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+# CONVOLUTIONAL NEURAL NETWORK DYNAMICS: A GRAPH PERSPECTIVE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The success of neural networks (NNs) in a wide range of applications has led to increased interest in understanding the underlying learning dynamics of these models. In this paper, we go beyond mere descriptions of the learning dynamics by taking a graph perspective and investigating the relationship between the graph structure of NNs and their performance. Specifically, we propose (1) representing the neural network learning process as a time-evolving graph (i.e., a series of static graph snapshots over epochs), (2) capturing the structural changes of the NN during the training phase in a simple temporal summary, and (3) leveraging the structural summary to predict the accuracy of the underlying NN in a classification or regression task. For the dynamic graph representation of NNs, we explore structural representations for fully-connected and convolutional layers, which are key components of powerful NN models. Our analysis shows that a simple summary of graph statistics, such as weighted degree and eigenvector centrality, over just a few epochs can be used to accurately predict the performance of NNs. For example, a weighted degree-based summary of the time-evolving graph that is constructed based on 5 training epochs of the LeNet architecture achieves classification accuracy of over $93 \%$ . Our findings are consistent for different NN architectures, including LeNet, VGG, AlexNet, and ResNet.
+
+# 1 INTRODUCTION
+
+Neural networks (NNs) have driven advancements in many domains, including computer vision and image processing (Hu et al., 2018), natural language processing (Sutskever et al., 2014; Bahdanau et al., 2015), and bioinformatics (Cao et al., 2020; Li et al., 2019). As task complexity increases, networks grow deeper and larger, consequently requiring more computational resources and training data, as well as sacrificing interpretability for improved task performance. Some works have focused on understanding and interpreting deep NNs (Raghu et al., 2017; Chakraborty & et al., 2017; Ioffe & Szegedy, 2015; He et al., 2016). One approach towards this goal involves representing the NN as its underlying graph structure, and studying selected graph properties, such as clustering coefficient, path length (You et al., 2020), modularity (Filan et al., 2021), persistence (Rieck et al., 2019). For example, You et al. (2020) represent NNs as relational graphs capturing the message passing process, and investigate the correlation between the predictive performance of NNs and architectural changes. However, the studies of NN structures as graphs are limited, and the structural changes of the underlying graph during the training process have been largely overlooked in the literature.
+
+To fill this gap, in this work, we take the graph perspective and aim to predict the performance of an NN by capturing early NN dynamics during the training phase. Successful performance prediction based on only a few epochs could be used for early stopping (Yu & Zhu, 2020), and thus, more efficient NN training. To solve the performance prediction problem, we propose a multi-step framework, depicted in Fig. 1. Specifically, we propose to represent the underlying graph structure of an NN as a time-evolving k-partite graph, where each part corresponds to a different NN layer, and each graph snapshot in the evolving graph maps to an NN instance at a specific epoch. We build on existing graph representations of fully-connected and convolutional layers, both of which are key components of popular NN architectures, and introduce a new, compact, efficient-to-compute (“rolled”) graph representation for convolutional layers. Then, we extract well-known node features (weighted degree and eigenvector centrality) from the time-evolving graph and construct temporal signatures by computing summary statistics on the node feature distributions. Finally, we cast NN performance prediction as a classification task and a regression task, each of which operates on the temporal structural signatures of the NN. Our main contributions are:
+
+![](images/e289bcecb54995782532600550750cd3ff3c42731d3d05d640a0ee59c5f43ee7.jpg)  
+Figure 1: Our proposed framework for predicting NN performance in a downstream image classification task, shown for one test instance. The input to our framework is an NN trained for a few epochs (3, in this example). Steps: (S1) the input NN is converted to three static graphs, each representing one training epoch; (S2) node features (e.g., degree) are extracted from each graph snapshot (i.e., one per epoch); (S3) in order to summarize the changes in the graph structure over time, a signature vector is constructed by aggregating the node features per graph snapshot and concatenating the individual snapshot signatures; (S4) a pre-trained classifier and regressor predict the performance of the input NN given the signature vector from (S3).
+
+• New graph-based NN representation: We introduce a new graph representation for convolutional layers that is more compact and efficient-to-compute than the existing unrolled representation (Rieck et al., 2019), while not sacrificing accuracy in the NN performance prediction task.
+
+• Framework for NN performance prediction: We propose a simple, multi-step graph-based framework to solve the NN performance prediction problem by capturing the early NN dynamics during the training phase.
+
+• Extensive empirical analysis: Using well-known image classification datasets (ImageNet and CIFAR-10), and a variety of NN architectures (AlexNet, VGG, LeNet, and ResNet), we show that our framework can effectively predict the performance of NNs by observing only a few epochs of training, well before their corresponding early stopping epochs. For instance, using our framework to capture the changes in the graph structure in only 5 training epochs of the ResNet architecture results in classification accuracy of over $90 \%$ .
+
+# 2 PRELIMINARIES
+
+We first present the key concepts that our work builds upon. Table 1 gives the major symbols and their descriptions.
+
+An NN model is a collection of connected units (neurons) that are organized in layers, and is defined by a set of parameters that adjust during the training process. We refer to the training process of a single architecture along with its respective hyperparameters as an ‘instance’. We focus on two types of layers that are the key components of many powerful NN models such as LeNet(Lecun et al., 1998), VGG (Simonyan & Zisserman, 2015) and ResNet(He et al., 2016): fully connected layers (fc) and convolutional (conv) layers.
+
+# 2.1 GRAPHS: TERMINOLOGY AND NOTATION
+
+Let $\mathbf { G } = ( \nu , \mathcal { E } )$ be an undirected, weighted graph with node set $\nu$ , edge set $\mathcal { E }$ , and weighted adjacency matrix $\mathbf { W } \in \mathbb { R } ^ { | \mathcal { V } | \times | \mathcal { V } | }$ . The neighbors of node $v$ are defined as $\mathcal { N } _ { G } ( v ) = \{ u : ( u , v ) \in \mathcal { E } \}$ ; i.e., the set of all nodes that connect directly to $v$ . Graph $\mathbf { G }$ is $k$ -partite if its nodeset $\nu$ can be partitioned into $k$ independent sets: $\textstyle \mathcal { V } = \bigcup _ { i = 1 } ^ { k } \mathcal { V } _ { i }$ and $\mathcal { V } _ { i } \cap \mathcal { V } _ { j } = \emptyset , i \neq j$ . A time-evolving graph is a series of static graph snapshots over time: $\mathcal { G } = \{ \mathbf { G } ^ { 1 } , \mathbf { G } ^ { 2 } , . . . , \mathbf { G } ^ { T } \}$ . The graphs in the series may have different nodesets and edgesets.
+
+Here we focus on two of the most commonly-used node features in graph mining and network science, degree and eigenvector centrality, which capture different types of node importance or influence. We discuss the interpretation of these features in the context of neural network dynamics in $\ S 3 . 2$ .
+
+• Degree: Degree is the simplest and most efficient to compute node feature, and captures the connectivity of a node. The weighted degree of node $v$ is defined as the sum of weights of the edges that are incident to $i$ : $\begin{array} { r } { d _ { i } ^ { w } = \sum _ { j } \mathbf { W } _ { i j } } \end{array}$ . Tracking changes in the degree of a node over time is the most direct way of capturing its structural evolution.
+
+• Eigenvector centrality: This centrality is a sophisticated extension of degree centrality, related to Google’s PageRank, which indicates the influence of a node in $\mathbf { G }$ (Bonacich, 1972). If a node is connected to several nodes with high eigenvector centrality, then that node will have high centrality. The eigenvector centrality of node $i$ is defined as the $i ^ { t h }$ element in the principal eigenvector $\mathbf { v }$ of W (i.e., the eigenvector that corresponds to the largest eigenvalue $\lambda$ ): $\mathbf { W } \mathbf { v } = \lambda _ { m a x } \mathbf { v }$ .
+
+# 2.2 FULLY-CONNECTED LAYERS: GRAPH REPRESENTATION
+
+Let $\mathbf { x } \in \mathbb { R } ^ { D _ { i n } }$ be the input vector, $\mathbf { W } \in \mathbb { R } ^ { D _ { i n } \times D _ { o u t } }$ the learnable weight matrix, $\mathbf { b } \in \mathbb { R } ^ { D _ { o u t } }$ the bias vector and $z$ a non-linear activation functions. Then, the output of this layer is given as $\mathbf { y } = z ( \mathbf { x } \mathbf { W } + \mathbf { b } )$ , where $D _ { i n }$ and $D _ { o u t }$ correspond to the dimension of input and output layers. Fully-connected layers are straightforward to represent with a weighted, undirected graph (Filan et al., 2021). Each neuron, including those in the input and output layers, corresponds to a node. Two neurons are connected via an edge if they appear in consecutive layers. More formally, let nodeset $\nu _ { i }$ be the set of all neurons at layer $i$ , and nodeset $\nu _ { j }$ the set of all neurons at layer $j$ (which follows layer $i$ ). The edges that connect all nodes across the two nodesets are defined by the learnable weight matrix $\mathbf { W } \in \mathbb { R } ^ { | \mathcal { V } _ { i } | \times | \mathcal { V } _ { j } | }$ .
+
+Table 1: Major symbols and definitions   
+
+<table><tr><td>Symbol</td><td>Definition</td></tr><tr><td>G= (V,ε) 9={Gi,.,G𝑇}</td><td>a graph with its nodeset and edgeset a time-evolving graph,i.e.,series of graph snapshots</td></tr><tr><td>du，du W</td><td>degree and weighted degree of node u learnableweightmatrixinanNN,and</td></tr><tr><td></td><td>the weighted adj matrix of G</td></tr><tr><td>K</td><td>convolutional kernel</td></tr><tr><td>Din/out</td><td>dimension of the input/output vectors</td></tr><tr><td>c,f</td><td>number of channels and filters</td></tr><tr><td>h,w</td><td>height and width of an input image</td></tr><tr><td>hker,Wker</td><td>height and width of a kernel</td></tr><tr><td>x</td><td>input tensor of conv layer</td></tr></table>
+
+# 2.3 CONVOLUTIONAL LAYERS: UNROLLED GRAPH REPRESENTATION
+
+Let $\begin{array} { r l r } { { \pmb x } } & { { } \in } & { \mathbb { R } ^ { c \times h \times w } } \end{array}$ be the input and $\kappa \in \mathbb { R } ^ { f \times c \times h _ { k e r } \times w _ { k e r } }$ the convolutional kernel, where $c$ is the number of channels, and $h _ { k e r } , w _ { k e r }$ are the height and width of the kernel respectively. After performing the convolution, we have $\pmb { y } ^ { } \in \mathsf { \tilde { R } } ^ { F \times h _ { o u t } \times \mathsf { \tilde { w } } _ { o u t } }$ , where $h _ { o u t } = h - h _ { k e r } + 1$ and $w _ { o u t } = w - w _ { k e r } + 1$ Typical multilayer CNNs consist of convolutional layers followed by fully-connected layers. The representation of fully connected layers is straightforward but cannot be used to model
+
+![](images/50157361f7d0836e8518212df6cdf24242bb9454432482bd5a371b0fc0aeee97.jpg)  
+Figure 2: Unrolled graph representation example. The two typed nodes in the resultant bipartite graph map to the filtration operation and the output. For the stride in this example, the output node $o _ { 1 }$ (in dark blue) is the output of 4 operation-typed nodes (in light blue).
+
+convolutional layers. Rieck et al. (2019) proposed to "unroll" the convolution to convert conv layers into graphs. In Sec. 3, we introduce a new, compact, rolled representation that is both efficient to compute and effective at predicting the NN performance, as we show empirically in Sec. 4.
+
+For a convolutional layer, we first unroll the convolutional operation and then represent the graph as in the case of a fully-connected layer. In this representation, the nodes and edges of the graph are defined through the convolutional operation as matrix multiplication (Gebhart et al., 2019). Specifically, for an input image $_ { x }$ and kernel $\boldsymbol { \kappa } _ { i } \in \mathbb { R } ^ { f _ { i } \times c _ { i } \times h _ { i } \times \overset { . } { w } _ { i } }$ , each node in layer $l$ is defined to be the output of the mapped feature of that input for each filter $f _ { i }$ . Edges connect each of these nodes to the corresponding nodes of the output neuron in the next layer. These edges are weighted by the activation value of that neuron (i.e., the input image for a specific stride is multiplied by the filter value at that location in the image). For an input image filter size $( c _ { i } \times h _ { i } \times w _ { i } )$ that results in feature-map (output) of size $1 \times o _ { h } \times o _ { w }$ , the number of nodes of output layer of the graph representation is a function of $o _ { h } \times o _ { w }$ , while the number of nodes in the conv operation layer of graph is a function of $c _ { i } \times h _ { i } \times w _ { i }$ .
+
+As an example, Fig. 2 depicts the unrolled graph representation of a toy convolutional layer. It shows one conv operation on a small 2-dimensional input image $\left( 4 \times 4 \right)$ and filter $( 2 \times 2 )$ . The nodes and edges of the represented graph are shown based on one stride (of size 2) of convolution operation.
+
+# 3 NEURAL NETWORK PERFORMANCE PREDICTION: A TEMPORALGRAPH-BASED APPROACH
+
+In this section, we first formally introduce the problem that we seek to solve. Then, we present a new, compact, and efficient-to-compute graph representation for convolutional layers $( \ S 3 . 1 )$ , and describe our proposed temporal graph-based framework that captures the NN dynamics during training $( \ S \ 3 . 2 )$
+
+Problem 1 (NN Performance Prediction) Let $\mathcal { N } = \{ N _ { t r _ { 1 } } , N _ { t r _ { 2 } } , . . . , N _ { t r _ { n } } \}$ be a training set of n NNs trained for $T$ epochs and $\mathcal { A } = \{ \alpha _ { 1 } , \alpha _ { 2 } , . . . , \alpha _ { n } \}$ their corresponding downstream task accuracies (e.g., for image classification). We seek to predict the accuracy $\alpha _ { t s t }$ of a new instance $N _ { t s t }$ trained for a very small number of $t \ll T$ epochs by using $t$ epochs for the trained NNs in $\mathcal { N }$ .
+
+Our proposed solution takes the graph perspective, and its first step is to represent each NN as a temporal graph. In addition to the graph representations that we presented in Sec. 2, we introduce a new, efficient representation for convolutional layers, which we describe next.
+
+# 3.1 ROLLED GRAPH REPRESENTATION FOR CONVOLUTIONAL LAYERS
+
+We call our proposed graph representation for conv layers “rolled,” since it avoids unrolling the convolutional operations introduced in (Rieck et al., 2019).
+
+Overview $\pmb { \& }$ Motivation. To preserve the semantic meaning of conv layers, we represent each filter as a node, and link filters in consecutive layers via weighted edges (we define the weights below), as shown in Fig. 3. The motivation behind this approach is that for larger networks, there is an explosion of nodes by unrolling the convolutions (unrolled representation explained in $\ S \ 2 . 3 \AA ,$ ), and the integrity of a unit (a single filter or kernel) becomes untenable. Also, by mapping the nodes to specific entities of NNs, our proposed graph model is more interpretable than the unrolled model, and thus it is easier to interpret the outputs of downstream graph analysis on our graph representation (e.g., computing node features, tracking the evolution of the graph).
+
+![](images/7051065a2d20e1f400b4423d47355d2f618c86e339e0360238b85cb8ec3b9b1a.jpg)  
+Figure 3: Rolled graph representation example. The resultant graph is a tri-partite graph with three node types (Conv1, Conv2, FC) corresponding to two convolutional layers and one FC layer. Gray and light blue nodes represent filters in the conv layers and dark blue nodes represent neurons in the FC layer. The red edge between nodes 6 (gray) and 16 (light blue) is weighted by the $\mathrm { { N o r m ( 6 ^ { t h } } }$ channel of filter 16). Due to dropout, the maximum number of edges is $6 \times 1 6$ between the conv layers, and $1 6 \times 1 2 0$ between the Conv2 and FC layers.
+
+Detailed Description. Formally, let tensor $\kappa _ { i }$ be a kernel in layer $i$ with $f _ { i }$ filters, each with $c _ { i }$ nels and dim indexes the sions filter $h _ { i } \times w _ { i }$ . el use bracket. We create tation to index into the kernel: nodes representing each filter $\kappa _ { i } [ l , :$ $, : , : ]$ $l ^ { \mathrm { t h } }$ $\kappa _ { i }$ $f _ { i }$ $\{ v _ { 1 } ^ { ( i ) } , v _ { 2 } ^ { ( i ) } , . . . v _ { f _ { i } } ^ { ( i ) } \}$ with features defined as the corresponding biases in that layer. Let be the next convolutional layer defined analogously. While edge weights between neurons in FC layers are defined in the standard way, we define the edge weights between conv layers as the norm over each kernel’s channels. The edge between node $v _ { k } ^ { \left( i \right) }$ representing the $k ^ { t h }$ filter in layer $i$ (i.e., $\kappa _ { i } [ k , : , : , : ] )$ and node/filter $v _ { l } ^ { ( j ) }$ in layer $j$ (i.e. $\kappa _ { j } [ l , \vdots , \vdots , \vdots )$ has weight $w _ { v _ { k } ^ { ( i ) } , v _ { l } ^ { ( j ) } } = \mathrm { n o r m } ( \mathcal { K } _ { j } [ l , k , : , : ] )$ , which is the norm of the $k ^ { \mathrm { { t h } } }$ channel of the $l ^ { \mathrm { t h } }$ filter in the $j ^ { \mathrm { t h } }$ layer. Though the proposed representations enable alternative edge weight configurations, we focus on the norm of filters as other studies, including those on pruning NNs (Li et al., 2016), have demonstrated that this metric has strong correlation with filter importance. In the case of two conv layers, the resultant graph is an attributed bipartite graph with $f _ { i } + f _ { j }$ nodes and $f _ { i } \times f _ { j }$ edges, where node attributes include flattened weight vectors or filter maps. Other information such as average gradients or the bias vector can also be used for node features.
+
+# 3.2 PROPOSED TEMPORAL GRAPH-BASED FRAMEWORK
+
+A key objective of our work is to test whether the introduced graph representation of NNs is informative for predicting the performance of NNs. Next, we describe the steps of our proposed framework for solving Problem 1, namely performance prediction from the NN training dynamics.
+
+As shown in Fig. 1, our method consists of four steps: (S1) generation of a temporal graph for the training phase of each NN; (S2) extraction of node features; (S3) construction of a feature-based graph signature that captures the NN dynamics; (S4) prediction of NN performance by training a classifier or regressor on the constructed signatures. We describe these steps in more detail next.
+
+(S1) Graph generation. The first step involves converting the training process of each input NN into a time-evolving graph. Each NN $N _ { t r _ { i } } \in \mathcal { N }$ , is represented as a series of checkpoints saved for $t$ training epochs. Based on these checkpoints and the graph representation approaches for fc and conv layers presented in 2.2, 3.1 and 2.3, we first convert the $\tau ^ { t \bar { h } }$ NN checkpoint into weighted graph $G _ { i } ^ { ( \tau ) }$ at timestamp/epoch $\tau$ . We note that although our proposed graph representation involves node features, our framework does not leverage them, and thus we consider the generated graphs unattributed. Therefore, each NN $N _ { t r _ { i } }$ is mapped to a time-evolving graph $\mathcal { G } _ { t r _ { i } } = \{ \mathbf { \bar { G } } _ { i } ^ { 1 } , \mathbf { G } _ { i } ^ { 2 } , . . . , \mathbf { \bar { G } } _ { i } ^ { t } \}$ The output of this step is a set of $n$ time-evolving graphs $\{ \mathcal { G } _ { t r _ { 1 } } , \mathcal { G } _ { t r _ { 2 } } , . . . , \mathcal { G } _ { t r _ { n } } \}$ corresponding to the original $n$ NNs in $\mathcal { N }$ .
