in (4.2).
+
+The intuition for this proof is similar to Theorem 3.6: the soft prompt conditions the memory $M$ to concentrate on $\operatorname { s u p p } ( \mu )$ . As a result, all irrelevant information to the task is removed from $\overline { { G } } _ { i } ( \widehat { e } ( x ) )$ , making it easier to recover the task-specific information about the posterior of $M$ . A more general theorem statement for the multiple memories setting, and the full proof, is provided in Section D.3
+
+# 5 Simulations
+
+We empirically evaluate our theoretical results by pretraining a BERT-like masked language model (MLM) [4] on synthetic data generated by an HMM. Our goal is to verify key implications of our theory in a more realistic setting where some assumptions, such as that $G$ outputs exact conditional probabilities, may not hold. First, we compare head and prompt tuning and show that prompt tuning improves downstream performance, especially when the recovery problem is degenerate. Second, we compare the effect of changing the data distribution from vanilla HMMs to memory-augmented HMMs on head tuning with an attention layer. We find that the downstream performance improves when the data has a long-term memory component. These observations support our theory.
+
+Pretraining data and downstream task. We generate pretraining data from an HMM with randomly generated transition matrix, emission probabilities, and start distributions. In all experiments, the HMMs have 10 vocabulary symbols, while the hidden state size varies. The downstream task uses input sequences $X _ { 1 : T }$ of length 129, where the first token $X _ { 1 } ~ = ~ [ \mathrm { { M A S K } ] }$ . We consider binary classifcation where labels are generated using linear functions of the analytically-computed posteriors in the HMMs. In all experiments, the ground truth linear weight is sparse with 6 nonzero entries at uniformly random locations with Gaussian values. More details are in Appendix E.
+
+![](images/90c6c6c133241c188eafac3f327482674ef97dc68c975ab5708e5bd785d64deb.jpg)  
+Figure 3: Left: Head vs. prompt tuning with a linear head on synthetically-generated HMM data, with varying hidden state sizes. Prompt tuning improves downstream accuracy especially when the problem is degenerate $( | \mathcal { H } | > | \mathcal { X } | )$ . Right: Downstream accuracy of head tuning on data from vanilla HMM vs. memory-augmented HMM, across varying values of $| { \mathcal { M } } | | { \mathcal { H } } |$ . Long-term dependencies in the memory-augmented HMM data improve downstream recovery with attention. We average over 20 trials (left) and 5 trials (right) of pretraining and finetuning, with $9 5 \%$ intervals shown.
+
+Head vs. prompt tuning. We compare head and prompt tuning as the hidden state size of the HMM varies. The downstream label is computed via $\mu ^ { \top } \bar { P } [ H _ { 1 } | \bar { X _ { - 1 } } = x _ { - 1 } ]$ , where $\mu$ is a random ground-truth linear weight. Head tuning learns a linear head on top of the softmax probabilities predicted by the pretrained model for filling in the first [MASK] token. Prompt tuning uses the same setup but also optimizes a length 20 continuous embedding prepended to the input sequence.
+
+Figure 3 (left) shows that prompt tuning improves downstream performance substantially across all hidden state sizes ({4,8,10,15,25,30}). Prompt tuning improves especially when the hidden state size increases beyond the vocabulary size, which makes the recovery problem degenerate. Thus, as suggested by Theorem 3.6, prompt tuning helps relax the non-degeneracy conditions.
+
+Memory-augmented HMMs. We investigate the effect of augmenting the data-generating HMM with a long-term memory. We consider the single memory case with $| \mathcal { H } | = 4$ and varying memory sizes $| \mathcal { M } | \in \{ 2 , 3 , 5 , 7 \}$ . The downstream label is generated by computing $\mu ^ { \top } P [ M | X _ { - 1 } = x _ { - 1 } ]$ where $\mu$ denotes the ground-truth weights. Viewing the memory HMM as a HMM where the component on $\mathcal { M }$ never changes, we can compare against the vanilla HMMs from the previous setting. For the memory-augmented HMM, we use head tuning with a single-cell attention layer on the entire sequence of softmax probability outputs. For the vanilla HMM in the comparison, we use a linear head on the output at the first position, as an attention head would perform worse since the downstream task depends only on $H _ { 1 }$ and not any other timesteps.
+
+Figure 3 (right) verifies that head tuning recovers the downstream task better when there is more structure in the data, as predicted by Theorem 4.3. Head tuning achieves near $100 \%$ downstream accuracy on all hidden state sizes.
+
+# 6 Conclusion
+
+We analyze how pretraining on generic language modeling tasks can improve performance on diverse downstream tasks. In our analysis framework, the downstream task requires predicting properties of the posterior distribution over latent variables in an underlying generative model. When the generative model is a standard HMM, downstream recovery is possible with a simple classification head under strong non-degeneracy assumptions. We also show that we can relax the non-degeneracy conditions by changing the generative model to a memory-augmented HMM or using prompt tuning. The distributions studied here are meant to provide a first-cut result – we also expect similar theorems to hold for other generative models, which we leave as an interesting direction for future work.
+
+# Acknowledgements
+
+We thank Percy Liang, Tianyi Zhang, and Nelson Liu for helpful discussions.
+
+# Funding statement
+
+CW was supported by a NSF Graduate Research Fellowship. SMX was supported by a NDSEG Fellowship. TM acknowledges support of Google Faculty Award, NSF IIS 2045685, and JD.com. Additional revenue: CW received an honorarium for a talk at G-Research.
+
+# References
+
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+        "type": "text",
+        "text": "Why Do Pretrained Language Models Help in Downstream Tasks? An Analysis of Head and Prompt Tuning ",
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+        "text": "Colin Wei Sang Michael Xie Tengyu Ma ",
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+    },
+    {
+        "type": "text",
+        "text": "Department of Computer Science Stanford University ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "{colinwei,xie,tengyuma}@cs.stanford.edu ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+            535,
+            387
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Pretrained language models have achieved state-of-the-art performance when adapted to a downstream NLP task. However, theoretical analysis of these models is scarce and challenging since the pretraining and downstream tasks can be very different. We propose an analysis framework that links the pretraining and downstream tasks with an underlying latent variable generative model of text — the downstream classifier must recover a function of the posterior distribution over the latent variables. We analyze head tuning (learning a classifier on top of the frozen pretrained model) and prompt tuning in this setting. The generative model in our analysis is either a Hidden Markov Model (HMM) or an HMM augmented with a latent memory component, motivated by long-term dependencies in natural language. We show that 1) under certain non-degeneracy conditions on the HMM, simple classification heads can solve the downstream task, 2) prompt tuning obtains downstream guarantees with weaker non-degeneracy conditions, and 3) our recovery guarantees for the memory-augmented HMM are stronger than for the vanilla HMM because task-relevant information is easier to recover from the long-term memory. Experiments on synthetically generated data from HMMs back our theoretical findings. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+            310,
+            679
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Natural language processing (NLP) has been revolutionized by large-scale pretrained language models such as BERT [4] and GPT [25], which are adapted to a variety of downstream NLP tasks. Although a large body of empirical work seeks to understand the effectiveness of pretrained models [7, 5, 12, 35, 34, 11, 27, 15], theoretical understanding is scarce. Theoretically analyzing the relationship between the pretraining and downstream tasks is challenging because pretraining and downstream settings can greatly differ. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The key starting point for our analysis is to link the pretraining and downstream settings through an underlying generative model of the data. We model the data distribution as a latent variable model and the downstream task as a function of the latent variables. Assuming that pretraining on a large corpus allows us to learn the generative model, the conditional token probabilities predicted by the pretrained model carry information about the hidden variables. In downstream adaptation, we aim to recover this information to solve the downstream task. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Though full finetuning is the de facto empirical standard, analyzing it is challenging because it requires characterizing the weights of the pretrained model. In this paper, we focus on head tuning and prompt tuning, which both freeze all pretrained parameters and allow us to treat the pretrained model as a black box. Head tuning [22] trains task-specific heads on top of the pretrained model outputs. Prompt tuning [31, 19, 9, 21] optimizes a task-specific “prompt” that is concatenated to the model input. Studying prompt tuning is particularly interesting since it can match the performance of full finetuning with less computation time [19, 9, 21]. ",
+        "bbox": [
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+        "type": "text",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our work contrasts with prior theoretical work [28], which assumes that downstream labels are recoverable via a linear head applied to the conditional token probabilities, and analyze how errors in pretraining or model misspecification propagate downstream. We consider specific generative distributions for which we can prove these assumptions, showing that head and prompt tuning can recover the downstream labels. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our analysis considers two data-generating distributions with increasing realism. First, we consider data generated from a Hidden Markov Model (HMM), where the downstream task is to learn a linear classifier on the posterior distribution over the hidden states (Section 3). We prove that, under strong non-degeneracy conditions on token emission probabilities, a linear head applied to a pretrained model $G$ which outputs exact conditional token probabilities $( G _ { i } ( x ) = P [ X _ { i } \\bar { | } \\bar { x } _ { - i } ] )$ can recover the downstream label (Theorem 3.3). Furthermore, we can prove better recovery guarantees with relaxed non-degeneracy assumptions (Assumption 3.1) by using continuous prompt tuning (Theorem 3.6), reflecting the strong empirical performance of prompt tuning [19, 9, 21]. Intuitively, prompt tuning conditions the latent variables so that nonessential information for the downstream task can be ignored during the tuning phase, making task-essential information easier to recover. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Second, we also strengthen our analysis by leveraging additional structure in the data. Motivated by long-range dependences in natural language, we analyze HMM variants with additional latent “memory” variables that can store long-term information more easily than vanilla HMMs (Section 4). Here, the downstream task is to learn a linear classifier on the posterior distribution of the memory variables. We show that, under weaker non-degeneracy conditions than the first setting, an attentionbased classification head can recover ground-truth downstream labels from pretrained model outputs (Theorem 4.3). Intuitively, our recovery guarantees improve because the classification head can focus on the persistent, task-essential information in the memory while ignoring other transient and nonessential aspects of the latent variables. As with the vanilla HMM, we analyze prompt tuning for relaxing the non-degeneracy conditions even further (Theorem 4.6). ",
+        "bbox": [
+            173,
+            387,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In summary, we relate the pretraining and downstream tasks by assuming that the downstream task is to learn a classifier on the posterior distributions of the latent variables defined by an underlying generative model of text. Our theoretical contributions are: 1) in this setting we analyze an HMM generative model show that simple classification heads can recover the true downstream labels under certain non-degeneracy assumptions, 2) we prove that soft prompt tuning can relax the non-degeneracy assumptions needed for downstream recovery making it easier to extract task-specific information, and 3) our recovery guarantees are stronger for memory-augmented HMMs in comparison to the vanilla HMM when tuning an attention-based classfication head. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We empirically evaluate our theoretical results with language models pretrained on synthetically generated data from HMMs. We find that prompt tuning obtains good downstream performance when our non-degeneracy conditions are relaxed, whereas head tuning performs poorly. Furthermore, we show that head tuning obtains better downstream performance when data is generated from a memory-augmented HMM, compared to a vanilla HMM, as is predicted by our theory. ",
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+    {
+        "type": "text",
+        "text": "1.1 Related works ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            738,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The black box nature of BERT and related models has inspired a variety of empirical works which seek to understand them. Probing papers study whether a pretrained model computes various types of structured information (e.g., syntactic [35, 11]) by evaluating the performance of simple classifiers, or probes, on the representations [7, 12, 34, 27, 15]. Other papers ablate various aspects of pretraining, such as changing the masking scheme [14, 20, 40] or permuting the word order [32]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In comparison, theoretical analysis of pretrained language models is limited. Besides [28], which we discussed in Section 1, Zhang and Hashimoto [40] analyze using a linear classifier to approximately recover the latent variable in a Gaussian graphical model with sparse dependencies between observed variables. However, their analysis and setting are focused towards understanding syntactic dependencies between tokens, whereas we directly model and analyze downstream performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Prompt-based tuning [31, 19, 9, 21, 13, 6, 41, 2, 23], which has improved empirical downstream performance for lightweight adaptation methods beyond head tuning to approach full finetuning, is an important focus of our theoretical analysis. Shin et al. [31] employ task-specific prompts that are optimized over the discrete token space. Schick and Schütze [29, 30] reformulate natural language tasks as cloze-style phrases to enable few-shot learning. Subsequent methods [19, 9, 21] optimize “soft” prompts, or continuous embedding vectors. Lester et al. [19] employ soft prompts on pretrained large-scale T5 [26] models and show that as the model size increases, prompt tuning performance can eventually match finetuning. Hambardzumyan et al. [9] applies a variant of soft prompt tuning to MLM models. Li and Liang [21] propose prefix tuning, which prepends a trainable prefix embedding sequence to all layers of the transformer. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "More broadly, Lee et al. [18] analyze reconstruction-based self-supervised learning methods in a general setting and show that under certain conditional independence assumptions, predicting one observed variable from another allows recovery of the latent with a linear head. Other theoretical works analyzing self-supervised or constrastive learning include [1, 10, 36, 38, 37], but they do not directly relate to our particular setting. ",
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+    {
+        "type": "text",
+        "text": "2 Formulations and notations ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We analyze models pretrained on masked language modeling (MLM) objectives. Let $\\mathcal { X }$ denote a finite vocabulary of input tokens, $\\mathcal { X } ^ { \\ast }$ the set of variable-length sequences of tokens, and $X =$ $( X _ { 1 } , \\dots , X _ { T } ) \\in { \\mathcal { X } } ^ { * }$ a random sequence of $T$ tokens. Let $\\Delta ^ { | \\bar { x } | }$ denote the space of probability distributions over tokens. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Pretraining and downstream task. Let $G ( x ) = ( G _ { 1 } ( x ) , G _ { 2 } ( x ) , . . . )$ denote the masked language model which predicts a probability vector for each timestep in the input $x$ . Our theoretical abstraction is that $G _ { i }$ perfectly computes the distribution of $X _ { i }$ , the $i$ -th token, conditioned on all other tokens: $G _ { i } ( x ) = P [ X _ { i } | X _ { - i } = x _ { - i } ]$ . Here $P [ X _ { i } \\mid X _ { - i } = x _ { - i } ] \\in \\Delta ^ { | \\mathcal { X } | }$ is a probability vector. In particular, $G _ { i } ( x )$ does not depend on $x _ { i }$ . The downstream task involves labeled examples $( x , F ^ { \\star } ( x ) ) \\in \\mathcal { X } ^ { * } \\times \\mathcal { Y }$ , where $F ^ { \\star } : \\mathcal { X } ^ { * }  \\mathcal { Y }$ provides ground-truth downstream labels and $\\mathcal { V }$ is a discrete set of labels for classification. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Head and prompt tuning. Head tuning trains a classification head $f$ on top of fixed model outputs, resulting in the classifier $\\mathsf { \\bar { F } } ( x ) = \\mathbb { 1 } ( f ( G ( x ) ) \\geqslant 0 )$ . We expect $f$ to be a simple function such as a linear or one layer attention model. We also analyze variants where $f$ also takes the tokens $x$ or embeddings of $x$ as input, which provides additional information. Soft prompt tuning requires viewing the pretrained model $G$ as a function of the token embeddings; we refer to this model by $\\overline { { G } }$ . Letting $e ( \\boldsymbol { x } ) = e ( x _ { 1 } ) , \\ldots , e ( x _ { t } )$ denote the token embeddings, we have $\\overline { { G } } ( e ( x ) ) = G ( x )$ . Soft prompt tuning concatenates a trainable prompt $u$ so that the model output is $\\overline { { G } } ( ( u , e ( x ) )$ . We consider simultaneously training the prompt parameter $u$ and a classification head to fit the downstream task. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Notations. Let $\\Delta ^ { d }$ denote the space of $d$ -dimensional probability vectors. We work with discrete random variables $V$ taking values in a finite set $\\nu$ . We use $P [ V ] \\in \\Delta ^ { | \\nu | }$ to denote the distribution of $V$ and $P [ U | V = v ] \\in \\mathbb { R } ^ { | \\mathcal { U } | }$ the conditional distribution of $U$ given $V = v$ . $\\operatorname* { P r } ( V = v ) \\in [ 0 , 1 ]$ will denote the probability that $V$ takes values $v$ . We also let $P [ U = u \\vert V ] \\in \\mathbb { R } ^ { \\vert \\nu \\vert }$ denote the vector with entries $\\operatorname* { P r } ( U = u \\vert V = v )$ . $P [ U | V ] \\in \\mathbb { R } ^ { | \\mathcal { U } | \\times | \\mathcal { V } | }$ will describe the matrix with entries $P [ U | V ] _ { u , v } = \\operatorname* { P r } ( U \\dot { = } u | V \\dot { = } v )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For a sequence $v = ( v _ { 1 } , \\ldots , v _ { t } )$ , we use the notation $v _ { i : j }$ for $i \\leqslant j$ to denote $( v _ { i } , \\ldots , v _ { j } )$ , and $v _ { - i }$ to denote $( v _ { 1 : i - 1 } , v _ { i + 1 : t } )$ . We let 1 denote the indicator function. For set $\\nu$ , we let $\\mathcal { V } ^ { * } = \\mathcal { \\bar { V } } ^ { 1 } \\cup \\mathcal { V } ^ { 2 } \\cup \\cdots$ denote variable-length sequences of elements of $\\nu$ . Let $\\odot$ denote elementwise product. Let $\\mathbf { 1 } _ { d } , \\mathbf { 0 } _ { d }$ denote the $d$ -dimensional all-1’s and all-0’s vector. We omit the subscript if the dimension is clear from context. For two vectors $a , b \\in \\mathbb { R } ^ { d }$ , we let $a / b$ denote their element-wise division. We use $\\operatorname { s u p p } ( a )$ to denote the set of indices where vector $a$ is non-zero. ",
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+    {
+        "type": "text",
+        "text": "3 Analysis for Hidden Markov Models ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Defining a relation between pretraining and downstream tasks is the foremost challenge for analysis. We propose to link the two via latent variable generative assumptions on the input distribution. We model the downstream task as a function of the posterior distribution of the latent variables. Towards a first result, this section studies the case where inputs are generated by HMMs (see Figure 1 (left)), which have been well-studied in the context of language and speech processing (see e.g. [24, 17, 3]). ",
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+    {
+        "type": "image",
+        "img_path": "images/565f61d89a958502b132a1b5d80068eabd4d909632a3aadb21ffc7dd9929aad0.jpg",
+        "image_caption": [
+            "Figure 1: Left: Illustration of HMM graphical model. Right: Overview of the formulation and analysis setting for prompt (and head) tuning. To abstractify soft prompt tuning, we note that every token has a natural embedding, the corresponding row of the emission probability matrix. We view prompt tuning as adding a fake token $\\widetilde { z }$ to the vocabulary, assigning it a row $u$ in the emission matrix, and prepending it to the input embedding sequence. More details are provided in Section 3.1. "
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Data distribution. Let $\\mathcal { H }$ denote the hidden state space of the HMM. We use $\\begin{array} { r l } { H } & { { } = } \\end{array}$ $( H _ { 0 } , H _ { 1 } , \\ldots , H _ { T } ) \\in \\mathcal { H } ^ { * }$ to denote the sequence of hidden states. For all timesteps $i > 0$ , the transition probabilities are time-invariant, i.e. $P [ H _ { i } | H _ { i - 1 } ] = A$ for $A \\in \\mathbb { R } ^ { | \\mathcal { H } | \\times | \\mathcal { H } | }$ . For each timestep $i \\geqslant 1$ , tokens $X _ { i }$ are emitted following some time-invariant probability: $P [ X _ { i } \\mid H _ { i } ] = W$ for $W \\in \\mathbb { R } ^ { | \\mathcal { X } | \\times | \\mathcal { H } | }$ . The joint probability of $X , H$ is ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/5b407ed6b7b8653b37a95e9df92a4fd83b6fb665d7ac046108064d7f8a52492b.jpg",
+        "text": "$$\n\\operatorname* { P r } ( X , H = x , h \\mid T = t ) = \\operatorname* { P r } ( H _ { 0 } = h _ { 0 } ) \\prod _ { i = 1 } ^ { t } \\operatorname* { P r } ( H _ { i } = h _ { i } \\mid H _ { i - 1 } = h _ { i - 1 } ) \\operatorname* { P r } ( X _ { i } = x _ { i } \\mid H _ { i } = h _ { i } ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Downstream tasks. We assume that $H _ { 0 }$ has the meaningful information for the downstream task, which is a binary classification task where the ground-truth labeling $F ^ { \\star }$ is assumed to be a linear classifier on the posterior $P [ H _ { 0 } | X _ { 1 : T } = x ]$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b7fdeea1e041267a66f5bf88466ecd6e1ad341f540e12c94fc3c604dd2141ff.jpg",
+        "text": "$$\nF ^ { \\star } ( x ) = \\mathbb { 1 } ( \\mu ^ { \\top } P [ H _ { 0 } | X _ { 1 : T } = x ] \\geqslant 0 )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for $\\mu \\in \\mathbb { R } ^ { | \\mathcal { H } | }$ . Our results are easily extended to the multiclass setting. We consider tuning a linear head for the downstream classifier, which formally computes $\\mathbb { 1 } ( b ^ { \\top } G _ { 1 } ( x ) \\geqslant 0 )$ for $b \\in \\mathbb { R } ^ { | \\mathcal { X } | }$ . The following non-degeneracy condition is crucial for our recovery result in this setting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Assumption 3.1 (Non-degeneracy, vanilla HMM). The token emission probability matrix $W$ has linearly independent columns. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We also require the following regularity conditions on $H _ { 0 }$ and the state transitions. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3.2 (Regularity). The Markov chain $H _ { 0 } , H _ { 1 } , \\ldots$ is ergodic, and $P [ H _ { 0 } ]$ has full support ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We show that if $W$ has linearly independent columns, a linear head fits downstream labels. ",
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+        "type": "text",
+        "text": "Theorem 3.3. Assume that non-degeneracy (Assumption 3.1) and regularity (Assumption 3.2) hold. Then any downstream task $F ^ { \\star } ( x )$ of the form (3.1) can be computed by a linear head on $G$ applied to $a$ shifted sequence. That is, there exists linear head weights $b \\in \\mathbb { R } ^ { | \\mathcal { X } | }$ such that for all $x \\in \\operatorname { s u p p } ( P [ X ] )$ , ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/d75d0e86f39fbe2da2dd7c9d84e198ba0f998de42d30b9b197a03cfa9c8245eb.jpg",