+
+(S2) Feature extraction. Next, the goal is to capture the structural dynamics of the NN training process. We aim to select graph measures that can capture changes during the training process, take into account the edge weight of graphs and can be calculated efficiently. In order to do that in an interpretable way, we extract two well-known node centralities from each snapshot of each generated time-evolving graph $\mathcal { G } _ { i }$ : weighted degree centrality and eigenvector centrality $( \ S 2 . 1 )$ . The weighted degree is a simple function of the learnable weight matrix W during the training phase of NN, therefore it gives us insights into the training dynamics at the node/neuron/filter level. The eigenvector centrality is an extension of the degree centrality, which captures the highly influential nodes, and has been successfully used in neuroscience to capture the dynamic changes of real neural networks (or connectomes) (Lohmann et al., 2010). Eigenvector centrality can be used to capture importance and connectivity of filters/neurons (i.e., the nodes in our graph representation). Also, eigenvector centrality has been used for detecting communities (Newman, 2006) or clusters (Wu et al., 2013), and thus provides structural information about the clusterability of the NN, which is complementary to that provided by the simpler and more efficient-to-compute degree centrality.
+
+Our choice of features is also guided by the inherent $k$ -partite structure of our proposed graph representation, which cannot be represented well by several other commonly-used graph features. For example, clustering-based features (e.g., number of triangles, transitivity, clustering coefficient) and cycle-based metrics which account for closed paths in a graph are always equal to 0 for $k$ -partite graphs. Moreover, connected component-related features (i.e., strong/weak connectivity) do not capture the learned edge weights, which are important for modeling the training dynamics of NNs. Other features (e.g., betweenness centrality) tend to be computationally expensive, and would add significant overhead compared to early stopping methods.
+
+(S3) Graph signature construction. In order to be able to compare NN-based graphs (with different number of nodes and edges), we summarize the structural changes in the generated time-evolving graphs at the graph level (rather than the node level, as in (S2)), and construct a statistical summary of the extracted node centralities (signature) per time-evolving graph $\mathcal { G } _ { i }$ . For each snapshot $\mathbf { G } _ { i } ^ { ( \tau ) }$ of $\mathcal { G } _ { i }$ , we create a signature vector using five node feature aggregators, which were introduced in (Berlingerio et al., 2012) for graph similarity: median, mean, standard deviation, skewness, and kurtosis, where all but the median are moments of the corresponding distribution. Thus, $\mathbf { G } _ { i } ^ { ( \tau ) }$ is mapped to a (static) signature vector $s _ { i } ^ { ( \tau ) } \in \mathbb { R } ^ { 5 }$ , representing the statistical summary of its node features (i.e., degree or eigenvector centrality) at time $\tau$ . To put more emphasis on the most recent timestamp, we can redefine the signature at time $\tau$ as the linear weighted average of the signatures up to that point, $\mathbf { s } _ { i } ^ { ( \tau ) } \gets \frac { \sum _ { j } ^ { \tau } j * \mathbf { s } ^ { ( j ) } } { \sum j }$ s  , or an exponential function of the previous signatures, s(τ)i ← αs(τ)i + (1 − α)s(τ−1)i . To obtain the temporal signature of the evolving graph $\mathcal { G } _ { i }$ , we aggregate the (static) signatures up to timestamp/epoch $t$ : $\mathbf { s } _ { i } ^ { \tilde { t } } = \mathbf { s } _ { i } ^ { 1 } \oplus \mathbf { s } _ { i } ^ { 2 } \oplus \ldots \oplus \bar { \mathbf { s } _ { i } ^ { t } }$ , where $\oplus$ denotes concatenation. We note that global features such as algebraic connectivity, modularity, and average shortest paths may be seen as alternative ways for constructing global graph signatures while circumventing the local feature extraction step (S2); however, these features fail to capture the structural changes in our proposed graph representations (they remain (near-)constant over time) and lead to poor performance.
+
+(S4) Performance prediction. For the last step of performance prediction, we consider two tasks:
+
+Table 2: Information for the generated NNs: range for early stopping epoch, range for accuracy, and accuracy threshold used for defining the class labels for classification task (predicting the accuracy level of NNs).   
+
+<table><tr><td rowspan="2"></td><td colspan="5">CIFAR-10</td><td colspan="3">ImageNet</td></tr><tr><td>LeNet</td><td>AlexNet</td><td>VGG</td><td>ResNet-32</td><td>ResNet-44</td><td>LeNet</td><td>AlexNet</td><td>ResNet-50</td></tr><tr><td>Early stopping</td><td>11~50</td><td>30~50</td><td>45~50</td><td>16~120</td><td>16~120</td><td>16~50</td><td>16~50</td><td>16~120</td></tr><tr><td>Acc. range</td><td>9.4~ 73.8</td><td>5.5~82.4</td><td>8.8~87.6</td><td>8.4~90.0</td><td>9.9~89.8</td><td>0.6~14.4</td><td>0.6~20.1</td><td>0.86~41.66</td></tr><tr><td>Acc. thres.</td><td>40</td><td>40</td><td>40</td><td>40</td><td>40</td><td>9</td><td>10</td><td>25</td></tr></table>
+
+• Classification: We train a classifier (e.g., SVM, MLP) using the training graphs $\{ \mathcal { G } _ { t r _ { 1 } } , \mathcal { G } _ { t r _ { 2 } } , . . . , \mathcal { G } _ { t r _ { n } } \}$ represented by their temporal signatures $\{ \mathbf { s } _ { t r _ { 1 } } ^ { t } , \mathbf { s } _ { t r _ { 2 } } ^ { t } , . . . , \mathbf { s } _ { t r _ { n } } ^ { t } \}$ , and their corresponding accuracies $\mathcal { A } = \{ \alpha _ { 1 } , \alpha _ { 2 } , . . . , \alpha _ { n } \}$ mapped to labels $\mathcal { L = } \{ l _ { 1 } , \bar { l } _ { 2 } , \dots , \bar { l _ { n } } \}$ (e.g., high/low accuracy) based on some threshold. Any test NN instance is then classified using the trained classifier.
+
+• Linear regression: We perform linear regression to estimate the actual accuracy value $\alpha _ { t s t }$ of a new test instance $N _ { t s t }$ based on its signature obtained through steps (S1)-(S3).
+
+# 4 EMPIRICAL ANALYSIS
+
+In this section, we empirically evaluate the effectiveness and efficiency of our framework in the classification and regression tasks for different graph representations (rolled and unrolled graphs for conv layers) and different feature-based signatures (degree- vs. eigenvector centrality-based).
+
+Data. We investigate NN dynamics using our framework on two well-known image classification datasets, CIFAR-10 (Krizhevsky, 2009) and ImageNet (Russakovsky et al., 2015). CIFAR-10 consists of 50K training images and 10K test images. For ImageNet, we use a sample that has 50K training images and 5K validation images used as the test set. The sample is obtained by randomly selecting 100 classes from Tiny ImageNet (tin) and downsizing the images to $3 2 \times 3 2$ colored images.
+
+Configuration. The configuration of training different NN models, and the early stopping method are described in App. A.1. For the unrolled graph representation, which is signed, we consider different graph types (e.g., positive, negative), which we describe in App. A.2.
+
+# 4.1 CLASSIFICATION: PREDICTING NN ACCURACY RANGE FROM NN TRAINING DYNAMICS
+
+Task setup. We cast the NN performance prediction as a classification task. Specifically, the generated time-evolving graphs are labeled as high and low accuracy based on the performance of their corresponding NNs; Table 2 lists the threshold value chosen for low and high accuracy labels based on the final accuracy range of trained NNs, as well as the early stopping epochs for each architecture. Five-fold cross validation is used to predict the label of the test graphs in a binary classification task using SVM and MLP, where the input is the set of temporal signatures $\{ \mathbf { s } _ { t r _ { 1 } } ^ { t } , \mathbf { \bar { s } } _ { t r _ { 2 } } ^ { t } , . . . , \mathbf { s } _ { t r _ { n } } ^ { t } \}$ . We report the classification accuracy. Since the sample of NNs (App. A.1) is randomly selected with balanced high/low accuracy instances, the accuracy of a random classifier as the baseline is $50 \%$ (omitted from the charts to avoid clutter). Additionally, to show that our proposed graph representation and signatures are general and can be useful across different NN architectures, we consider the following setup: we train the classifier on a small set of NN models (i.e., different architectures—such as LeNet, AlexNet, and VGG—and hyperparameters), and predict the performance on unseen architectures (e.g., ResNet). We describe these experiments in App. A.4.
+
+Results. Figure 4 illustrates the performance of SVM and MLP classifiers operating on weighted degree-based signatures of the rolled and unrolled graph representations of the LeNet and AlexNet architectures, trained on the CIFAR-10 dataset. We omit the results on VGG and ResNet, as well as image classification on ImageNet, because the unrolled graph generation process is prohibitively expensive, both in terms of time and space. Overall, the rolled and unrolled graph representations show similar trends in classifying NNs by effectively capturing their early training dynamics: the structural changes in the training NN architectures during the first 6-15 iterations are sufficient to classify the performance of NN instances with over $90 \%$ accuracy. However, as we discuss in $\ S 4 . 3$ , our proposed rolled representation is significantly more efficient than the unrolled representation, and can generalize to deeper and larger NNs. We provide more details for these experiments in App. A.2.
+
+![](images/474b2bc179ef6780beb4b2042007fa98943c8f1ff7ba0dbec980e12613a49e0f.jpg)  
+Figure 4: CIFAR-10: NN performance classification for different NN architectures, graph representations, and features for the temporal signatures. Shorthands: ‘deg’ for degree-based and ‘evec’ for eigenvector centralitybased temporal signature. (a)–(d): Accuracy based on weighted degree-based signature vectors for both the rolled and unrolled graph representations. Our rolled graph representation is as effective as the unrolled representation in predicting the image classification performance of NNs, while being significantly more efficient. For the unrolled representation ((c) and (d)), the negative subgraph (solid lines) results in the most accurate performance prediction among the three subgraphs. (e)–(h): Accuracy based on the weighted degree- and eigenvector centrality-based signature vectors of the rolled graph representations of VGG and ResNet-44. For both architectures, SVM performs best when leveraging the degree-based signatures ( $90 \%$ accuracy after 5 training epochs), while MLP outperforms SVM when operating on the eigenvector centrality-based signatures.
+
+OBSERVATION 1 Both the rolled and unrolled time-evolving graph representations of NNs are effective in capturing the changes in the NN dynamics during the training phase, and can be used to predict the accuracy of an NN instance after observing only a few training epochs. Our proposed rolled representation is also space- and time-efficient, unlike the unrolled representation.
+
+In the remainder of this analysis, we focus on the rolled representation, which is more efficient for larger NN models and datasets. We present the classification results for both types of signatures for the temporal graphs corresponding to the training dynamics of VGG and ResNet-44 (CIFAR-10 dataset) in Fig. 4(e)-(h) and LeNet, AlexNet and ResNet-50 (ImageNet datatset) in Fig. 5. In addition to the results discussed above for LeNet and AlexNet, we provide the NN classification accuracy for the eigenvector-based signatures in Fig. 11 in the appendix. In all the cases, classification accuracy of $80 \%$ is achieved in less than 10 training epochs. For degree-based signatures, SVM tends to outperform MLP, while the trend is reversed for eigenvector-based signatures. For example, for both VGG and AlexNet for the CIFAR-10 image classification task, MLP can predict the performance with accuracy ${ \sim } 9 5 \%$ using the eigenvector-based signatures from the first 6 training epochs; the same trend is observed on ImageNet for the LeNet and AlexNet architectures. For ResNet on the ImageNet classification task, SVM tends to perform well for both degree and eigenvector centrality signature vectors; while MLP tends to perform poorly in this case, the MLP variant operating on the exponential average of signature vectors outperforms the original MLP and all SVM variants. In all the cases, both classifiers reach performance over $80 \%$ - $90 \%$ significantly before the early stopping point for all the architectures. In general, for ImageNet, we observe that the signatures based on eigenvector centrality are more effective in the NN performance prediction task compared to the weighted degree-based signatures—irrespective of NN architecture.
+
+OBSERVATION 2 For the CIFAR-10 image classification task, the rolled graph representation for all NN architectures and both signature types achieve accuracy of ${ > } 9 0 \%$ . For ImageNet, the eigenvector centrality-based signatures tend to yield higher performance compared to the weighted degree-based signatures, though both achieve accuracy of $80 \%$ .
+
+4.2 REGRESSION ANALYSIS: PREDICTING NN ACCURACY FROM NN TRAINING DYNAMICS
+
+In this section, we discuss the results and key findings of predicting the actual performance (accuracy) of an NN using a linear regression model, instead of treating this task as a classification problem as in the previous section.
+
+Table 4: Regression Analysis on CIFAR-10: Accuracy prediction based on the rolled graph representation and the degree-based signature.   
+
+<table><tr><td></td><td colspan="2">LeNet</td><td colspan="2">AlexNet</td><td colspan="2">VGG</td><td colspan="2">ResNet</td></tr><tr><td>Time</td><td>MAE</td><td>R²</td><td>MAE</td><td>R²</td><td>MAE</td><td>R²</td><td>MAE</td><td>R²</td></tr><tr><td>0-2</td><td>14.95</td><td>0.16</td><td>18.712</td><td>0.62</td><td>17.72</td><td>0.62</td><td>19.84</td><td>0.60</td></tr><tr><td>1-3</td><td>14.67</td><td>0.3</td><td>18.57</td><td>0.66</td><td>11.98</td><td>0.81</td><td>19.41</td><td>0.62</td></tr><tr><td>3-5</td><td>17.09</td><td>0.67</td><td>18.4</td><td>0.59</td><td>11.91</td><td>0.81</td><td>17.63</td><td>0.67</td></tr><tr><td>5-7</td><td>12.49</td><td>0.75</td><td>16.74</td><td>0.66</td><td>6.87</td><td>0.94</td><td>18.28</td><td>0.68</td></tr></table>
+
+Task setup. Following the same experimental methodology as in the classification task, we use the degree-based signatures of the generated time-evolving graph as the predictor and the overall accuracy of the corresponding NN as the dependent variable. We use $20 \%$ of our observations as a test set, and the rest as a training set. We report the testing mean absolute error (MAE) and coefficient of determination $( R ^ { 2 } )$ .
+
+Table 3 displays the results for regression prediction on ImageNet dataset, the prediction model show similar trend as CIFAR-10 on this datatset. The MAE values are smaller for this dataset since the accuracy range of both architectures are much narrower than the CIFAR-10 dataset (See Table 2 for accuracy ranges of different architectures and datasets.)
+
+Results. Table 4 summarizes the regression results for four architectures where the independent variable is degree features of rolled graph representation. For each step of this experiment, we use a concatenation of two consecutive feature vectors as independent variable of regression model. For LeNet, at timestamp 5-7 the regression model error (MAE for test set) is 5.74, while the accuracy range of LeNet for this dataset is wide (0.9-73.8), and based on $R ^ { 2 } = 0 . 9 1$ , we can interpret that $91 \%$ of variant in accuracy of NN can be explained by the de
+
+![](images/3d32b327795b2241af9c59f39e8bf38b9bc7d42232d19e73538ebeac97b39dbd.jpg)  
+Figure 5: ImageNet: NN classification using the weighted degree-based and eigenvector centrality-based signature vectors for LeNet, AlexNet and ResNet. Eigenvector centrality-based signatures (b), (d), (f) yield higher performance compared to the weighted degreebased signatures (a), (c), (e).
+
+gree changes of the rolled graph between timestamp 5-7. A similar trend is observed for AlexNet, VGG and ResNet, where after observing a few epochs, the prediction model shows a very low MAE with high $R ^ { 2 }$ .
+
+OBSERVATION 3 In sum, the weighted degree-based signature vector of time-evolving graphs generated based on the rolled graph representation is a strong predictor of the actual accuracy value of NNs. For both CIFAR-10 and ImageNet, we show that, by observing only a subset of early training epochs, we can effectively predict the accuracy value of NNs with a small MAE and a high coefficient of determination $R ^ { 2 } > 0 . 5$ .
+
+# 4.3 DISCUSSION
+
+Time efficiency and early stopping. Figure 6 represents the total average runtime of NN training for early stopping of each architecture (green bars) with comparison of average run-time of graph generation, degree calculation and eigenvector centrality calculation for the number of epochs that were needed rolled graph representation can achieve a high accuracy prediction much faster than the early stopping of NN training. But for unrolled representation, this is only true for LeNet architecture. As the size of NN increases, the unrolled graph generation gets slower and training with early stopping method is faster than our prediction framework.
+
+Table 3: Regression Analysis on ImageNet: Accuracy prediction based on the rolled graph representation and the degree-based signature.   
+
+<table><tr><td></td><td colspan="2">AlexNet</td><td colspan="2">LeNet</td><td colspan="2">ResNet</td></tr><tr><td>Time</td><td>MAE</td><td>R²</td><td>MAE</td><td>R²</td><td>MAE</td><td>R²</td></tr><tr><td>0-3</td><td>1</td><td>1</td><td>4.04</td><td>0.30</td><td>7.99</td><td>0.57</td></tr><tr><td>3-5</td><td>4.10</td><td>0.65</td><td>3.94</td><td>0.34</td><td>8.36</td><td>0.60</td></tr><tr><td>5-7</td><td>3.96</td><td>0.70</td><td>3.58</td><td>0.46</td><td>8.02</td><td>0.62</td></tr><tr><td>7-9</td><td>2.70</td><td>0.84</td><td>3.24</td><td>0.54</td><td>7.49</td><td>0.67</td></tr></table>
+
+for that architecture to achieve the highest accuracy in classification task. For all the 3 architectures
+
+![](images/82123f4b0801d63b1f28e0a64dbf2d7e4ae1386bbc2659e72c8ed030c20b36c6.jpg)  
+Figure 6: Average total runtime for the NN training, and the rolled/unrolled graph generation (S1) and feature extraction (S2) on CIFAR-10 dataset ((a)-(c)), ResNet architecture on both CIFAR-10 (ResNet-44) and ImageNet (ResNet-50)(d). Rolled graph representation is more efficient/faster than early stopping to generate graph and calculate signature vector for all the architectures.
+
+Size analysis. Table 5 shows the number of nodes and edges of the rolled graph representations compared with the unrolled representation, our proposed rolled representation of NN reduces the complexity of NN architectures in terms of #nodes and #edges in the graph while retaining comparable performance on the prediction tasks. Our proposed rolled graph representation has big advantage in terms of size over the unrolled graph representation, especially for deeper networks.
+
+Limitations. The unrolled representation is not scalable therefore cannot be applicable for larger networks and datasets. The main advantage of this representation is that there is no loss of information during the training phase of NNs, and we can interpret the learning process easier than the rolled method. For example, we showed that the negative subgraph alone would be a stronger predictor of the accuracy of NNs, which can lead us to another direction of signed graph analysis in further study of the learning process of NNs.
+
+Complexity analysis of the rolled graph representation. For an NN with $n _ { f }$ total number of filters in all the conv layers and $n _ { n }$ total number of neurons in the fully connected layers, modeling the nodes takes a constant time $O ( | \nu | )$ , where $\nu$ is the set of generated nodes in the graph and $| \mathcal { V } | < = n _ { f } + n _ { n }$ (due to dropout some nodes are removed). The computation of the edge weights is $O ( \sum _ { l } | \mathcal { V } | _ { l } * | \mathcal { V } | _ { l + 1 } )$ , where $| \nu | _ { l }$ are the nodes at layer $l$ .
+
+# 5 RELATED WORK
+
+We cover the most relevant work here, and dynamic graph mining in App. A.5. Some recent research efforts are devoted to modeling the NN architectures as graphs due to their topological identity and study their graph properties Rieck et al. (2019); Filan et al. (2020); You et al. (2020). For example, You et al. You et al. (2020) propose a relational graph representation to model message exchange between layers, and empirically show the common properties shared by NNs with significantly improved predictive performance in terms of graph clustering coefficients and average path lengths. Rieck et al. Rieck et al. (2019) propose a complexity measure related to NN performance—neural persistence—based on topological data analysis on weighted stratified graph. Filan et al. Filan et al. (2020) present an exploratory study of the NN modularity. Gebhart et al. (2019) proposes to compute persistent homology over the activation graph of an NN. The output is a graded set of subgraphs, which are shown to be related to the task-specific semantic that are captured by original NN.
+
+# 6 CONCLUSION
+
+In this work, we investigated the early training dynamics of NNs from a time-evolving graph perspective. We proposed a new graph representation to efficiently convert convolutional layers into compact and intuitive graph structures. Then, we showed that a simple, temporal graph signature based on summary statistics of the degree or eigenvector centrality distributions over only a few epochs can be used as a strong predictor variable to estimate the accuracy of NNs in downstream tasks (e.g., image classification). Exploring the role of our efficient proposed framework for early stopping is a promising future direction.
+
+# 7 REPRODUCIBILITY STATEMENT
+
+For reproducibility, we provide the references to the datasets and existing (rolled) graph representation of convolutional layers in $\ S 4$ . In App. $\ S \operatorname { A } . 1$ , we provide the detailed hyperparameter settings for NN training. We will also make our code publicly available upon acceptance.
+
+# REFERENCES
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+
+# A APPENDIX
+
+# A.1 EXPERIMENTAL SETUP: CONFIGURATION
+
+We first train five different NN architectures, LeNet-5 (Lecun et al., 1998), VGG13 (Simonyan & Zisserman, 2015), AlexNet (Krizhevsky et al., 2012), ResNet-32 and ResNet-44 (He et al., 2016) , on CIFAR-10 dataset and three architectures, LeNet-5, AlexNet and ResNet-50, on ImageNet. During training, the NN parameters are updated using stochastic gradient descent. On each dataset, we train NNs by combining 48 learning rates $\{ 1 , \bar { 1 } . 5 , \ldots , 4 . 5 \bar  \} \times \{ 1 0 ^ { - 6 } , \ldots , 1 0 ^ { - 1 } \}$ , and 10 dropout rates $\{ 0 , 0 . 1 , \ldots , 0 . 9 \}$ . All the models are trained for 50 epochs with batch size 128 and an early stopping method by which the training stops when the testing accuracy does not increase for 10 consecutive epochs. Consequently, we obtain $4 8 0 \mathrm { N N s }$ per architecture and dataset, with diverse performance. Information for the generated NNs is summarized in Table 2: the range of epochs at which the training stops early, the range of final testing accuracy for the trained NNs, and the accuracy threshold used to map actual NN performance to ‘low/high accuracy’ labels for the classification task. We trained the NNs on an Nvidia 1080Ti GPU with 11G memory, and we conducted all the other experiments on 2.60GHz Intel Xeon E5-2697 v3 platform with 1024G memory.
+
+For computational efficiency and to avoid having a largely imbalanced dataset, out of the $4 8 0 \mathrm { N N s }$ , we randomly sampled 250 NNs with an even split of high- and low-accuracy networks. For this sample, the first $t$ epochs of training for each configuration were saved as checkpoints to be converted to time-evolving graphs (step (S1)).
+
+# A.2 UNROLLED GRAPH REPRESENTATION
+
+In these experiments, we split the represented graph into two subgraphs with positive and negative edge weight 1. Then we used the two graphs $\mathbf { G } _ { p o s }$ and ${ \bf G } _ { n e g }$ to calculate the degree and eigenvector centrality feature vector summary for both of them, and concatenate those feature vectors to represent the whole graph $\mathbf { G } _ { p o s - n e g }$ . According to Figure 4, weighted degree signature vector of ${ \bf G } _ { n e g }$ is a good descriptive variable to predict performance of NNs. For LeNet, we achieve an accuracy of $90 \%$ with only 3-5 epochs. For AlexNet, since the graph generation process was slow, we constructed the time-evolving graph based on a few sub-samples of training epochs where $T = 1 , 5 , 1 0 , 1 5$ . The results show that even with a fewer sequence of observations the classifier can achieve a high value of accuracy. For instance, at timestamp 10, in which the feature vector is the concatenation of degree features at epochs 1, 5, and 10, the accuracy of the classifier is more than $8 5 \%$ .
+
+Based on results in Figure 11a, signature vector calculated based on eigenvector centrality of ${ \bf G } _ { n e g }$ is also very good descriptive variable to predict performance of NNs and can achieve accuracy of $90 \%$ with only 3-5 epochs for LeNet.
+
+For all the experiments on unrolled representation, the feature vector only based on ${ \bf G } _ { n e g }$ achieves very high accuracy similar to or higher than that of $\mathbf { G } _ { p o s - n e g }$ , while feature vectors only based on $\mathbf { G } _ { p o s }$ result in the poorest accuracy among them.
+
+# A.3 FEATURE ANALYSIS
+
+Classification feature analysis. Figure 7a shows the weight of each statistical aggregator for graph degree of VGG architecture for CIFAR-10. The top three highly weighted features are mean, standard deviation and median of degree. Figures 7b, 7c and 7d represent the change in value of those features respectively over time for high and low accuracy instances. We observe that the statistical aggregator of degree for high accuracy graphs tends to increase over time while for most of low accuracy one these values do not change over time or the change is very small. Figure 8 depicts similar trend for the LeNet architecture, while the values of degree mean and standard deviation for low accuracy cases tend to have a very big spike in the first 3 epochs of training but after that they follow a flat line trend and tend to not change. Figure 9 illustrates similar trend for ResNet-44 architecture and CIFAR-10 dataset. The mode degree of graphs based on high accuracy cases tends to change drastically over time while the low accuracy cases do not change and show a flat line pattern.
+
+![](images/d672de29cb178d51564999d9e15647b734d50e10211ba9dea7981fac942251bc.jpg)  
+Figure 7: VGG on CIFAR-10: (a) The top three most important features in SVM classification are the mean, stdev, and median of the node degrees. (b) The change of the average node degree over time shows that, in graphs corresponding to high-accuracy NNs (blue lines), the average degree exhibits an increasing pattern, while it has a flat pattern for low-accuracy cases (red lines). The few cases of low-accuracy NNs with increasing trend may be miss-classified in the classification task. (c) (d) The changes of the standard deviation and median of node degrees follow a similar pattern to mean for both low- and high-accuracy cases.
+
+![](images/a9aed3cd6ccbaad1d34e703358f6379219eb34f83b4cf11ff909db1d5fcdcd5d.jpg)  
+Figure 8: LeNet on CIFAR-10: (a) The top three most important features in SVM classification are the mean, standard deviation and median of the node degrees. (b)(d) The change of average node degree over time shows that in graphs corresponding to high-accuracy NNs (blue lines) the average degree tends to change over time with a smooth pattern, while it tends to follow a flat line pattern (after an early extreme spike) for low-accuracy cases (red lines).
+
+![](images/7582bf7c2059c82a11f7ed783a27c075047d3ab80c8141018cfdf58e0a3a461b.jpg)  
+Figure 9: ReNet-44 on CIFAR-10: (a) The top three most important features in SVM classification are the mean, standard deviation and median of the node degrees. (b)(d) The change of average node degree over time shows that in graphs corresponding to high-accuracy NNs (blue lines) the average degree tends to change over time drastically, while it tends to follow a flat line pattern for low-accuracy cases (red lines).
+
+# A.4 GENERALIZING TO UNSEEN ARCHITECTURES
+
+In this section, we broaden the scope of our empirical setup to show that our proposed graph representation and signatures are general and can be useful for performance prediction across different NN architectures. In this setup, we fully train just a small set of different NN architectures (i.e., different architectures and hyperparameters), and predict the performance on unseen NN architectures. We consider two sets of experiments:
+
+![](images/1b429c8a5279688fe049c63b437d6c09ae77533e96ad356d830c88071b1ede3d.jpg)  
+(a) CIFAR-10, ResNet-44 (b) ImageNet, ResNet-50 (c) CIFAR-10, ResNet-32 (d) CIFAR-10, ResNet-44   
+Figure 10: NN classification based on degree signatures for the empirical setup that tests generalization to unseen architectures. (a) Train on ResNet-32, test on ResNet-44. (b) Train on ResNet-34, test on ResNet-50. (c)-(d) Train on LeNet, AlexNet and VGG, test on ReNet-32 and ResNet-44, respectively. Our proposed framework is able to accurately predict the performance level on previously unseen architectures based on the NN structural dynamics in a small number of epochs $( < 1 0 )$ .
+
+1. We train our classifier on the smaller ResNet architectures (ResNet-32 for CIFAR-10, and ResNet34 for ImageNet), and test the performance of the larger ResNet architectures (ResNet-44 for CIFAR-10, and ResNet-50 for ImageNet). In Figs. 10a and 10b, we observe that our proposed framework is able to accurately predict the performance level of the previously unseen, large ResNet architectures. Our results show that the proposed temporal signatures can be used in a generalized scenario to predict the accuracy level of the same architecture with different numbers of layers (on the same dataset). This generalization from small to bigger architectures is important since it is faster to train the smaller architectures.
+
+2. In Figs. 10c and 10d, we see that our proposed method also successfully predicts the performance level of a new architecture (i.e. ResNet) when the training set is a combination of older architectures (LeNet, VGG, AlexNet).
+
+For both of the experiments, we set a universal threshold value to label graphs in the train and test set. For the training set of classifiers, we randomly choose a subset of the NNs with balanced high/low accuracy labels. The size of the training set in the first experiment is 250 and in the second experiment is 400. The experiments were repeated 5 times and the average accuracy of classification is reported.
+
+# A.5 ADDITIONAL RELATED WORK
+
+Dynamic Graph Mining. Dynamic graphs are mostly modeled as a sequence of edge additions and/or edge deletions. To mine the dynamic graphs, traditional approaches leverage graph properties such as node centralities or motifs Riondato et al. (2017); Paranjape et al. (2017); Kovanen et al. (2011). Despite their simplicity, these approaches show effectiveness in temporal tasks such as event detection Paranjape et al. (2017) and structural prediction Aggarwal & Subbian (2014). Recent embedding-based approaches tend to model dynamic graphs as a sequence of discrete-time snapshots and simultaneously represent the graph structure of each snapshot as well as the temporal evolution using deep neural networks such as GRU/LSTM Sankar et al. (2020); Singer et al. (2019); Pareja et al. (2020). DySAT Sankar et al. (2020) leverages self-attention to compute node representations by jointly modeling graph structural property and temporal dynamics. EvolveGCN Pareja et al. (2020) uses GCN to generate node embeddings for the past snapshots, and learns parameters of the hidden layer for the next snapshot using GRU/LSTM. Unlike these methods, we propose two different approaches to represent the underlying graph structure of NN, and use a very simple and efficient dynamic graph signature feature vector to predict the accuracy of corresponding NNs.
+
+![](images/759d5efdb9b81d03f670814fcb3444eaa800fe2cdcda54829e3fe34ca2c19135.jpg)  
+Figure 11: CIFAR-10: NN classification based on eigenvector centrality-based signatures. The eigenvector centrality is a strong predictor of NN performance after observing a few epochs of training. The MLP classifier (red lines) outperforms SVM for all the architectures.
+
+![](images/e6d91b80109e8cb353b1584b2663c89452d35eb021d661fddad8ecd097e55f78.jpg)  
+Figure 12: CIFAR-10, ResNet-32: NN classification based on degree and eigenvector centrality-based signatures. The both degree and eigenvector centrality are strong predictor of NN performance after observing a few epochs of training.
+
+Table 5: Size of a generated graph snapshot based on the rolled and unrolled conv layer representation.   
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="5">CIFAR-10</td><td colspan="3">ImageNet</td></tr><tr><td>LeNet</td><td>AlexNet</td><td>VGG</td><td>ResNet32</td><td>ResNet44</td><td>LeNet</td><td>AlexNet</td><td>ResNet50</td></tr><tr><td></td><td>[V|,rolled</td><td>239</td><td>2925</td><td>1613</td><td>1149</td><td>1597</td><td>713</td><td>3015</td><td>22823</td></tr><tr><td></td><td>, rolled</td><td>12954</td><td>1160480</td><td>304576</td><td>51888</td><td>73392</td><td>115 362</td><td>1206560</td><td>10 821824</td></tr><tr><td>V</td><td>,unrolled</td><td>11166</td><td>54986</td><td>205066</td><td></td><td></td><td>11640</td><td>55076</td><td></td></tr><tr><td></td><td>|ε, unrolled</td><td>658024</td><td>45 997 696</td><td>119 714496</td><td></td><td></td><td>843376</td><td>46043 776</td><td></td></tr></table>

parse/dev/EMLJ_mTz_z/EMLJ_mTz_z_content_list.json

@@ -0,0 +1,1542 @@
+[
+    {
+        "type": "text",
+        "text": "CONVOLUTIONAL NEURAL NETWORK DYNAMICS: A GRAPH PERSPECTIVE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The success of neural networks (NNs) in a wide range of applications has led to increased interest in understanding the underlying learning dynamics of these models. In this paper, we go beyond mere descriptions of the learning dynamics by taking a graph perspective and investigating the relationship between the graph structure of NNs and their performance. Specifically, we propose (1) representing the neural network learning process as a time-evolving graph (i.e., a series of static graph snapshots over epochs), (2) capturing the structural changes of the NN during the training phase in a simple temporal summary, and (3) leveraging the structural summary to predict the accuracy of the underlying NN in a classification or regression task. For the dynamic graph representation of NNs, we explore structural representations for fully-connected and convolutional layers, which are key components of powerful NN models. Our analysis shows that a simple summary of graph statistics, such as weighted degree and eigenvector centrality, over just a few epochs can be used to accurately predict the performance of NNs. For example, a weighted degree-based summary of the time-evolving graph that is constructed based on 5 training epochs of the LeNet architecture achieves classification accuracy of over $93 \\%$ . Our findings are consistent for different NN architectures, including LeNet, VGG, AlexNet, and ResNet. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+            539,
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+            554
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural networks (NNs) have driven advancements in many domains, including computer vision and image processing (Hu et al., 2018), natural language processing (Sutskever et al., 2014; Bahdanau et al., 2015), and bioinformatics (Cao et al., 2020; Li et al., 2019). As task complexity increases, networks grow deeper and larger, consequently requiring more computational resources and training data, as well as sacrificing interpretability for improved task performance. Some works have focused on understanding and interpreting deep NNs (Raghu et al., 2017; Chakraborty & et al., 2017; Ioffe & Szegedy, 2015; He et al., 2016). One approach towards this goal involves representing the NN as its underlying graph structure, and studying selected graph properties, such as clustering coefficient, path length (You et al., 2020), modularity (Filan et al., 2021), persistence (Rieck et al., 2019). For example, You et al. (2020) represent NNs as relational graphs capturing the message passing process, and investigate the correlation between the predictive performance of NNs and architectural changes. However, the studies of NN structures as graphs are limited, and the structural changes of the underlying graph during the training process have been largely overlooked in the literature. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To fill this gap, in this work, we take the graph perspective and aim to predict the performance of an NN by capturing early NN dynamics during the training phase. Successful performance prediction based on only a few epochs could be used for early stopping (Yu & Zhu, 2020), and thus, more efficient NN training. To solve the performance prediction problem, we propose a multi-step framework, depicted in Fig. 1. Specifically, we propose to represent the underlying graph structure of an NN as a time-evolving k-partite graph, where each part corresponds to a different NN layer, and each graph snapshot in the evolving graph maps to an NN instance at a specific epoch. We build on existing graph representations of fully-connected and convolutional layers, both of which are key components of popular NN architectures, and introduce a new, compact, efficient-to-compute (“rolled”) graph representation for convolutional layers. Then, we extract well-known node features (weighted degree and eigenvector centrality) from the time-evolving graph and construct temporal signatures by computing summary statistics on the node feature distributions. Finally, we cast NN performance prediction as a classification task and a regression task, each of which operates on the temporal structural signatures of the NN. Our main contributions are: ",
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+    {
+        "type": "image",
+        "img_path": "images/e289bcecb54995782532600550750cd3ff3c42731d3d05d640a0ee59c5f43ee7.jpg",
+        "image_caption": [
+            "Figure 1: Our proposed framework for predicting NN performance in a downstream image classification task, shown for one test instance. The input to our framework is an NN trained for a few epochs (3, in this example). Steps: (S1) the input NN is converted to three static graphs, each representing one training epoch; (S2) node features (e.g., degree) are extracted from each graph snapshot (i.e., one per epoch); (S3) in order to summarize the changes in the graph structure over time, a signature vector is constructed by aggregating the node features per graph snapshot and concatenating the individual snapshot signatures; (S4) a pre-trained classifier and regressor predict the performance of the input NN given the signature vector from (S3). "
+        ],
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+        "bbox": [