+        "text": "$$\nF ^ { \\star } ( x ) = \\mathbb { 1 } ( b ^ { \\top } G _ { 1 } ( x ^ { \\prime } ) \\geqslant 0 )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { x } ^ { \\prime } = \\left( \\boldsymbol { \\mathcal { O } } , x _ { 1 : t } \\right)$ is the concatenation of a special token $\\mathcal { D }$ with $x$ .1",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "The key for the proof is to leverage the following general statement about random variables $U , V , Z$ such that $U \\perp V | Z$ , which decomposes the expression for $P [ U | V ]$ . ",
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+    {
+        "type": "text",
+        "text": "Proposition 3.4. Let $U , V , Z$ be random variables such that $U \\perp V | Z$ . Then for any $v$ , $P [ U | V =$ $v ] = P [ U | Z ] \\cdot P [ Z | V = v ]$ . Thus, i $f P [ U | Z ]$ has a left inverse $( P [ U | Z ] ) ^ { \\dagger }$ , then $P [ Z | V =$ $v ] = ( P [ U \\mid Z ] ) ^ { \\dagger } P [ U \\mid V = v ]$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By the conditional independence structure of the HMM, Proposition 3.4 immediately implies ",
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+        "page_idx": 4
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+        "type": "equation",
+        "img_path": "images/4085826973020cb388046c2e1b46b629d8c6654bd17c69b8885e04fc2c5edbea.jpg",
+        "text": "$$\nG _ { 1 } ( x ^ { \\prime } ) = W P [ H _ { 1 } | X _ { 2 : T + 1 } = x ] \\implies P [ H _ { 1 } | X _ { 2 : T + 1 } = x ] = W ^ { \\dagger } G _ { 1 } ( x ^ { \\prime } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $W ^ { \\dagger }$ is the left inverse for $W$ , guaranteed to exist by Assumption 3.1. This lets us recover $P [ H _ { 1 } | X _ { 2 : T + 1 } = x ]$ by applying a linear function to $G _ { 1 } ( \\bar { x } ^ { \\prime } )$ . Additional linear functions will be sufficient to obtain $\\mathrm { \\bar { \\mu } } ^ { \\top } P [ H _ { 0 } | X _ { 1 : T } = x ]$ from $P [ H _ { 1 } | X _ { 2 : T + 1 } = x ]$ . We provide the full proof in Section B. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.4 is reminiscent of the arguments of [18], which leverages the independence structure in the same way. Subsequent sections will require more complicated analyses and recovery procedures. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A drawback of Theorem 3.3 is that it relies heavily on assuming $W$ has full column rank, which implies the necessary condition that $| { \\mathcal { H } } | \\leqslant | { \\mathcal { X } } |$ . Without this assumption, it is unclear how to recover $P [ \\tilde { H } _ { 0 } | X _ { 1 : T } = x ]$ from $G ( x )$ alone. However, in realistic settings we would expect $| { \\mathcal { H } } | > | { \\mathcal { X } } |$ , as increasing the size of the hidden state space improves language modeling capabilities of HMMs [3]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1 Relaxed non-degeneracy assumptions via prompt tuning ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we study applying soft, or continuous, prompt tuning [19, 9] to the setting above. We show that by using soft prompt tuning, we can recover $F ^ { \\star }$ using a linear head on $G$ for HMMs where the non-degeneracy assumptions on $W$ are relaxed. Our analysis provides insight into the empirical successes of prompt-tuning: intuitively, prompt tuning enables better recovery of the downstream task by conditioning the output of $G$ to only contain task-specific information. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Soft prompt tuning trains task-specific embedding vectors, but analyzing how the model processes embedding vectors is challenging because it requires opening up the black box of the pretrained model. Thus, we require additional abstractions about how the pretrained model processes the embedding vectors. We will extend the mask language model $G$ to a model $\\overline { { G } }$ that maps a sequence of embeddings $e _ { 1 } , \\ldots , e _ { t }$ to conditional probabilities $G _ { 1 } ( x ) , \\dots , G _ { t } ( x )$ as follows. We observe that each token $z$ in the vocabulary $\\mathcal { X }$ naturally corresponds to a $| \\mathcal { H } |$ -dimensional vector: the $z$ -th row of the emission probability matrix $W$ , or equivalently, $P [ X _ { i } = z \\mid H _ { i } ]$ . We denote this embedding by $e ( z )$ and call the family of embeddings $\\bar { \\{ e ( z ) : z \\in \\mathcal { X } \\} }$ proper embeddings. A fundamental property of HMMs is that the conditional probability $P [ X _ { i } \\mid X _ { - i } = x _ { - i } ]$ only depends on $x _ { 1 } , \\ldots , x _ { t }$ through their embeddings $e ( x ) = ( e ( x _ { 1 } ) , \\ldots , e ( x _ { t } ) )$ . In other words, there exists a function $\\overline { { G } } _ { i }$ such that ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/40c9144f49d6c50acc9898c856a3d28a5a993c1812564f9e77493d1cceef70dc.jpg",
+        "text": "$$\nG _ { i } ( x _ { 1 } , \\dots , x _ { t } ) = { \\overline { { G } } } _ { i } ( e ( x _ { 1 } ) , \\dots , e ( x _ { t } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In particular, we let $\\overline { { G } } _ { i }$ compute the standard message passing algorithm [16] that computes the conditional probability of HMMs. This ensures that $\\overline { { G } } _ { i }$ is well defined on all sequences of nonnegative vectors in $[ 0 , 1 ] ^ { | \\mathcal { H } | }$ , beyond sequences of proper embeddings.We assume that pretraining produces this $\\overline { { G } } _ { i }$ , which we treat as a blackbox for prompt tuning. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In particular, for prompt tuning we can consider the case where we pass an arbitrary nonnegative vector $u \\in [ 0 , 1 ] ^ { | \\mathcal { H } | }$ to $\\overline { G }$ in the first argument and proper embeddings at positions $i > 1$ . We can interpret $u$ as the embedding of a fake token $\\widetilde { z }$ . Concretely, consider adding a new token $\\widetilde { z }$ to the vocabulary $\\mathcal { X }$ , and changing the emission probability at position 1 to satisfy $P [ X _ { 1 } = \\tilde { z } | H _ { 1 } ] = u$ and for all $z \\neq \\widetilde { z }$ , $P [ X _ { 1 } = z | H _ { 1 } ] { \\propto } ( 1 - u ) \\odot e ( z )$ . Then $\\overline { { G } } _ { i } ( u , e ( x _ { 1 } ) , \\ldots , e ( x _ { t } ) )$ precisely computes the conditional probability $P [ X _ { i } | X _ { - i } = ( { \\tilde { z } } , x _ { 1 } , \\dots , x _ { t } ) _ { - i } ]$ under the modified HMM. We refer the readers to Section C for the formal definition of $\\overline { { G } } _ { i }$ and formal proofs of the interpretation above. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We consider a downstream training algorithm which trains the prompt tuning parameter $u$ described above and a linear classification head. Letting $u$ denote the trainable prompt parameter and $b \\in \\mathbb { R } ^ { | \\mathcal { X } | }$ the trainable linear head weights, the model uses the embedding sequence ",
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+        "img_path": "images/b170e7f0f88ac0c93dc7b05b631a9a34a40da93d9dd01dee12ab53eab2f74f07.jpg",
+        "text": "$$\n{ \\widehat { e } } ( x ) \\triangleq ( u , e ( \\emptyset ) , e ( x _ { 1 } ) , \\dots , e ( x _ { t } ) )\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "and outputs the prediction $F ( x ) = \\mathbb { 1 } ( b ^ { \\top } G _ { 2 } ( { \\widehat { e } } ( x ) ) \\geqslant 0 )$ . We can provide recovery guarantees for this model if the ground-truth classifier weights $\\mu$ (defined in (3.1)) and columns of the HMM transition matrix $A$ satisfy the following relaxation of the requirement in Theorem 3.3 that $W$ is nondegenerate. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3.5 (Relaxed non-degeneracy condition). There exists a set of essential hidden states $\\mathcal { H } ^ { \\star } \\subseteq \\mathcal { H }$ , so that the columns of $W$ corresponding to $\\mathcal { H } ^ { \\star }$ , $\\{ W _ { : , h } \\} _ { h \\in \\mathcal { H } ^ { \\star } }$ , are linearly independent. Furthermore, $\\mathcal { H } ^ { \\star }$ covers all meaningful information for the downstream tasks: $\\operatorname { s u p p } ( \\mu ) \\subseteq { \\mathcal { H } } ^ { \\star }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In addition, a last technical requirement on $\\mathcal { H } ^ { \\star }$ is as follows: there exists a set $B \\subseteq { \\mathcal { H } }$ such that ${ \\mathcal { H } } ^ { \\star } = \\cup _ { h \\in B } \\operatorname { s u p p } ( A _ { : , h } )$ . In other words, $\\mathcal { H } ^ { \\star }$ must be the set of all states reachable by starting from some state in $\\boldsymbol { B }$ and transitioning one step in the hidden Markov chain. ",
+        "bbox": [
+            176,
+            138,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Compared to Assumption 3.1, which required that all columns of $W$ are linearly independent, Assumption 3.5 only requires linear independence on a subset $\\mathcal { H } ^ { \\star }$ of essential states. In the setting where $| { \\mathcal { H } } | > | { \\mathcal { X } } |$ , the condition for Theorem 3.3 can never hold. On the other hand, Assumption 3.5 could still hold, for example, if $| \\operatorname { s u p p } ( \\mu ) | < | \\chi |$ and the set of columns of $W$ corresponding to hidden states in $\\operatorname { s u p p } ( \\mu )$ is linearly independent. The last technical requirement in Assumption 3.5 is also required, which could be satisfied if columns of $A$ are sparse. The following theorem shows that when Assumption 3.5 holds, we can recover $F ^ { \\star }$ using soft prompt tuning with a linear head. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.6. In the above setting, assume that Assumptions 3.2 and 3.5 hold. Then $F ^ { \\star }$ can be computed using soft prompt tuning with a linear head on $\\overline { { G } }$ . Concretely, there is a continuous prompt parameter $u \\in \\mathbb { R } ^ { | \\mathcal { H } | }$ and weight vector $b \\in \\mathbb { R } ^ { | \\mathcal { X } | }$ , such that for all $x \\in \\operatorname { s u p p } ( P [ X ] )$ , ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/245d30d0909ee9b6cbbeb60308204c22465b7d79253277f00d8658da71fb4faa.jpg",
+        "text": "$$\nF ^ { \\star } ( x ) = \\mathbb { 1 } ( b ^ { \\top } \\overline { { G } } _ { 2 } ( \\hat { e } ( x ) ) \\geqslant 0 )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            345,
+            598,
+            366
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where ep prepends u to the input embedding sequence, as defined in (3.2). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.6 provides a stronger recovery result than Theorem 3.3, which only used a linear head. This is also reflected in our synthetic experiments (Section 5), and prior work which shows that variants of prompt tuning can perform much better than only training the last few layers of the model [21]. Our theory suggests that prompt tuning could help by conditioning the hidden variables to remove nonessential information for the task from the output of $G$ . This makes task-essential information easier to recover. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The key proof intuition is that although recovering $P [ H _ { 0 } | X _ { 1 : T } = x ]$ is impossible without strong non-degeneracy conditions (Assumption 3.1), we can aim to recover $\\bar { P [ \\cal H _ { 0 } | \\bar { X } _ { 1 : T } = x ] }$ on the subset of essential states $\\mathcal { H } ^ { \\star }$ defined in Assumption 3.5, which suffices for computing $\\mu ^ { \\top } P [ H _ { 0 } | X _ { 1 : T } = x ]$ , since $\\mathcal { H } ^ { \\star } \\supseteq \\mathsf { s u p p } ( \\mu )$ . To recover $\\bar { P [ H _ { 0 } | X _ { 1 : T } \\ = \\ x ] }$ on $\\mathcal { H } ^ { \\star }$ , we observe in Lemma C.2 that prepending the prompt $u$ is equivalent to introducing a modified random sequence $\\hat { X }$ and fake token $\\widetilde { z }$ which influences the posterior of $H _ { 2 }$ as follows: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cce86b24d7c731e1deb7e23cdd2ea63d33007f2f95782180cc0461735a018af5.jpg",
+        "text": "$$\n\\overline { { G } } _ { 2 } ( \\widehat { e } ( x ) ) = r _ { x } W D ( P [ H _ { 2 } | \\widehat { X } _ { 1 } = \\widetilde { z } ] \\odot P [ H _ { 0 } | X _ { 1 : T } = x ] )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            583,
+            696,
+            604
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for invertible diagonal matrix $D$ and positive scalar $r _ { x }$ . We choose $u$ so $P [ H _ { 2 } | \\hat { X } _ { 1 } = \\tilde { z } ] \\odot$ $P [ H _ { 0 } | X _ { 1 : T } = \\bar { x ] }$ is supported only on $\\mathcal { H } ^ { \\star }$ . As corresponding columns of $W$ are linearly independent (Assumption 3.5), we recover $\\mathrm { P r } ( H _ { 0 } = h | X _ { 1 : T } = x )$ for $h \\in \\mathcal { H } ^ { \\star }$ via a linear function of $\\overline { { G } } _ { 2 } ( \\widehat { e } ( x ) )$ . This suffices for computing $\\mu ^ { \\top } P [ H _ { 0 } | X _ { 1 : T } = x ]$ . For more details, see Section $\\textrm { C }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 Analysis for memory-augmented Hidden Markov Models ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We study a memory-augmented HMM which explicitly disentangles the evolution of hidden states from a persistent “memory” variable. Inspired by natural sentences, this model is intended to better capture the distinction between syntax, which constantly evolves, and semantics, which changes less. This additional structure in the generative model allows us to strengthen our results by relaxing the non-degeneracy conditions on $W$ , the token emission probabilities. Thus, both head and prompt tuning are more powerful in this setting compared to Section 3 and can recover the downstream label with weaker non-degeneracy assumptions on $W$ . In Section 4.2, we show that soft prompt tuning also provides an advantage over head tuning alone. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Data distribution. The memory-augmented HMM, depicted in Figure 2, can be viewed as a generative variant of memory networks [39, 33] and is closely related to Hidden Topic Markov Models [8]. There are two sets of latent variables in the memory-augmented HMM: a Markov chain on hidden states $H _ { 0 } , H _ { 1 } , . . . ,$ meant to model the evolution of syntax, and a persistent “memory” $M = ( M _ { 1 } , \\dots , M _ { N } )$ with $N$ total cells, where each $M _ { i }$ takes values in a finite set $\\mathcal { M }$ . The full joint ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/c5f0766cf68bc506c99186f4cb7010d27a8e6453ddbd36505507a0cd3ad19b4e.jpg",
+        "image_caption": [
+            "Figure 2: Left: Memory-augmented HMM with a single memory cell. The memory $M$ and hidden state $H _ { i }$ determine the emission probabilities for each state $X _ { i }$ . Right: Memory-augmented HMM with multiple memories $M _ { 1 } , \\dots , M _ { N }$ . The hidden state $H _ { i }$ consists of a cell index $J _ { i }$ and syntax state $S _ { i }$ . To sample $X _ { i }$ , we first look up the $J _ { i }$ -th memory cell $M _ { J _ { i } }$ . The token emission probability is then determined by the tuple $( M _ { J _ { i } } , J _ { i } , S _ { i } )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "probability is as follows: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9504ad8a5f1a666b666f51d023b4e9b9fbf173cf3ea8e57fffaed794383f3731.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\sf P r } ( X , H , M = x , h , m | T = t ) = } \\ ~ } \\\\ { { \\displaystyle ~ { \\sf P r } ( M = m ) { \\bf P r } ( H _ { 0 } = h _ { 0 } ) \\prod _ { i = 1 } ^ { t } { \\bf P r } ( H _ { i } = h _ { i } | H _ { i - 1 } = h _ { i - 1 } ) { \\bf P r } ( X _ { i } = x _ { i } | M = m , H _ { i } = h _ { i } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            364,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The hidden state is modified to explicitly consist of a disentangled cell index $J \\in [ N ]$ and syntax state $S \\in S$ , such that $H _ { i } = ( J _ { i } , S _ { i } )$ and $\\mathbf { \\dot { \\mathcal { H } } } = [ N ] \\times \\mathbf { \\mathcal { S } }$ . To sample the token at timestep $i$ given the hidden state $H _ { i } = ( J _ { i } , S _ { i } )$ , we first use $J _ { i }$ to index the memory $M$ , obtaining the random variable $M _ { J _ { i } }$ . $X _ { i }$ is then sampled according to some time-invariant probability depending on $M _ { J _ { i } } , J _ { i } , S _ { i }$ : ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4bdaca6579571674643c282035bb3997d521a6142d90d83ab0c92a28fab1eab9.jpg",
+        "text": "$$\nP [ X _ { i } \\mid M = m , H _ { i } = ( j , s ) ] = P [ X _ { i } \\mid M _ { J _ { i } } = m _ { j } , H _ { i } = ( j , s ) ] = W _ { : , ( m _ { j } , j , s ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            486,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here $W \\in \\mathbb { R } ^ { | \\mathcal { X } | \\times | \\mathcal { M } | | \\mathcal { H } | }$ stores the emission probabilities for each choice of memory cell value and hidden state. Note that in particular, the conditional probabilities for $X _ { i }$ only depend on a single memory cell for each timestep. We also note that memory-augmented HMMs can be viewed as vanilla HMMs with structured transitions because $( H _ { 0 } , M ) , \\bar { ( } H _ { 1 } , \\bar { M } ) , . . .$ can be viewed as a Markov chain where the memory component does not change. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Example 4.1 (Generating natural sentence with memory-augmented HMM). We consider how this model may generate the sentence “The cow in the pasture rolled on the grass’ happily.” $M _ { 1 }$ could store the subject (“cow”), $M _ { 2 }$ the location (“pasture”), $M _ { 3 }$ the sentiment (“happily”), and $S _ { i }$ could determine part-of-speech. For timesteps where “cow” and “rolled” are emitted $J _ { i } = 1$ because we emit information related to the sentence subject. Timesteps for “pasture” and “grass” have $J _ { i } = 2$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Downstream tasks. We consider downstream tasks where ground-truth labels are obtained via a linear classifier on the posterior distribution of a particular memory cell $j ^ { \\star } \\in [ N ]$ : $F ^ { \\star } ( x ) =$ $\\begin{array} { r } { \\mathbb { 1 } ( \\mu ^ { \\top } P [ M _ { j ^ { \\star } } | X _ { 1 : T } = x ] \\geqslant 0 ) } \\end{array}$ q, where $\\mu \\in \\mathbb { R } ^ { | \\mathcal { M } | }$ . Intuitively, this formulation models downstream tasks which depend on a particular aspect of the semantics but not on syntax (e.g. in the setting of Example 4.1, if $j ^ { \\star } = 3$ , the task is sentiment analysis). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Tuning attention head for recovering ground-truth downstream labels ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "To recover the downstream labeling, we require an attention-based classification head, which is a function of both the input embeddings and outputs of $G$ . Formally, let $q \\in \\mathbb { R } ^ { | \\mathcal { H } | + 1 }$ denote a query parameter and $\\beta _ { 1 } , \\dots , \\beta _ { t } \\in \\mathbb { R } ^ { | \\mathcal { H } | + 1 }$ denote trainable position embeddings. Given pretrained model outputs $G _ { i } ( x )$ and trainable token embeddings $e ( x _ { i } )$ , the attention head $\\mathrm { A t t n } ( \\cdot )$ applies key and value functions $K , V$ to compute the output as follows: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/01ca557807bc811f590ee8c6b03fdf3abf4fa8f0c73c54e401ff606eb8f30f50.jpg",
+        "text": "$$\n\\begin{array} { c } { \\displaystyle \\mathcal { Z } \\triangleq \\arg \\underset { i } { \\operatorname* { m a x } } \\{ q ^ { \\top } ( K ( G _ { i } ( x ) ) + \\beta _ { i } ) \\} } \\\\ { \\displaystyle \\mathrm { A t t n } ( ( G _ { i } ( x ) , e ( x _ { i } ) ) _ { i = 1 } ^ { t } ) \\triangleq \\frac { 1 } { | \\mathcal { Z } | } \\sum _ { i \\in \\mathcal { Z } } V ( G _ { i } ( x ) , e ( x _ { i } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where arg max refers to the set of indices achieving the maximum in (4.1). We note that standard attention heads in practice rely on the softmax function, but the expression based on arg max above captures the limiting behavior as $\\| q \\| _ { 2 } \\to \\infty$ . We consider linear key functions given by ${ \\bar { K } } ( G _ { i } ( x ) ) =$ $\\Theta ^ { ( K ) } G _ { i } ( x )$ . The value function $V : \\mathbb { R } ^ { | \\mathcal { X } | } \\times \\mathbb { R } ^ { | \\mathcal { M } | | \\mathcal { H } | } \\to \\mathbb { R }$ uses parameters $\\Theta ^ { ( V ) } \\in \\mathbb { R } ^ { | \\mathcal { M } | | \\mathcal { H } | \\times | \\mathcal { X } | }$ and $b \\in \\mathbb { R } ^ { | \\mathcal { M } | | \\mathcal { H } | }$ and computes $V ( G _ { i } ( x ) , e ( x _ { i } ) ) = b ^ { \\top } ( ( \\Theta ^ { ( V ) } G _ { i } ( x ) ) \\odot e ( x _ { i } ) )$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Because our generative model disentangles $H$ and $M$ , we can relax the non-degeneracy assumption on the token emission probabilities $W$ , compared to Theorem 3.3. The relaxed assumption only requires the columns $\\{ \\bar { W } _ { : , ( m , h ) } \\} _ { m \\in \\mathcal { M } , h \\in \\mathcal { H } ^ { \\star } }$ to be linearly independent in a subset $\\mathcal { H } ^ { \\star }$ of “recoverable” hidden states, whereas Assumption 3.1 required all columns to be linearly independent. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.2 (Existence of “recoverable” hidden states). There exists a set of recoverable hidden states ${ \\bar { \\mathcal { H } } } ^ { \\star } = \\{ j ^ { \\star } \\} \\times S ^ { \\star }$ , such that the collection of token emission probabilities from $\\mathcal { M } \\times \\mathcal { H } ^ { \\star }$ $\\{ W _ { : , ( m , h ) } \\} _ { m \\in { \\mathcal { M } } , h \\in { \\mathcal { H } } ^ { \\star } }$ , is a linearly independent set of vectors. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, the span of these vectors must be disjoint from the span of token emission probabilities from $\\mathcal { M } \\times ( \\mathcal { H } \\backslash \\mathcal { H } ^ { \\star } )$ $\\begin{array} { r } { \\colon \\mathrm { s p a n } \\big ( \\{ W _ { : , ( m , h ) } \\} _ { m \\in \\mathcal { M } , h \\in \\mathcal { H } ^ { \\star } } \\big ) \\cap \\mathrm { s p a n } \\big ( \\{ W _ { : , ( m , h ^ { \\prime } ) } \\} _ { m \\in \\mathcal { M } , h \\in \\mathcal { H } \\backslash \\mathcal { H } ^ { \\star } } \\big ) = \\{ \\mathbf { 0 } _ { | \\mathcal { X } | } \\} \\mathrm { . ~ } } \\end{array}$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Note that the non-degeneracy condition of Theorem 3.3 would require $\\{ W _ { : , ( m , h ) } \\} _ { m \\in { \\mathcal { M } } , h \\in { \\mathcal { H } } }$ to be linearly independent, whereas Assumption 4.2 only requires linear independence for $h \\in \\mathcal { H } ^ { \\star }$ . The second condition states that $\\mathcal { H } ^ { \\star }$ and $\\mathcal { H } \\backslash \\mathcal { H } ^ { \\star }$ are distinguishable by the token emission probabilities. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We explain Assumption 4.2 in the setting of Example 4.1. For natural language, there might be choices of $h = ( j _ { i } , s _ { i } )$ for which the set $\\{ W _ { : , ( m , h ) } \\} _ { m \\in \\mathcal { M } }$ of token emission probabilities is fundamentally not very diverse, and therefore not linearly independent. For example, if the syntax $s _ { i }$ indicates “article”, i.e. words such as “a”, “an”, and “the”, the token emission probabilities would carry little information about $M _ { j _ { i } }$ because the choice of article does not depend much on semantics, so columns corresponding to $s _ { i } =$ “article” would not be linearly independent, violating Assumption 3.1. However, Assumption 4.2 allows us to avoid this issue by placing such $h$ in $\\mathcal { H } \\backslash \\mathcal { H } ^ { \\star }$ , a set of hidden states which we can ignore, and only including hidden states which carry a lot of information about $M$ in $\\mathcal { H } ^ { \\star }$ . In Example 4.1, when $J _ { i } = 2$ (location), $S _ { i } = \\mathrm { \\ \" n o u n } ^ { \\mathrm { * } }$ , the position $i$ should convey a lot about the location (in this case, “pasture”), so it is more reasonable to assume that $\\{ W _ { : , m , h } \\} _ { m \\in \\mathcal { M } }$ is linearly independent for this hidden state. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Thus, our aim is to focus on recovering information for the downstream task from positions $i$ where $H _ { i } \\in { \\mathcal { H } } ^ { \\star }$ . Formally, we define the following set of input sequences containing positions $i$ where the posterior of $H _ { i }$ given $x _ { - i }$ concentrates on $\\mathcal { H } ^ { \\star }$ : ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/add73bfaef2e3dd42c379a55fce85691cb5dc7daa33bd75695659b931e6058ee.jpg",
+        "text": "$$\n{ \\mathcal { R } } \\triangleq \\{ ( x _ { 1 } , \\dots , x _ { t } ) \\in \\operatorname { s u p p } ( P [ X ] ) : \\exists i { \\mathrm { ~ w i t h ~ } } \\operatorname { s u p p } ( P [ H _ { i } \\mid X _ { - i } = x _ { - i } ] ) \\subseteq { \\mathcal { H } } ^ { \\star } \\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            573,
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+            590
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The following theorem shows that under Assumption 4.2, we can recover $F ^ { \\star }$ using the attention head described above, if $x \\in \\mathcal { R }$ is nonempty. Note that $\\mathcal { R }$ is nonempty if the posterior of $H _ { i }$ concentrates on $\\mathcal { H } ^ { \\star }$ for some $i$ . For natural language, it is realistic to assume this can occur because syntactic aspects of a sentence are typically low-entropy when the full sentence is observed. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.3. Assume that non-degeneracy (Assumption 4.2) and regularity (Assumption 3.2) hold. Define $\\mathcal { R }$ as in (4.3). Then there exist an attention head on $G ( x )$ and token embeddings $e ( x _ { i } )$ such that the following holds for any $x \\in \\mathcal { R }$ : ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/739c8926b469b0e01458db2c246136d0bae1d631d5427bbf0cd5415edb4da20e.jpg",
+        "text": "$$\nF ^ { \\star } ( x ) = \\mathbb { 1 } \\bigl ( \\mathrm { A t t n } ( ( G _ { i } ( x ) , e ( x _ { i } ) ) _ { i = 1 } ^ { t } ) \\geqslant 0 \\bigr )\n$$",
+        "text_format": "latex",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where the function Attn is in the form described in (4.2). ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "The idea is to use the attention mechanism to attend to positions $i$ where ${ \\mathrm { s u p p } } ( P [ H _ { i } | X _ { - i } = x _ { - i } ] ) \\subseteq$ $\\mathcal { H } ^ { \\star }$ . The intuition of Assumption 4.2 is that such positions are more informative for recovering the latent posteriors; indeed, from the outputs $G _ { i } ( x )$ at such $i$ , the value function in the attention will be able to recover $P [ M _ { j ^ { \\star } } | X _ { 1 : T } = x ]$ . A full proof is provided in Section D.1. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 Guarantees for prompt-tuning ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+        "type": "text",
+        "text": "Though the generative modeling assumptions in this section already allowed relaxed non-degeneracy assumptions, applying soft prompt tuning allows us to relax them even further. For simplicity, we consider the setting where there is a single memory cell, so $M \\in \\mathcal { M }$ , and the downstream task is a linear classifier on the posterior of the memory: $F ^ { \\star } ( x ) = \\mathbb { 1 } ( \\mu ^ { \\top } P [ M | X _ { 1 : T } = x ] \\geqslant 0 )$ q. This simplified setting doesn’t require the explicit disentanglement between $J _ { i }$ and $S _ { i }$ in $H _ { i }$ . We analyze continuous prompt-tuning in a setting where the pretrained model $\\overline { { G } }$ follows the same abstraction as in Section 3.1. We modify the model to take $| \\mathcal { M } | | \\mathcal { H } |$ -dimensional vectors, so the proper embedding for token $z$ is given by $e ( z ) = P [ X _ { i } = z | M , H _ { i } ] = W _ { z , : } ^ { \\top }$ . In Section D.3, we describe the formal construction and interpretation of $\\overline { { G } }$ in the more general setting with more memories. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Letting $u \\in \\mathbb { R } ^ { | \\mathcal { M } | | \\mathcal { H } | }$ denote the trainable prompt parameter, we define the input embeddings ",
+        "bbox": [
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+            174,
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dbc809c1153f622ca6e0781ce806e2591e8028ba935f8d33fc147650d7fc9320.jpg",
+        "text": "$$\n{ \\widehat { e } } ( x ) \\triangleq ( u , e ( x _ { 1 } ) , \\dots , e ( x _ { t } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The downstream model applies an attention head to the output of $\\overline { G }$ : $\\begin{array} { r l r l } { F ( x ) } & { { } = } & { } \\end{array}$ $\\mathbb { 1 } \\left( \\mathrm { A t t n } ( ( \\overline { G } _ { i } ( \\widehat { e } ( x ) ) , \\widehat { e } _ { i } ( x ) ) _ { i = 1 } ^ { t + 1 } ) \\ \\geqslant \\ 0 \\right)$ q, where Attn is defined in (4.2). An additional stationarity assumption on $P [ H _ { 0 } ]$ will simplify the recovery procedure (though it can be removed). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.4 (Stationarity). Assumption 3.2 holds on the Markov chain $H _ { 0 } , H _ { 1 } , \\dots$ . Furthermore, $P [ H _ { 0 } ]$ is the stationary distribution: $P [ H _ { 0 } ] = A P [ H _ { 0 } ]$ , where $A$ is the transition matrix. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As before, we assume sparsity of $\\mu$ and some non-degeneracy of $W$ , though the assumption is more relaxed and easier to state compared to the vanilla HMM setting. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Assumption 4.5 (Relaxed version of Assumption 4.2). Let $\\mathcal { M } ^ { \\star } \\triangleq \\operatorname { s u p p } ( \\mu )$ denote the set of non-zero coordinates in $\\mu$ . There exists a set of recoverable hidden states $\\mathcal { H } ^ { \\star }$ , such that the collection of token emission probabilities from $\\mathcal { M } ^ { \\star } \\times \\mathcal { H } ^ { \\star }$ , $\\{ W _ { : , ( m , h ) } \\} _ { m \\in { \\mathcal { M } } ^ { \\star } , h \\in { \\mathcal { H } } ^ { \\star } }$ , is linearly independent. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, the span of these vectors must be disjoint from the span of token emission probabilities from $\\mathcal { M } ^ { \\star } \\times ( \\mathcal { H } \\backslash \\mathcal { H } ^ { \\star } )$ $) \\colon \\operatorname { s p a n } ( \\{ W _ { : , ( m , h ) } \\} _ { m \\in \\mathcal { M } ^ { \\star } , h \\in \\mathcal { H } ^ { \\star } } ) \\cap \\operatorname { s p a n } ( \\{ W _ { : , ( m , h ^ { \\prime } ) } \\} _ { m \\in \\mathcal { M } ^ { \\star } , h \\in \\mathcal { H } \\backslash \\mathcal { H } ^ { \\star } } ) = \\{ \\mathbf { 0 } _ { | \\mathcal { X } | } \\}$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We note that Assumption 4.5, and Assumption D.5 for multiple memories, are relaxations of Assumption 4.2, as they only consider memory values in $\\operatorname { s u p p } ( \\mu )$ , whereas Assumption 4.2 considers all $m \\in \\mathcal { M }$ . An additional advantage of the memory-augmented HMM is that Assumption 4.2 is simpler than Assumption 3.1 and does not require any conditions on the transition matrix $A$ . We now state our result for recovering $F ^ { \\star }$ with soft prompt tuning and an attention head. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.6. In the setting above, suppose that non-degeneracy Assumption 4.5 and stationarity Assumption 4.4 hold. Then there exists a prompt u and attention head on $\\overline { { G } } ( \\widehat { e } ( x ) )$ and the token embeddings which can compute the ground-truth $F ^ { \\star } ( x )$ for any $x \\in \\mathcal { R }$ , defined in (4.3): ",
+        "bbox": [
+            173,
+            496,
+            825,
+            537
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1cdbf60ceaa3a0cd22e2c1a9a027de34b022be105de1c52e40afad4dda4d3684.jpg",
+        "text": "$$\nF ^ { \\star } ( x ) = \\mathbb { 1 } \\left( \\mathrm { A t t n } ( ( \\overline { G } _ { i } ( \\widehat e ( x ) ) , \\widehat e _ { i } ( x ) ) _ { i = 1 } ^ { t + 1 } ) \\geqslant 0 \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            542,
+            648,
+            561
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { e }$ is the embedding in (4.4) and Attn is defined in (4.2). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The intuition for this proof is similar to Theorem 3.6: the soft prompt conditions the memory $M$ to concentrate on $\\operatorname { s u p p } ( \\mu )$ . As a result, all irrelevant information to the task is removed from $\\overline { { G } } _ { i } ( \\widehat { e } ( x ) )$ , making it easier to recover the task-specific information about the posterior of $M$ . A more general theorem statement for the multiple memories setting, and the full proof, is provided in Section D.3 ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 Simulations ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We empirically evaluate our theoretical results by pretraining a BERT-like masked language model (MLM) [4] on synthetic data generated by an HMM. Our goal is to verify key implications of our theory in a more realistic setting where some assumptions, such as that $G$ outputs exact conditional probabilities, may not hold. First, we compare head and prompt tuning and show that prompt tuning improves downstream performance, especially when the recovery problem is degenerate. Second, we compare the effect of changing the data distribution from vanilla HMMs to memory-augmented HMMs on head tuning with an attention layer. We find that the downstream performance improves when the data has a long-term memory component. These observations support our theory. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Pretraining data and downstream task. We generate pretraining data from an HMM with randomly generated transition matrix, emission probabilities, and start distributions. In all experiments, the HMMs have 10 vocabulary symbols, while the hidden state size varies. The downstream task uses input sequences $X _ { 1 : T }$ of length 129, where the first token $X _ { 1 } ~ = ~ [ \\mathrm { { M A S K } ] }$ . We consider binary classifcation where labels are generated using linear functions of the analytically-computed posteriors in the HMMs. In all experiments, the ground truth linear weight is sparse with 6 nonzero entries at uniformly random locations with Gaussian values. More details are in Appendix E. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/90c6c6c133241c188eafac3f327482674ef97dc68c975ab5708e5bd785d64deb.jpg",
+        "image_caption": [
+            "Figure 3: Left: Head vs. prompt tuning with a linear head on synthetically-generated HMM data, with varying hidden state sizes. Prompt tuning improves downstream accuracy especially when the problem is degenerate $( | \\mathcal { H } | > | \\mathcal { X } | )$ . Right: Downstream accuracy of head tuning on data from vanilla HMM vs. memory-augmented HMM, across varying values of $| { \\mathcal { M } } | | { \\mathcal { H } } |$ . Long-term dependencies in the memory-augmented HMM data improve downstream recovery with attention. We average over 20 trials (left) and 5 trials (right) of pretraining and finetuning, with $9 5 \\%$ intervals shown. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            90,
+            807,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Head vs. prompt tuning. We compare head and prompt tuning as the hidden state size of the HMM varies. The downstream label is computed via $\\mu ^ { \\top } \\bar { P } [ H _ { 1 } | \\bar { X _ { - 1 } } = x _ { - 1 } ]$ , where $\\mu$ is a random ground-truth linear weight. Head tuning learns a linear head on top of the softmax probabilities predicted by the pretrained model for filling in the first [MASK] token. Prompt tuning uses the same setup but also optimizes a length 20 continuous embedding prepended to the input sequence. ",
+        "bbox": [
+            174,
+            362,
+            823,
+            431
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 (left) shows that prompt tuning improves downstream performance substantially across all hidden state sizes ({4,8,10,15,25,30}). Prompt tuning improves especially when the hidden state size increases beyond the vocabulary size, which makes the recovery problem degenerate. Thus, as suggested by Theorem 3.6, prompt tuning helps relax the non-degeneracy conditions. ",
+        "bbox": [
+            174,
+            438,
+            825,
+            494
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Memory-augmented HMMs. We investigate the effect of augmenting the data-generating HMM with a long-term memory. We consider the single memory case with $| \\mathcal { H } | = 4$ and varying memory sizes $| \\mathcal { M } | \\in \\{ 2 , 3 , 5 , 7 \\}$ . The downstream label is generated by computing $\\mu ^ { \\top } P [ M | X _ { - 1 } = x _ { - 1 } ]$ where $\\mu$ denotes the ground-truth weights. Viewing the memory HMM as a HMM where the component on $\\mathcal { M }$ never changes, we can compare against the vanilla HMMs from the previous setting. For the memory-augmented HMM, we use head tuning with a single-cell attention layer on the entire sequence of softmax probability outputs. For the vanilla HMM in the comparison, we use a linear head on the output at the first position, as an attention head would perform worse since the downstream task depends only on $H _ { 1 }$ and not any other timesteps. ",
+        "bbox": [
+            174,
+            500,
+            825,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 (right) verifies that head tuning recovers the downstream task better when there is more structure in the data, as predicted by Theorem 4.3. Head tuning achieves near $100 \\%$ downstream accuracy on all hidden state sizes. ",
+        "bbox": [
+            174,
+            632,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            696,
+            299,
+            713
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We analyze how pretraining on generic language modeling tasks can improve performance on diverse downstream tasks. In our analysis framework, the downstream task requires predicting properties of the posterior distribution over latent variables in an underlying generative model. When the generative model is a standard HMM, downstream recovery is possible with a simple classification head under strong non-degeneracy assumptions. We also show that we can relax the non-degeneracy conditions by changing the generative model to a memory-augmented HMM or using prompt tuning. The distributions studied here are meant to provide a first-cut result – we also expect similar theorems to hold for other generative models, which we leave as an interesting direction for future work. ",
+        "bbox": [
+            173,
+            729,
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+            840
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            863,
+            338,
+            881
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We thank Percy Liang, Tianyi Zhang, and Nelson Liu for helpful discussions. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Funding statement ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            89,
+            333,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "CW was supported by a NSF Graduate Research Fellowship. SMX was supported by a NDSEG Fellowship. TM acknowledges support of Google Faculty Award, NSF IIS 2045685, and JD.com. Additional revenue: CW received an honorarium for a talk at G-Research. ",
+        "bbox": [
+            176,
+            121,
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+            162
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            183,
+            266,
+            199
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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+    }
+]

parse/train/S1lqMn05Ym/S1lqMn05Ym.md

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+# INFORMATION ASYMMETRY IN KL-REGULARIZED RL
+
+Alexandre Galashov, Siddhant M. Jayakumar, Leonard Hasenclever, Dhruva Tirumala,   
+Jonathan Schwarz, Guillaume Desjardins, Wojciech M. Czarnecki, Yee Whye Teh,   
+Razvan Pascanu, Nicolas Heess   
+DeepMind   
+London, UK   
+{agalashov,sidmj,leonardh,dhruvat,schwarzjn,gdesjardins,   
+lejlot,ywteh,razp,heess}@google.com
+
+# ABSTRACT
+
+Many real world tasks exhibit rich structure that is repeated across different parts of the state space or in time. In this work we study the possibility of leveraging such repeated structure to speed up and regularize learning. We start from the KL regularized expected reward objective which introduces an additional component, a default policy. Instead of relying on a fixed default policy, we learn it from data. But crucially, we restrict the amount of information the default policy receives, forcing it to learn reusable behaviours that help the policy learn faster. We formalize this strategy and discuss connections to information bottleneck approaches and to the variational EM algorithm. We present empirical results in both discrete and continuous action domains and demonstrate that, for certain tasks, learning a default policy alongside the policy can significantly speed up and improve learning.
+
+# 1 INTRODUCTION
+
+For many interesting reinforcement learning tasks, good policies exhibit similar behaviors in different contexts, behaviors that need to be modified only slightly or occasionally to account for the specific task at hand or to respond to information becoming available. For example, a simulated humanoid in navigational tasks is usually required to walk – independently of the specific goal it is aiming for. Similarly, an agent in a simulated maze tends to primarily move forward with occasional left/right turns at intersections.
+
+This intuition has been explored across multiple fields, from cognitive science (e.g. Kool & Botvinick, 2018) to neuroscience and machine learning. For instance, the idea of bounded rationality (e.g. Simon, 1956) emphasizes the cost of information processing and the presence of internal computational constraints. This implies that the behavior of an agent minimizes the need to process information, and more generally trades off task reward with computational effort, resulting in structured repetitive patterns. Computationally, these ideas can be modeled using tools from information and probability theory (e.g. Tishby & Polani, 2011; Ortega & Braun, 2011; Still & Precup, 2012; Rubin et al., 2012; Ortega & Braun, 2013; Tiomkin & Tishby, 2017), for instance, via constraints on the channel capacity between past states and future actions in a Markov decision process.
+
+In this paper we explore this idea, starting from the KL regularized expected reward objective (e.g. Todorov, 2007; Toussaint, 2009; Kappen et al., 2012; Rawlik et al., 2012; Levine & Koltun, 2013; Teh et al., 2017), which encourages an agent to trade off expected reward against deviations from a prior or default distribution over trajectories. We explore how this can be used to inject subjective knowledge into the learning problem by using an informative default policy that is learned alongside the agent policy This default policy encodes default behaviours that should be executed in multiple contexts in absence of additional task information and the objective forces the learned policy to be structured in alignment with the default policy.
+
+![](images/90eed7428aeee80f1544fed9e6516e59ffba29b19ad30e4040d85602c4b1ab12.jpg)  
+Figure 1: Default policy-agent architecture.
+
+To render this approach effective, we introduce an information asymmetry between the default and agent policies, preventing the default policy from accessing certain information in the state. This prevents the default policy from collapsing to the agent’s policy. Instead, the default policy is forced to generalize across a subset of states, implementing a form of default behavior that is valid in the absence of the missing information, and thereby exerting pressure that encourages sharing of behavior across different parts of the state space.
+
+Figure 1 illustrates the proposed setup, with asymmetry imposed by hiding parts of the state from the default policy. We investigate the proposed approach empirically on a variety of challenging problems including both continuous action problems such as controlling simulated high-dimensional physical embodied agents, as well as discrete action visual mazes. We find that even when the agent and default policies are learned at the same time, significant speed-ups can be achieved on a range of tasks. We consider several variations of the formulation, and discuss its connection to several ideas in the wider literature, including information bottleneck, and variational formulations of the EM algorithm for learning generative models.
+
+# 2 KL AND ENTROPY REGULARIZED REINFORCEMENT LEARNING
+
+Throughout this paper we use $s _ { t }$ and $a _ { t }$ to denote the state and action at time step $t$ , and $r ( s , a )$ the instantaneous reward for the agent if it executes action $a$ in state $s$ . We denote the history up to time $t$ by $x _ { t } = ( s _ { 1 } , a _ { 1 } , \dotsc , s _ { t } )$ , and the whole trajectory by $\tau = ( s _ { 1 } , a _ { 1 } , s _ { 2 } , . . . )$ . Our starting point is the KL regularized expected reward objective
+
+$$
+\begin{array} { r } { \mathcal { L } ( \pi , \pi ^ { 0 } ) = \mathbb { E } _ { \pi _ { \tau } } \left[ \sum _ { t } \gamma ^ { t } r ( s _ { t } , a _ { t } ) - \alpha \gamma ^ { t } \mathsf { K L } \left[ \pi ( a _ { t } | x _ { t } ) \| \pi ^ { 0 } ( a _ { t } | x _ { t } ) \right] \right] , } \end{array}
+$$
+
+where $\pi$ is the agent policy (parameterized by $\theta$ and to be learned), $\pi ^ { 0 }$ the default policy, and $\mathbb { E } _ { \pi _ { \tau } } [ \cdot ]$ is taken with respect to the distribution ś $\pi _ { \tau }$ over trajectories defined by the agent policy and system dynamics: $\begin{array} { r } { \pi _ { \tau } ( \tau ) = p ( s _ { 1 } ) \prod _ { t } \pi ( a _ { t } | x _ { t } ) p ( s _ { t + 1 } | s _ { t } , \bar { a _ { t } } ) } \end{array}$ . Note that our policies are history-dependent. ${ \mathsf { K L } } [ \pi ( a _ { t } | x _ { t } ) \| \pi ^ { 0 } ( a _ { t } | x _ { t } ) ]$ is the Kullback-Leibler (KL) divergence between the agent policy $\pi$ and a default or prior policy $\pi ^ { 0 }$ given history $x _ { t }$ . The discount factor is $\gamma \in [ 0 , 1 ]$ and $\alpha$ is a hyperparameter scaling the relative contributions of both terms.
+
+Intuitively, this objective expresses the desire to maximize the reward while also staying close to a reference behaviour defined by $\pi ^ { 0 }$ . As discussed later, besides being a convenient way to express a regularized RL problem, it also has deep connections to probabilistic inference. One particular instantiation of eq. (1) is when $\pi ^ { 0 }$ is the uniform distribution (assuming a compact action space). In this case one recovers, up to a constant, the entropy regularized objective (e.g. Ziebart, 2010; Fox et al., 2015; Haarnoja et al., 2017; Schulman et al., 2017a; Hausman et al., 2018):
+
+$$
+\begin{array} { r } { \mathcal { L } _ { H } ( \pi ) = \mathbb { E } _ { \pi _ { \tau } } \left[ \sum _ { t } \gamma ^ { t } r ( s _ { t } , a _ { t } ) + \alpha \gamma ^ { t } \mathsf { H } [ \pi ( a _ { t } | x _ { t } ) ] \right] . } \end{array}
+$$
+
+This objective has been motivated in various ways: it prevents the policy from collapsing to a deterministic solution thus improving exploration, it encourages learning of multiple solutions to a task which can facilitate transfer, and it provides robustness to perturbations and model mismatch. One approximation of the entropy regularized objective is for the history dependent entropy to be used as an additional (auxiliary) loss to the RL loss; this approach is widely used in the literature (e.g. Williams & Peng, 1991; Mnih et al., 2016). While the motivations for considering the entropy regularized objective are intuitive and reasonable, the choice of regularizing towards an uniform policy is less obvious, particularly in cases with large or high dimensional action spaces. In this work we explore whether regularization towards more sophisticated default policies can be advantageous.
+
+Both objectives (1) and (2) can be generalized beyond the typical Markov assumption in MDPs. In particular, additional correlations among actions can be introduced, e.g. using latent variables Hausman et al. (2018). This can be useful when, as discussed below, either $\bar { \pi } ^ { 0 }$ or $\pi$ are not given full access to the state, rendering the setup partially observed. In the following we will not explore such extensions, though note that we do work with policies $\pi ( \boldsymbol { a } _ { t } | \boldsymbol { x } _ { t } )$ and $\pi ^ { 0 } ( a _ { t } | \boldsymbol x _ { t } )$ that depend on history $x _ { t }$ .
+
+# 3 LEARNING DEFAULT POLICIES
+
+Many works that consider the KL regularized objective either employ a simple or fixed default policy or directly work with the entropy formulation (e.g. Rubin et al., 2012; Fox et al., 2015; Haarnoja et al., 2017; Hausman et al., 2018). In contrast, here we will be studying the possibility of learning the default policy itself, and the form of the subjective knowledge that this introduces to the learning system. Our guiding intuition, as described earlier, is the notion of a default behaviour that is executed in the absence of additional goal-directed information. Instances which we explore in this paper include a locomotive body navigating to a goal location where the locomotion pattern depends largely on the body configuration and less so on the goal, and a 3D visual maze environment with discrete actions, where the typical action includes forward motion, regardless of the specific task at hand.