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• New graph-based NN representation: We introduce a new graph representation for convolutional layers that is more compact and efficient-to-compute than the existing unrolled representation (Rieck et al., 2019), while not sacrificing accuracy in the NN performance prediction task. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Framework for NN performance prediction: We propose a simple, multi-step graph-based framework to solve the NN performance prediction problem by capturing the early NN dynamics during the training phase. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Extensive empirical analysis: Using well-known image classification datasets (ImageNet and CIFAR-10), and a variety of NN architectures (AlexNet, VGG, LeNet, and ResNet), we show that our framework can effectively predict the performance of NNs by observing only a few epochs of training, well before their corresponding early stopping epochs. For instance, using our framework to capture the changes in the graph structure in only 5 training epochs of the ResNet architecture results in classification accuracy of over $90 \\%$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We first present the key concepts that our work builds upon. Table 1 gives the major symbols and their descriptions. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "An NN model is a collection of connected units (neurons) that are organized in layers, and is defined by a set of parameters that adjust during the training process. We refer to the training process of a single architecture along with its respective hyperparameters as an ‘instance’. We focus on two types of layers that are the key components of many powerful NN models such as LeNet(Lecun et al., 1998), VGG (Simonyan & Zisserman, 2015) and ResNet(He et al., 2016): fully connected layers (fc) and convolutional (conv) layers. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "2.1 GRAPHS: TERMINOLOGY AND NOTATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let $\\mathbf { G } = ( \\nu , \\mathcal { E } )$ be an undirected, weighted graph with node set $\\nu$ , edge set $\\mathcal { E }$ , and weighted adjacency matrix $\\mathbf { W } \\in \\mathbb { R } ^ { | \\mathcal { V } | \\times | \\mathcal { V } | }$ . The neighbors of node $v$ are defined as $\\mathcal { N } _ { G } ( v ) = \\{ u : ( u , v ) \\in \\mathcal { E } \\}$ ; i.e., the set of all nodes that connect directly to $v$ . Graph $\\mathbf { G }$ is $k$ -partite if its nodeset $\\nu$ can be partitioned into $k$ independent sets: $\\textstyle \\mathcal { V } = \\bigcup _ { i = 1 } ^ { k } \\mathcal { V } _ { i }$ and $\\mathcal { V } _ { i } \\cap \\mathcal { V } _ { j } = \\emptyset , i \\neq j$ . A time-evolving graph is a series of static graph snapshots over time: $\\mathcal { G } = \\{ \\mathbf { G } ^ { 1 } , \\mathbf { G } ^ { 2 } , . . . , \\mathbf { G } ^ { T } \\}$ . The graphs in the series may have different nodesets and edgesets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Here we focus on two of the most commonly-used node features in graph mining and network science, degree and eigenvector centrality, which capture different types of node importance or influence. We discuss the interpretation of these features in the context of neural network dynamics in $\\ S 3 . 2$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Degree: Degree is the simplest and most efficient to compute node feature, and captures the connectivity of a node. The weighted degree of node $v$ is defined as the sum of weights of the edges that are incident to $i$ : $\\begin{array} { r } { d _ { i } ^ { w } = \\sum _ { j } \\mathbf { W } _ { i j } } \\end{array}$ . Tracking changes in the degree of a node over time is the most direct way of capturing its structural evolution. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• Eigenvector centrality: This centrality is a sophisticated extension of degree centrality, related to Google’s PageRank, which indicates the influence of a node in $\\mathbf { G }$ (Bonacich, 1972). If a node is connected to several nodes with high eigenvector centrality, then that node will have high centrality. The eigenvector centrality of node $i$ is defined as the $i ^ { t h }$ element in the principal eigenvector $\\mathbf { v }$ of W (i.e., the eigenvector that corresponds to the largest eigenvalue $\\lambda$ ): $\\mathbf { W } \\mathbf { v } = \\lambda _ { m a x } \\mathbf { v }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 FULLY-CONNECTED LAYERS: GRAPH REPRESENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            253,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathbf { x } \\in \\mathbb { R } ^ { D _ { i n } }$ be the input vector, $\\mathbf { W } \\in \\mathbb { R } ^ { D _ { i n } \\times D _ { o u t } }$ the learnable weight matrix, $\\mathbf { b } \\in \\mathbb { R } ^ { D _ { o u t } }$ the bias vector and $z$ a non-linear activation functions. Then, the output of this layer is given as $\\mathbf { y } = z ( \\mathbf { x } \\mathbf { W } + \\mathbf { b } )$ , where $D _ { i n }$ and $D _ { o u t }$ correspond to the dimension of input and output layers. Fully-connected layers are straightforward to represent with a weighted, undirected graph (Filan et al., 2021). Each neuron, including those in the input and output layers, corresponds to a node. Two neurons are connected via an edge if they appear in consecutive layers. More formally, let nodeset $\\nu _ { i }$ be the set of all neurons at layer $i$ , and nodeset $\\nu _ { j }$ the set of all neurons at layer $j$ (which follows layer $i$ ). The edges that connect all nodes across the two nodesets are defined by the learnable weight matrix $\\mathbf { W } \\in \\mathbb { R } ^ { | \\mathcal { V } _ { i } | \\times | \\mathcal { V } _ { j } | }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/1b0a0163439aaa784c656b5c4cf20aed0bf469dca30bf8aed2ac7c546adc435a.jpg",
+        "table_caption": [
+            "Table 1: Major symbols and definitions "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Symbol</td><td>Definition</td></tr><tr><td>G= (V,ε) 9={Gi,.,G𝑇}</td><td>a graph with its nodeset and edgeset a time-evolving graph,i.e.,series of graph snapshots</td></tr><tr><td>du，du W</td><td>degree and weighted degree of node u learnableweightmatrixinanNN,and</td></tr><tr><td></td><td>the weighted adj matrix of G</td></tr><tr><td>K</td><td>convolutional kernel</td></tr><tr><td>Din/out</td><td>dimension of the input/output vectors</td></tr><tr><td>c,f</td><td>number of channels and filters</td></tr><tr><td>h,w</td><td>height and width of an input image</td></tr><tr><td>hker,Wker</td><td>height and width of a kernel</td></tr><tr><td>x</td><td>input tensor of conv layer</td></tr></table>",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 CONVOLUTIONAL LAYERS: UNROLLED GRAPH REPRESENTATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            496,
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+            508
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $\\begin{array} { r l r } { { \\pmb x } } & { { } \\in } & { \\mathbb { R } ^ { c \\times h \\times w } } \\end{array}$ be the input and $\\kappa \\in \\mathbb { R } ^ { f \\times c \\times h _ { k e r } \\times w _ { k e r } }$ the convolutional kernel, where $c$ is the number of channels, and $h _ { k e r } , w _ { k e r }$ are the height and width of the kernel respectively. After performing the convolution, we have $\\pmb { y } ^ { } \\in \\mathsf { \\tilde { R } } ^ { F \\times h _ { o u t } \\times \\mathsf { \\tilde { w } } _ { o u t } }$ , where $h _ { o u t } = h - h _ { k e r } + 1$ and $w _ { o u t } = w - w _ { k e r } + 1$ Typical multilayer CNNs consist of convolutional layers followed by fully-connected layers. The representation of fully connected layers is straightforward but cannot be used to model ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/50157361f7d0836e8518212df6cdf24242bb9454432482bd5a371b0fc0aeee97.jpg",
+        "image_caption": [
+            "Figure 2: Unrolled graph representation example. The two typed nodes in the resultant bipartite graph map to the filtration operation and the output. For the stride in this example, the output node $o _ { 1 }$ (in dark blue) is the output of 4 operation-typed nodes (in light blue). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "convolutional layers. Rieck et al. (2019) proposed to \"unroll\" the convolution to convert conv layers into graphs. In Sec. 3, we introduce a new, compact, rolled representation that is both efficient to compute and effective at predicting the NN performance, as we show empirically in Sec. 4. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For a convolutional layer, we first unroll the convolutional operation and then represent the graph as in the case of a fully-connected layer. In this representation, the nodes and edges of the graph are defined through the convolutional operation as matrix multiplication (Gebhart et al., 2019). Specifically, for an input image $_ { x }$ and kernel $\\boldsymbol { \\kappa } _ { i } \\in \\mathbb { R } ^ { f _ { i } \\times c _ { i } \\times h _ { i } \\times \\overset { . } { w } _ { i } }$ , each node in layer $l$ is defined to be the output of the mapped feature of that input for each filter $f _ { i }$ . Edges connect each of these nodes to the corresponding nodes of the output neuron in the next layer. These edges are weighted by the activation value of that neuron (i.e., the input image for a specific stride is multiplied by the filter value at that location in the image). For an input image filter size $( c _ { i } \\times h _ { i } \\times w _ { i } )$ that results in feature-map (output) of size $1 \\times o _ { h } \\times o _ { w }$ , the number of nodes of output layer of the graph representation is a function of $o _ { h } \\times o _ { w }$ , while the number of nodes in the conv operation layer of graph is a function of $c _ { i } \\times h _ { i } \\times w _ { i }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As an example, Fig. 2 depicts the unrolled graph representation of a toy convolutional layer. It shows one conv operation on a small 2-dimensional input image $\\left( 4 \\times 4 \\right)$ and filter $( 2 \\times 2 )$ . The nodes and edges of the represented graph are shown based on one stride (of size 2) of convolution operation. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 NEURAL NETWORK PERFORMANCE PREDICTION: A TEMPORALGRAPH-BASED APPROACH",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first formally introduce the problem that we seek to solve. Then, we present a new, compact, and efficient-to-compute graph representation for convolutional layers $( \\ S 3 . 1 )$ , and describe our proposed temporal graph-based framework that captures the NN dynamics during training $( \\ S \\ 3 . 2 )$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Problem 1 (NN Performance Prediction) Let $\\mathcal { N } = \\{ N _ { t r _ { 1 } } , N _ { t r _ { 2 } } , . . . , N _ { t r _ { n } } \\}$ be a training set of n NNs trained for $T$ epochs and $\\mathcal { A } = \\{ \\alpha _ { 1 } , \\alpha _ { 2 } , . . . , \\alpha _ { n } \\}$ their corresponding downstream task accuracies (e.g., for image classification). We seek to predict the accuracy $\\alpha _ { t s t }$ of a new instance $N _ { t s t }$ trained for a very small number of $t \\ll T$ epochs by using $t$ epochs for the trained NNs in $\\mathcal { N }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our proposed solution takes the graph perspective, and its first step is to represent each NN as a temporal graph. In addition to the graph representations that we presented in Sec. 2, we introduce a new, efficient representation for convolutional layers, which we describe next. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 ROLLED GRAPH REPRESENTATION FOR CONVOLUTIONAL LAYERS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We call our proposed graph representation for conv layers “rolled,” since it avoids unrolling the convolutional operations introduced in (Rieck et al., 2019). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Overview $\\pmb { \\& }$ Motivation. To preserve the semantic meaning of conv layers, we represent each filter as a node, and link filters in consecutive layers via weighted edges (we define the weights below), as shown in Fig. 3. The motivation behind this approach is that for larger networks, there is an explosion of nodes by unrolling the convolutions (unrolled representation explained in $\\ S \\ 2 . 3 \\AA ,$ ), and the integrity of a unit (a single filter or kernel) becomes untenable. Also, by mapping the nodes to specific entities of NNs, our proposed graph model is more interpretable than the unrolled model, and thus it is easier to interpret the outputs of downstream graph analysis on our graph representation (e.g., computing node features, tracking the evolution of the graph). ",
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+    {
+        "type": "image",
+        "img_path": "images/7051065a2d20e1f400b4423d47355d2f618c86e339e0360238b85cb8ec3b9b1a.jpg",
+        "image_caption": [
+            "Figure 3: Rolled graph representation example. The resultant graph is a tri-partite graph with three node types (Conv1, Conv2, FC) corresponding to two convolutional layers and one FC layer. Gray and light blue nodes represent filters in the conv layers and dark blue nodes represent neurons in the FC layer. The red edge between nodes 6 (gray) and 16 (light blue) is weighted by the $\\mathrm { { N o r m ( 6 ^ { t h } } }$ channel of filter 16). Due to dropout, the maximum number of edges is $6 \\times 1 6$ between the conv layers, and $1 6 \\times 1 2 0$ between the Conv2 and FC layers. "
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+        "type": "text",
+        "text": "Detailed Description. Formally, let tensor $\\kappa _ { i }$ be a kernel in layer $i$ with $f _ { i }$ filters, each with $c _ { i }$ nels and dim indexes the sions filter $h _ { i } \\times w _ { i }$ . el use bracket. We create tation to index into the kernel: nodes representing each filter $\\kappa _ { i } [ l , :$ $, : , : ]$ $l ^ { \\mathrm { t h } }$ $\\kappa _ { i }$ $f _ { i }$ $\\{ v _ { 1 } ^ { ( i ) } , v _ { 2 } ^ { ( i ) } , . . . v _ { f _ { i } } ^ { ( i ) } \\}$ with features defined as the corresponding biases in that layer. Let be the next convolutional layer defined analogously. While edge weights between neurons in FC layers are defined in the standard way, we define the edge weights between conv layers as the norm over each kernel’s channels. The edge between node $v _ { k } ^ { \\left( i \\right) }$ representing the $k ^ { t h }$ filter in layer $i$ (i.e., $\\kappa _ { i } [ k , : , : , : ] )$ and node/filter $v _ { l } ^ { ( j ) }$ in layer $j$ (i.e. $\\kappa _ { j } [ l , \\vdots , \\vdots , \\vdots )$ has weight $w _ { v _ { k } ^ { ( i ) } , v _ { l } ^ { ( j ) } } = \\mathrm { n o r m } ( \\mathcal { K } _ { j } [ l , k , : , : ] )$ , which is the norm of the $k ^ { \\mathrm { { t h } } }$ channel of the $l ^ { \\mathrm { t h } }$ filter in the $j ^ { \\mathrm { t h } }$ layer. Though the proposed representations enable alternative edge weight configurations, we focus on the norm of filters as other studies, including those on pruning NNs (Li et al., 2016), have demonstrated that this metric has strong correlation with filter importance. In the case of two conv layers, the resultant graph is an attributed bipartite graph with $f _ { i } + f _ { j }$ nodes and $f _ { i } \\times f _ { j }$ edges, where node attributes include flattened weight vectors or filter maps. Other information such as average gradients or the bias vector can also be used for node features. ",
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+        "type": "text",
+        "text": "3.2 PROPOSED TEMPORAL GRAPH-BASED FRAMEWORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A key objective of our work is to test whether the introduced graph representation of NNs is informative for predicting the performance of NNs. Next, we describe the steps of our proposed framework for solving Problem 1, namely performance prediction from the NN training dynamics. ",
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+        "type": "text",
+        "text": "As shown in Fig. 1, our method consists of four steps: (S1) generation of a temporal graph for the training phase of each NN; (S2) extraction of node features; (S3) construction of a feature-based graph signature that captures the NN dynamics; (S4) prediction of NN performance by training a classifier or regressor on the constructed signatures. We describe these steps in more detail next. ",
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+        "type": "text",
+        "text": "(S1) Graph generation. The first step involves converting the training process of each input NN into a time-evolving graph. Each NN $N _ { t r _ { i } } \\in \\mathcal { N }$ , is represented as a series of checkpoints saved for $t$ training epochs. Based on these checkpoints and the graph representation approaches for fc and conv layers presented in 2.2, 3.1 and 2.3, we first convert the $\\tau ^ { t \\bar { h } }$ NN checkpoint into weighted graph $G _ { i } ^ { ( \\tau ) }$ at timestamp/epoch $\\tau$ . We note that although our proposed graph representation involves node features, our framework does not leverage them, and thus we consider the generated graphs unattributed. Therefore, each NN $N _ { t r _ { i } }$ is mapped to a time-evolving graph $\\mathcal { G } _ { t r _ { i } } = \\{ \\mathbf { \\bar { G } } _ { i } ^ { 1 } , \\mathbf { G } _ { i } ^ { 2 } , . . . , \\mathbf { \\bar { G } } _ { i } ^ { t } \\}$ The output of this step is a set of $n$ time-evolving graphs $\\{ \\mathcal { G } _ { t r _ { 1 } } , \\mathcal { G } _ { t r _ { 2 } } , . . . , \\mathcal { G } _ { t r _ { n } } \\}$ corresponding to the original $n$ NNs in $\\mathcal { N }$ . ",
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+        "type": "text",
+        "text": "(S2) Feature extraction. Next, the goal is to capture the structural dynamics of the NN training process. We aim to select graph measures that can capture changes during the training process, take into account the edge weight of graphs and can be calculated efficiently. In order to do that in an interpretable way, we extract two well-known node centralities from each snapshot of each generated time-evolving graph $\\mathcal { G } _ { i }$ : weighted degree centrality and eigenvector centrality $( \\ S 2 . 1 )$ . The weighted degree is a simple function of the learnable weight matrix W during the training phase of NN, therefore it gives us insights into the training dynamics at the node/neuron/filter level. The eigenvector centrality is an extension of the degree centrality, which captures the highly influential nodes, and has been successfully used in neuroscience to capture the dynamic changes of real neural networks (or connectomes) (Lohmann et al., 2010). Eigenvector centrality can be used to capture importance and connectivity of filters/neurons (i.e., the nodes in our graph representation). Also, eigenvector centrality has been used for detecting communities (Newman, 2006) or clusters (Wu et al., 2013), and thus provides structural information about the clusterability of the NN, which is complementary to that provided by the simpler and more efficient-to-compute degree centrality. ",
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+        "type": "text",
+        "text": "Our choice of features is also guided by the inherent $k$ -partite structure of our proposed graph representation, which cannot be represented well by several other commonly-used graph features. For example, clustering-based features (e.g., number of triangles, transitivity, clustering coefficient) and cycle-based metrics which account for closed paths in a graph are always equal to 0 for $k$ -partite graphs. Moreover, connected component-related features (i.e., strong/weak connectivity) do not capture the learned edge weights, which are important for modeling the training dynamics of NNs. Other features (e.g., betweenness centrality) tend to be computationally expensive, and would add significant overhead compared to early stopping methods. ",
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+        "type": "text",
+        "text": "(S3) Graph signature construction. In order to be able to compare NN-based graphs (with different number of nodes and edges), we summarize the structural changes in the generated time-evolving graphs at the graph level (rather than the node level, as in (S2)), and construct a statistical summary of the extracted node centralities (signature) per time-evolving graph $\\mathcal { G } _ { i }$ . For each snapshot $\\mathbf { G } _ { i } ^ { ( \\tau ) }$ of $\\mathcal { G } _ { i }$ , we create a signature vector using five node feature aggregators, which were introduced in (Berlingerio et al., 2012) for graph similarity: median, mean, standard deviation, skewness, and kurtosis, where all but the median are moments of the corresponding distribution. Thus, $\\mathbf { G } _ { i } ^ { ( \\tau ) }$ is mapped to a (static) signature vector $s _ { i } ^ { ( \\tau ) } \\in \\mathbb { R } ^ { 5 }$ , representing the statistical summary of its node features (i.e., degree or eigenvector centrality) at time $\\tau$ . To put more emphasis on the most recent timestamp, we can redefine the signature at time $\\tau$ as the linear weighted average of the signatures up to that point, $\\mathbf { s } _ { i } ^ { ( \\tau ) } \\gets \\frac { \\sum _ { j } ^ { \\tau } j * \\mathbf { s } ^ { ( j ) } } { \\sum j }$ s  , or an exponential function of the previous signatures, s(τ)i ← αs(τ)i + (1 − α)s(τ−1)i . To obtain the temporal signature of the evolving graph $\\mathcal { G } _ { i }$ , we aggregate the (static) signatures up to timestamp/epoch $t$ : $\\mathbf { s } _ { i } ^ { \\tilde { t } } = \\mathbf { s } _ { i } ^ { 1 } \\oplus \\mathbf { s } _ { i } ^ { 2 } \\oplus \\ldots \\oplus \\bar { \\mathbf { s } _ { i } ^ { t } }$ , where $\\oplus$ denotes concatenation. We note that global features such as algebraic connectivity, modularity, and average shortest paths may be seen as alternative ways for constructing global graph signatures while circumventing the local feature extraction step (S2); however, these features fail to capture the structural changes in our proposed graph representations (they remain (near-)constant over time) and lead to poor performance. ",
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+        "type": "text",
+        "text": "(S4) Performance prediction. For the last step of performance prediction, we consider two tasks: ",
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+    {
+        "type": "table",
+        "img_path": "images/047e49e054b9df6aed62b7de57cecf5975ba4fe8f3ada94e37a07a41a7e8f9af.jpg",
+        "table_caption": [
+            "Table 2: Information for the generated NNs: range for early stopping epoch, range for accuracy, and accuracy threshold used for defining the class labels for classification task (predicting the accuracy level of NNs). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"5\">CIFAR-10</td><td colspan=\"3\">ImageNet</td></tr><tr><td>LeNet</td><td>AlexNet</td><td>VGG</td><td>ResNet-32</td><td>ResNet-44</td><td>LeNet</td><td>AlexNet</td><td>ResNet-50</td></tr><tr><td>Early stopping</td><td>11~50</td><td>30~50</td><td>45~50</td><td>16~120</td><td>16~120</td><td>16~50</td><td>16~50</td><td>16~120</td></tr><tr><td>Acc. range</td><td>9.4~ 73.8</td><td>5.5~82.4</td><td>8.8~87.6</td><td>8.4~90.0</td><td>9.9~89.8</td><td>0.6~14.4</td><td>0.6~20.1</td><td>0.86~41.66</td></tr><tr><td>Acc. thres.</td><td>40</td><td>40</td><td>40</td><td>40</td><td>40</td><td>9</td><td>10</td><td>25</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "• Classification: We train a classifier (e.g., SVM, MLP) using the training graphs $\\{ \\mathcal { G } _ { t r _ { 1 } } , \\mathcal { G } _ { t r _ { 2 } } , . . . , \\mathcal { G } _ { t r _ { n } } \\}$ represented by their temporal signatures $\\{ \\mathbf { s } _ { t r _ { 1 } } ^ { t } , \\mathbf { s } _ { t r _ { 2 } } ^ { t } , . . . , \\mathbf { s } _ { t r _ { n } } ^ { t } \\}$ , and their corresponding accuracies $\\mathcal { A } = \\{ \\alpha _ { 1 } , \\alpha _ { 2 } , . . . , \\alpha _ { n } \\}$ mapped to labels $\\mathcal { L = } \\{ l _ { 1 } , \\bar { l } _ { 2 } , \\dots , \\bar { l _ { n } } \\}$ (e.g., high/low accuracy) based on some threshold. Any test NN instance is then classified using the trained classifier. ",