+
+To express the notion of a default behavior, which we also refer to as “goal-agnostic” (although the term should be understood very broadly), we consider the case where the default policy $\pi ^ { 0 }$ is a function (parameterized by $\phi$ ) of a subset of the interaction history up to time $t$ , i.e. $\pi ^ { 0 } \bar { ( } a _ { t } | \dot { x _ { t } } ) = \pi ^ { 0 } ( a _ { t } | x _ { t } ^ { \mathcal { D } } )$ where $x _ { t } ^ { \mathcal { D } }$ is a subset of the full history $x _ { t }$ t and is the goal-agnostic information that we allow the default policy to depend on. We denote by $x _ { t } ^ { \mathcal { G } }$ the other (goal-directed) information in $x _ { t }$ and assume that the full history is the disjoint union of both. The objective (1) specializes to:
+
+$$
+\begin{array} { r } { \mathcal { L } ( \pi , \pi ^ { 0 } ) = \mathbb { E } _ { \pi _ { \tau } } \left[ \sum _ { t } \gamma ^ { t } r ( s _ { t } , a _ { t } ) - \alpha \gamma ^ { t } \mathsf { K L } \left[ \pi ( a _ { t } | x _ { t } ) \| \pi ^ { 0 } ( a _ { t } | x _ { t } ^ { D } ) \right] \right] , } \end{array}
+$$
+
+To give a few examples: If $x _ { t } ^ { \mathcal { D } }$ is empty then the default policy does not depend on the history at all (e.g. uniform policy). If $x _ { t } ^ { \tilde { D } } = a _ { 1 : t - 1 }$ then it depends only on past actions. In multitask learning $x _ { t } ^ { \mathcal { G } }$ can be the task identifier, while $x _ { t } ^ { \mathcal { D } }$ the state history. And finally, in continuous control $x _ { t } ^ { \mathcal { D } }$ can contain proprioceptive information about the body, while $x _ { t } ^ { \mathcal { G } }$ contains exteroceptive (goal-directed) information (e.g. vision).
+
+By hiding information from the default policy, the system forces the default policy to learn the average behaviour over histories $x _ { t }$ with the same value of $x _ { t } ^ { \mathcal { D } }$ . If $x _ { t } ^ { \mathcal { D } }$ hides goal-directed information, the default policy will learn behaviour that is generally useful regardless of the current goal. We can make this precise by noting that optimizing the objective (1) with respect to $\pi ^ { 0 }$ amounts to supervised learning of $\pi ^ { 0 }$ on trajectories generated by $\pi _ { \tau }$ , i.e. this is a distillation process from $\pi _ { \tau }$ to $\pi ^ { 0 }$ (Hinton et al., 2015; Rusu et al., 2016; Parisotto et al., 2016; Teh et al., 2017). In the nonparametric case, the optimal default policy $\pi _ { * } ^ { 0 }$ can be derived as:
+
+$$
+\pi _ { * } ^ { 0 } ( a _ { t } | x _ { t } ^ { \mathcal { D } } ) = \frac { \sum _ { \tilde { t } } \gamma ^ { \tilde { t } } \int \left( \mathbb { 1 } ( x _ { t } ^ { \mathcal { D } } = \tilde { x } _ { \tilde { t } } ^ { \mathcal { D } } ) \pi ( a _ { t } | \tilde { x } _ { \tilde { t } } ) \right) \pi _ { \tau } ( \tilde { x } _ { \tilde { t } } ) d \tilde { x } _ { \tilde { t } } } { \sum _ { \tilde { t } } \gamma ^ { \tilde { t } } \int \left( \mathbb { 1 } ( x _ { t } ^ { \mathcal { D } } = \tilde { x } _ { \tilde { t } } ^ { \mathcal { D } } ) \right) \pi _ { \tau } ( \tilde { x } _ { \tilde { t } } ) d \tilde { x } _ { \tilde { t } } } ,
+$$
+
+where $\pi _ { \tau } ( \tilde { x } _ { \tilde { t } } )$ is the probability of seeing history $\tilde { x } _ { \tilde { t } }$ at time step $\tilde { t }$ under the policy $\pi$ , and the indicator $\mathbb { 1 } ( x _ { t } ^ { \mathcal { D } } = \tilde { x } _ { \tilde { t } } ^ { \mathcal { D } } ,$ q is 1 if the goal-agnostic information of the two histories matches and 0 otherwise.
+
+It is also worth considering the effect of the objective eq. (3) on the learned policy $\pi$ . Since $\pi ^ { 0 }$ is learned alongside $\pi$ and not specified in advance, this objective does not favor any particular behavior a priori. Instead it will encourage a solution in which similar behavior will be executed in different parts of the state space that are similar as determined by $x _ { t } ^ { \mathcal { D } }$ , since the policy $\pi$ is regularized towards the default policy $\mathbf { \bar { \boldsymbol { \pi } } } ^ { 0 }$ . More generally, during optimization of $\pi$ the default policy effectively acts like a shaping reward while the entropy contained in the KL discourages deterministic solutions.
+
+# 4 CONNECTION TO INFORMATION BOTTLENECK AND VARIATIONAL EM
+
+# 4.1 INFORMATION BOTTLENECK
+
+Reinforcement learning objectives with information theoretic constraints have been considered by multiple authors (Tishby & Polani, 2011; Still & Precup, 2012; Tiomkin & Tishby, 2017). Such constraints can be motivated by the internal computational limitations of the agent, which limit the rate with which information can be extracted from states (or observations) and translated into actions. Such capacity constraints can be expressed via an information theoretic regularization term that is added to the expected reward. Specializing to our scenario, where the “information flow” to be controlled is between the goal-directed history information $x _ { t } ^ { G }$ and action $a _ { t }$ (so that the agent prefers default, goal-agnostic, behaviour), consider the objective:“
+
+$$
+\begin{array} { r } { \mathcal { L } _ { I } = \mathbb { E } _ { \pi _ { \tau } } \left[ \sum _ { t } \gamma ^ { t } r ( s _ { t } , a _ { t } ) - \alpha \gamma ^ { t } \mathsf { M } \mathsf { I } \big [ x _ { t } ^ { \mathcal { G } } , a _ { t } \big | x _ { t } ^ { \mathcal { D } } \big ] \right] , } \end{array}
+$$
+
+where $\mathsf { M I } [ x _ { t } ^ { \mathcal { G } } , a _ { t } | x _ { t } ^ { \mathcal { D } } ]$ is the conditional mutual information between $x _ { t } ^ { \mathcal { G } }$ and $a _ { t }$ given $x _ { t } ^ { \mathcal { D } }$ . The conditional mutual information can be upper bounded:
+
+$$
+\mathbb { E } _ { \pi _ { \tau } } \big [ \boldsymbol { \mathsf { M } } \boldsymbol { \mathsf { I } } [ \boldsymbol { x } _ { t } ^ { \mathcal { G } } , a _ { t } \big | \boldsymbol { x } _ { t } ^ { \mathcal { D } } ] \big ] = \mathbb { E } _ { \pi _ { \tau } } \left[ \log \frac { \pi _ { \tau } \big ( x _ { t } ^ { \mathcal { G } } \big | \boldsymbol { x } _ { t } ^ { \mathcal { D } } \big ) \pi \big ( a _ { t } \big | x _ { t } ^ { \mathcal { G } } , x _ { t } ^ { \mathcal { D } } \big ) } { \pi _ { \tau } \big ( x _ { t } ^ { \mathcal { G } } \big | x _ { t } ^ { \mathcal { D } } \big ) \pi _ { \tau } \big ( a _ { t } \big | x _ { t } ^ { \mathcal { D } } \big ) } \right] \leqslant \mathbb { E } _ { \pi _ { \tau } } \left[ \log \frac { \pi \big ( a _ { t } \big | x _ { t } ^ { \mathcal { G } } , x _ { t } ^ { \mathcal { D } } \big ) } { \pi ^ { 0 } \big ( a _ { t } \big | x _ { t } ^ { \mathcal { D } } \big ) } \right]
+$$
+
+where the inequality is from the fact that the KL divergence $\mathsf { K L } [ \pi _ { \tau } ( a _ { t } | x _ { t } ^ { \mathcal { D } } ) \| \pi ^ { 0 } ( a _ { t } | x _ { t } ^ { \mathcal { D } } ) ]$ is positive (see Alemi et al., 2016). Re-introducing this into (5) we find that the KL regularized objective in eq. (3) can be seen as a lower bound to eq. (5), where the agent has a capacity constraint on the channel between goal-directed history information and (future) actions. See section A in the appendix for a generalization including latent variables. In this light, we can see our work as a particular implementation of the information bottleneck principle, where we penalize the dependence on the information that is hidden from the default policy.
+
+# 4.2 VARIATIONAL EM
+
+The above setup also bears significant similarity to the training of variational autoencoders (Kingma & Welling, 2013; Rezende et al., 2014) and, more generally the variational EM framework for learning latent variable models (Dempster et al., 1977; Neal & Hinton, 1999). The setup is as follows. Given observations ř $\mathcal { X } = \{ x _ { 1 } , . . . x _ { N } \}$ the goal is to maximize the log marginal likelihood $\begin{array} { r } { \log p _ { \theta } ( \mathcal { X } ) = \sum _ { i } \log p _ { \theta } ( x _ { i } ) } \end{array}$ where $\dot { p } _ { \theta } ( x ) = \int \dot { p } _ { \theta } ( x , z ) d z$ . This marginal likelihood can be bounded from below by $\begin{array} { r } { \sum _ { i } \mathbb { E } _ { q _ { \phi } ( z | x _ { i } ) } [ \log p _ { \theta } ( x _ { i } | z ) - \log \frac { q _ { \phi } ( z | x _ { i } ) } { p _ { \theta } ( z ) } ] } \end{array}$ qφpz|xiqpθpzq s with qφpz|xiq being a learned approximation to the true posterior $p _ { \theta } ( z | x _ { i } )$ . This lower bound exhibits a similar information asymmetry between $q$ and $p$ as the one introduced between $\pi$ and $\pi ^ { 0 }$ in the objective in eq. (3). In particular, in the multi-task case discussed in section 3 with one task per episode, $x _ { i }$ can be seen to take the role of the task, $\log p ( x _ { i } | \boldsymbol { z } )$ that of the task reward, $q ( \boldsymbol { z } | \boldsymbol { x } _ { i } )$ that of task conditional policy, and $p ( z )$ the default policy. Therefore maximizing eq. (3) can then be thought of as learning a generative model of behaviors that can explain the solution to different tasks.
+
+# 5 ALGORITHM
+
+In practice the objective in eq. 3 can be optimized in different ways. A simple approach is to perform alternating gradient ascent in $\pi ^ { 0 }$ and $\pi$ . Optimizing $\mathcal { L }$ with respect to $\bar { \pi } ^ { 0 }$ amounts to supervised learning with $\pi$ as the data distribution (distilling $\pi$ into $\pi ^ { 0 }$ ). Optimizing $\pi$ given $\pi ^ { 0 }$ requires solving a regularized expected reward problem which can be achieved with a variety of algorithms (Schulman et al., 2017a; Teh et al., 2017; Haarnoja et al., 2017; Hausman et al., 2018; Haarnoja et al., 2018).
+
+The specific algorithm choice in our experiments depends on the type of environment. For the continuous control domains we use SVG(0) (Heess et al., 2015) with experience replay and a modification for the KL regularized setting (Hausman et al., 2018; Haarnoja et al., 2018). The SVG(0) algorithm learns stochastic policies by backpropagation from the action-value function. We estimate the action value function using $K$ -step returns and the Retrace operator for low-variance off-policy correction (see Munos et al. (2016); as well as Hausman et al. (2018); Riedmiller et al. (2018b)). For discrete action spaces we use a batched actor-critic algorithm (see Espeholt et al. (2018)). The algorithm employs a learned state-value function and obtains value estimates for updating the value function and advantages for computing the policy gradient using $K$ -step returns in combination with the V-trace operator for off-policy correction. All algorithms are implemented in batched distributed fashion with a single learner and multiple actors. In algorithm 1 we provide pseudo-code for actor-critic version of the algorithm with $K$ -step returns. Details of the off-policy versions of the algorithms for continuous and discrete action spaces can be found in the appendix (section D).
+
+# 6 RELATED WORK
+
+There are several well established connections between certain formulations of the reinforcement learning literature and concepts from the probabilistic modeling literature. The formalisms are often closely related although derived from different intuitions, and with different intentions.
+
+policy: $\pi _ { \theta }$ , initial parameters $\theta ^ { 0 }$   
+default policy: $\pi _ { \phi } ^ { 0 }$ ; initial parameters $\phi ^ { 0 }$   
+Q-function: $Q _ { \psi }$ ; initial parameters $\psi ^ { 0 }$   
+for $\mathrm { j } { = } 1$ , . . . do for $\mathrm { { t } } = 0$ , K, 2K, . . . T do rollout partial trajectory: $\tau _ { t : t + K } = ( s _ { t } , a _ { t } , r _ { t } \ldots r _ { t + K } )$ compute KL: $\widehat { \mathsf { K L } } _ { t ^ { \prime } } = \mathsf { K L } [ \pi ( \cdot | s _ { t ^ { \prime } } ) \| \pi ^ { 0 } ( \cdot | s _ { t ^ { \prime } } ) ]$ Estimate boostrap value: $\hat { V } = \mathbb { E } _ { \pi ( \cdot | s _ { t + K } ) } [ Q ( s _ { t + K } , a ) ] - \alpha \widehat { \mathsf { K L } } _ { t + K }$ Estimate Q targets: $\begin{array} { r } { \hat { Q } _ { t ^ { \prime } } = \sum _ { t ^ { \prime \prime } = t ^ { \prime } } ^ { t + K - 1 } ( r _ { t ^ { \prime \prime } } - \alpha \widehat { \mathsf { K L } } _ { t ^ { \prime \prime } } ) + \hat { V } } \end{array}$ Agent policy Q-value loss: $\begin{array} { r } { \hat { L } _ { \pi } = \sum _ { t ^ { \prime } = t } ^ { t + K - 1 } \mathbb { E } _ { \pi ( \cdot | s _ { t ^ { \prime } } ) } \big [ Q ( s _ { t ^ { \prime } } , a ) \big ] - \alpha \widehat { \sf K L } _ { t ^ { \prime } } } \end{array}$ $\begin{array} { r } { \hat { L } _ { Q } = \sum _ { t ^ { \prime } = t } ^ { t + K - 1 } \| \hat { Q } _ { t ^ { \prime } } - Q ( s _ { t ^ { \prime } } , a _ { t ^ { \prime } } ) \| ^ { 2 } } \end{array}$ Default policy loss: $\begin{array} { r } { \widehat { L } _ { \pi ^ { 0 } } = \sum _ { t ^ { \prime } = t } ^ { t + K - 1 } \widehat { \mathsf { K L } } _ { t ^ { \prime } } } \end{array}$ $\begin{array} { r l r l r l } { \theta  \theta + \beta _ { \pi } \nabla _ { \theta } \hat { L } _ { \pi } } & { } & { \phi  \phi - \beta _ { \pi ^ { 0 } } \nabla _ { \phi } \hat { L } _ { \pi ^ { 0 } } } & { } & { \psi  \psi - \beta _ { Q } \nabla _ { \psi } \hat { L } _ { Q } } \end{array}$ end for   
+end for
+
+Maximum entropy reinforcement learning, stochastic optimal control, and related approaches build on the observation that some formulation of the reinforcement learning problem can be interpreted as exact or approximate variational inference in a probabilistic graphical model in which the reward function takes the role of log-likelihood (e.g. Ziebart, 2010; Kappen et al., 2012; Toussaint, 2009). While the exact formulation and algorithms vary, they result in an entropy or KL regularized expected reward objective. These algorithms were originally situated primarily in the robotics and control literature but there has been a recent surge in interest in deep reinforcement learning community (e.g. Fox et al., 2015; Schulman et al., 2017a; Nachum et al., 2017a; Haarnoja et al., 2017; Hausman et al., 2018; Haarnoja et al., 2018).
+
+Related but often seen as distinct is the familiy of expectation maximization policy search algorithms (e.g. Peters et al., 2010; Rawlik et al., 2012; Levine & Koltun, 2013; Montgomery & Levine, 2016; Chebotar et al., 2016; Abdolmaleki et al., 2018). These cast policy search as an alternating optimization problem similar to the EM algorithm for learning probabilistic models. They differ in the specific implementation of the equivalents of the E and M steps; intuitively the default policy is repeatedly replaced by a new version of the policy.
+
+The DISTRAL algorithm (Teh et al., 2017) as well as the present paper can be seen as taking an intermediate position: unlike in the class of RL-as-inference algorithms the default policy is not fixed but learned, but unlike in the classical EM policy search the final result of the optimization remains regularized since the default policy is constrained relative to the policy. As explained above this can be seen as analogous to the relative roles of learned model and observation specific posterior in fitting a generative model. Similar to DISTRAL, Divide and Conquer (Ghosh et al., 2018) learns an ensemble of policies, each specializing to a particular context, which are regularized towards one another via a symmetric KL penalty, with the behavior of the ensemble distilled to a single fixed policy. In concurrent work Goyal et al. (2019) propose an information bottleneck architecture for policies with latent variables that leads to a KL-regularized formulation similar to the one described in Appendix A.2. The information bottleneck is implemented in latent space and the default policy is obtained by marginalization with a goal-agnostic prior.
+
+An important feature of EM policy search and other policy gradient algorithms is the presence of a KL constraint that limits the relative change of the policy to some older version across iterations to control for the rate of change in the policy (e.g. Schulman et al., 2015; Heess et al., 2015; Schulman et al., 2017b; Heess et al., 2017; Nachum et al., 2017b). The constraint can be implemented in different ways, and collectively the algorithms are often classified as “trust region” methods. Note that for a KL regularized objective to be a trust region (Nocedal & Wright, 2006), additional assumptions need to hold. In principle, as an optimization technique, the critical points of the KL regularized objective for some function $f ( \theta )$ have to be, provably, the same as for the non-regularized objective. This is not trivial to show unless the trust region for step $k$ is around $\theta _ { k }$ . In our case, there is no such guarantee even if we remove the asymmetry in information between default policy and policy or make the default policy be an old copy of the policy.
+
+Other related works motivated from an optimization perspective include Deep Mutual Learning (Zhang et al., 2018) applied in supervised learning, where KL-regularization is used with a learned prior that receives the same amount of information as the trained model. Kirkpatrick et al. (2017) introduces EWC to address catastrophic forgetting, where a second order Taylor expansion of the KL, in a KL-regularized objective, forces the main policy to stay close to solutions of previously encountered tasks. Czarnecki et al. (2018) also relies on a KL-regularized objective to ensure policies explored in a curriculum stay close to each other.
+
+Conceptually distinct but formally closely related to maximum entropy and KL-regularized formulations are computational models of bounded rationality (e.g. Tishby & Polani, 2011; Ortega & Braun, 2011; Still & Precup, 2012; Rubin et al., 2012; Tiomkin & Tishby, 2017) which introduce information constraints to account for the agent’s internal computational constraints on its ability to process information. As discussed in section 4 the present formulation can be seen as a more general formulation of the idea.
+
+# 7 CONTINUOUS CONTROL EXPERIMENTS
+
+In our experiments, we study the effect of using a learned default policy to regularize the behavior of our agents, across a wide range of environments spanning sparse and dense reward tasks. In particular, we evaluate the impact of conditioning the default policy on various information sets $x ^ { \mathcal { D } }$ on the learning dynamics, and evaluate the potential of pretrained default policies for transfer learning. In these experiments, we consider two streams of information which are fed to our agents: task specific information (task) and proprioception (proprio), corresponding to walker (body) specific observations (joint angles etc.).
+
+![](images/8f237a49d8bdf0b10cf100766c9c71f14c4568d5ee1381b1db9554bfabf50323.jpg)  
+Figure 2: Tasks visualization. (a): Go to one of K target tasks, with quadruped; (b): Move one box to one of K targets task, with jumping ball (red); (c): Foraging in the maze task, with quadruped; (d): Walls task with humanoid, where the goal is avoid walls while running through a terrain.
+
+We consider three walkers: jumping ball with 3 degrees of freedom (DoF) and 3 actuators; quadruped with $1 2 \mathrm { D o F }$ and 8 actuators; humanoid with $2 8 \mathrm { D o F }$ and 21 actuators. The task is specified to the agent either via an additional feature vector (referred to as feature-tasks) or in the form of visual input (vision-task). The tasks differ in the type of reward: in sparse reward tasks a non-zero reward is only given when a (sub-)goal is achieved (e.g. the target was reached); in dense reward tasks smoothly varying shaping reward is provided (e.g. negative distance to the target). We consider the following tasks.
+
+Walking task, a dense-reward task based on features. The walker needs to move in one of four randomly sampled directions, with a fixed speed; the direction being resampled half-way through the episode. Walls task, a dense-reward vision-task. Here the walker has to traverse a corridor while avoiding walls. Go to one of K targets task, a sparse-reward feature-based task. The walker has to go to one of K randomly sampled targets. For $\mathbf { K } { = } 1$ , the target can either reappear within the episode (referred to as the moving target task) or the episode can end upon reaching the target. Move one box to one of K targets, a sparse-reward feature-based-task. The walker has to move a box to one of K targets, and optionally, go on to one of the remaining targets. The latter is referred to as the move one box to one of $K$ targets and go to another target). Foraging in the maze task, a sparse-reward vision-task. The walker collects apples in a maze. Figure 2 shows visualizations of the walkers and some of the tasks. Refer to appendix C for more details.
+
+Experimental Setup As baseline, we consider policies trained with standard entropy regularization. When considering the full training objective of eq. 1, the default policy network shares the same structure as the agent’s policy. In both cases, hyper-parameters are optimized on a per-task basis. We employ a distributed actor-learner architecture (Espeholt et al., 2018): actors execute recent copies of the policy and send data to a replay buffer of fixed size; while the learner samples short trajectory windows from the replay and computes updates to the policy, value, and default policy. We experimented with a number of actors in t32, 64, 128, 256u (depending on the task) and a single learner. Results with a single actor are presented in appendix B. Unless otherwise mentioned, we plot average episodic return as a function of the number of environment transitions processed by the learner1. Each experiment is run with five random seeds. For more details, see appendix D.2
+
+We consider three information sets passed to the default policy: proprioceptive, receiving only proprioceptive information; task-subset, receiving proprioceptive and a subset of task-specific information; full-information, receiving the same information as the policy.
+
+![](images/aef497a4c416561e4a5f9c96b13e57e5466b10045e646e30959092b56d14f6a8.jpg)  
+Figure 3: Results for the sparse-reward tasks with complex walkers. Left: go to moving target task with humanoid. Center: foraging in the maze results with quadruped. Right: moving one box to one of two targets and go to another target task with quadruped. The legends denote additional to the proprioception, information passed to the default policy (except baseline, where we do not use default policy).
+
+The main finding of our experiments is that the default policy with limited task information provides considerable speed-up in terms of learner steps for the sparse-reward tasks with complex walkers (quadruped, humanoid). The results on these tasks are presented in figure 3. More cases are covered in the appendix E.
+
+Overall, the proprioceptive default policy is very effective and gives the biggest gains in the majority of tasks. Providing additional information to the default policy, leads to an improvement only in a small number of cases (figure 3, right and appendix E.3). In these cases, the additional information (e.g. box position), adds useful inductive bias for the policy learning. For the dense-reward tasks or for a simple walker body adding the default policy has limited or no effect (see appendix E.1, E.2). We hypothesize that the absence of gain is due to the relative simplicity of the regular policy learning versus the KL-regularized setup. In the case of dense-reward tasks the agent has a strong reward signal. For simple walkers, the action space is too simple to require sophisticated exploration provided by the default policy. Finally, with full information in the default policy, the optimal default policy would exactly copy the agent policy, which would not provide additional learning signal beyond the regular policy learning. In all these cases, the default policy will not be forced to generalize across different contexts and hence not provide a meaningful regularization signal.