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+        "type": "text",
+        "text": "• Linear regression: We perform linear regression to estimate the actual accuracy value $\\alpha _ { t s t }$ of a new test instance $N _ { t s t }$ based on its signature obtained through steps (S1)-(S3). ",
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+    {
+        "type": "text",
+        "text": "4 EMPIRICAL ANALYSIS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we empirically evaluate the effectiveness and efficiency of our framework in the classification and regression tasks for different graph representations (rolled and unrolled graphs for conv layers) and different feature-based signatures (degree- vs. eigenvector centrality-based). ",
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+        "type": "text",
+        "text": "Data. We investigate NN dynamics using our framework on two well-known image classification datasets, CIFAR-10 (Krizhevsky, 2009) and ImageNet (Russakovsky et al., 2015). CIFAR-10 consists of 50K training images and 10K test images. For ImageNet, we use a sample that has 50K training images and 5K validation images used as the test set. The sample is obtained by randomly selecting 100 classes from Tiny ImageNet (tin) and downsizing the images to $3 2 \\times 3 2$ colored images. ",
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+    {
+        "type": "text",
+        "text": "Configuration. The configuration of training different NN models, and the early stopping method are described in App. A.1. For the unrolled graph representation, which is signed, we consider different graph types (e.g., positive, negative), which we describe in App. A.2. ",
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+    {
+        "type": "text",
+        "text": "4.1 CLASSIFICATION: PREDICTING NN ACCURACY RANGE FROM NN TRAINING DYNAMICS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Task setup. We cast the NN performance prediction as a classification task. Specifically, the generated time-evolving graphs are labeled as high and low accuracy based on the performance of their corresponding NNs; Table 2 lists the threshold value chosen for low and high accuracy labels based on the final accuracy range of trained NNs, as well as the early stopping epochs for each architecture. Five-fold cross validation is used to predict the label of the test graphs in a binary classification task using SVM and MLP, where the input is the set of temporal signatures $\\{ \\mathbf { s } _ { t r _ { 1 } } ^ { t } , \\mathbf { \\bar { s } } _ { t r _ { 2 } } ^ { t } , . . . , \\mathbf { s } _ { t r _ { n } } ^ { t } \\}$ . We report the classification accuracy. Since the sample of NNs (App. A.1) is randomly selected with balanced high/low accuracy instances, the accuracy of a random classifier as the baseline is $50 \\%$ (omitted from the charts to avoid clutter). Additionally, to show that our proposed graph representation and signatures are general and can be useful across different NN architectures, we consider the following setup: we train the classifier on a small set of NN models (i.e., different architectures—such as LeNet, AlexNet, and VGG—and hyperparameters), and predict the performance on unseen architectures (e.g., ResNet). We describe these experiments in App. A.4. ",
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+        "type": "text",
+        "text": "Results. Figure 4 illustrates the performance of SVM and MLP classifiers operating on weighted degree-based signatures of the rolled and unrolled graph representations of the LeNet and AlexNet architectures, trained on the CIFAR-10 dataset. We omit the results on VGG and ResNet, as well as image classification on ImageNet, because the unrolled graph generation process is prohibitively expensive, both in terms of time and space. Overall, the rolled and unrolled graph representations show similar trends in classifying NNs by effectively capturing their early training dynamics: the structural changes in the training NN architectures during the first 6-15 iterations are sufficient to classify the performance of NN instances with over $90 \\%$ accuracy. However, as we discuss in $\\ S 4 . 3$ , our proposed rolled representation is significantly more efficient than the unrolled representation, and can generalize to deeper and larger NNs. We provide more details for these experiments in App. A.2. ",
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+    {
+        "type": "image",
+        "img_path": "images/474b2bc179ef6780beb4b2042007fa98943c8f1ff7ba0dbec980e12613a49e0f.jpg",
+        "image_caption": [
+            "Figure 4: CIFAR-10: NN performance classification for different NN architectures, graph representations, and features for the temporal signatures. Shorthands: ‘deg’ for degree-based and ‘evec’ for eigenvector centralitybased temporal signature. (a)–(d): Accuracy based on weighted degree-based signature vectors for both the rolled and unrolled graph representations. Our rolled graph representation is as effective as the unrolled representation in predicting the image classification performance of NNs, while being significantly more efficient. For the unrolled representation ((c) and (d)), the negative subgraph (solid lines) results in the most accurate performance prediction among the three subgraphs. (e)–(h): Accuracy based on the weighted degree- and eigenvector centrality-based signature vectors of the rolled graph representations of VGG and ResNet-44. For both architectures, SVM performs best when leveraging the degree-based signatures ( $90 \\%$ accuracy after 5 training epochs), while MLP outperforms SVM when operating on the eigenvector centrality-based signatures. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "OBSERVATION 1 Both the rolled and unrolled time-evolving graph representations of NNs are effective in capturing the changes in the NN dynamics during the training phase, and can be used to predict the accuracy of an NN instance after observing only a few training epochs. Our proposed rolled representation is also space- and time-efficient, unlike the unrolled representation. ",
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+    {
+        "type": "text",
+        "text": "In the remainder of this analysis, we focus on the rolled representation, which is more efficient for larger NN models and datasets. We present the classification results for both types of signatures for the temporal graphs corresponding to the training dynamics of VGG and ResNet-44 (CIFAR-10 dataset) in Fig. 4(e)-(h) and LeNet, AlexNet and ResNet-50 (ImageNet datatset) in Fig. 5. In addition to the results discussed above for LeNet and AlexNet, we provide the NN classification accuracy for the eigenvector-based signatures in Fig. 11 in the appendix. In all the cases, classification accuracy of $80 \\%$ is achieved in less than 10 training epochs. For degree-based signatures, SVM tends to outperform MLP, while the trend is reversed for eigenvector-based signatures. For example, for both VGG and AlexNet for the CIFAR-10 image classification task, MLP can predict the performance with accuracy ${ \\sim } 9 5 \\%$ using the eigenvector-based signatures from the first 6 training epochs; the same trend is observed on ImageNet for the LeNet and AlexNet architectures. For ResNet on the ImageNet classification task, SVM tends to perform well for both degree and eigenvector centrality signature vectors; while MLP tends to perform poorly in this case, the MLP variant operating on the exponential average of signature vectors outperforms the original MLP and all SVM variants. In all the cases, both classifiers reach performance over $80 \\%$ - $90 \\%$ significantly before the early stopping point for all the architectures. In general, for ImageNet, we observe that the signatures based on eigenvector centrality are more effective in the NN performance prediction task compared to the weighted degree-based signatures—irrespective of NN architecture. ",
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+        "type": "text",
+        "text": "OBSERVATION 2 For the CIFAR-10 image classification task, the rolled graph representation for all NN architectures and both signature types achieve accuracy of ${ > } 9 0 \\%$ . For ImageNet, the eigenvector centrality-based signatures tend to yield higher performance compared to the weighted degree-based signatures, though both achieve accuracy of $80 \\%$ . ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "4.2 REGRESSION ANALYSIS: PREDICTING NN ACCURACY FROM NN TRAINING DYNAMICS ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we discuss the results and key findings of predicting the actual performance (accuracy) of an NN using a linear regression model, instead of treating this task as a classification problem as in the previous section. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c87143333a84787713c814028844a15b4379f17cef114dc0cde351003861db40.jpg",
+        "table_caption": [
+            "Table 4: Regression Analysis on CIFAR-10: Accuracy prediction based on the rolled graph representation and the degree-based signature. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">LeNet</td><td colspan=\"2\">AlexNet</td><td colspan=\"2\">VGG</td><td colspan=\"2\">ResNet</td></tr><tr><td>Time</td><td>MAE</td><td>R²</td><td>MAE</td><td>R²</td><td>MAE</td><td>R²</td><td>MAE</td><td>R²</td></tr><tr><td>0-2</td><td>14.95</td><td>0.16</td><td>18.712</td><td>0.62</td><td>17.72</td><td>0.62</td><td>19.84</td><td>0.60</td></tr><tr><td>1-3</td><td>14.67</td><td>0.3</td><td>18.57</td><td>0.66</td><td>11.98</td><td>0.81</td><td>19.41</td><td>0.62</td></tr><tr><td>3-5</td><td>17.09</td><td>0.67</td><td>18.4</td><td>0.59</td><td>11.91</td><td>0.81</td><td>17.63</td><td>0.67</td></tr><tr><td>5-7</td><td>12.49</td><td>0.75</td><td>16.74</td><td>0.66</td><td>6.87</td><td>0.94</td><td>18.28</td><td>0.68</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Task setup. Following the same experimental methodology as in the classification task, we use the degree-based signatures of the generated time-evolving graph as the predictor and the overall accuracy of the corresponding NN as the dependent variable. We use $20 \\%$ of our observations as a test set, and the rest as a training set. We report the testing mean absolute error (MAE) and coefficient of determination $( R ^ { 2 } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 3 displays the results for regression prediction on ImageNet dataset, the prediction model show similar trend as CIFAR-10 on this datatset. The MAE values are smaller for this dataset since the accuracy range of both architectures are much narrower than the CIFAR-10 dataset (See Table 2 for accuracy ranges of different architectures and datasets.) ",
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+    {
+        "type": "text",
+        "text": "Results. Table 4 summarizes the regression results for four architectures where the independent variable is degree features of rolled graph representation. For each step of this experiment, we use a concatenation of two consecutive feature vectors as independent variable of regression model. For LeNet, at timestamp 5-7 the regression model error (MAE for test set) is 5.74, while the accuracy range of LeNet for this dataset is wide (0.9-73.8), and based on $R ^ { 2 } = 0 . 9 1$ , we can interpret that $91 \\%$ of variant in accuracy of NN can be explained by the de",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/3d32b327795b2241af9c59f39e8bf38b9bc7d42232d19e73538ebeac97b39dbd.jpg",
+        "image_caption": [
+            "Figure 5: ImageNet: NN classification using the weighted degree-based and eigenvector centrality-based signature vectors for LeNet, AlexNet and ResNet. Eigenvector centrality-based signatures (b), (d), (f) yield higher performance compared to the weighted degreebased signatures (a), (c), (e). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "gree changes of the rolled graph between timestamp 5-7. A similar trend is observed for AlexNet, VGG and ResNet, where after observing a few epochs, the prediction model shows a very low MAE with high $R ^ { 2 }$ . ",
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+    {
+        "type": "text",
+        "text": "OBSERVATION 3 In sum, the weighted degree-based signature vector of time-evolving graphs generated based on the rolled graph representation is a strong predictor of the actual accuracy value of NNs. For both CIFAR-10 and ImageNet, we show that, by observing only a subset of early training epochs, we can effectively predict the accuracy value of NNs with a small MAE and a high coefficient of determination $R ^ { 2 } > 0 . 5$ . ",
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+    {
+        "type": "text",
+        "text": "4.3 DISCUSSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Time efficiency and early stopping. Figure 6 represents the total average runtime of NN training for early stopping of each architecture (green bars) with comparison of average run-time of graph generation, degree calculation and eigenvector centrality calculation for the number of epochs that were needed rolled graph representation can achieve a high accuracy prediction much faster than the early stopping of NN training. But for unrolled representation, this is only true for LeNet architecture. As the size of NN increases, the unrolled graph generation gets slower and training with early stopping method is faster than our prediction framework. ",
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+    {
+        "type": "table",
+        "img_path": "images/e761ba8781610aa2d87db432d547ed36ed554de89f134b64882dcb8ac1aed36a.jpg",
+        "table_caption": [
+            "Table 3: Regression Analysis on ImageNet: Accuracy prediction based on the rolled graph representation and the degree-based signature. "
+        ],
+        "table_footnote": [
+            "for that architecture to achieve the highest accuracy in classification task. For all the 3 architectures "
+        ],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">AlexNet</td><td colspan=\"2\">LeNet</td><td colspan=\"2\">ResNet</td></tr><tr><td>Time</td><td>MAE</td><td>R²</td><td>MAE</td><td>R²</td><td>MAE</td><td>R²</td></tr><tr><td>0-3</td><td>1</td><td>1</td><td>4.04</td><td>0.30</td><td>7.99</td><td>0.57</td></tr><tr><td>3-5</td><td>4.10</td><td>0.65</td><td>3.94</td><td>0.34</td><td>8.36</td><td>0.60</td></tr><tr><td>5-7</td><td>3.96</td><td>0.70</td><td>3.58</td><td>0.46</td><td>8.02</td><td>0.62</td></tr><tr><td>7-9</td><td>2.70</td><td>0.84</td><td>3.24</td><td>0.54</td><td>7.49</td><td>0.67</td></tr></table>",
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+    {
+        "type": "image",
+        "img_path": "images/82123f4b0801d63b1f28e0a64dbf2d7e4ae1386bbc2659e72c8ed030c20b36c6.jpg",
+        "image_caption": [
+            "Figure 6: Average total runtime for the NN training, and the rolled/unrolled graph generation (S1) and feature extraction (S2) on CIFAR-10 dataset ((a)-(c)), ResNet architecture on both CIFAR-10 (ResNet-44) and ImageNet (ResNet-50)(d). Rolled graph representation is more efficient/faster than early stopping to generate graph and calculate signature vector for all the architectures. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Size analysis. Table 5 shows the number of nodes and edges of the rolled graph representations compared with the unrolled representation, our proposed rolled representation of NN reduces the complexity of NN architectures in terms of #nodes and #edges in the graph while retaining comparable performance on the prediction tasks. Our proposed rolled graph representation has big advantage in terms of size over the unrolled graph representation, especially for deeper networks. ",
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+    {
+        "type": "text",
+        "text": "Limitations. The unrolled representation is not scalable therefore cannot be applicable for larger networks and datasets. The main advantage of this representation is that there is no loss of information during the training phase of NNs, and we can interpret the learning process easier than the rolled method. For example, we showed that the negative subgraph alone would be a stronger predictor of the accuracy of NNs, which can lead us to another direction of signed graph analysis in further study of the learning process of NNs. ",
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+    {
+        "type": "text",
+        "text": "Complexity analysis of the rolled graph representation. For an NN with $n _ { f }$ total number of filters in all the conv layers and $n _ { n }$ total number of neurons in the fully connected layers, modeling the nodes takes a constant time $O ( | \\nu | )$ , where $\\nu$ is the set of generated nodes in the graph and $| \\mathcal { V } | < = n _ { f } + n _ { n }$ (due to dropout some nodes are removed). The computation of the edge weights is $O ( \\sum _ { l } | \\mathcal { V } | _ { l } * | \\mathcal { V } | _ { l + 1 } )$ , where $| \\nu | _ { l }$ are the nodes at layer $l$ . ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We cover the most relevant work here, and dynamic graph mining in App. A.5. Some recent research efforts are devoted to modeling the NN architectures as graphs due to their topological identity and study their graph properties Rieck et al. (2019); Filan et al. (2020); You et al. (2020). For example, You et al. You et al. (2020) propose a relational graph representation to model message exchange between layers, and empirically show the common properties shared by NNs with significantly improved predictive performance in terms of graph clustering coefficients and average path lengths. Rieck et al. Rieck et al. (2019) propose a complexity measure related to NN performance—neural persistence—based on topological data analysis on weighted stratified graph. Filan et al. Filan et al. (2020) present an exploratory study of the NN modularity. Gebhart et al. (2019) proposes to compute persistent homology over the activation graph of an NN. The output is a graded set of subgraphs, which are shown to be related to the task-specific semantic that are captured by original NN. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we investigated the early training dynamics of NNs from a time-evolving graph perspective. We proposed a new graph representation to efficiently convert convolutional layers into compact and intuitive graph structures. Then, we showed that a simple, temporal graph signature based on summary statistics of the degree or eigenvector centrality distributions over only a few epochs can be used as a strong predictor variable to estimate the accuracy of NNs in downstream tasks (e.g., image classification). Exploring the role of our efficient proposed framework for early stopping is a promising future direction. ",
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+    },
+    {
+        "type": "text",
+        "text": "7 REPRODUCIBILITY STATEMENT ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For reproducibility, we provide the references to the datasets and existing (rolled) graph representation of convolutional layers in $\\ S 4$ . In App. $\\ S \\operatorname { A } . 1$ , we provide the detailed hyperparameter settings for NN training. We will also make our code publicly available upon acceptance. ",
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+    {
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A.1 EXPERIMENTAL SETUP: CONFIGURATION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We first train five different NN architectures, LeNet-5 (Lecun et al., 1998), VGG13 (Simonyan & Zisserman, 2015), AlexNet (Krizhevsky et al., 2012), ResNet-32 and ResNet-44 (He et al., 2016) , on CIFAR-10 dataset and three architectures, LeNet-5, AlexNet and ResNet-50, on ImageNet. During training, the NN parameters are updated using stochastic gradient descent. On each dataset, we train NNs by combining 48 learning rates $\\{ 1 , \\bar { 1 } . 5 , \\ldots , 4 . 5 \\bar  \\} \\times \\{ 1 0 ^ { - 6 } , \\ldots , 1 0 ^ { - 1 } \\}$ , and 10 dropout rates $\\{ 0 , 0 . 1 , \\ldots , 0 . 9 \\}$ . All the models are trained for 50 epochs with batch size 128 and an early stopping method by which the training stops when the testing accuracy does not increase for 10 consecutive epochs. Consequently, we obtain $4 8 0 \\mathrm { N N s }$ per architecture and dataset, with diverse performance. Information for the generated NNs is summarized in Table 2: the range of epochs at which the training stops early, the range of final testing accuracy for the trained NNs, and the accuracy threshold used to map actual NN performance to ‘low/high accuracy’ labels for the classification task. We trained the NNs on an Nvidia 1080Ti GPU with 11G memory, and we conducted all the other experiments on 2.60GHz Intel Xeon E5-2697 v3 platform with 1024G memory. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For computational efficiency and to avoid having a largely imbalanced dataset, out of the $4 8 0 \\mathrm { N N s }$ , we randomly sampled 250 NNs with an even split of high- and low-accuracy networks. For this sample, the first $t$ epochs of training for each configuration were saved as checkpoints to be converted to time-evolving graphs (step (S1)). ",