+
+We analyze the agent behavior on the go to moving target task with a quadruped walker. We illustrate the agent trajectory for this task in figure 4, left. The red dot corresponds to the agent starting position. The green stars on the left and central figures correspond to the locations of the targets with blue numbers indicating the order of achieving the targets. The yellow dots on the left and central curves indicate the segment (of 40 time steps) near the target. In figure 4, center, we show the KL divergence, $K L [ \pi \| \pi ^ { 0 } ]$ , from the agent policy to the proprioceptive default policy. We observe that for the segments which are close to the target (yellow dots near green star), the value of the KL divergence is high. In these segments the walker has to stop and turn in order to go to another target. It represents a deviation from the standard, walking behavior, and we can observe it as spikes in the KL. Furthermore, for the segments between the targets, e.g. $4 \mathrm { - } > 5$ , the KL is much lower.
+
+![](images/82eb13b7757453a3e0b25f2f64cc81a0b22e1e217e8233cd2de250b91de4c8b9.jpg)  
+Figure 4: Behavior analysis and transfer results. Left: the trajectory of the agent on go to moving target task with quadruped. Center: KL divergence from the agent policy to the proprioceptive default policy plotted over time for the same trajectory. Right: Performance of the transfer on move one box to one of 3 targets task with quadruped. The legend whether the default policy is learned or is transferred. Furthermore, it specifies the task from which the default policy is transferred as well as additional information other than the proprioceptive information that the default policy is conditioned on, if any.
+
+Default Policy Transfer We additionally explore the possibility of reusing pretrained default policies to regularize learning on new tasks. Our transfer task is moving one box to one of 2 targets and going to another target task with the quadruped. We consider different default policies: GTT proprio: proprioceptive information only trained on going to moving target task (GTT); MB proprio: proprioceptive information only trained on moving one box to one target task (MB); MB box: similar MB proprio, but with box position information as additional input. The results are given in figure 4, right. We observe a significant improvement in learning speed transferring the pretrained default policies to the new task. Performance improves as the trajectory distribution modeled by the default policy is closer to the one appropriate for the transfer task (compare GTT proprio with MB proprio; and MB proprio with MB box).
+
+![](images/38f9d0af5239994bdc5ac28d6ce8b03c8b05e94a2d3766d8162502ff4e4e9705.jpg)  
+Figure 5: Ablations. Left: Comparing various regularization schemes. Center: Benefits of default policy vanish when using (dense) shaping rewards. Right: Optimistic baselines comparing pretrained default policies.
+
+Ablative Analysis To gain deeper insights into our method, we compare different forms of regularization of the standard RL objective: entropy bonus - adding an entropy term $H ( \pi ( \cdot | s _ { t } ) )$ to the per-timestep actor loss; entropy regularization - optimizing the objective (2); “ ‰ $K L$ bonus - adding the KL-divergence term KL $\cdot \left[ \dot { \pi } ( a _ { t } | \dot { s } _ { t } ) \| \pi ^ { 0 } ( a _ { t } | s _ { t } ) \right]$ from the agent policy to the default one to the per-timestep actor loss; KL-regularization - optimizing the objective (1); $K L$ -regularization to the old policy - optimization of the objective 1 where regularization is done wrt. an older version of the main policy (updated every 100 steps). The default policy receives only proprioceptive information in these experiments. The task is go to moving target. As can be seen in Figure 5 left, all three KL-based variants improve performance over the baseline, but regularizing against the information restricted default policy outperforms regularization against an old version of the policy.
+
+Figure 5 center, demonstrates that the benefit of the default policy depends on the reward structure. When replacing the sparse reward with a dense shaping reward, proportional to the inverse distance from the walker to the target, our method and the baseline perform similarly, which is consistent with dense-reward results.
+
+Finally, we assess the benefit of the KL-regularized objective 1 when used with an idealized default policy. We repeat the go-to-target experiment with a pretrained default policy on the same task. Figure 5 right, shows a significant difference between the baseline and different regularization variants: using the pretrained default policy, learning the default policy alongside the main policy or using a pretrained expert (default policy with access to the full state). This suggests that large gains may be achievable in situations when a good default policy is known a priori. We performed the same analysis for the dense reward but we did not notice any gain. The speed-up from regularizing to the pretrained expert is significant, however it corresponds to regularizing against an existing solution and can thus primarily be used as a method to speed-up the experiment cycles, as it was demonstrated in kickstarting framework (Schmitt et al., 2018).
+
+Finally, we study impact of the direction of the KL in objective 1 on the learning dynamics. Motivated“ ‰ by the work in policy distillation (Rusu et al., 2016) we flip the KL and use “ ‰ ${ \mathsf { K L } } \left[ { \check { \pi } } ^ { 0 } ( a _ { t } | s _ { t } ) \| { \boldsymbol { \pi } } ( a _ { t } | s _ { t } ) \right]$ instead of the described before ${ \mathsf { K L } } \left[ \pi ( a _ { t } | s _ { t } ) \| \pi ^ { 0 } ( a _ { t } | s _ { t } ) \right]$ . The experiments showed that there was no significant difference between these regularization schemes, which suggests that the idea of learned default policy can be viewed from student-teacher perspective, where default policy plays the role of the teacher. This teacher can be used in a new task. For the details, please refer to the appendix E.6.
+
+# 8 DISCRETE ACTION SPACES EXPERIMENTS
+
+We also evaluate our method on the DMLab-30 set of environments. DMLab (Beattie et al., 2016) provides a suite of rich, first-person environments with tasks ranging from complex navigation and laser-tag to language-instructed goal finding. Recent works on multitask training (Espeholt et al., 2018) in this domain have used a form of batched-A2C with the V-trace algorithm to maximize an approximation of the entropy regularized objective described earlier, where the default policy is a uniform distribution over the actions.
+
+Typically, the agent receives visual information at each step, along with an instruction channel used in a subset of tasks. The agent receives no task identifier. We adopt the architecture employed in previous work (Espeholt et al., 2018) in which frames, past actions and rewards are passed successively through a deep residual network and LSTM, finally predicting a policy and value function. All our experiments are tuned with population-based training (Jaderberg et al., 2017). Further details are provided in appendix D.1.
+
+DMLab exposes a large action space, specifically the cartesian product of atomic actions along seven axes. However, commonly a human-engineered restricted subset of these actions is used at training and test time, simplifying the exploration problem for the agent. For example, the used action space has a forward bias, with more actions resulting in the agent moving forward rather than backwards. This helps with exploration in navigation tasks, where even a random walk can get the agent to move away from the starting position. The uniform default policy is used on top of this human engineered small action space, where its semantics are clear.
+
+In this work, we instead consider a much larger combinatorial space of actions. We show that a pure uniform default policy is in fact unhelpful when human knowledge is removed from defining the right subset of actions to be uniform over, and the agent under-performs. Learning the default policy, even in the extreme case when the default policy is not conditioned on any state information, helps recovering which actions are worth exploring and leads to the emergence of a useful action space without any hand engineering.
+
+Figure 6 shows the results of our experiments. We consider a flat action space of 648 actions, each moving the agent in different spatial dimensions. We run the agent from (Espeholt et al., 2018) as baseline which is equivalent to considering the default policy to be a uniform distribution over the 648 actions, and three variants of our approach, where the default policy is actually learnt.
+
+![](images/334aa676d579a0f6b0544fa569463e0086111d2e21a08c76ad3cad63866c8130.jpg)  
+Figure 6: DMLab30. Left, comparison between baseline (same as Espeholt et al. (2018)) that uses uniform distribution over actions as a default policy and three different possible default policies. Center, the entropy for the vector default policy over learning. Right, marginalized distribution over few actions of interest for the vector default policy.
+
+For feed forward default policy, while the agent is recurrent, the default policy is not. That is the policy $\pi$ is conditioned on the full trace of observed states $s _ { 1 } , a _ { 1 } , . . s _ { t }$ , while the default policy $\pi ^ { 0 }$ is conditioned only on the current frame $a _ { t - 1 } , s _ { t }$ . Given that most of the 30 tasks considered require memory in order to be solvable, the default policy has to generalize over important task details. LSTM default policy on the other hand, while being recurrent as the agent, it observes only the previous action $a _ { t - 1 }$ and does not receive any other state information. In this instance, the default policy can only model the most likely actions given recent behaviour $a _ { 1 } , . . a _ { t - 1 }$ in absence of any visual stimuli. For example, if previous actions are moving forward, the default policy might predict moving forward as the next action too. This is because the agent usually moves consistently in any given direction in order to navigate efficiently. Finally, the vector default policy refers to a default policy that is independent of actions and states (i.e. average behaviour over all possible histories of states and actions).
+
+Using any of the default policies outperforms the baseline, with LSTM default policy slightly underperforming compared with the others. The vector default policy performs surprisingly well, highlighting that for DMLab defining a meaningful action space is extremely important for solving the task. Our approach can provide a mechanism for identifying this action space without requiring human expert knowledge on the tasks. Note in middle plot, figure 6, that the entropy of the default policy over learning frames goes down, indicating that the default policy becomes peaky and is quite different from the uniform distribution which the baseline assumes. Note that when running the same experiments with the original human-engineered smaller action space, no gains are observed. This is similar to the continuous control setup, corresponding to changing the walker to a simple one and hence converting the task into a denser reward one.
+
+Additionally, in figure 6 right, for the vector default policy, we show the probability of a few actions of interest by marginalizing over all other actions. We notice that the agent has a tendency of moving forward $7 0 \%$ , while moving backwards is quite unlikely $1 0 \%$ . The default policy discovers one element of the human defined action space, namely forward-bias which is quite useful for exploring the map. The uniform bias would put same weight for moving forward as for moving backwards, making exploration harder. We also note that the agent has a tendency to turn right and look right. Given that each episode involves navigating a new sampled map, such a bias provides a meaningful exploration boost, as it suggest a following the wall strategy, where at any new intersection the agent always picks the same turning direction (e.g. right) to avoid moving in circles. But as expected, since neither looking up or looking down provides any advantage, these actions are equally probable.
+
+# 9 DISCUSSION AND CONCLUSIONS
+
+In this work we studied the influence of learning the default policy in the KL-regularized RL objective. Specifically we looked at the scenario where we enforce information asymmetry between the default policy and the main one. In the continuous control, we showed empirically that in the case of sparse-reward tasks with complex walkers, there is a significant speed-up of learning compared to the baseline. In addition, we found that there was no significant gain in dense-reward tasks and/or with simple walkers. Moreover, we demonstrated that significant gains can be achieved in the discrete action spaces. We provided evidence that these gains are mostly due to the information asymmetry between the agent and the default policy. Best results are obtained when the default policy sees only a subset of information, allowing it to learn task-agnostic behaviour. Furthermore, these default polices can be reused to significantly speed-up learning on new tasks.
+
+# 10 ACKNOWLEDGMENTS
+
+The authors would like to thank Abbas Abdolmaleki, Arun Ahuja, Jost Tobias Springenberg, Siqi Liu for their help on experimental side. Furthermore, The authors would like to thank Greg Wayne for useful discussions. Finally, the authors are very grateful to Simon Osindero and Phil Blunsom for their insightful feedback on the paper.
+
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+# A KL-REGULARIZED RL AND INFORMATION BOTTLENECK
+
+In this appendix we derive the connection between KL-regularized RL and information bottleneck in detail. For simplicity we assume that $x _ { t } ^ { \mathcal { D } }$ is empty, consider dependence only on current state $s _ { t }$ and do not use subscript by $t$ in detailed derivations for notational convenience. We also apologize for some notational inconsistencies, and will fix them in a later draft.
+
+A.1 MINIMIZING INFORMATION FLOW FROM $S _ { t }$ TO $A _ { t }$ FOR UNSTRUCTURED POLICIES
+
+The simple formulation of the information bottleneck corresponds to maximizing reward while minimizing the per-timestep information between actions and state (or a subset of state, like the goal):
+
+$$
+\mathcal { L } = \mathbb { E } _ { \pi } [ \sum _ { t } ( r ( s _ { t } , a _ { t } ) - \mathsf { M } \mathsf { I } [ A _ { t } ; S _ { t } ] ) ]
+$$
+
+Upper-bounding the mutual information term:
+
+$$
+\begin{array} { l } { { \displaystyle \mathbb { M } [ A ; S ] = \int \pi ( s ) \pi ( a | s ) \log \frac { \pi ( s ) \pi ( a | s ) } { \pi ( s ) \pi ( a ) } } } \\ { ~ = \int \pi ( s ) \pi ( a | s ) \log \frac { \pi ( a | s ) } { \pi ( a ) } } \\ { ~ \leqslant \int \pi ( s ) \pi ( a | s ) \log \frac { \pi ( a | s ) } { \pi ^ { 0 } ( a ) } } \\ { ~ = \mathbb { E } _ { \pi } [ \mathsf { K L } [ \pi ( A | s ) \| \pi ^ { 0 } ( A ) | s ] ] , } \end{array}
+$$
+
+since
+
+$$
+\begin{array} { c l l } { 0 \leqslant \pi \mathsf { K L } [ \pi ( a ) \| \pi ^ { 0 } ( a ) ] = \mathbb { E } _ { \pi } [ \log \displaystyle \frac { \pi ( a ) } { \pi ^ { 0 } ( a ) } ] = \mathbb { E } _ { \pi } [ \log \pi ( a ) ] - \mathbb { E } _ { \pi } [ \log \pi ^ { 0 } ( a ) ] } \\ { \iff \mathbb { E } _ { \pi } [ \log \pi ( a ) ] \geqslant \mathbb { E } _ { \pi } [ \log \pi ^ { 0 } ( a ) ] . } \end{array}
+$$
+
+Thus
+
+$$
+\begin{array} { r l r } {  { \mathcal { L } = \mathbb { E } _ { \boldsymbol { \pi } } [ \sum _ { t } ( \boldsymbol { r } ( s _ { t } , a _ { t } ) - \mathsf { M } \mathsf { I } [ \boldsymbol { A } _ { t } ; \boldsymbol { S } _ { t } ] ) ] } } \\ & { } & { \geqslant \mathbb { E } _ { \boldsymbol { \pi } } [ \sum _ { t } ( \boldsymbol { r } ( s _ { t } , a _ { t } ) - \mathsf { K L } [ \pi _ { t } \| \pi _ { t } ^ { 0 } | s _ { t } ] , } \end{array}
+$$
+
+i.e. the problem turns into one of KL-regularized RL.
+
+A.2 MINIMIZING INFORMATION FLOW FROM $S _ { t }$ TO $A _ { t }$ FOR POLICIES WITH LATENT VARIABLES
+
+For policies with latent variables such as $\pi ( a | s ) = \int \pi ( a | z ) \pi ( z | s ) d z$ we obtain:
+
+$$
+\begin{array} { l } { { \displaystyle { \sf M I } [ A ; S ] = \int \pi ( a , s ) \log \pi ( a | s ) d a d s - \int \pi ( a ) \log \pi ( a ) d a } } \\ { { \displaystyle ~ \leqslant \pi \int \pi ( a , s ) \log \pi ( a | s ) d a d s - \int \pi ( a ) \log \bar { \pi ^ { 0 } } ( a ) d a } } \end{array}
+$$
+
+as before.
+
+We choose $\pi ^ { 0 } ( a ) = \int \pi ( a | z ) \pi ^ { 0 } ( z ) d z$ , then:
+
+$$
+\begin{array} { r } { \{ \pi ( a ) \log \tilde { \mathfrak { n } } ^ { 0 } ( a ) d a - \displaystyle \int \pi ( a ) \log \Bigg \int \pi ( a | z ) \pi ^ { 0 } ( z ) d z d a } \\ { = \int \pi ( a , s ) \log \Bigg \{ \pi ( a | z ) \pi ^ { 0 } ( z ) d z d a d s } \\ { = \int \pi ( a , s ) \log \Bigg \{ \pi ( a ) \sigma \frac { \pi ( z ) \pi ^ { 0 } ( s , a ) } { \pi ( z ) s , a } \pi ^ { 0 } ( z ) d z d a d s } \\ { \geqslant \displaystyle \int \pi ( a , s , z ) \log \frac { \pi ( a ) \pi ^ { 0 } ( s , a ) } { \pi ( z ) s , a } \Bigg \} z d a d s } \\ { = \displaystyle \int \pi ( a , s , z ) \log \frac { \pi ( a ) \pi ^ { 0 } ( z ) \pi ( a ) ( s , a ) } { \pi ( a ) ( z ) \pi ( z ) ( s ) } d z d a d s } \\ { = \displaystyle \int \pi ( a , s ) \log \pi ( a | s ) d a d s + \displaystyle \int \pi ( z , s ) \log \frac { \pi ^ { 0 } ( z ) } { \pi ( z ) ( s , a ) } d z d s , } \end{array}
+$$
+
+and thus
+
+$$
+\begin{array} { l } { { { \displaystyle { \sf M I } [ A ; S ] \leqslant \int \pi ( a , s ) \log q ( a | s ) d a d s - \int \pi ( a ) \log \bar { \pi ^ { 0 } } ( a ) d a } } } \\ { { \displaystyle ~ \leqslant \int \pi ( a , s ) \log \pi ( a | s ) d a d s - \int \pi ( a , s ) \log \pi ( a | s ) d a d s - \int \pi ( z , s ) \log \displaystyle \frac { \pi ^ { 0 } ( z ) } { \pi ( z | s ) } d z } } \\ { { \displaystyle ~ = \int \pi ( z , s ) \log \displaystyle \frac { \pi ( z | s ) } { \pi ^ { 0 } ( z ) } d z = \mathbb { E } _ { \pi } [ \mathsf { K L } [ \pi ( Z | s ) \| \pi ^ { 0 } ( Z ) | s ] ] . } } \end{array}
+$$
+
+Therefore:
+
+$$
+\begin{array} { r l } & { \mathcal { L } = \mathbb { E } _ { \boldsymbol { \pi } } [ \displaystyle \sum _ { t } ( r ( s _ { t } , a _ { t } ) - \mathsf { M I } [ A _ { t } ; S _ { t } ] ) ] } \\ & { ~ \geqslant \mathbb { E } _ { \boldsymbol { \pi } } [ \displaystyle \sum _ { t } ( r ( s _ { t } , a _ { t } ) - \mathsf { K L } [ \pi ( Z _ { t } | s ) \| \pi ^ { 0 } ( Z _ { t } ) | s _ { t } ] , } \end{array}
+$$
+
+Thus, the KL regularized objective discussed above can be seen as implementing an information bottleneck. Different forms of the default policy correspond to restricting the information flow between different components of the interaction history (past states or observations), and to different approximations to the resulting mutual information penalties.
+
+This perspective suggests two different interpretations of the KL regularized objective discussed above: We can see the role of the default policy implementing a way of restricting information flow between (past) states and (future) actions. An alternative view, more consistent with the analogy between RL and probabilistic modeling invoked above is that of learning a “default” behavior that is independent of some aspect of the state. (Although the information theoretic view has recently gained more hold in the probabilistic modeling literature, too (e.g. Alemi et al., 2016; 2017)).
+
+# B DISTRIBUTED LEARNING SETUP
+
+We use a distributed off-policy setup similar to Riedmiller et al. (2018a). There is one learner and multiple actors. These are essentially the instantiations of the main agent used for different purposes. Each actor is the main agent version which receives the copy of parameters from the learner and unrolls the trajectories in the environment, saving it to the replay buffer of fixed size 1e6. The learner is the agent version which samples a batch of short trajectories windows (window size is defined by unroll length) from the replay buffer, calculates the gradients and updates the parameters. The updated parameters are then communicated to the actors. Such a setup speeds-up learning significantly and makes the final performance of the policy better. We compare the performance of on go to moving target task with 1 and 32 actors. From figure 7, we see that the effect of the default policy does not disappear when the number of actor decreases to 1, but the learning becomes much slower, noisier and weaker.
+
+![](images/6ebd06a735f0ea0317987c1c05dae8537f05270de46c917ee2ae01457a9ee0c4.jpg)  
+Figure 7: Single versus multiple actors comparison on go to moving target task. Left: 32 actors. Right: 1 actor.
+
+![](images/b5fbd32af3c69ba714a9c609d23c37f28f0dda13d455c1831bf24fc80300d9db.jpg)  
+Figure 8: Walkers visualization.
+
+# C CONTINUOUS CONTROL: WALKERS AND TASK DETAILS
+
+Walkers visualization is provided in figure 8. Below we give a detaatiled description of each continuous control task we studied.
+
+# Walking task.
+
+Type. Dense-reward feature-based-task.
+
+Description. Each half of the episode, a random direction among 4 (left, right, forward and backwards) is sampled. Task information is specified via a one-hot encoding of the required direction. The walker is required to move in this direction with the target speed $v _ { t }$ and receives the reward $r$ .
+
+Reward. $r = \exp ^ { - | v _ { c u r } - v _ { t } | ^ { 2 } }$
+
+Technical details. Target speed, $v _ { t } = 3$ . The episode length is 10 seconds. For the humanoid task we use the absolute head height termination criteria: $h < 0 . 9 5$ .
+
+# Walls.
+
+Type. Dense-reward vision-task.
+
+Description. Walker is required to run through a terrain and avoid the walls. The task-specific information is a vision input. It receives the reward $r$ defined as a difference between the current walker speed $v _ { c u r }$ and the target speed $v _ { t }$ along the direction of the track.
+
+Reward. $r = \exp ^ { - | v _ { c u r } - v _ { t } | ^ { 2 } }$
+
+Technical details. Target speed, $v _ { t } = 3$ . The episode length is 45 seconds. For the humanoid task we use the absolute head height termination criteria: $h < 0 . 9$ .
+
+# Go to one of K single targets.
+
+Type. Sparse-reward feature-based-task.
+
+Description. On an infinite floor, there is a finite area of size $8 \mathrm { x } 8$ with K randomly placed targets. The walker is also randomly placed in a finite area. The walker’s initial position is also randomly placed on the finite area. The walker is required to one of the K targets, specified via command vector. Once it achieves the target, the episode terminates and the walker receives the reward $r$ .
+
+Technical details. The episode length is 20 seconds.
+
+# Go to one moving target.
+
+Type. Sparse-reward feature-based-task.
+
+Description. Similar to the previous one, but there is only one target and once the walker achieves it, the target reappears in a new random place. The walker receives $r$ for 10 consecutive steps staying on the target before the target reappears in a new random position.
+
+Reward. $r = 1$
+
+Technical details. The episode length is 25 seconds.
+
+# Move one box to one of the K targets.
+
+Type. Sparse-reward feature-based-task.
+
+Description. There is a finite floor of size 3x3 padded with walls with K randomly placed targets and one box. The walker is required to move this box to one of the specified targets. Once the box is placed on the target, the episode terminates and the walker receives the reward $r$ .
+
+Technical details. The episode length is 30 seconds. Control timestep is 0.05 for quadruped and 0.025 for jumping ball.
+
+# Move one box to one of the K targets and go to another.
+
+Type. Sparse-reward feature-based-task.
+
+Description. Similar to the previous one, but the walker is also required to go to another target (which is different from the one where it must place the box on). The walker receives the a $r _ { t a s k }$ for each task solved, and a $r _ { e n d }$ if it solves both tasks. The other parameters are the same.
+
+Reward. $r _ { t a s k } = 1 0$ , $r _ { e n d } = 5 0$ .
+
+Technical details. Same as in the previous task.
+
+# Foraging in the maze.
+
+Type. Sparse-reward vision-task.
+
+Description. There is a maze with 8 apples which walker must collect. For each apple, it receives reward $r$ . The episode terminates once the walker collects all the apples or the time is elapsed. Reward. $r = 1$ .
+
+Technical details. The episode length is 90 seconds. Control timestep is 0.025 for jumping ball, and 0.05 for quadruped.
+
+# D ALGORITHMS, BASELINE AND HYPERPARAMETERS
+
+Our agents run in off-policy regime sampling the trajectories from the replay buffer. In practice, it means that the trajectories are coming from the behavior (replay buffer) policy $\pi _ { b }$ , and thus, the correction must be applied (specified below). Below we provide architecture details, baselines, hyperparmaeters as well as algorithm details for discrete and continuous control cases.
+
+# D.1 DISCRETE CASE
+
+In discrete experiments, we use V-trace off-policy correction as in Espeholt et al. (2018). We reuse all the hyperparameters for DMLab from the mentionned paper. At the top of that, we add default policy network and optimize the corresponding $\alpha$ parameter using population-base training. The difference with the setup in Espeholt et al. (2018) is that they use the human prior over actions (table D.2 in the mentionned paper), which results in 9-dimensional action space. In our work, we take the rough DMLab action space, consisting of all possible rotations, and moving forward/backward, and "fire" actions. It results in the action space of dimension 648. It make the learning much more challenging, as it has to explore in much larger space.
+
+# D.2 CONTINUOUS CASE
+
+The agent network (see figure 1) is divided into actor and critic networks without any parameter sharing. In the case of feature-based-task, the task-specific information is encoded by one layer MLP with ELU activations. For the vision-task, we use a 3-layer ResNet He et al. (2015). The encoded task information is then concatenated with the proprioceptive information and passed to the agent network. The actor network encodes a Gaussian policy, $\mathcal { N } ( \tilde { \mu } , \tilde { \sigma } )$ , by employing a two-layer MLP, with mean $\mu$ and log variance $\log \sigma$ as outputs and applying the following processing procedures:
+
+$$
+\tilde { \mu } = t a n h ( \mu ) ,
+$$
+
+$$
+\tilde { \sigma } = 0 . 1 + ( \sigma _ { m a x } - 0 . 1 ) f ( \log \sigma ) ,
+$$
+
+where $f$ is a sigmoid function:
+
+$$
+f ( x ) = { \frac { 1 } { 1 + \exp ^ { - x } } } .
+$$
+
+The critic network is a two-layer MLP and a linear readout. The default policy network has the same structure as actor network, but receives a concatenation of the proprioceptive information with only a subset (potentially, empty) of a task-specific information. There is no parameter sharing between the agent and the default policy. ELU is used as activation everywhere. The exact actor, critic and default policy network architectures are described below. We tried to use LSTM for the default policy network instead of MLP, but did not see a difference. We use separate optimizers and learning rates $\beta _ { \pi } , \beta _ { Q } , \beta _ { \pi ^ { 0 } }$ for the actor, critic and default policy networks correspondingly. For each network (which we call online), we also define the target network, similar to the target $Q$ -networks (Mnih et al., 2015). The target networks are updated are updated in a slower rate than the online ones by copying their parameters.
+
+We assume that the trajectories are coming from the replay buffer $\boldsymbol { B }$ . To correct for being off-policy, we make use of the Retrace operator (see Munos et al. (2016)). This operator is applied to the $Q$ function essentially introducing the importance weights. We will note $\mathcal { R } Q$ the action for this operator. Algorithm 2 is an off-policy version with retraced $Q$ function of the initial algorithm 1.
+
+We use the same update period for actor and critic networks, $P _ { a }$ and a different period for the default network $P _ { d }$ . The baseline is the agent network (see figure 1) without the default policy with an entropy bonus $\lambda$ . All the hyperparameters of the baseline are tuned for each task. For each best baseline hyperparameters configuration, we tune the default policy parameters. When we use the default policy, we do not have the entropy bonus. Instead, we have a regularisation parameter $\alpha$ . The other parameteres which we consider are: batch size, unroll length. Below we provide the hyperparameters for each of the task. The following default hyperparameters are used unless some particular one is specified.
+
+# Default hyperparameters.
+
+Actor learning rate, $\beta _ { \pi } = 0 . 0 0 0 5$ .   
+Critic learning rate, $\beta _ { Q } = 0 . 0 0 0 5$ .   
+Default policy learning rate, $\beta _ { \pi ^ { 0 } } = 0 . 0 0 0 5$ .   
+Agent target network update period: $P _ { a } = 1 0 0$ .   
+Default policy target network update period: $P _ { d } = 1 0 0$ .   
+Actor network: MLP with sizes p300, 200q.   
+Critic network: MLP with sizes p400, 300q.   
+Default policy network: MLP with sizes p300, 200q.   
+Command encoder network: 1-layer MLP of size 50.   
+Image encoder: ResNet with filter sizes p16, 32, 32q.   
+Gaussian policy maximum noise: $\sigma _ { m a x } = 1 . 0$ .   
+Batch size: 512.   
+Unroll length: 10.   
+Entropy bonus: $\lambda = 0 . 0 0 0 1$ .   
+Regularization constant: $\alpha = 0 . 0 1$ .   
+Number of actors: 128.
+
+<table><tr><td>online policy: πO,0o, initial parameters 00 target policy: πT,0r, initial parameters 0T online default policy: πO; initial parameters target default policy: πT,r; nitial parameters online Q-function: Qo,o; initial parameters γo target Q-function: QT,ψr; initial parameters T target update period: P replay buffer: B unroll length: K for j=1,... do</td></tr><tr><td>Sample partial trajectory from replay buffer B: Tt:t+K = (St, at,rt ... Tt+K) compute online KL: KLo,t&#x27; = KL[πo(:|st)|Iπ(*|st)] compute target KL: KLT,t&#x27; = KL[πo(-|st&#x27;)π♀(*|st&#x27;)] Estimate boostrap value: V = Eπr(*|st+κ)[Qr(St+K,α)] -aKLT,t+K</td></tr><tr><td>Estimate Q targets: Qt = rt + V Apply Retrace operator: QR = RQt Q-value loss: LQ = t+K-1 Q-Qo(s,av)²</td></tr><tr><td>∑t&#x27;=t t+K-1KLo,t Default policy loss: Lπo = ∑t&#x27;=t 00←00+βπVθLπ Φ←Φ+βπ∀Lπ 4o ←ψo-βQVLQ</td></tr></table>
+
+# Walking quadruped
+
+Actor network: MLP with sizes p400, 300, 200q.   
+Critic network: MLP with sizes p400, 400, 300q.   
+default policy network: MLP with sizes p400, 300, 200q.   
+Regularization constant: $\alpha = 0 . 0 0 0 1$ .   
+Number of actors: 256.
+
+Walking humanoid Entropy bonus: $\lambda = 0 . 0 0 5$ . Regularization constant: $\alpha = 0 . 0 0 0 1$ . The rest is similar to Walking quadruped.
+
+Walls quadruped Actor and critic learning rate: $\beta _ { \pi } , \beta _ { Q } = 5 e - 5$ .   
+Batch size: 48.   
+Regularization constant: $\alpha = 0 . 0 0 1$ .   
+Number of actors: 64. Walls humanoid Actor and critic learning rate: $\beta _ { \pi } , \beta _ { Q } = 0 . 0 0 0 1$ .   
+Batch size: 48.   
+Regularization constant: $\alpha = 0 . 0 0 1$ .   
+Number of actors: 64.
+
+Go to moving target quadruped Regularization constant: $\alpha = 0 . 0 0 6$ . Number of actors: 32.
+
+Go to moving target humanoid Regularization constant: $\alpha = 0 . 1$
+
+Go to K targets quadruped Actor and critic learning rate: $\beta _ { \pi } , \beta _ { Q } = 0 . 0 0 0 1$ . default policy target network update period: $P _ { d } = 5 0$ . Regularization constant: $\alpha = 0 . 0 0 6$ .
+
+Move 1 box to 1 target jumping ball Default
+
+Move 1 box to 1 target quadruped Actor and critic learning rate: $\beta _ { \pi } , \beta _ { Q } = 0 . 0 0 0 1$ .
+
+Move 1 box to one of 2 targets quadruped Actor and critic learning rate: $\beta _ { \pi } , \beta _ { Q } = 0 . 0 0 0 1$ . default policy target network update period: $P _ { d } = 5 0$ .
+
+Move 1 box to one of 2 targets with go to another one quadruped Same as previous task.
+
+Move 1 box to one of 3 targets quadruped Same as previous task.
+
+# Foraging jumping ball
+
+Actor and critic learning rate: $\beta _ { \pi } , \beta _ { Q } = 0 . 0 0 0 1$ .   
+Actor network: LSTM with one hidden unit of size 128.   
+Critic network: LSTM with one hidden unit of size 128.   
+Batch size: 48.   
+Number of actors: 64. Foraging quadruped   
+Unroll length: 20   
+Regularization constant: $\alpha = 0 . 0 0 6$ .
+
+For the Foraging quadruped task, the initial agent did not learn, so we used a slightly different version of the agent. In algorithm 2, we essentially learn a $Q$ function and update the policy by sampling actions from it and backpropagating through $Q$ . In this algorithm, we learn a value function $V$ using V-trace (Espeholt et al., 2018) and the policy is updated using an off-policy corrected policy gradient with empirical returns.
+
+# E CONTINUOUS CONTROL: ADDITIONAL RESULTS
+
+# E.1 DENSE REWARD TASKS
+
+![](images/6104c75932d43113755c7961fdf2201e5dd1b47f1a99c41b2b89cc98e97ca9e7.jpg)  
+Figure 9: Results for the dense-reward tasks. Starting from left. First: walking quadruped task. Second: walking humanoid task. Third: walls quadruped task. Forth: walls humanoid task. The legends denote additional to the proprioception, information passed to the default policy (except baseline, where we do not use default policy).
+
+The results for the dense-reward tasks are given in figure 9. We observe little difference of using default policy comparing to the baseline. In the walls task, we also consider the default policy with global information, such as the orientation, position and speed in the global coordinates. We do not observe a significant difference between using the default policy and the baseline. The reason for this, we believe, is that the agent is being trained very quickly by seeing a strong reward signal, so the default policy cannot catch it up.
+
+# E.2 SPARSE REWARD JUMPING BALL
+
+The results for the sparse reward tasks with jumping ball are given in figure 9. We see little difference of using default policy comparing to the baseline. Our hypothesis consists in the fact that since the default policy affects the policy by regularizing the state-conditional action distribution (policy), for too simple actions space such is given here (3 actions), this effect is not strong enough.
+
+![](images/4d911905eee69c7611698278497097e40d1a0142303943e57407905b34ff459b.jpg)  
+Figure 10: Results for sparse-reward tasks with jumping ball walker.Left: go to moving target. Center: moving one box to one target. Right: foraging in the maze. The legends denote additional to the proprioception, information passed to the default policy (except baseline, where we do not use default policy).
+
+# E.3 SPARSE REWARD WITH QUADRUPED ADDITIONAL RESULTS
+
+In this section, we provide more results for the sparse reward tasks. In figure 11 the results for going to one of $\mathbf { K }$ targets task with quadruped are presented. The proprioceptive default policy gives significant gains comparing to others. What interesting is that when the number of targets $K$ increases, the baseline performance drops dramatically, whereas the proprioceptive default policy solve the task reliably. Our hypothesis is that the default policy learns quickly the default walking behavior which becomes very helpful for the agent to explore the floor and search for the target.
+
+![](images/a4cb31268dfc077a9c903d583736a2c92830c40f4bf490f6c7e302bb7c6a5812.jpg)  
+Figure 11: Results for go to one of K targets tasks with quadruped. Left: go to 1 target. Center: go to one of 2 targets. Right: go to one of 3 targets. The legends denote additional to the proprioception, information passed to the default policy (except baseline, where we do not use default policy).
+
+We also provide the results for move box to one of K targets task, where $K = 1 , 2 , 3$ , and move box to one of two targets task with go to another. The results are given in figure 12. Similar effect occurs here.
+
+![](images/f0bd84d898001429eade9156f541936109f9431ab0893daf30042f87faf7b4c9.jpg)  
+Figure 12: Results for box pushing tasks with quadruped. Starting from left, first: move one box to one of 2 targets with go to another. Second: move one box to 1 target. Third: move one box to one of 2 targets. Forth: move one box to one of 3 targets. The legends denote additional to the proprioception, information passed to the default policy (except baseline, where we do not use default policy).
+
+# E.4 ADDITIONAL TRANSFER RESULTS
+
+In this section, we provide additional transfer experiment results for the range of the tasks. They are given in figure 13. In the first two cases we see that proprioceptive default policy from the go to target task gives a significant boost to the performance comparing to the learning from scratch. We also observe, that for the box pushing tasks, the default policy with the box position significantly speeds up learning comparing to other cases. We believe it happens because this default policy learns the best default behavior for these tasks possible: going to the box and push it. For the most complicated task, move one box to one of two targets and go to another one, 13, right, the box default policy makes a big difference: it makes the policy avoid being stuck in go to target behavior (line with reward of 10).
+
+Additional results for the transfer experiments are given in figure 13. We observe the same effect happening: whereas the baseline performance drops significantly, the agent with default policy stays
+
+# E.5 ABLATION WALLS QUADRUPED
+
+Ablations for the walls quadruped are given in figure 14.
+
+![](images/e9bad263ceeccd6fa20b10cd75ccab9dc38e1a7c2f5b92487574fa2b87087d25.jpg)  
+Figure 13: Performance of the transfer with quadruped walker. Left: Go to one of 3 targets. Center: move one box to one of two targets. Right: move one box to one of two targets and go to another one. The legend whether the default policy is learned or is transferred. Furthermore, it specifies the task from which the default policy is transferred as well as additional information other than the proprioceptive information that the default policy is conditioned on, if any.
+
+![](images/9d95082258933e9c66a0155ede943cfdb3b41c6896fd53320a99a96d9bb803c0.jpg)  
+Figure 14: Ablations for walls task with quadruped. Left: Comparing various regularization schemes. Right: Optimistic baselines comparing pretrained default policies.
+
+# E.6 ORDER OF THE DEFAULT POLICY IN THE KL-TERM
+
+The results for having the different order of the default policy in the KL-term $( K L [ \pi | | \pi ^ { 0 } ]$ or $K L [ \pi ^ { 0 } | | \pi ] )$ for go to moving target task with quadruped walker are shown in figure 15. We use this term either in per time step actor loss (auxiliary loss) or as a regularizer by optimizing the objective 1 (with different order of KL). We do not observe significant difference.
+
+![](images/3073797c7c0fc741f2c198b1eb30d969837b32660ed89b705946972b4af12feb.jpg)  
+Figure 15: KL direction results for go to moving target task with quadruped.

parse/train/iAmZUo0DxC0/iAmZUo0DxC0.md

@@ -0,0 +1,389 @@
+# UNLEARNABLE EXAMPLES: MAKING PERSONALDATA UNEXPLOITABLE
+
+Hanxun Huang1 Xingjun $\mathbf { M } \mathbf { a } ^ { 2 \dagger }$ Sarah Monazam Erfani1 James Bailey1 Yisen Wang3†
+
+1The University of Melbourne, VIC, Australia   
+2Deakin University, Geelong, VIC, Australia   
+3Key Lab. of Machine Perception (MoE), School of EECS, Peking University, Beijing, China
+
+# ABSTRACT
+
+The volume of “free” data on the internet has been key to the current success of deep learning. However, it also raises privacy concerns about the unauthorized exploitation of personal data for training commercial models. It is thus crucial to develop methods to prevent unauthorized data exploitation. This paper raises the question: can data be made unlearnable for deep learning models? We present a type of error-minimizing noise that can indeed make training examples unlearnable. Error-minimizing noise is intentionally generated to reduce the error of one or more of the training example(s) close to zero, which can trick the model into believing there is “nothing” to learn from these example(s). The noise is restricted to be imperceptible to human eyes, and thus does not affect normal data utility. We empirically verify the effectiveness of error-minimizing noise in both sample-wise and class-wise forms. We also demonstrate its flexibility under extensive experimental settings and practicability in a case study of face recognition. Our work establishes an important first step towards making personal data unexploitable to deep learning models. Code is available at https://github.com/HanxunH/Unlearnable-Examples.
+
+# 1 INTRODUCTION
+
+In recent years, deep learning has had groundbreaking successes in several fields, such as computer vision (He et al., 2016) and natural language processing (Devlin et al., 2018). This is partly attributed to the availability of large-scale datasets crawled freely from the Internet such as ImageNet (Russakovsky et al., 2015) and ReCoRD (Zhang et al., 2018b). Whilst these datasets provide a playground for developing deep learning models, a concerning fact is that some datasets were collected without mutual consent (Prabhu & Birhane, 2020). Personal data has also been unconsciously collected from the Internet and used for training commercial models (Hill, 2020). This has raised public concerns about the “free” exploration of personal data for unauthorized or even illegal purposes.
+
+In this paper, we address this concern by introducing unlearnable examples, which aims at making training examples unusable for Deep Neural Networks (DNNs). In other words, DNNs trained on unlearnable examples will have a performance equivalent to random guessing on normal test examples. Compared with preserving an individual’s privacy by obfuscating information from the dataset, what we aim to achieve here is different but more challenging. First, making an example unlearnable should not affect its quality for normal usage. For instance, an unlearnable “selfie” photo should be free from obvious visual defects so it can be used as a social profile picture. Ideally, this can be achieved by using imperceptible noise. In our setting, the noise can only be added to training examples on a single occasion (when the data is uploaded to the internet) prior to model training. However, DNNs are known to be robust to small noise either random (Fawzi et al., 2016) or adversarial (Szegedy et al., 2013; Goodfellow et al., 2014; Ma et al., 2018). It is still not clear whether small, imperceptible noise can stop the training of high-performance DNNs.
+
+The development of unlearnable examples should take full advantage of the unique characteristics, and more importantly, the weaknesses of DNNs. One well-studied characteristic of DNNs is that they tend to capture more of the high-frequency components of the data (Wang et al., 2020a). Surprisingly, by exploiting this characteristic, we find that small random noise when applied in a class-wise manner to the training data can easily fool DNNs to overfit to such noise (shown in Section 4). However, early stopping can effectively counteract this type of noise. DNNs are also known to be vulnerable to adversarial (or error-maximizing) noise, which are small perturbations crafted to maximize the model’s error at the test time (Szegedy et al., 2013; Goodfellow et al., 2014). We find that error-maximizing noise cannot stop DNN learning when applied in a sample-wise manner to the training examples. This motivates us to explore the opposite direction to error-maximizing noise. Specifically, we propose a type of error-minimizing noise that can prevent the model from being penalized by the objective function during training, and thus can trick the model into believing there is “nothing” to learn from the example(s). We refer to an example that contains the errorminimizing noise as an unlearnable example. Error-minimizing noise can be generated in different forms: sample-wise and class-wise. Class-wise error-minimizing noise is superior to random noise and cannot be circumvented by early stopping. Sample-wise error-minimizing noise is the only effective noise that can make training examples unlearnable compared to random (Fawzi et al., 2016) or error-maximizing noise (Munoz-Gonz ˜ alez et al., 2017). Our main contributions are: ´
+
+• We present a type of error-minimizing noise that can create unlearnable examples to prevent personal data from being freely exploited by deep learning models. The noise is small, imperceptible to human eyes, thus it does not reduce general data utility. We propose a bi-level optimization process to effectively generate different forms of errorminimizing noise: sample-wise and class-wise. We empirically verify the effectiveness and flexibility of error-minimizing noise for creating unlearnable examples. We also demonstrate the practical application of unlearnable examples in real-world scenarios via a case study on face recognition.
+
+# 2 RELATED WORK
+
+In this section, we briefly review most relevant works in data privacy, data poisoning, adversarial attacks against deep learning models.
+
+Data Privacy. Privacy issues have been extensively studied in the field of privacy-preserving machine learning (Shokri & Shmatikov, 2015; Abadi et al., 2016; Phan et al., 2016; 2017; Shokri et al., 2017). While these works have made significant progress towards protecting data privacy, they are developed based on the assumption that the model can freely explore the training data and turn to protect the model from leaking sensitive information about the training data. In this paper, we consider a more challenging scenario where the goal of the defender is to make personal data completely unusable by unauthorized deep learning models. Fawkes (Shan et al., 2020) has made the first attempt towards this type of strict situation. By leveraging the targeted adversarial attack, Fawkes prevents unauthorized face tracker from tracking a person’s identity. This work is similar to ours as we share a common objective that prevents unauthorized data usage. In contrast to the targeted adversarial attack, we propose a novel error-minimizing noise to produce unlearnable examples which can be used as a generic framework for a wide range of data protection tasks.
+
+Data Poisoning. Data poisoning attacks aim to degrade the model’s performance on clean examples by modifying the training examples. Previous work has demonstrated a poisoning attack on SVM (Biggio et al., 2012). Koh & Liang (2017) proposed to poison the most influential training examples using adversarial (error-maximizing) noise against DNNs, which has also been integrated into an endto-end framework (Munoz-Gonz ˜ alez et al., 2017). Although data poisoning attacks can potentially ´ prevent free data exploitation, these approaches are quite limited against DNNs and hard to operate in real-world scenarios. For example, poisoned examples can only slightly decrease DNNs’ performance (Munoz-Gonz ˜ alez et al., 2017), and often appear distinguishable to clean examples (Yang et al., ´ 2017) which will reduce normal data utility. The backdoor attack is another type of attack that poisons training data with a stealthy trigger pattern (Chen et al., 2017; Liu et al., 2020). However, the backdoor attack does not harm the model’s performance on clean data (Chen et al., 2017; Shafahi et al., 2018; Barni et al., 2019; Liu et al., 2020; Zhao et al., 2020). Thus, it is not a valid method for data protection. Different from these works, we generate unlearnable examples with invisible noise to “bypass” the training of DNNs.
+
+Adversarial Attack. It has been found that adversarial examples (or attacks) can fool DNNs at the test time (Szegedy et al., 2013; Goodfellow et al., 2014; Kurakin et al., 2016; Carlini & Wagner, 2017; Madry et al., 2018; Jiang et al., 2019; Wu et al., 2020a; Bai et al., 2020; Croce & Hein, 2020; Wang et al., 2020b; Duan et al., 2020; Ma et al., 2020). The adversary finds an error-maximizing noise that maximizes the model’s prediction error, and the noise can be crafted universally for the entire test set (Moosavi-Dezfooli et al., 2017). Adversarial training has been shown to be the most robust training strategy against error-maximizing noise (Madry et al., 2018; Zhang et al., 2019; Wang et al., 2019; Wu et al., 2020b; Wang et al., 2020c). Adversarial training can be formulated as a min-max optimization problem. In this paper, we explore the opposite direction of error-maximizing noise, i.e., finding small noise that minimizes the model’s error via a min-min optimization process.
+
+# 3 UNLEARNABLE EXAMPLES AND ERROR-MINIMIZING NOISE
+
+# 3.1 PROBLEM STATEMENT
+
+Assumptions on Defender’s Capability. We assume the defender has full access to the portion of data which they want to make unlearnable. However, the defender cannot interfere with the training process and does not have access to the full training dataset. In other words, the defender can only transform their portion of data into unlearnable examples. Moreover, the defender cannot further modify their data once the unlearnable examples are created.
+
+Objectives. We formulate the problem in the context of image classification with DNNs. Given a typical $K$ -class classification task, we denote the clean training and test datasets as $\mathcal { D } _ { c }$ and $\mathcal { D } _ { t }$ respectively, and the classification DNN trained on $\mathcal { D } _ { c }$ as $f _ { \theta }$ where $\theta$ are the parameters of the network∗. Our goal is to transform the training data $\mathcal { D } _ { c }$ into unlearnable dataset $\mathcal { D } _ { u }$ such that DNNs trained on the $\mathcal { D } _ { u }$ will perform poorly on the test set $\mathcal { D } _ { t }$ .