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+    },
+    {
+        "type": "text",
+        "text": "A.2 UNROLLED GRAPH REPRESENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In these experiments, we split the represented graph into two subgraphs with positive and negative edge weight 1. Then we used the two graphs $\\mathbf { G } _ { p o s }$ and ${ \\bf G } _ { n e g }$ to calculate the degree and eigenvector centrality feature vector summary for both of them, and concatenate those feature vectors to represent the whole graph $\\mathbf { G } _ { p o s - n e g }$ . According to Figure 4, weighted degree signature vector of ${ \\bf G } _ { n e g }$ is a good descriptive variable to predict performance of NNs. For LeNet, we achieve an accuracy of $90 \\%$ with only 3-5 epochs. For AlexNet, since the graph generation process was slow, we constructed the time-evolving graph based on a few sub-samples of training epochs where $T = 1 , 5 , 1 0 , 1 5$ . The results show that even with a fewer sequence of observations the classifier can achieve a high value of accuracy. For instance, at timestamp 10, in which the feature vector is the concatenation of degree features at epochs 1, 5, and 10, the accuracy of the classifier is more than $8 5 \\%$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Based on results in Figure 11a, signature vector calculated based on eigenvector centrality of ${ \\bf G } _ { n e g }$ is also very good descriptive variable to predict performance of NNs and can achieve accuracy of $90 \\%$ with only 3-5 epochs for LeNet. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For all the experiments on unrolled representation, the feature vector only based on ${ \\bf G } _ { n e g }$ achieves very high accuracy similar to or higher than that of $\\mathbf { G } _ { p o s - n e g }$ , while feature vectors only based on $\\mathbf { G } _ { p o s }$ result in the poorest accuracy among them. ",
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+    },
+    {
+        "type": "text",
+        "text": "A.3 FEATURE ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Classification feature analysis. Figure 7a shows the weight of each statistical aggregator for graph degree of VGG architecture for CIFAR-10. The top three highly weighted features are mean, standard deviation and median of degree. Figures 7b, 7c and 7d represent the change in value of those features respectively over time for high and low accuracy instances. We observe that the statistical aggregator of degree for high accuracy graphs tends to increase over time while for most of low accuracy one these values do not change over time or the change is very small. Figure 8 depicts similar trend for the LeNet architecture, while the values of degree mean and standard deviation for low accuracy cases tend to have a very big spike in the first 3 epochs of training but after that they follow a flat line trend and tend to not change. Figure 9 illustrates similar trend for ResNet-44 architecture and CIFAR-10 dataset. The mode degree of graphs based on high accuracy cases tends to change drastically over time while the low accuracy cases do not change and show a flat line pattern. ",
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+    {
+        "type": "image",
+        "img_path": "images/d672de29cb178d51564999d9e15647b734d50e10211ba9dea7981fac942251bc.jpg",
+        "image_caption": [
+            "Figure 7: VGG on CIFAR-10: (a) The top three most important features in SVM classification are the mean, stdev, and median of the node degrees. (b) The change of the average node degree over time shows that, in graphs corresponding to high-accuracy NNs (blue lines), the average degree exhibits an increasing pattern, while it has a flat pattern for low-accuracy cases (red lines). The few cases of low-accuracy NNs with increasing trend may be miss-classified in the classification task. (c) (d) The changes of the standard deviation and median of node degrees follow a similar pattern to mean for both low- and high-accuracy cases. "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 8: LeNet on CIFAR-10: (a) The top three most important features in SVM classification are the mean, standard deviation and median of the node degrees. (b)(d) The change of average node degree over time shows that in graphs corresponding to high-accuracy NNs (blue lines) the average degree tends to change over time with a smooth pattern, while it tends to follow a flat line pattern (after an early extreme spike) for low-accuracy cases (red lines). "
+        ],
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+        "img_path": "images/7582bf7c2059c82a11f7ed783a27c075047d3ab80c8141018cfdf58e0a3a461b.jpg",
+        "image_caption": [
+            "Figure 9: ReNet-44 on CIFAR-10: (a) The top three most important features in SVM classification are the mean, standard deviation and median of the node degrees. (b)(d) The change of average node degree over time shows that in graphs corresponding to high-accuracy NNs (blue lines) the average degree tends to change over time drastically, while it tends to follow a flat line pattern for low-accuracy cases (red lines). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            590,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 GENERALIZING TO UNSEEN ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            825,
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+            839
+        ],
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we broaden the scope of our empirical setup to show that our proposed graph representation and signatures are general and can be useful for performance prediction across different NN architectures. In this setup, we fully train just a small set of different NN architectures (i.e., different architectures and hyperparameters), and predict the performance on unseen NN architectures. We consider two sets of experiments: ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/1b429c8a5279688fe049c63b437d6c09ae77533e96ad356d830c88071b1ede3d.jpg",
+        "image_caption": [
+            "(a) CIFAR-10, ResNet-44 (b) ImageNet, ResNet-50 (c) CIFAR-10, ResNet-32 (d) CIFAR-10, ResNet-44 ",
+            "Figure 10: NN classification based on degree signatures for the empirical setup that tests generalization to unseen architectures. (a) Train on ResNet-32, test on ResNet-44. (b) Train on ResNet-34, test on ResNet-50. (c)-(d) Train on LeNet, AlexNet and VGG, test on ReNet-32 and ResNet-44, respectively. Our proposed framework is able to accurately predict the performance level on previously unseen architectures based on the NN structural dynamics in a small number of epochs $( < 1 0 )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            92,
+            812,
+            185
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "1. We train our classifier on the smaller ResNet architectures (ResNet-32 for CIFAR-10, and ResNet34 for ImageNet), and test the performance of the larger ResNet architectures (ResNet-44 for CIFAR-10, and ResNet-50 for ImageNet). In Figs. 10a and 10b, we observe that our proposed framework is able to accurately predict the performance level of the previously unseen, large ResNet architectures. Our results show that the proposed temporal signatures can be used in a generalized scenario to predict the accuracy level of the same architecture with different numbers of layers (on the same dataset). This generalization from small to bigger architectures is important since it is faster to train the smaller architectures. ",
+        "bbox": [
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+            290,
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+            401
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "2. In Figs. 10c and 10d, we see that our proposed method also successfully predicts the performance level of a new architecture (i.e. ResNet) when the training set is a combination of older architectures (LeNet, VGG, AlexNet). ",
+        "bbox": [
+            176,
+            407,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For both of the experiments, we set a universal threshold value to label graphs in the train and test set. For the training set of classifiers, we randomly choose a subset of the NNs with balanced high/low accuracy labels. The size of the training set in the first experiment is 250 and in the second experiment is 400. The experiments were repeated 5 times and the average accuracy of classification is reported. ",
+        "bbox": [
+            174,
+            460,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.5 ADDITIONAL RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            534,
+            428,
+            547
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Dynamic Graph Mining. Dynamic graphs are mostly modeled as a sequence of edge additions and/or edge deletions. To mine the dynamic graphs, traditional approaches leverage graph properties such as node centralities or motifs Riondato et al. (2017); Paranjape et al. (2017); Kovanen et al. (2011). Despite their simplicity, these approaches show effectiveness in temporal tasks such as event detection Paranjape et al. (2017) and structural prediction Aggarwal & Subbian (2014). Recent embedding-based approaches tend to model dynamic graphs as a sequence of discrete-time snapshots and simultaneously represent the graph structure of each snapshot as well as the temporal evolution using deep neural networks such as GRU/LSTM Sankar et al. (2020); Singer et al. (2019); Pareja et al. (2020). DySAT Sankar et al. (2020) leverages self-attention to compute node representations by jointly modeling graph structural property and temporal dynamics. EvolveGCN Pareja et al. (2020) uses GCN to generate node embeddings for the past snapshots, and learns parameters of the hidden layer for the next snapshot using GRU/LSTM. Unlike these methods, we propose two different approaches to represent the underlying graph structure of NN, and use a very simple and efficient dynamic graph signature feature vector to predict the accuracy of corresponding NNs. ",
+        "bbox": [
+            173,
+            560,
+            825,
+            753
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/759d5efdb9b81d03f670814fcb3444eaa800fe2cdcda54829e3fe34ca2c19135.jpg",
+        "image_caption": [
+            "Figure 11: CIFAR-10: NN classification based on eigenvector centrality-based signatures. The eigenvector centrality is a strong predictor of NN performance after observing a few epochs of training. The MLP classifier (red lines) outperforms SVM for all the architectures. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            133,
+            795,
+            281
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/e6d91b80109e8cb353b1584b2663c89452d35eb021d661fddad8ecd097e55f78.jpg",
+        "image_caption": [
+            "Figure 12: CIFAR-10, ResNet-32: NN classification based on degree and eigenvector centrality-based signatures. The both degree and eigenvector centrality are strong predictor of NN performance after observing a few epochs of training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            352,
+            751,
+            517
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/b274539cc9e9d5c301db785382be2b330b85762add08209ffdbb789fd6159e6d.jpg",
+        "table_caption": [
+            "Table 5: Size of a generated graph snapshot based on the rolled and unrolled conv layer representation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"5\">CIFAR-10</td><td colspan=\"3\">ImageNet</td></tr><tr><td>LeNet</td><td>AlexNet</td><td>VGG</td><td>ResNet32</td><td>ResNet44</td><td>LeNet</td><td>AlexNet</td><td>ResNet50</td></tr><tr><td></td><td>[V|,rolled</td><td>239</td><td>2925</td><td>1613</td><td>1149</td><td>1597</td><td>713</td><td>3015</td><td>22823</td></tr><tr><td></td><td>, rolled</td><td>12954</td><td>1160480</td><td>304576</td><td>51888</td><td>73392</td><td>115 362</td><td>1206560</td><td>10 821824</td></tr><tr><td>V</td><td>,unrolled</td><td>11166</td><td>54986</td><td>205066</td><td></td><td></td><td>11640</td><td>55076</td><td></td></tr><tr><td></td><td>|ε, unrolled</td><td>658024</td><td>45 997 696</td><td>119 714496</td><td></td><td></td><td>843376</td><td>46043 776</td><td></td></tr></table>",
+        "bbox": [
+            207,
+            614,
+            789,
+            691
+        ],
+        "page_idx": 14
+    }
+]

parse/dev/Wjp1AYB8lH/Wjp1AYB8lH.md

@@ -0,0 +1,435 @@
+# Large Language Models as Commonsense Knowledge for Large-Scale Task Planning
+
+Zirui Zhao Wee Sun Lee David Hsu National University of Singapore {ziruiz, leews, dyhsu}@comp.nus.edu.sg
+
+# Abstract
+
+Large-scale task planning is a major challenge. Recent work exploits large language models (LLMs) directly as a policy and shows surprisingly interesting results. This paper shows that LLMs provide a commonsense model of the world in addition to a policy that acts on it. The world model and the policy can be combined in a search algorithm, such as Monte Carlo Tree Search (MCTS), to scale up task planning. In our new LLM-MCTS algorithm, the LLM-induced world model provides a commonsense prior belief for MCTS to achieve effective reasoning; the LLM-induced policy acts as a heuristic to guide the search, vastly improving search efficiency. Experiments show that LLM-MCTS outperforms both MCTS alone and policies induced by LLMs (GPT2 and GPT3.5) by a wide margin for complex, novel tasks. Further experiments and analyses on multiple tasks—multiplication, travel planning, object rearrangement—suggest minimum description length (MDL) as a general guiding principle: if the description length of the world model is substantially smaller than that of the policy, using LLM as a world model for model-based planning is likely better than using LLM solely as a policy.1
+
+# 1 Introduction
+
+Consider, for example, an autonomous robot butler in a household environment. The human user sits in the living room and asks the robot to "Put fruits into the fridge." The robot looks for fruits, such as apples, peaches, etc., which may be on a table in the dining room, on the kitchen counter, but unlikely in a wardrobe in the bedroom. To complete the task, the robot has to consider fruits’ likely locations as well as their spatial relations, traverse these locations efficiently, and finally put them into the fridge. A household environment typically contains hundreds of movable items and locations, resulting in a huge search space that makes the task very challenging for the robot.
+
+Recently, multiple attempts exploiting pre-trained large language models (LLMs) show surprisingly interesting results for such tasks [22, 18, 2, 19, 5, 44]. Their underlying idea is simple: treat the LLM as a policy and query it directly for the next actions, given the history of past actions and observations. We call this strategy L-Policy, which exploits LLMs’ vast commonsense knowledge to circumvent the challenge of searching a very large space. In our example task, L-Policy may simply instruct the robot to move to the hallway and then to the kitchen. Even though LLMs are trained on internet-scale data, this strategy shows limits in generalization [10, 4], especially when encountering uncommon, complex tasks. Alternatively, we may use LLMs’ knowledge to build a world model and apply a planning algorithm to the model. The world model may contain, e.g., a belief over the target object’s location, which biases the search and drastically improves search efficiency. We call this strategy L-Model. L-Model’s performance depends on two critical preconditions: the accuracy of the world model and the efficiency of the planning algorithm. The former is a question of sample complexity for learning, and the latter is that of computational complexity.
+
+This paper presents LLM-MCTS (shown in Fig 1), which combines the ideas of L-Model and L-Policy for large-scale task planning. Like L-Model, LLM-MCTS uses an LLM to build a commonsense world model; it then uses the model to perform Monte Carlo Tree Search (MCTS) [8] online for the next actions. During the tree search, LLM-MCTS chooses the promising action branches heuristically by querying the LLM. This is similar in spirit to L-Policy. While L-Policy commits to the actions chosen by the LLM for execution, LLM-MCTS uses these choices only as a search heuristic.
+
+We evaluate LLM-MCTS in VirtualHome [24], a standard household activity simulation platform widely used in earlier work [18, 22, 25, 33]. The evaluation set consists of 800 randomly generated large-scale, partially observable object rearrangement tasks. In each task, a robot aims to fetch a common household item with an unknown location and place it in a designated container. Our main experimental findings are summarized below:
+
+F1. L-Model performs poorly. There are two possible reasons. One is model inaccuracy: the robot has an incorrect belief of the target object’s location. The other is huge search space size, beyond the reach of even the state-of-the-art MCTS algorithm. Further experiments indicate that search space size is the main cause.   
+F2. L-Policy performs reasonably with both GPT2 and GPT3.5, but the performance degrades quickly for novel, complex tasks. This generally corroborates with earlier results [22, 18, 2, 19].   
+F3. LLM-MCTS outperforms L-Model. This clearly shows the benefit of using the LLM as a heuristic policy to guide the search.   
+F4. LLM-MCTS outperforms L-Policy, especially for novel, complex tasks. LLM-MCTS basically combines L-Model and L-Policy. Since the L-Model performs very poorly on its own, why does the combination outperform L-Policy? One explanation is that search space size is the main cause of L-Model’s poor performance. The LLM-induced world model is sufficiently accurate; tree search with this world model, when limited to the neighbourhood of the LLMinduced policy, provides improved performance over L-Policy.
+
+The explanation for (F4) begs a new question: with an efficient planning algorithm for large search space, would $L$ -Model outperform L-Policy? To answer this question, we study two related, simpler tasks: multiplication of two large numbers and multi-hop travel planning. We discuss multiplication here and travel planning in Section 4.1. A decimal number is described as a sequence of $n$ digits, $( d _ { n - 1 } , d _ { n - 2 } , \ldots , d _ { 0 } )$ . There are two methods of implementing multiplication with an LLM. The first one corresponds to L-Policy. We represent the multiplication function as a table. Each row or column corresponds to a number. The table entry is the multiplication of two numbers, obtained by querying an LLM. Experimentally, GPT4 performs single-digit multiplication perfectly with $1 0 0 \%$ accuracy, 2-digit multiplication with $9 9 \%$ accuracy, 4-digit multiplication with merely $\dot { 4 } \%$ accuracy, and fails almost completely on 5-digit multiplication [10]. The second approach uses LLM-derived small single-digit multiplication tables, which GPT4 performs with $100 \%$ accuracy. To multiply multi-digit numbers, it applies the long multiplication algorithm with the single-digit table. This method corresponds to L-Model. While long multiplication differs from planning in algorithmic details, it plays the same role in L-Model and is highly efficient. Clearly, this second method achieves $1 0 0 \%$ accuracy for arbitrarily large numbers, provided that the single-digit multiplication table is accurate. So, the L-Model outperforms L-Policy for the multiplication task, contrary to the finding for object rearrangement tasks.
+
+How do we choose between L-Model and L-Policy then? One idea is the minimum description length (MDL) principle. Theoretical analysis suggests that a hypothesis with a shorter description length has a smaller generalization error and is preferred [28]. For multiplication, the method corresponding to L-Policy uses a large table of $O ( 1 0 ^ { 2 n } )$ entries. It takes $O ( n 1 0 ^ { 2 n } )$ bits to represent it. The method corresponding to the L-Model uses a single-digit multiplication table of constant size. The long multiplication algorithm can be encoded in any reasonable programming language with constant size. So, the total representation size is constant. According to MDL, the L-Model has a smaller generalization error than the L-Policy for multiplication, with sufficiently large $n$ . This is fully consistent with experimental results [10]. The analysis of travel planning provides further evidence (Section 4.1).
+
+In summary, LLM-MCTS combines the ideas of L-Model and L-Policy, outperforming either alone for complex task planning, particularly, object rearrangement (Sections 2 and 3). To choose between L-Model and L-Policy, MDL provides a useful guiding principle (Section 4). In essence, simplicity is preferred, a well-known general principle in machine learning.
+
+![](images/d3a86493932917f60740a2a2f3e1eab1b1ad4a19df626a8233cb29f65da1584e.jpg)  
+Figure 1: Overview of LLM-MCTS. For each simulation in the MCTS, we sample from the commonsense belief to obtain an initial state of the world and use the LLM as heuristics to guide the trajectory to promising parts of the search tree.
+
+# 2 LLM-MCTS: Monte Carlo planning with commonsense knowledge
+
+We aim to solve task-planning problems in large-scale domains with partial observation. One example is object rearrangement tasks [3] in household environments. It is a meaningful and challenging problem with a large-scale and long-term planning horizon. It has many practical implications in everyday life [3, 34, 39, 20, 17], such as setting the table, tidying up the room, loading the dishwasher, etc.. To solve the problem, we present LLM-MCTS (shown in Fig 1), which combines the L-Model and L-Policy for large-scale planning. It uses LLM to build a commonsense world model to perform MCTS for reasoned planning and uses L-Policy to guide the MCTS and reduce the large search space.