+
+Suppose the clean training dataset consists of $n$ clean examples, that is, $\mathcal { D } _ { c } = \{ ( \boldsymbol { x } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ with $\pmb { x } \in \mathcal { X } \subset \mathbb { R } ^ { d }$ are the inputs and $y \in \mathcal { Y } = \{ 1 , \cdots , K \}$ are the labels and $K$ is the total number of classes. We denote its unlearnable version by $\mathcal { D } _ { u } \overset { ^ { \prime } } { = } \{ ( \mathbf { { x } } _ { i } ^ { \prime } , y _ { i } ) \} _ { i = 1 } ^ { n }$ , where ${ \pmb x } ^ { \prime } = { \pmb x } + { \pmb \delta }$ is the unlearnable version of training example $\mathbf { \boldsymbol { x } } \in \mathcal { D } _ { c }$ and $\pmb { \delta } \in \Delta \subset \mathbb { R } ^ { d }$ is the “invisible” noise that makes $_ { \textbf { \em x } }$ unlearnable. The noise $\delta$ is bounded by $\| \delta \| _ { p } \leq \epsilon$ with $\| \cdot \| _ { p }$ is the $L _ { p }$ norm, and $\epsilon$ is set to be small such that it does not affect the normal utility of the example.
+
+In the typical case, the DNN model will be trained on $\mathcal { D } _ { c }$ to learn the mapping from the input space to the label space: $f : \mathcal { X }  \mathcal { Y }$ . Our goal is to trick the model into learning a strong correlation between the noise and the labels: $f : \Delta \to \mathcal { V } , \Delta \neq \mathcal { X }$ , when trained on $\mathcal { D } _ { u }$ :
+
+$$
+\underset { \theta } { \arg \operatorname* { m i n } } \mathbb { E } _ { ( \pmb { x } ^ { \prime } , \pmb { y } ) \sim \mathcal { D } _ { \pmb { u } } } \mathcal { L } ( f ( \pmb { x } ^ { \prime } ) , \pmb { y } )
+$$
+
+where, $\mathcal { L }$ is the classification loss such as the commonly used cross entropy loss.
+
+Noise Form. We propose two forms of noise: sample-wise and class-wise. For sample-wise noise, $\pmb { x } _ { i } ^ { \prime } = \pmb { x } _ { i } + \pmb { \delta } _ { i } , \pmb { \delta } _ { i } \in \bar { \Delta } _ { s } = \{ \pmb { \delta } _ { 1 } , \cdots , \pmb { \delta } _ { n } \}$ , while for class-wise noise, $\pmb { x } _ { i } ^ { \prime } = \pmb { x } _ { i } + \delta _ { y _ { i } } ^ { - } , \delta _ { y _ { i } } \in \Delta _ { c } =$ $\{ \delta _ { 1 } , \cdots , \delta _ { K } \}$ . Sample-wise noise needs to generate noise separately for each example. This may have more limited practicality. In contrast to sample-wise noise, a class of examples can be made unlearnable by the addition of class-wise noise, where all examples in the same class have the same noise added. As such, class-wise noise can be generated more efficiently and more flexibly in practical usage. However, we will see that class-wise noise may get more easily exposed.
+
+# 3.2 GENERATING ERROR-MINIMIZING NOISE
+
+Ideally, the noise should be generated on an additional dataset that is different from $\mathcal { D } _ { c }$ . This will involve a class-matching process to find the most appropriate class from the additional dataset for each class to protect in $\mathcal { D } _ { c }$ . For simplicity, here we define the noise generation process on $\mathcal { D } _ { c }$ and will verify the effectiveness of using an additional dataset in the experiments. Given a clean example $_ { \textbf { \em x } }$ , we propose to generate the error-minimizing noise $\delta$ for training input $_ { \textbf { \em x } }$ by solving the following bi-level optimization problem:
+
+$$
+\arg \operatorname* { m i n } _ { \theta } \ \mathbb { E } _ { ( \pmb { x } , \pmb { y } ) \sim \mathcal { D } _ { c } } \Big [ \operatorname* { m i n } _ { \pmb { \delta } } \mathcal { L } \big ( f ^ { \prime } ( \pmb { x } + \pmb { \delta } ) , \pmb { y } \big ) \Big ] \ \mathrm { ~ s . t . ~ } \ \| \pmb { \delta } \| _ { p } \leq \epsilon
+$$
+
+where, $f ^ { \prime }$ denotes the source model used for noise generation. Note that this is a min-min bi-level optimization problem: the inner minimization is a constrained optimization problem that finds the
+
+$L _ { p }$ -norm bounded noise $\delta$ that minimizes the model’s classification loss, while the outer minimization problem finds the parameters $\theta$ that also minimize the model’s classification loss.
+
+Note that the above bi-level optimization has two components that optimize the same objective. In order to find effective noise $\pmb { \delta }$ and unlearnable examples, the optimization steps for $\theta$ should be limited, compared to standard or adversarial training. Specifically, we optimize $\pmb { \delta }$ over $\mathcal { D } _ { c }$ after every $M$ steps of optimization of $\theta$ . The entire bi-level optimization process is terminated once the error rate is lower than $\lambda$ . The detailed training pipeline is described in Algorithm 1 in the Appendix.
+
+Sample-wise Generation. We adopt the first-order optimization method PGD (Madry et al., 2018) to solve the constrained inner minimization problem as follows:
+
+$$
+\begin{array} { r } { \pmb { x } _ { t + 1 } ^ { \prime } = \Pi _ { \epsilon } \big ( \pmb { x } _ { t } ^ { \prime } - \alpha \cdot \mathrm { s i g n } ( \nabla _ { \pmb { x } } \mathcal { L } ( f ^ { \prime } ( \pmb { x } _ { t } ^ { \prime } ) , y ) ) \big ) } \end{array}
+$$
+
+where, $t$ is the current perturbation step $T$ steps in total), $\nabla _ { \pmb { x } } \mathcal { L } ( f ( \pmb { x } _ { t } ^ { \prime } ) , y )$ is the gradient of the loss with respect to the input, $\Pi$ is a projection function that clips the noise back to the $\epsilon$ -ball around the original example $_ { \textbf { \em x } }$ when it goes beyond, and $\alpha$ is the step size. The perturbation is iteratively applied for $T$ steps after each $M$ step of model training as we explained earlier. The final output is a unlearnable example $\mathbf { x } ^ { \prime }$ and the generated error-minimizing noise is $\delta = \boldsymbol { x } ^ { \prime } - \boldsymbol { x }$ .
+
+Class-wise Generation. Class-wise noise $\Delta _ { c }$ can be obtained by a cumulative perturbation on all examples in a given class. For each example in class $k$ at step $t$ , it applies $\delta _ { k }$ to the original example $_ { \textbf { \em x } }$ and follows Equation 3 to produce $\pmb { x } _ { t + 1 } ^ { \prime }$ . The $\delta _ { k }$ accumulates over every example for the corresponding class $k$ in the entire bi-level optimization process.
+
+# 4 EXPERIMENTS
+
+In this section, we first demonstrate the effectiveness in creating unlearnable examples using random noise, error-maximizing noise and our proposed error-minimizing noise in both sample-wise and class-wise forms. We further empirically verify the effectiveness of error-minimizing noise on 4 benchmark image datasets. We then conduct a set of stability and transferability analyses of the noise. Finally, we show effectiveness in real-world scenarios via a case study on face recognition. More analyses regarding the effectiveness of error-minimizing noise on small patches or a mixture of class-wise noise can be found in Appendix F.
+
+According to previous studies in adversarial research, small $L _ { \infty }$ -bounded noise within $\| \delta \| _ { \infty } < \epsilon =$ 8/255 on images are imperceptible to human observers. We consider the same constraint for all types and forms of the noise in our experiments, unless otherwise explicitly stated.
+
+Experimental Setting for Error-Minimizing Noise. We apply error-minimizing noise to the training set of 4 commonly used image datasets: SVHN (Netzer et al., 2011), CIFAR-10, CIFAR-100 (Krizhevsky, 2009), and ImageNet subset (the first 100 classes) (Russakovsky et al., 2015). The experiments on the ImageNet subset are to confirm the effectiveness of high-resolution images. For all experiments, the ResNet-18 (RN-18) (He et al., 2016) is used as the source model $f ^ { \prime }$ to generate the noise. We use $20 \%$ of the training dataset to generate the class-wise noise and the entire training dataset for the sample-wise noise except for ImageNet †. We transform the entire training dataset into the unlearnable datasets for experiments in section 4.1 and section 4.2. Different percentages of unlearnable examples are used for experiments in section 4.3. We train four different DNNs on the unlearnable training sets: VGG-11 (Simonyan & Zisserman, 2014), ResNet-18 (RN-18), ResNet-50 (RN-50) and DenseNet-121 (DN-121) (Huang et al., 2017). We also use clean training sets as a comparison. Detailed training configurations settings can be found in Appendix B. We evaluate the effectiveness of unlearnable examples by examining the model’s accuracy on clean test examples, i.e., the lower the clean test accuracy the better the effectiveness.
+
+Experimental Setting for Random and Error-Maximizing Noise. For random noise, we randomly sample the noise from $[ - \epsilon , \epsilon ]$ independently for each training example (eg. sample-wise) or each class (eg. class-wise). For error-maximizing (adversarial) noise, we generate the noise using PGD-20 attack (Madry et al., 2018) using a pre-trained ResNet-18 model on the training set. We generate sample-wise error-maximizing noise for each training example, and the class-wise noise based on $20 \%$ of the training set following the universal attack procedure in (Moosavi-Dezfooli et al., 2017).
+
+Both the random and error-maximizing noise are applied to the same amount of training examples as our error-minimizing noise.
+
+# 4.1 COMPARISONS OF RANDOM, ERROR-MAXIMIZING AND ERROR-MINIMIZING NOISE
+
+First, we examine an extreme case that applies different types of noise to the entire training set. Figure 1 illustrates the effectiveness of both sample-wise and class-wise noise. In the sample-wise case, the network is robust to random or error-maximizing noise. This is understandable since DNNs are known to be robust to small random (Fawzi et al., 2016) and error-maximizing noise (Munoz-Gonz ˜ alez et al., ´ 2017; Madry et al., 2018; Wang et al., 2019). Surprisingly, when applied in a class-wise manner, both types of noise can prevent the network from learning useful information from the data, especially after the 15-th epoch. This reveals that DNNs are remarkably vulnerable to class-wise noise. While effective in the middle and later training stages, class-wise random noise can still be circumvented by early stopping (eg. at epoch 15). This is also the case for error-maximizing noise, although not as easy as random noise since the highest clean test accuracy under error-maximizing noise is only $50 \%$ . From the perspective of making data unexploitable, both random and error-maximizing noise are only partially effective. In comparison, our error-minimizing noise is more flexible. As shown in Figure 1, the error-minimizing noise can reduce the model’s clean test accuracy to below $23 \%$ in both settings. Moreover, it remains effective across the entire training process.
+
+![](images/c8773dfab899b3f6cade9665512f2dc8aa5e200f407e61fbd6db9be9ab6a0e96.jpg)  
+Figure 1: The unlearnable effectiveness of different types of noise: random, adversarial (errormaximizing) and our proposed error-minimizing noise on CIFAR-10 dataset. The lower the clean test accuracy the more effective of the noise.
+
+The class-wise and sample-wise noises work in different ways. Class-wise noise has an explicit correlation with the label. Learning such correlation can effectively reduce the training error. Consequently, when there is class-wise noise, the model is tricked to learn the noise rather than the real content, reducing its generalization performance on clean data. The existence of class-wise noise only in the training data also breaks the i.i.d. assumption between the training and test data distribution. This also indicates that noises that can break the i.i.d. assumption might be effective techniques for data protection. However, in the sample-wise case, every sample has a different noise, and there is no explicit correlation between the noise and the label. In this case, only low-error samples can be ignored by the model, and normal and high-error examples have more positive impact on model learning than low-error examples. This makes error-minimizing noise more generic and effective in making data unlearnable.
+
+# 4.2 EFFECTIVENESS OF ERROR-MINIMIZING NOISE ON DIFFERENT DATASETS
+
+Table 1: The top-1 clean test accuracies $( \% )$ of DNNs trained on the clean training sets $( \mathcal { D } _ { c } )$ or their unlearnable ones $( \mathcal { D } _ { u } )$ made by sample-wise $( \Delta _ { s } )$ or class-wise $( \Delta _ { c } )$ error-minimizing noise.   
+
+<table><tr><td rowspan="2">Noise Form</td><td rowspan="2">Model</td><td colspan="2">SVHN</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td><td colspan="2">ImageNet*</td></tr><tr><td>Dc</td><td>Du</td><td>Dc</td><td>Du</td><td>Dc</td><td>Du</td><td>Dc</td><td>Du</td></tr><tr><td rowspan="4">△s</td><td>VGG-11</td><td>95.38</td><td>35.91</td><td>91.27</td><td>29.00</td><td>67.67</td><td>17.71</td><td>48.66</td><td>11.38</td></tr><tr><td>RN-18</td><td>96.02</td><td>8.22</td><td>94.77</td><td>19.93</td><td>70.96</td><td>14.81</td><td>60.42</td><td>12.20</td></tr><tr><td>RN-50</td><td>95.97</td><td>7.66</td><td>94.42</td><td>18.89</td><td>71.32</td><td>12.19</td><td>61.58</td><td>11.12</td></tr><tr><td>DN-121</td><td>96.37</td><td>10.25</td><td>95.04</td><td>20.25</td><td>74.15</td><td>13.71</td><td>63.76</td><td>15.44</td></tr><tr><td rowspan="4">△c</td><td>VGG-11</td><td>95.29</td><td>23.44</td><td>91.57</td><td>16.93</td><td>67.89</td><td>7.13</td><td>71.38</td><td>2.30</td></tr><tr><td>RN-18</td><td>95.98</td><td>9.05</td><td>94.95</td><td>16.42</td><td>70.50</td><td>3.95</td><td>76.52</td><td>2.70</td></tr><tr><td>RN-50</td><td>96.25</td><td>8.94</td><td>94.37</td><td>13.45</td><td>70.48</td><td>3.80</td><td>79.68</td><td>2.70</td></tr><tr><td>DN-121</td><td>96.36</td><td>9.10</td><td>95.12</td><td>14.71</td><td>74.51</td><td>4.75</td><td>80.52</td><td>3.28</td></tr></table>
+
+? ImageNet subset of the first 100 classes.
+
+Table 1 reports the effectiveness of both the sample-wise (eg. $\Delta _ { s }$ ) and the class-wise (eg. $\Delta _ { c , \ - }$ ) errorminimizing noise on datasets SVHN, CIFAR-10, CIFAR-100 and ImageNet subset. As shown in the table, our proposed method can reliably create unlearnable examples in both forms on all 4 datasets with images of different resolutions. Moreover, the noise generated on RN-18 works remarkably well to protect the data from other types of models. Compared to sample-wise noise, class-wise noise is particularly more effective, which can reduce the model’s performance to a level that is close to random guessing. These results clearly show that error-minimizing noise is a promising technique for preventing unauthorized data exploration.
+
+# 4.3 STABILITY ANALYSIS
+
+We run a set of experiments to analyze the stability of error-minimizing noise in creating unlearnable examples, and answer two key questions regarding its practical usage: 1) Is the noise still effective if only applied to a certain proportion or class of the data? and 2) Can the noise be removed by data augmentation or adversarial training? Experiments are conducted on CIFAR-10 with RN-18.
+
+Different Unlearnable Percentages. In practical scenarios, it is very likely that not all the training data need to be made unlearnable. For example, only a certain number of web users have decided to use this technique but not all users, or only a specific class of medical data should be kept unexploited. This motivates us to examine the effectiveness of error-minimizing noise when applied only on a proportion of randomly selected training examples. In other words, we make a certain percentage of the training data unlearnable while keeping the rest of the data clean. We train the model on this partially unlearnable and partially clean training set $\mathcal { D } _ { u } + \mathcal { D } _ { c }$ . As a comparison, we also train the model on only the clean proportion, which is denoted by $\mathcal { D } _ { c }$ .
+
+A quick glance at the $\mathcal { D } _ { u } + \mathcal { D } _ { c }$ results in Table 2 tells us that the effectiveness drops quickly when the data are not made $100 \%$ unlearnable. This is the case for both sample-wise and class-wise noise. The unlearnable effect is almost negligible even when the noise is applied to $40 \%$ of the data. Such a limitation against DNNs has also been identified in previous work for protecting face images using error-maximizing noise (Shan et al., 2020).
+
+To better understand the above limitation, we take $80 \%$ as an example and plot the learning curves of the RN-18 model trained on 1) only the $20 \%$ clean proportion, 2) only the $80 \%$ unlearnable proportion, or 3) both. The results are shown in Figure 2 (a-b). Interestingly, we find that the $80 \%$ data with the error-minimizing noise are still unlearnable to the model, whereas the rest of the $20 \%$ clean data are sufficient for the model to achieve a good performance. In other words, models trained only on $\mathcal { D } _ { c }$ demonstrate a similar performance as models trained on $\mathcal { D } _ { u } + \mathcal { D } _ { c }$ . This phenomenon is consistent across different unlearnable percentages. This indicates that the high performance of the model on $< 1 0 0 \%$ unlearnable dataset may not be a failure of the error-minimizing noise. We further verify this by investigating the scenario where only one class is made unlearnable.
+
+Table 2: Effectiveness under different unlearnable percentages on CIFAR-10 with RN-18 model: lower clean accuracy indicates better effectiveness. $\mathcal { D } _ { u } + \mathcal { D } _ { c }$ : a mix of unlearnable and clean data; $\mathcal { D } _ { c }$ : only the clean proportion of data. Percentage of unlearnable examples: $\frac { \mathcal { D } _ { u } } { \mathcal { D } _ { c } + \mathcal { D } _ { u } }$ .   
+
+<table><tr><td rowspan="2">Noise Type</td><td colspan="10">Percentage of unlearnable examples</td></tr><tr><td>0%</td><td colspan="2">20%</td><td colspan="2">40%</td><td colspan="2">60%</td><td colspan="2">80%</td><td>100%</td></tr><tr><td>△s</td><td></td><td>Du+Dc</td><td>Dc</td><td>Du+Dc</td><td>Dc</td><td>Du+Dc</td><td>Dc</td><td>Du+Dc</td><td>Dc</td><td></td></tr><tr><td>c</td><td>94.95 94.95</td><td>94.38 94.24</td><td>93.75 93.75</td><td>93.10 92.99</td><td>92.56 92.56</td><td>91.90 91.10</td><td>89.77 89.77</td><td>86.85 87.23</td><td>84.30 84.30</td><td>19.93 16.42</td></tr></table>
+
+One Single Unlearnable Class. We take the ‘bird’ class of CIFAR-10 as an example, and apply the error-minimizing noise (either sample-wise or class-wise) to all training images in the bird class. We train RN-18 on each of the unlearnable training set, and plot the prediction confusion matrix of the model on the clean test set in Figure 2 (c-d). The ‘bird’ class is indeed unlearnable when either the sample-wise or the class-wise error-minimizing noise is added to the class. Compared to sample-wise noise, class-wise noise is more effective with almost all the test ‘bird’ images being misclassified into other classes. Interestingly, this customized unlearnable class does not seem to influence much of the learning on other classes. Only the images from the unlearnable class are incorrectly predicted into other classes, not the other way around. This not only confirms that the unlearnable group of data is indeed unlearnable to the model and it suggests that our error-minimizing noise can be flexibly applied to suit different protection tasks. A similar result can also be demonstrated on more than one unlearnable class (see Appendix C). In summary, if an individual can only apply the noise to portion of his/her data, these data will not contribute to model training. If an individual can apply the noise to all his/her data, the model will fail to recognize this particular class of data. In other words, our method is effective for the defender to protect his/her own data.
+
+![](images/72eac76deabf416f20433e7e886f2b0800b71a8258e37dab283aff52f02ce0f6.jpg)  
+Figure 2: (a-b): For both sample-wise (a) and class-wise (b) noise, learning curves of RN-18 on CIFAR-10 dataset with different types of training data: 1) only $20 \%$ clean data, 2) only $80 \%$ unlearnable data, and 3) both clean and unlearnable data. (c-d): Prediction confusion matrices (on the clean test set) of two RN-18s trained on CIFAR-10 with the ‘Bird’ unlearnable class created by sample-wise (c) or class-wise (d) error-minimizing noise.
+
+Under Noise Filtering Techniques. The resistance of our error-minimizing noise to 4 types of data augmentation techniques and adversarial training is analyzed in Appendix D. The noise is fairly robust to all 4 data augmentation techniques, and the highest accuracy that can achieve using data augmentation is $5 8 . 5 1 \%$ on CIFAR-10. Comparing with data augmentation, the error-minimizing noise is less resistant to adversarial training. With slightly increased $\epsilon$ on CIFAR-10, the noise can only compromise the model’s clean test accuracy to $79 \%$ . Since adversarial training forces the model to learn only robust features (Ilyas et al., 2019), we believe our method can be improved by crafting the noise based on the robust features, which can be extracted from an adversarially pre-trained model on the clean data (Ilyas et al., 2019). We leave further explorations of these techniques as future work.
+
+# 4.4 TRANSFERABILITY ANALYSIS
+
+Another important question we haven’t explored so far is that Can error-minimizing noise be generated on a different dataset? This is also important as a positive answer to the question increases the practicability of using unlearnable examples to protect millions of web users. We examine this capability for both sample-wise noise and class-wise error-minimizing noise.
+
+![](images/73ad53bc84743f728f7193d93598faff5b11474ec00ed0813a1f8decdb8bbaed.jpg)  
+Figure 3: (a): Comparison between ‘bus’ (clean) class, ‘ship’ (unlearnable) class and the overall accuracy. (b): Clean test accuracy of RN-18/RN-50/DN-121 on unlearnable CIFAR-10 with errorminimizing noise crafted on ImageNet. (c): Prediction confusion matrix of RN-18 trained on CIFAR10 with only 4 classes (‘airplane’, ‘car’, ‘ship’, ‘truck’) are unlearnable by ImageNet transferred noise, and the confusion matrix is computed on CIFAR-10 clean test set.
+
+Sample-wise Noise. For $\Delta _ { s }$ , we generate the noise and create unlearnable ‘ship’ class examples on CIFAR-10 and add the unlearnable ‘ship’ class to CIFAR-100. For testing, we also include the clean test set of ‘ship’ class to the test set of CIFAR-100. Note the main difference of this experiment to the previous singe unlearnable class experiment is the unlearnable examples are generated on a different dataset. The class-wise and the overall accuracy on the clean test set are shown in Figure 3a. We find that the ‘ship’ class is indeed unlearnable to the model at the end, despite a small amount $( 6 0 \%$ clean test accuracy) of information being learned in the early stage. We suspect this is because the ‘ship’ class shares some common features with other classes in CIFAR-100. Overall, the generated unlearnable examples can transfer to a different dataset. Note that this experiment simulates the scenario where a user uses a different dataset (eg. CIFAR-10) to generate the sample-wise noise to make his/her personal images unlearnable before uploading them to the Internet, and these images are then collected into a large CIFAR-100 dataset.
+
+Class-wise Noise. For $\Delta _ { c }$ , we use the entire ImageNet as the source dataset to generate the noise using RN-18, then apply the noise to CIFAR-10. All ImageNet images are resized to $3 2 \times 3 2$ . For each CIFAR-10 class, we determine its corresponding class in ImageNet based on the visual similarities between the classes. The detailed class mapping can be found in Table 4 in the Appendix. We train RN-18, RN-50 and DN-121 on the unlearnable CIFAR-10 created with the ImageNet transferred noise, and show their learning curves in Figure 3b. As can be observed, the transferred class-wise noise works reasonably well on CIFAR-10 with the clean test accuracy of all three models being reduced to around $10 \%$ . Compared to the noise directly generated on CIFAR-10 (see Figure 1), here the models can still capture some useful features in the first epoch. We conjecture this is because CIFAR-10 classes are more generic than ImageNet classes, and noise that minimizes the error of fine-grained classes can become less effective on generic classes. We also conduct an experiment on the ImageNet transferred noise to make 4 classes unlearnable of CIFAR-10. We plot the prediction confusion matrix of the RN-18 trained on the 4-class unlearnable CIFAR-10 in Figure 3c. It shows that the 4 classes are indeed unlearnable, and the model’s performance on the other 6 classes is not affected. This confirms that transferred noise can also be used in customized scenarios.