+
+# 2.1 Task planning
+
+We focus on task-planning problems with partial observation and large-scale domains. The problem can be formulated as a Partially Observable Markov Decision Process (POMDP): $\bar { ( } S , A , \Omega , T , O , R , \gamma )$ . The state space $S$ define the state of the robot and its environment. The action space $A$ defines the action that the robot can do. $\Omega$ is the observation space. $T$ defines the transition function of states, which we assume to be given. $O$ is the observation function that provides partial information about a state. $R ( s , a )$ is the reward function determined by the action $a$ taken at the state $s$ . The discount factor is specified by $\gamma$ . The history trajectory $h _ { t }$ at time step $t$ consists of a sequence of executed actions and received observations up to time $t - 1$ , $h _ { t } = \left( o _ { 0 } , a _ { 0 } , o _ { 1 } , a _ { 1 } , \ldots , o _ { t - 1 } , a _ { t - 1 } \right)$ . The objective is to find an optimal policy $\pi ^ { * } ( h _ { t } )$ that maximize the expected cumulative rewards $\begin{array} { r } { \pi ^ { * } ( h _ { t } ) = \arg \operatorname* { m a x } _ { a \in A } \mathbb { E } \left[ \sum _ { i = 0 } ^ { \infty } \gamma ^ { i } R ( s _ { t + i } , a _ { t + i } ) | a _ { t } = a \right] . } \end{array}$ .
+
+In this work, we focus on the object rearrangement task, though our approach is a general method for large-scale task planning. Object rearrangement is a representative embodied AI task [3, 34, 39, 20, 17] with various daily applications, such as setting the table, tidying up the room, loading the dishwasher, and more. It is a challenging task [3] as the robot must navigate, locate target objects and positions, and execute multi-step planning. As with hundreds of items and containers in the domain, identifying target objects can be challenging. Even with known state information, it requires long-horizon planning to achieve the goal. In addition, the vast number of domain objects leads to a large action space, given the actions are to interact with objects. The vast action space produces an exponentially large search tree, making the planning extremely challenging.
+
+Following the approach of [22, 25, 33], we model the task as a POMDP as outlined above. The state $S$ comprises variables denoting the positions of the robot, movable items, and containers. Actions $A$ encompass five predefined actions from VirtualHome, parameterized by object/container/rooms: (1) pick(object), where the robot collects an observed, proximate object; (2) place(object,placement), allowing the robot to set a picked object nearby or inside an open container; (3) open(container) and (4) close(container), for interacting with an observed, nearby containers; and (5) move(room/object/container), where the robot relocates within a room or near an observed object/container. Given our assumption of the robot’s familiarity with house structures and the manageable size of the house, the robot can move directly to designated rooms. The deterministic transition $T$ is pre-defined by actions. Partial observation $O$ enables the robot to discern object/container positions within its room or an opened container at its location. The objective is to reorganize household items based on verbal instructions, represented by a reward of $R$ for achieving the desired item arrangement.
+
+# 2.2 LLM as a commonsense world model
+
+A commonsense prior belief of states can improve the effectiveness of object and location searches by prioritizing the search to appropriate locations. Our approach utilizes LLM’s commonsense knowledge to generate the initial belief of states, which is updated with each action and observation in the real world. MCTS samples from the belief in simulation to estimate the value of the action.
+
+Initial belief of state. We use object-centric state representation and categorize the objects in the house as moveable objects (e.g., apples), containers (e.g., fridge), and surfaces (e.g., kitchen table). The states of a moveable object might be inside the containers or on the surfaces. The containers and surfaces should be inside a room. Similar to [22, 25], we maintain the belief in object-centric graphs, where nodes are objects and edges describe abstract-level relationships (e.g., apples are inside fridge, fridge is inside kitchen) between objects and rooms. Details are in the Appendix C.2.
+
+Assume a dataset $\mathcal { D }$ is accessible, containing expert actions and observations in similar household environments to solve daily tasks. LLMs can use the observations in the data to know what are the objects in the house and predict their positions, forming the commonsense belief of the state. To achieve this, we find all the objects, containers, and surfaces that appeared in the dataset $\mathcal { D }$ to form a list of objects $\mathcal { D } _ { \mathrm { o b j } }$ using a unique name for all of them. To approximate $b ( s _ { 0 } )$ , we ask the LLMs to sample the positions of objects $M$ times. For each sample, we ask the LLM to predict the position of objects using $\mathcal { D } _ { \mathrm { o b j } }$ and a fixed prompt. For instance, we ask LLM to complete “The containers in the apartment are: fridge, . . . ; The surfaces in the apartment are: kitchen counter, . . . ; Question: what are the possible positions of strawberry? Answer: inside fridge, inside pantry. . . Question: what are the possible positions of apple? Answer:__.” We use three prompt examples to provide example formats of the response. The exact prompts we used are provided in the appendix. As the responses from LLM are free-form natural language, we have to precisely map those expressions to $\mathcal { D } _ { \mathrm { o b j } }$ for consistent state representation. Thus, we encode the names of objects in the LLM’s response into embeddings using sentence-BERT $f ( \cdot )$ [26] and examine their cosine similarity to the unique name of objects in $\mathcal { D } _ { \mathrm { o b j } }$ : $\begin{array} { r } { \mathrm { C o s i n e S i m } ( e _ { i } , e ) = \frac { f ( e _ { i } ) f ( e ) } { \| f ( e _ { i } ) \| \| f ( e ) \| } } \end{array}$ f(ei)f(e)∥f(ei)∥∥f(e)∥ , where e is the name of objects, containers, or surfaces in the LLM’s response, and $e _ { i } \in \mathcal { D } _ { \mathrm { o b j } }$ are the unique names in the object list. We select the most similar expressions in $\mathcal { D } _ { \mathrm { o b j } }$ to form the sampled state. For example, when querying the position of an apple, the LLM’s response is “on the kitchen table,” we use the above technique to translate “the kitchen table” to “kitchentable,” a unique name in $\mathcal { D } _ { \mathrm { o b j } }$ .
+
+Goal. Similar to [40], we use LLMs to translate the natural language goal into a formal goal for MCTS. We use a fixed set of prompt examples for LLM to interpret natural language goals, such as “put one apple into the fridge” is translated as a tuple “(apple, inside, fridge).” For compositional instructions, it will translate it into multiple tuples, such as “put one apple on the kitchen table and one plate inside the dishwasher” is translated as “(apple, on, kitchentable), (plate, inside, dishwasher).” We precisely map the LLM-generated goal into the admissible expressions in $\mathcal { D } _ { \mathrm { o b j } }$ for search using the same representation as the state. In MCTS, the goal is used to identify the reward. As the representations are the same, we can directly check whether the object’s state is the same as the goal by string matching. If the goal is reached, it will receive a large positive reward, or 0 otherwise.
+
+# 2.3 LLM as a heuristic policy
+
+We use LLMs to play the role of $\pi ( a | h )$ in PUCT to guide the action selection in the simulation procedure. In this procedure, the LLM takes as input the examples in the dataset, the goal description, the current observation, and the history of actions, and then outputs the suggested action plan (e.g., “Next actions: move to the kitchen, open the fridge, ...”). Similar to [22], the observations and goal description are translated into English sentences. As the answer of LLM is from the conditional distribution of the following words given the context, it can also be viewed as a commonsense policy of actions to take conditioned on the context of tasks, observations, and completed actions. However, direct implementation and access to the probability value of the GPT-3.5 is not available. Thus, we propose an empirical policy distribution $\hat { \pi }$ that uses sampling to approximate the policy distribution.
+
+We sample the LLM for $M$ times to approximate the policy probability distribution. For each sample, we query the LLM with prompt and trajectory history $h$ and receive an answer of the following actions to take $\alpha _ { i } \sim \mathrm { L L M } ( h , \mathrm { p r o m p t } )$ , where $\alpha _ { i }$ is the first action of the answer. The prompt examples are retrieved from the dataset according to the similarity to the current language instruction $\ell$ . We use [26] to translate the instructions in the dataset $\ell _ { i } \in \mathcal { D }$ into embedding and examine their
+
+1: procedure SEARCH(h, b, , N)   
+2: n 0   
+3: while n < N do   
+4: s ∼ b(s)   
+5: SIMULATE(s, h, False, 0,  )   
+6: n ← n + 1   
+7: end while   
+8: return $\operatorname { a r g m a x } _ { a \in A } Q ( h , a )$   
+9: end procedure   
+10: procedure ROLLOUT $( s , h , { \mathrm { d o n e } } , d )$   
+11: if $\gamma ^ { d } < \epsilon$ or done $=$ True then   
+12: return 0   
+13: end if   
+14: $a \sim \pi _ { \mathrm { r o l l o u t } } ( h , \cdot )$   
+15: $( s ^ { \prime } , o , r , \mathrm { d o n e } ) \sim \mathcal { G } ( s , a )$   
+16: $h ^ { \prime } \gets \mathrm { P U S H B A C K } ( h , [ a ^ { * } , o ] ) , d ^ { \prime } \gets d + 1$   
+17: return r + γ $\cdot \mathrm { R o L L O U T } ( s , h ^ { \prime } , \mathrm { d o n e } , d ^ { \prime } )$   
+18: end procedure   
+19: procedure SIMULAT $\mathrm { E } ( s , h , \mathrm { d o n e } , d , \mathcal { T } )$   
+20: if γd < ϵ or done $=$ True then   
+21: return 0   
+22: end if   
+23: if $h$ is not in $\tau$ then   
+24: h, N (h)  0   
+25: $\forall a \in A , N ( h , a ) \stackrel { \prime } {  } 0 , Q ( h , a )  0$   
+26: return ROLLOUT(s, h, done, d)   
+27: end if   
+28: $\hat { \pi } ( a | h ) \gets$ QUERYLLMPOLICY(h)   
+29: a∗←argmax Q(h, a)+cπˆ(a|h) N(h)N(h,a)+1   
+30: $( s ^ { \prime } , o , r , \mathrm { d o n e } ) \sim \mathcal { G } ( s , a ^ { * } )$   
+31: $h ^ { \prime } \gets \mathrm { P U S H B A C K } ( h , [ a ^ { * } , o ] ) , d ^ { \prime } \gets d + 1$   
+32: $R \gets r \mathrm { + } \gamma { \cdot } \operatorname { S I M U L A T E } ( s ^ { \prime } , h ^ { \prime } , \mathrm { d o n e } , d ^ { \prime } , \mathcal { T } )$   
+33: N (h, a∗) += 1, N (h) += 1   
+34: Q(h, a∗) ← Q(h, a∗) + R−Q(h,a∗)N(h,a∗)   
+35: return R   
+36: end procedure
+
+cosine similarity to the current instruction: $\mathrm { C o s i n e S i m } ( \ell _ { i } , \ell )$ . In experiments, we use a subset of $\mathcal { D }$ to show its performance when restricted to a small training set. We select the top $K$ similar instructions and use the corresponding expert trajectories as a $K$ -shot prompt. However, the answer $\alpha _ { i }$ is a free-formed natural language sentence that cannot be mapped to admissible actions for the agent directly. To ensure that the action can be executed, we follow the method in prior works [18] to represent the actions and admissible actions by embeddings from [26] and evaluate their cosine similarity $\mathrm { C o s i n e S i m } ( \alpha _ { i } , a )$ . The empirical policy distribution is formulated as follows: $\begin{array} { r } { \hat { \pi } ( a | h ) = \lambda \frac { 1 } { | A | } + ( 1 - \lambda ) \mathrm { S o f t m a x } \{ \sum _ { i = 1 } ^ { M } \mathrm { C o s i n e S i m } ( \alpha _ { i } , a ) - \eta \} } \end{array}$ , where $\eta$ is the average value of $\textstyle \sum _ { i } \mathrm { C o s i n e S i m } ( \alpha _ { i } , a )$ and $| A |$ is the size of the admissible action space. $\lambda$ is a hyper-parameter that adds randomness to the belief, as the sampled actions from LLM could be very deterministic. Therefore, the empirical policy distribution is a mixture of approximated policy from LLM and uniform distribution. The example prompts are provided in Appendix F.
+
+# 2.4 Monte Carlo tree search
+
+We integrate the commonsense world belief and policy from LLM in MCTS, presented in $\mathrm { A l g }$ . For each simulation, MCTS samples a state from the belief $b ( s )$ at the root (line 4). It independently samples one position for each object to construct a state $s$ . This sampled state $s$ is then employed in the simulation, generating a new tree trajectory. An action $a ^ { * }$ is chosen during the simulation based on the $Q$ value, visit counts, and LLM policy (lines 28 and 29). The observation and transition function, denoted as $\mathcal { G }$ (lines 15 and 30), predict the next state $s ^ { \prime }$ given the selected action $a ^ { * }$ and the sampled state $s$ , thus progressing to the subsequent step in the simulation (lines 30 and 31). When encountering leaf nodes in the tree, MCTS expands the tree and performs a random rollout for the corresponding node (lines 23 to 26). A uniform policy is employed to sample actions in the rollout, and the discounted reward is then returned (lines 14 to 17). Upon completing the task or reaching the maximum depth, the accumulated rewards are backpropagated, updating each node’s estimated $Q$ value (lines 32 to 35). Following $N$ simulations, the output action is determined based on the estimated $Q$ value (lines 3 to 8). Upon completion of the search process, the agent will execute an action and receive a new observation. For simplicity, we assume that the observation and transition functions are deterministic and known. In cases where an object is detected, its corresponding position within the belief will be updated with the observed position. Conversely, if the object remains undetected at certain positions, the belief regarding its presence in those positions will be rendered null, denoted by a zero value.
+
+# 3 Experiments
+
+# 3.1 Experimental setup
+
+VirtualHome. We proceed with our experiments in the VirtualHome [24], a large household simulated environment with a large domain, partial observations, and large action space. It contains hundreds of interactive items and containers with various types of rooms. It is a well-suited platform for evaluating embodied decision-making for solving daily tasks in household environments.
+
+Data. To generate data for prompting and baseline training, we follow [25] to create 2000 tasks with randomly initialized scenes and expert trajectories. There are several settings for the evaluation. Simple tasks are the tasks that only require the rearrangement of one item generated from the same distribution as the training dataset. Comp. refers to the composition of simple tasks in order to rearrange multiple objects sampled from the same distribution as the dataset (e.g., “Put plate on kitchen table and chicken inside fridge” is the composition of “put plate on kitchen table” and “put chicken inside fridge,” ). The composition of tasks increases the planning horizon, making it more challenging to complete. In evaluation, we also use the Novel Simple tasks with seen items (e.g., in the dataset, we have “put one plate on the kitchen table” and “put one chicken inside the fridge,” and we use “put one plate inside the fridge” and “put one chicken on the kitchen table” to evaluate; these tasks are not included in the training dataset). For compositional tasks, we include Novel Compositional tasks, with 2 or 3 primary tasks composed, denoted as NovelComp(2) and NovelComp(3) (e.g., we have “put plate on kitchen table” and “put chicken inside fridge,” in dataset but their composition “Put plate on kitchen table and chicken inside fridge” is not.) We also generate scenes at a Novel Apartment for testing, where the distribution of object positions differs from the dataset.
+
+The expert data are generated by an Oracle agent implemented in [25]. The expert has the full knowledge of the environment (hence, does not need to understand where objects are likely to be placed) and uses handcrafted heuristics for completing various tasks. It uses regression planning to search for solutions to a task. We collect the actions and observations of the expert completing the tasks in the VirtualHome simulator as the dataset. There are 10,000 trajectories in total for training the baseline. To show the capability of LLMs when using a small training set, we only select 200 instances uniformly at random from the dataset as prompt candidates for the LLM model and policy when used in a few-shot mode with no fine-tuning. We also generated 800 tasks in total for evaluation.
+
+Evaluation. We evaluate the success rate of completing the tasks within 30 steps, while a typical task can be finished within at most 15 steps. The task is considered successful if all the requirements of object positions are satisfied. For example, given the instruction “Put one apple inside the fridge,” the task is successful if any apple is in the fridge. For simplicity, we don’t consider the task of rearranging a very specific object, e.g., putting the leftmost apple in the fridge.
+
+Baselines. We evaluate several baselines to compare. UCT [21]: We use the UCT algorithm to conduct planning without commonsense knowledge and use the ground-truth reward function in simulation. We use uniform distribution as the initial belief for states of objects. It is to provide evidence that commonsense knowledge improves planning efficiency. Finetuned GPT2 policy [22]: we use the training dataset with 10000 trajectories to fine-tune GPT-2 as the planning policy. This is to show that larger pre-trained LLM without fine-tuning outperforms the smaller model fine-tuned in specific tasks. GPT3.5 Policy [18]: LLM takes as input the instructions and history of actions and the currently visible objects to generate the next action. We use the LLM as the policy only, with a few examples as prompts to interact with the environments.
+
+This baseline demonstrates the benefits of additional information from the commonsense model and algorithmic benefits from MCTS.
+
+# 3.2 Results
+
+Table 1: Main results: mean $\pm$ standard error of success rate $( \% )$   
+
+<table><tr><td rowspan="2"></td><td colspan="4">Seen Home</td></tr><tr><td>Simple</td><td>Comp.</td><td>NovelSimple NovelComp.(2) NovelComp.(3)</td><td></td></tr><tr><td>UCT [21]</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0 0.0±0.0</td><td>0.0±0.0</td></tr><tr><td>finetuned GPT2 policy [22]</td><td>81.3±2.4 59.0±6.7</td><td></td><td>41.2±7.1 30.9±2.8</td><td>2.3±1.5</td></tr><tr><td>GPT3.5Policy[18]</td><td>83.4±6.8 47.0±7.8</td><td></td><td>74.3±4.0 48.2±8.8</td><td>5.4±2.0</td></tr><tr><td>GPT3.5-MCTS (Ours)</td><td>91.4±3.3 71.2±6.2</td><td></td><td>88.1±4.3 72.6±6.9</td><td>33.6±3.1</td></tr><tr><td></td><td colspan="4">Unseen Home</td></tr><tr><td>Method</td><td>Simple</td><td>Comp.</td><td>NovelSimple NovelComp.(2) NovelComp.(3)</td><td></td></tr><tr><td>UCT[21]</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0</td></tr><tr><td>finetuned GPT2 policy [22]</td><td>65.5±3.4 39.9±5.2</td><td>33.4±6.4</td><td>0.0±0.0 12.8±3.9</td><td>1.1±0.9</td></tr><tr><td>GPT3.5Policy[18]</td><td>74.3±5.0 43.3±4.0</td><td>67.8±4.9</td><td>54.0±3.0</td><td>6.9±2.1</td></tr><tr><td>GPT3.5-MCTS(Ours)</td><td>82.9±3.2 71.9±5.6</td><td>79.3±3.3</td><td>70.4±6.4</td><td>38.8±3.4</td></tr></table>
+
+Main result. The main results of the experiments are shown in Table 1, reporting the success rate of our method
+
+and baselines in completing the tasks in VirtualHome environments. In this result, GPT3.5-MCTS outperforms all the compared baselines, especially for unseen situations. UCT works poorly in all conditions, as the poor model and the huge search tree make the planning intractable. Thus, we focus our discussion on comparing the finetuned GPT2 policy and GPT3.5 policy. For Simple, in-distribution tasks, the planning horizon is relatively short. Finetuned GPT2 policy, GPT3.5 Policy, and our method work reasonably well, but our method still outperforms the baselines. For Novel Simple tasks, finetuned GPT2 policy works significantly worse than GPT3.5 Policy and GPT3.5-MCTS. This is because the fine-tuning of narrow tasks results in a biased distribution of the policy and compromises generalizability. GPT3.5 Policy and GPT3.5-MCTS work better due to the LLM’s few-shot planning capability. GPT3.5-MCTS works better for both situations. It benefits from the MCTS’ look-ahead search that explore possible states for potential outcomes in order to make reasoned decisions.