+
+![](images/c926e81a65a819cea75e041294d5a74f763c7773404d209bd5d4b6a5ef3fb3a6.jpg)  
+4.5 REAL-WORLD SCENARIO: A CASE STUDY ON FACE RECOGNITION   
+Figure 4: Preventing exploitation of face data using error-minimizing noise.
+
+Here, we conduct a case study to apply error-minimizing noise to personal face images, which is arguably one most important real-world scenarios. In this scenario, the defender wants to prevent his/her face images from being exploited to train face recognition or verification systems. The setting is illustrated in Figure 4a. We assume the user has access to a small dataset of facial images and will use this small dataset to generate and apply the error-minimizing noise to his/her own face images before sharing them on online social media platforms. After this, the unlearnable version of face images on social media gets collected to train a DNN based facial recognition or verification system. The system will then be used to recognize one of the user’s clean face image captured somewhere by a camera. The goal is to prevent the DNN from learning the defender’s face images and make it perform poorly on his/her clean face images.
+
+We conduct the experiments under two different settings: 1) partially unlearnable where a small subset of identities in the training set are unlearnable; or 2) fully unlearnable where the entire training set is unlearnable. We test our error-minimizing noise against both face recognition and verification models. Face recognition model classifies the identity (class) of a face image, while face verification model verifies whether two face images belong to the same identity. In the face verification problem, two face images are determined to be of the same identity if the cosine similarity between the features (extracted from a recognition model) of the two images is below a certain threshold.
+
+Experimental Settings. We randomly split the WebFace dataset (Yi et al., 2014) into $80 \%$ for training and $20 \%$ for testing, according to each identity. In the partially unlearnable setting, we randomly select 50 identities from the training set of WebFace into a subset WebFace-50 as the users who want to hide their identities. We also randomly select 100 identities from CelebA (Liu et al., 2015) into a subset CelebA-100 as the small dataset used to generate the unlearnable images for WebFace-50. The clean WebFace-50 part of the WebFace training data will be replaced by its unlearnable version for model training. This partially unlearnable version of WebFace training set consists of 10,525 clean identities and 50 unlearnable identities. In the fully unlearnable setting, the entire WebFace is made unlearnable. In both settings, the error-minimizing noises are generated using RN-18 and CelebA-100, following the procedure described in Section 4. We train Inception-ResNet models (Szegedy et al., 2016) following a standard training procedure (Taigman et al., 2014; Parkhi et al., 2015).
+
+Effectiveness Against Face Recognition. Here, we adopt the partially unlearnable setting. The accuracy of the Inception-ResNet model on the clean WebFace test set is reported in the Figure 4b, separately for the 50 unlearnable identities and the 10,525 clean identities. The result confirms that the 50 identities with the error-minimizing noise are indeed unlearnable and the recognition accuracy of their clean face images is only $16 \%$ , a much lower than the rest of the identities $( 8 6 \% )$ .
+
+Effectiveness Against Face Verification. Here, we adopt both the partially and the fully unlearnable settings. In the partially unlearnable setting, we evaluate the model performance on 6000 (3000 positives and 3000 negatives) face image pairs randomly sampled from the clean test set of WebFace. The pair is labeled as positive if the two face images are of the same identity, negative otherwise. In the fully unlearnable setting, we independently train a second Inception-ResNet model on the fully unlearnable WebFace. To further eliminate possible (dis)similarities shared across the images from the same dataset (e.g., WebFace), we follow the standard face verification evaluation protocol (Deng et al., 2019) and use the 6000 face pairs from LFW (Huang et al., 2008) for testing.
+
+Figure 7 in Appendix G shows the Receiver Operating Characteristic (ROC) performance of the models trained in both settings. The model trained in the partially unlearnable setting still has a good verification performance on unlearnable identities, although the Area Under the Curve (AUC) is lower than the clean identities. This is because a small subset of unlearnable examples is not powerful enough to stop the learning of a feature extractor. In fact, 2,622 (approximately equal to $2 5 \%$ of WebFace) clean identities are sufficient to train a high-quality facial feature extractor (Parkhi et al., 2015). This indicates that, while it is easy to make data unlearnable to classification models, it is much more challenging to stop the learning of a feature extractor. Nevertheless, the proposed error-minimizing noise is still effective in the fully unlearnable setting, reducing the AUC to 0.5321 (the clean setting AUC is 0.9975). Whilst there are still many unexplored factors, we believe our proposed error-minimization noise introduces a new practical tool for protecting personal data from unauthorized exploitation.
+
+# 5 CONCLUSION
+
+In this paper, we have explored the possibility of using invisible noise to prevent data from being freely exploited by deep learning. Different from existing works, our method makes data unlearnable to deep learning models. The effectiveness of this approach could have a broad impact for both the public and the deep learning community. We propose a type of error-minimizing noise and empirically demonstrate its effectiveness in creating unlearnable examples. The noise is effective in different forms (eg. sample-wise or class-wise), sizes (eg. full image or small patch), and is resistant to common data filtering methods. It can also be customized for a certain proportion of the data, one single class or for multiple classes. Furthermore, the noise can be easily transferred from existing public datasets to make private datasets unlearnable. Finally, we verify the usefulness for real-world scenarios via a case study on face recognition. Our work opens up a new direction of preventing free exploitation of data. Although there are still many practical obstacles for a large-scale application of the error-minimizing noise, we believe this study establishes an important first step towards preventing personal data being freely exploited by deep learning.
+
+# ACKNOWLEDGEMENT
+
+Yisen Wang is partially supported by the National Natural Science Foundation of China under Grant 62006153, and CCF-Baidu Open Fund (OF2020002). This research was undertaken using the LIEF HPC-GPGPU Facility hosted at the University of Melbourne, which was established with the assistance of LIEF Grant LE170100200.
+
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+# A ALGORITHM OF ERROR-MINIMIZATION GENERATION
+
+# Algorithm 1 Error-minimizing Perturbations
+
+1: Input: Source Model $\theta$ , Perturbations $\delta , L _ { p } \epsilon , ( \pmb { x } , y ) \in D _ { c }$ , Stop Error $\lambda$ , Optimization steps M   
+2: Output: δ   
+3: repeat   
+4: for $m$ in $1 \cdots M$ do   
+5: $i , \pmb { x } _ { i } , y _ { i } = \mathrm { N e x t } ( \pmb { x } , y )$   
+6: if sample-wise then   
+7: $\delta \stackrel { \cdot } { = } \delta _ { i }$   
+8: else if class-wise then   
+9: δ = δyi   
+10: end if   
+11: $\mathrm { O p t i m i z e } ( x _ { i } + \delta , y _ { i } , \theta )$   
+12: end for   
+13: for $\begin{array} { l } { { \pmb x } _ { j } , y _ { j } \ \mathbf { i n } \ x , y \ \mathbf { d o } } \\ { \delta = \mathrm { P e r t u r b a t i o n } ( x _ { i } , y _ { i } , \theta , \delta ) } \\ { \mathrm { C i l p } ( \delta , - \epsilon , \epsilon ) } \end{array}$   
+14: . Follow Equation 3   
+15:   
+16: end for   
+17: $e r r o r = \mathrm { E v a l } ( x , y , \delta , \theta )$   
+18: until error $< \lambda$
+
+# B MORE EXPERIMENT SETTING
+
+For all experiments, we use $L _ { p }$ -norm with to regularize the imperceptibility, $\epsilon = 8 / 2 5 5$ for CIFAR and SVHN, $\epsilon = 1 6 / 2 5 5$ for ImageNet, different $\epsilon$ is used in Appendix D for additional understandings. The iterative steps $T$ for Equation 3 is set to 20 steps for sample-wise noise and 1 for class-wise, $\alpha$ is set to $\epsilon / 1 0$ . Since the class-wise noise is generated universally for each class, small iterative steps avoid overfitting to the specific example. For the class-wise experiments, we only use $20 \%$ of the training data $\mathcal { D } _ { c }$ to generate the noise $\pmb { \delta }$ and apply to entire training dataset $\mathcal { D } _ { c } \to \mathcal { D } _ { u }$ . The stop condition error rate is $\lambda = 0 . 1$ for $\Delta _ { c }$ and $\lambda = 0 . 0 1$ for $\Delta _ { s }$ . For SVHN and CIFAR-10, we set the $M = 1 0$ for CIFAR-10, $M = 2 0$ for CIFAR-100 and $M = 1 0 0$ for ImageNet in the mini-setting. For all models and experiments, we use the Stochastic Gradient Descent (SGD) (LeCun et al., 1998) optimizer with momentum 0.9, initial learning rate 0.025 and cosine scheduler (Loshchilov & Hutter, 2017) without the restart. We train all DNN models for 30 epochs on SVHN, 60 epochs on CIFAR-10, 100 epochs on CIFAR-100 and ImageNet. To generated fully unlearnable setting CIFAR-10 dataset, the computational cost is roughly $10 \%$ of the standard model training time for class-wise noise and $60 \%$ of standard model training time for sample-wise noise. Overall, the cost of generating unlearnable examples is far less than training a model on the data to be protected.
+
+# C STABILITY ANALYSIS
+
+![](images/9339f032ff7082f5b5f2000a73d6a94bb472541eb95b36ec9b54a21c3252774f.jpg)  
+Figure 5: Prediction confusion matrix (on the clean test set) of two RN-18s trained on CIFAR-10 with the unlearnable classes in bold by (a) sample-wise or (b) class-wise error-minimizing noise.
+
+# D RESISTANCE TO DATA AUGMENTATION AND ADVERSARIAL TRAINING
+
+Resistance to Data Augmentation. Another important property of the unlearnable examples created by error-minimizing noise is its resistance to data augmentations. Since standard data augmentation techniques like random shift, crop, flip and rotation have already been applied in previous experiments, here we consider 4 more advanced data augmentation techniques: Cutout (DeVries & Taylor, 2017), Mixup (Zhang et al., 2018a), Cutmix (Yun et al., 2019) and Fast Au
+
+Table 3: Clean test accuracy of RN-18 models trained on unlearnable CIFAR-10 with Cutout, Mixup, Cutmix and FA. Test accuracy of RN-18 trained on the clean CIFAR-10 is $9 4 . 9 5 \%$ .   
+
+<table><tr><td>Noise</td><td>Cutout</td><td>Mixup</td><td>CutMix</td><td>FA</td></tr><tr><td>△s</td><td>19.30</td><td>58.51</td><td>22.40</td><td>42.70</td></tr><tr><td>△c</td><td>14.62</td><td>17.63</td><td>16.19</td><td>22.89</td></tr></table>
+
+toaugment (FA) (Lim et al., 2019). For Cutout, we set the cutout length to 16 pixels. For Mixup, we apply linear mixup of random pairs of training examples and their labels during the standard training process. For Cutmix, we apply linear mixup on the cutout region. For FA, we use the fixed augmentation policy, which consists of change contrast, brightness, sharpness, rotations and cutout. It is observed that advanced data augmentation techniques including Cutout, Mixup, Cutmix and FA can indeed remove the sample-wise noise to some extent, but far less effective on class-wise noise.
+
+Adversarial Training. We also consider the Adversarial Training (AdvTrain), which is an augmentation based defence method against error-maximizing noise (Goodfellow et al., 2014; Madry et al., 2018). AdvTrain has been shown can effectively remove the non-robust features from the input and force the model to learn only the robust features (Ilyas et al., 2019). Compared to data augmentation techniques, adversarial training as a type of robust training technique is indeed more effective against both sample-wise and class-wise noise. However, adversarial training is known to suffer from a trade-off between robustness and accuracy (Zhang et al., 2019). For example, adversarial training only achieved $85 \%$ accuracy on the unlearnable CIFAR-10, there is still a roughly $10 \%$ $9 4 . 9 5 \%$ vs $8 5 \%$ ) performance drop compared to standard training on the clean CIFAR-10. Moreover, due to the difficulty of min-max optimization, current adversarial training methods are still limited to small error-maximizing noise. To test this, we fix the maximum adversarial perturbation used by adversarial training to $8 / 2 5 5$ , while increasing the perturbation of error-minimizing noise to $\epsilon = 2 4 / 2 5 5$ . Note that adversarial training with perturbation $1 6 / 2 5 5$ or $2 4 / 2 5 5$ still suffers from convergence issues, even on small datasets like CIFAR-10. The results are shown in Figure 6. As can be confirmed, the clean test accuracy will drop to $79 \%$ on $\epsilon = 2 4 / 2 5 5$ error-minimizing noise.
+
+The above results indicate that 1) error-minimizing noise is resistant to standard data filtering especially the class-wise noise; 2) although it is less resistant to adversarial training, the model’s performance can still be significantly compromised by our error-minimizing noise noise. In future works, it is possible to develop more advanced unlearnable examples that can further decrease the performance of adversarial training. We believe our method can be improved by crafting the noise based on only the robust features, since adversarial training forces the model to learn only robust features (Ilyas et al., 2019).
+
+![](images/e5ad7003fdeac70441ef3108613b7a96272cfb2970f59f9a2d9fe50f1dda3b17.jpg)  
+Figure 6: Clean test accuracy of adversarially trained RN-18 on unlearnable CIFAR-10 by different sizes () of error-minimizing noise, and the dashed line indicates the performance of RN-18 trained on clean CIFAR-10.
+
+# E CLASS-WISE NOISE TRANSFER FROM IMAGENET TO CIFAR-10
+
+Table 4: Detailed class mapping from ImageNet classes to CIFAR-10 classes. This mapping is used for the second transfer experiment in Section 4.4.   
+
+<table><tr><td>CIFAR-10 Class</td><td>ImageNet Class</td></tr><tr><td>Airplane</td><td>Airliner</td></tr><tr><td>Car</td><td>Wagon</td></tr><tr><td>Bird</td><td>Humming Bird</td></tr><tr><td>Cat</td><td>Siamese Cat</td></tr><tr><td>Deer</td><td>Ox</td></tr><tr><td>Dog</td><td>Golden Retriever</td></tr><tr><td>Frog</td><td>Tailed Frog</td></tr><tr><td>Horse</td><td>Zebra</td></tr><tr><td></td><td></td></tr><tr><td>Ship Truck</td><td>Container Ship Trailer Truck</td></tr></table>
+
+# F FLEXIBILITY ANALYSIS OF ERROR-MINIMIZING NOISE
+
+Here, we focus on the class-wise error-minimizing noise and further explore its flexibility from two perspectives: 1) effectiveness when applied to smaller patches rather than the entire image; and 2) effectiveness of two sets of mixed class-wise noise. These experiments are also conducted with RN-18 on the CIFAR-10 dataset. For all the error-minimizing noise, we fix the maximum perturbation to $\epsilon = 8 / 2 5 5$ .
+
+Table 5: Clean test accuracy of RN-18 models trained on unlearnable CIFAR-10 by either small patch noise applied at random location or a mixture of two type of noises.   
+
+<table><tr><td rowspan="2">Methods</td><td colspan="4">Patch Size</td><td colspan="2">Noise Mixture</td></tr><tr><td>8×8</td><td>16 ×16</td><td>24×24</td><td>32×32</td><td>△c1 V△c2</td><td>△c1+△u</td></tr><tr><td>Standard Training</td><td>87.23</td><td>19.19</td><td>26.42</td><td>16.42</td><td>25.41</td><td>14.94</td></tr><tr><td>Fast Autoaugment</td><td>90.66</td><td>36.69</td><td>50.60</td><td>22.89</td><td>56.20</td><td>30.40</td></tr></table>
+
+Effectiveness on Smaller Patches. The unlearnable example can be very flexible and hard to be detected if the noise remains effective on smaller patches. To test this, we set the patch size to $3 2 \times 3 2$ , $2 4 \times 2 4$ , $1 6 \times 1 6$ and $8 \times 8$ (CIFAR-10 image size is $3 2 \times 3 2$ ). During the noise generation process when solving equation Equation 2, a random patch is selected and perturbed in each perturbation step and for each training example. Once a class-wise patch noise is generated, it then attached to a randomly selected location of a training example. The effectiveness of small patch noise is reported in Table 5. The effectiveness is still very high for patch size as small as $1 6 \times 1 6$ , although it is indeed decreased on smaller patches. An interesting observation is that the FA augmentation becomes less effective on $1 6 \times 1 6$ noise than $2 4 \times 2 4$ noise. We suspect this is because smaller patch noise can more easily escape the augmentation operations and stays unchanged after the augmentation.
+
+Effectiveness of Mixed Noises. We conduct this mix noise to simulate two possible real-world scenarios: 1) different users apply different noise apply to their own data; or 2) one set of noise is exposed and upgraded to a new set of noise. In this experiment, we apply the noise on the full image. We repeat the class-wise noise generation twice and generate two sets of noise: $\Delta _ { c 1 }$ and $\Delta _ { c 2 }$ For each CIFAR-10 training example, we randomly apply one of the above two class-wise noise to create unlearnable example. We note this mixture by $\Delta _ { c 1 } \lor \Delta _ { c 2 }$ . Note that the class-wise mixture will eventually become the sample-wise noise if we keep mixing more class-wise noise sets. We also perform another mixture between $\Delta _ { c 1 }$ and a random noise $\Delta _ { u }$ sampled from $[ - \epsilon , \epsilon ]$ mixed by element-wise addition. We denote this mixture by $\Delta _ { c 1 } + \Delta _ { u }$ . The effectiveness of mixed noise is also reported in Table 5. For both mixtures, the proposed error-minimizing noise remains highly effective, although there is a slight decrease. Surprisingly, the mixture of one class-wise noise with random noise is even more effective than the mixture of two class-wise noise, especially against the FA data augmentation. This makes the error-minimizing noise even more flexible in practical usage.
+
+![](images/b78d5f599a2a5243774f4fa367895301dc3c51d1586b80748d23de3bf0882ce4.jpg)
+
+(a) Partially Unlearnable Setting. Evaluated on WebFace test set against different identities.
+
+![](images/a6821f10328fabeefb16c3e392a5f302a7be8973cb6a7f4c9505cad5eea9844f.jpg)  
+(b) Fully Unlearnable Setting. Evaluated on LFW using unlearnable and clean training data.   
+Figure 7: Results on face verification: the Receiver Operating Characteristic (ROC) of two InceptionResNet models trained on partially unlearnable WebFace (left) and fully unlearnable WebFace (right). The ROC curves in the left plot are computed based on the clean WebFace test set while the ones in the right plot are computed based on LFW. Both Inception-ResNet models are trained as classifiers and tested as feature extractors. Clean: clean identities; Unlearnable: unlearnable identities. Partially Unlearnable Setting: 50 out of 10,575 identities are unlearnable.
+
+# H CLASS-WISE NOISE VS. BACKDOOR ATTACK
+
+![](images/cd71f68220d4b55f881ce75d33e321974d6dbbadf91d45a3a186debfcdf6253d.jpg)  
+Figure 8: Backdoor attack success rates when 3 types of class-wise noises including random, errormaximizing and error-minimizing noise are applied as the backdoor triggers. The x-axis shows 6 RN-18 models trained on CIFAR-10 training set with different percentages of the data were poisoned by the class-wise noise. The y-axis shows the attack success rate: the percentage of all non-target class test images are predicted to the target class when attached with the target-class class-wise noise.
+
+Here, we test if class-wise noise can be used as a trigger for the backdoor attack. Figure 8 shows the backdoor attack success rate when different types of class-wise noises are applied as the backdoor trigger. At the training time, we train RN-18 models on CIFAR-10 with $0 \% { - } 1 0 0 \%$ of the training samples were made unlearnable by three types of class-wise noise: random noise, error-minimizing and our error-maximizing noise. At the test (attack) time, we compute the attack success rate as follows. There are 6 models for each type of class-wise corresponding the 6 unlearnable percentages. For each type of class-wise noise and each RN-18 model, we iteratively take a clean test image from the CIFAR-10 test set, then randomly select a target class (which is different from the true class of the test image). Then, we attach the target class class-wise noise to the clean test image. If the new test image is predicted by the model as the target class, then the attack is successful.
+
+As shown in Figure 8, the class-wise error-maximizing (adversarial) noise is a highly effective trigger for backdoor attack, yet neither random nor our error-minimizing noise can be considered to be effective. The error-maximizing can achieve more than $7 5 \%$ attack success rate when only applied to $20 \%$ the training data. To achieve a similar level of attack success rate, both random and our error-minimizing noise should be added to at least $80 \%$ of the training data. This indicates that our method is different from backdoor attacks. The high attack success rate of adversarial noise may due to the fact that adversarial examples are hard (high error) examples, which force the model to pay more attention to the examples and remember more of the adversarial noise. Note that adversarial perturbations have also been used in backdoor research to enhance backdoor triggers (Turner et al., 2018; Zhao et al., 2020).
+
+# I DIFFERENT GENERALIZATION METHODS FOR ERROR-MINIMIZATION NOISE
+
+The proposed error-minimizing noise is generated using PGD. Specifically, PGD is applied to solve the inner minimization problem in Equation 2. Since it is a typical constrained optimization problem, it can also be solved by other optimization (attack) methods such as FGSM (Goodfellow et al., 2014), L-BFGS (Szegedy et al., 2013) and C&W (Carlini & Wagner, 2017). However, the objective needs to be reformulated to apply some of these attacks. Here, we take L-BFGS (Szegedy et al., 2013) as an example to reformulate and solve the inner minimization problem in Equation 2. In (Szegedy et al., 2013), the adversarial attack problem is formulated as:
+
+$$
+\begin{array} { r l } { \mathrm { m i n i m i z e } } & { { } c \left\| \pmb { \delta } \right\| _ { p } + \mathcal { L } ( \pmb { x } + \pmb { \delta } , \pmb { y } ^ { \prime } ) \quad \pmb { y } ^ { \prime } \neq \pmb { y } } \end{array}
+$$
+
+where $c$ is a hyperparameter and the $\mathcal { L }$ is the objective function (e.g. the cross entropy adversarial loss). This form has been extended to other objective functions in C&W (Carlini & Wagner, 2017). The main objective of adversarial attack is to find small $L _ { p }$ (i.e. $\| \cdot \| _ { p } )$ bounded noise $\delta$ that can trick the model to output a wrong label $y ^ { \prime } \ne y$ . For unlearnable examples, we want to trick the model to predict the correct label $y$ with the highest confidence (i.e. lowest error). Following Equation 4, the inner minimization problem in Equation 2 can be reformulated as:
+
+$$
+\begin{array} { r l } { \operatorname * { m i n i m i z e } } & { { } c \left\| \pmb { \delta } \right\| _ { 2 } ^ { 2 } + \mathcal { L } ( \pmb { x } + \pmb { \delta } , y ) . } \end{array}
+$$
+
+We set $c = 1 . 0$ and use the cross entropy loss for $\mathcal { L }$ . We apply $\mathrm { L }$ -BFGS to solve Equation 5 using Adam (Kingma & Ba, 2015) optimizer and generate sample-wise noise $\Delta _ { s }$ on CIFAR-10. The number of optimization steps for Equation 5 is set to $T = 2 0 0$ , while the number of optimization steps for the outer minimization in Equation 2 is set to $M = 1 0 0$ (see more details about $T$ and $M$ in Section 3.2). As shown in Figure 9, error-minimizing noise can also be generated using L-BFGS, and the noise is very effective in making the training examples unlearnable. However, compared to PGD, L-BFGS is slightly less effective.
+
+![](images/d0ba78efd35c9e06a14659ae76aab0f95416bc1586ce53c2fee5a903ca8476cc.jpg)  
+Figure 9: The unlearnable effectiveness of different types of noises on CIFAR-10 dataset: random, error-maximizing (adversarial), error-minimizing noise generated using PGD and error-minimizing noise generated using L-BFGS. The lower the clean test accuracy the more effective the noise in making training examples unlearnable. This is tested in the $100 \%$ unlearnable setting with the sample-wise noise.