+
+For the Compositional, in-distribution tasks, the finetuned GPT2 policy and GPT3.5 policy get significantly worse performance, while GPT3.5-MCTS works far better. The finetuned GPT2 policy is trained by behavior cloning that suffers from compounding errors. Therefore, when the planning horizon gets longer, the influence of the errors accumulates and compromises the overall performance significantly. As for GPT3.5 Policy, the longer horizon potentially introduces more errors during planning, which might not be included in the prompt examples. Without suitable guidance from prompt, we cannot guarantee the GPT3.5 Policy will carry out suitable replanning when encountering errors. MCTS encourages exploration to a certain extent of different possible actions during searching, introducing additional guidance to the GPT3.5 policy to look into other possible solutions. This is because the action selection procedure in GPT3.5-MCTS is not purely determined by GPT3.5 Policy but also by the $Q$ value and visit counts. Thus, MCTS encourages GPT3.5 Policy to explore other possible search directions instead of excessively applying certain actions sampled by itself.
+
+Ablation study. We conduct ablation studies to see the individual contributions of different components within the GPT3.5-MCTS framework. The No Heuristic Policy version of GPT3.5-MCTS refers to the absence of PUCT guided by the GPT3.5 Policy for action selection. Instead, it solely relies on UCT with an initial commonsense belief derived from LLM. The variant employing the Uniform State Prior utilizes a uniform prior belief regarding states, in contrast to the LLM-generated initial belief employed during the search process. Lastly, the variant operating in a Fully Observable environment aims to assess the accuracy of LLM’s knowledge in modeling the world.
+
+Table 2 shows the results of our ablation study. The outcomes obtained under the No Heuristic Policy version highlight the significance of heuristic policies in facilitating MCTS to conduct efficient searches for complex and large-scale planning tasks. Con
+
+Table 2: Ablation Study: mean $\pm$ standard error of success rate $( \% )$   
+
+<table><tr><td rowspan="2">Method</td><td colspan="5">Seen Home</td></tr><tr><td>Simple</td><td>Comp.</td><td>NovelSimple NovelComp.(2) NovelComp.(3)</td><td></td><td></td></tr><tr><td>GPT3.5-MCTS (No Heuristic Policy)</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0</td></tr><tr><td>GPT3.5-MCTS(Uniform State Prior)</td><td>3.2±1.1</td><td>0.0±0.0</td><td>1.1±0.4</td><td>0.0±0.0</td><td>0.0±0.0</td></tr><tr><td>GPT3.5-MCTS (Fully Observable)</td><td>94.0±2.1</td><td>80.7±3.3</td><td>94.3±2.4</td><td>78.5±4.0</td><td>34.0±4.4</td></tr><tr><td>GPT3.5-MCTS (Ours)</td><td></td><td>91.4±3.3 71.2±6.2</td><td>88.1±4.3</td><td>72.6±6.9</td><td>33.6±3.1</td></tr><tr><td></td><td colspan="5">Unseen Home</td></tr><tr><td>Method</td><td>Simple</td><td>Comp.</td><td></td><td>NovelSimple NovelComp.(2) NovelComp.(3)</td><td></td></tr><tr><td>GPT3.5-MCTS (No Heuristic Policy)</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0</td></tr><tr><td>GPT3.5-MCTS (Uniform State Prior)</td><td>1.1±0.2</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0</td><td>0.0±0.0</td></tr><tr><td>GPT3.5-MCTS(Fully Observable)</td><td>85.1±5.0 77.5±3.2</td><td></td><td>82.2±3.3</td><td>76.6±3.1</td><td>37.9±2.9</td></tr><tr><td>GPT3.5-MCTS (Ours)</td><td>82.9±3.2 71.9±5.6</td><td></td><td>79.3±3.3</td><td>70.4±6.4</td><td>38.8±3.4</td></tr></table>
+
+versely, the results of the Uniform State Prior row indicate that incorrect world models compromise search performance. This is because the model of the world determines the $Q$ value. The wrong model results in an inaccurate estimation of the $Q$ value, misleading the search process toward irrelevant locations. The Fully Observable results demonstrate that GPT3.5-MCTS with perfect knowledge of the environment only slightly outperforms its counterpart without it, implying that the commonsense knowledge of LLM regarding world modelling suffices for practical purposes.
+
+Failure analysis. Policy, model, and translation errors are the primary causes of failures. Among these, policy errors are responsible for the majority of the failures. Oftentimes, the policy produces unreasonable behaviours that mislead the search procedure. For example, it usually outputs inadmissible actions, such as “walk to the cutleryfork” where the “cutleryfork” is not in the observation. It also produces back-and-forth behaviours, resulting in an unreasonable heuristic and slowing the search procedure. For example, when putting objects inside the microwave, it is sometimes struck by repeatedly opening and closing the microwave. For model error, the predicted positions of objects are not always correct. Since a random rollout policy is employed, incorrect object states can result in higher $Q$ -values than correct states, leading to misguided exploration. The wrong translation also compromises the performance as we translate the response from LLM to admissible action or object names to ensure executability. This is partly caused by the VirtualHome environments, as the policy might not understand the underlying logic of the actions in VirtualHome, such as you have to walk close to interact with the object. Thus, if the LLM outputs “open fridge” but is not close enough to the fridge, the action will be translated to other admissible actions (“open fridge” is not inside the admissible actions for this case as it is invalid due to the setting of VirtualHome).
+
+# 4 LLM as a model or a policy?
+
+When would using LLM as a model outperform using LLM as a policy, and vice versa? We propose using the minimum description length (MDL) principle, also known as Occam’s Razor from the philosophy of science, to gain insights into the issue. The MDL principle suggests choosing the method that has a shorter description when both methods fit the training data well. MDL has been formalized in various ways. One formal statement (from section 7.3 of [28]) is provided here:
+
+Theorem 4.1 (Occam’s Razor). Let $\mathcal { H }$ be a hypothesis class and let $d : \mathcal { H } \to \{ 0 , 1 \} ^ { * }$ be a prefix-free description language for $\mathcal { H }$ . Then, for every sample size, $m$ , every confidence parameter, $\delta > 0$ , and every probability distribution, $D$ , with probability greater than $1 - \delta$ over the choice of $S \sim D ^ { m }$ we have that, $\forall h \in \mathcal { H } , L _ { D } ( h ) \leq L _ { S } ( h ) + \sqrt { ( | h | + \ln { ( 2 / \delta ) } ) / 2 m }$ where $L _ { S } ( h )$ is the empirical loss of $h$ on the $S$ , $L _ { D } ( h )$ is the expected loss of $h$ , and $| h |$ is the length of $d ( h )$ .
+
+According to Theorem 4.1, we can bound the expected loss of a solution $h$ by the description length $| h |$ and the training loss $L _ { S } ( h )$ . We do not know the LLM training loss for using it as a model or as a policy, but for the purpose of gaining insights, it is reasonable to assume that they are both small. In that case, the MDL principle suggests selecting between a model or policy depending on which of them has the smaller description length given the description language.
+
+Numerous caveats should be observed when using Theorem 4.1 to gain insights into the behaviour of LLM as a model and policy. The theorem assumes that the training data is independent and identically distributed (iid), which is likely not true in practice. Nonetheless, the qualitative behaviour is often similar for non-iid data. As the training data of GPT is unknown, when comparing the description length, we also assume that the training data for each subproblem is roughly the same. In addition, using the theorem to gain insights requires major assumptions on the predictor classes $\mathcal { H }$ for model and policy; here, we assume that $\mathcal { H }$ is the model (or policy) class that we are analysing and assume that LLM training using powerful approximators such as transformers has similar behaviour to training using the model (or policy) class. Finally, depending on how the model and policy are used, error propagation may need to be analysed separately.
+
+Besides the multiplication example described earlier, we discuss an air travel planning task and the VirtualHome object rearrangement task examined earlier.
+
+# 4.1 Travel planning
+
+Consider planning for air travel from a starting city to the destination city. To solve the problem through L-Model, we need to have the model – the direct flights out of each city – together with a shortest path algorithm. The model can be represented as a graph, which is likely to be sparse in the real world. For a sparse graph, an adjacency list will give a compact representation. Assuming that the total number of edges grows proportionally to the number of cities, $O ( n \log n )$ bits would be sufficient to describe a graph with $n$ cities, with approximately $\log n$ bits used to describe each city in the adjacency list structure. The shortest path algorithm can be described by any reasonable programming language with a constant size. The method regarding L-Policy can be represented as a 2-dimensional table, where each row and column denotes the current and destination city, and the table entry describes the next city to fly to on the shortest path from the current city to the destination. The next city in the table can be described with $\log n$ bits. As such, with $n$ cities in the rows and columns, there should be approximately $n ^ { 2 } \log n$ bits in total. Thus, according to MDL, the L-Model has a shorter description length and should make fewer generalization errors than the L-Policy for travel planning with sufficiently large $n$ .
+
+Experiment. We conducted experiments about planning for air travel from a starting city to a destination city, which we analyzed above. We utilized GPT-3.5 to generate flight paths between cities. We compare it to the GPT-3.5 model-based approach: we use GPT-3.5 to predict neighbouring cities connected by a direct flight, which feeds into the uniform-cost search (i.e., replace node expansion by GPT-3.5 as the world model).
+
+We use the data from the Kaggle World cities database, select 68 cities with populations exceeding 5 million in different countries and 62 middle-size cities with populations between 500 thousand and 2 million, and use the Virtual Radar Server to get the flight routes dataset as ground truth. In our tests, we sampled 400 city pairs among large cities and 400 pairs among mid-size cities, evaluating path accuracy by verifying each direct flight exists. Paths were accepted if all flights were valid, even if they extended beyond our selected source and target cities. We evaluate the methods in two settings: predicting the flight route given two large cities and two mid-size cities.
+
+Results. The main result shown in Fig 2 suggests that the LLM model $^ +$ search algorithm consistently outperforms the LLM policy, supporting our analysis2. Furthermore, the performance gaps for mid-size cities are larger than for large cities. This is consistent with the fact that there are more midsize cities than large cities (the gap between the description lengths of $n ^ { 2 } \log { n }$ for policies vs $n \log n$ for models grows with the number of cities). Note that performance decreases as the path length increases for both methods. As the number of predictions required for each path increases with path length, the probability of incorrect path prediction also increases.
+
+![](images/b64826072bda0ef9f00d63e5f49bc978d52958ce604d2214b6c8047bb969e4bd.jpg)  
+Figure 2: Flight planning results.
+
+# 4.2 Object rearrangement task
+
+Consider a house with $n$ movable objects, $m$ containers, and $k$ rooms. If we use L-Model, we must describe the model and search algorithm (i.e., MCTS). The model can be represented as a sparse graph with objects, containers, and rooms as nodes, and weighted directed edges signifying a "located in" relation between the nodes, with the weights specifying the probabilities. Assume that each weight is specified with a constant number of bits. Each of the $n$ objects can be located within the $m + k$ containers or rooms, so each object edge would require approximately $\log ( m + k )$ bits to describe. Each of the $m$ containers can be located in $k$ rooms, so each container edge would require approximately $\log ( k )$ bits to describe. Assume further that the degree of each object and container node in the graph is bounded by a constant; the entire graph then requires $O ( n \log ( \bar { m } + k ) +$ $m \log ( k ) ) = O ( ( m + n ) \log ( m + k ) )$ bits to describe. The MCTS algorithm should be described by a programming language with a constant size. For the L-Policy, tasks can be designated using object-container pairs. Each object-container policy can be defined by a sequence of actions, e.g. “walk to the fridge, open the fridge,” until the object is found, followed by a sequence of actions until the destination container is found. Each action takes $O ( m + k )$ bits to describe. Assuming search sequences and the size of each object-container policy are bounded by a constant. Describing the policies for all mn object-container pairs requires $O ( { \dot { m } } n \log ( m + k ) )$ bits.
+
+The composed tasks provide further challenges for L-Policy. Composing tasks increases the description complexity of policies to $O ( ( m n ) ^ { N } \log ( m + k ) )$ , where $N$ is the number of composed tasks if the composition is not exploited in the policies. For the L-Model, decomposition is automatically done in MCTS at the expense of more computation, whereas for the L-Policy, the LLM must learn to do the decomposition. This may make the problem of learning the decomposed policy computationally more difficult and less likely to be approximated by the LLM in practice.
+
+This analysis indicates that the L-Model has a shorter description length than the L-Policy. According to the MDL principle, the L-Model will likely have a lower error rate than the L-Policy. However, in this case, we do not have an algorithm that is guaranteed to be efficient for solving L-Model. Instead, we use the L-Policy as a search heuristic to obtain a practically effective search algorithm. As predicted by the MDL principle, the search with the LLM-induced world model, when limited to the neighbourhood of the LLM-induced policy, provides improved performance over L-Policy.
+
+# 4.3 Discussion
+
+While we have discussed problems where the L-Model outperforms the L-Policy, we would also expect the L-Policy to outperform the L-Model when the description length of policies is shorter than the models. For example, when recommending a tourist itinerary for a city, the description length of the itinerary should be shorter than describing all the places of interest plus the reward and the length of time recommended for visiting each location. In such a case, the LLM may be able to do better in recommending an itinerary than in providing accurate information on all places of interest for planning itineraries.
+
+In the multiplication problem, the model-based approach used a known efficient multiplication algorithm. In the air travel planning task, an efficient shortest-path algorithm is used. However, for the object rearrangement problem, we do not have an efficient search algorithm. In this case, we demonstrate another strength of LLMs – as a policy, it can be used as a heuristic for improving the efficiency of search algorithms.
+
+# 5 Related work
+
+Task planning has a long-standing history in the AI research community. In the early stage, many studies [1, 15, 14, 12] focused on task planning in a discrete state space with deterministic transitions. These methods are intractable for large-scale, long-horizon problems. Recently, researchers have used learning-based methods to learn planning policies directly [22] or to learn search heuristics to accelerate planning [32, 31, 41, 7, 29]. Those policies or heuristics are not generalizable to other unseen settings. Most recently, the pre-trained LLMs have been applied as few-shot policies for task planning [18, 2, 19]. However, the planning policy may not have good compositional generalizablity. In the robotics community, many studies proposed that task planning should be integrated with physical-level motion planning, i.e., task and motion planning (TAMP) [11, 12, 9]. This paper focuses on large-scale task planning with partial observations and large, object-centric domains, in which classical planning is intractable and learning-based methods require massive data.
+
+Various approaches are proposed to scale up to large-scale planning problems. Initially, the Monte Carlo method [21, 8, 6] is proposed to tackle intractable large-scale problems using random sampling in the tree search. However, further scaling up to the problem with a large domain with sparse rewards requires massive sampling. Silver et al. [32, 31] integrate MCTS with deep learning to bias the action selection and reduce sampling. The idea has been successfully applied in large-scale planning scenarios [32, 31, 7, 27]. Most recently, studies show that LLM can be a few-shot open-loop [18, 2] and closed-loop [19, 37] planning policy in the large domain, while it may suffer from hallucinations. In this paper, we show that LLMs’ commonsense knowledge can guide a search algorithm, reducing the search space sufficiently for practical planning and producing reasoned results.
+
+How to use LLMs for a planning problem? LLMs have been used as a few-shot policy for languageconditioned task planning [18, 2, 19], or a policy for multi-step reasoning problems [38, 42]. However, recent research [10] also suggests that transformer LLMs are inherently limited for solving multi-step reasoning problems. In addition, some studies try to use LLMs as heuristics [43] or transition function [13] in MCTS, boosting the performance in coding or small-scale reasoning. However, the literature has not discussed utilizing LLMs as a world model in depth. We show the benefits of using LLM to model the world, as well as using MDL analysis to decide how to use LLM for planning problems.
+
+# 6 Conclusion
+
+We use LLMs as the commonsense world model and the heuristic policy within MCTS to achieve better-reasoned decision-making for daily tasks. MCTS enables LLM to leverage its world modelling knowledge for informed reasoning and explore new combinations of actions to tackle novel tasks. LLM helps MCTS through the biased sampling of states and actions, improving its efficiency in resolving complex task-planning problems. Our analysis and empirical evidence suggest that, for certain real-world domains, if the description length of the world is substantially shorter than policy, using LLM as a model in a model-based approach is a better option than using LLM as policy.
+
+Limitations. The runtime of LLM-MCTS is currently hindered by multiple LLM calls. The details of runtime performance are in Appendix E. While our method requires multiple LLM calls, it provides substantially improved results. There are also various ways to enhance runtime performance like using smaller LLMs like Llama [35, 36] or distilling LLM’s knowledge into a smaller model [30, 16, 23]. Those are interesting avenues for future research.
+
+Broader impact. There might be concerns about the inherent biases of LLMs that may lead to unfair or risky decisions in some domains. Further study about the fairness and bias of LLMs’ knowledge would be beneficial.
+
+# Acknowledgments and Disclosure of Funding
+
+This research is supported in part by the National Research Foundation (NRF), Singapore and DSO National Laboratories under the AI Singapore Program (No. AISG2-RP-2020-016) and the Agency of Science, Technology and Research (A\*STAR), Singapore, under the National Robotics Program (No. M23NBK0053).
+
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+
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+
+# Appendix
+
+# A Virtualhome experimental environments
+
+We use the VirtualHome simulator [24] to evaluate our approach as well as the baseline methods. VirtualHome is a 3D household environment with partial observation, large action space, and a long planning horizon. It contains hundreds of interactive objects and containers, allowing it to perform various household object rearrangement tasks. This section introduces details of the tasks, the goal specifications, the actions, and the observations in our experimental settings.
+
+# A.1 List of objects, containers, surfaces, and rooms in the apartment
+
+We list all the objects that are included in our experimental environment. Here, we can put moveable objects into the Containers or on the Surfaces. The Containers and Surfaces are located at a Room in the apartment.
+
+• Containers: bathroom cabinet, kitchen cabinet, bathroom counter, fridge, oven, dishwasher, microwave, stove, bathroom cabinet   
+• Surfaces: bed, bookshelf, cabinet, coffee table, cutting board, floor, fryingpan, kitchen counter, kitchen table, nightstand, sofa, stove   
+moveable objects: alcohol, apple, banana, bar soap, bell pepper, boardgame, book, box, bread slice, bucket, candle, candy bar, carrot, cellphone, cereal, chicken, Chinese food, chips, chocolate syrup, clock, clothes pants, clothes pile, clothes shirt, coatrack, coffeepot, condiment bottle, condiment shaker, cooking pot, crackers, crayons, creamy buns, cupcake, cutlery fork, cutlery knife, cutlets, cutting board, dish bowl, dishwashing liquid, face cream, folder, fryingpan, glasses, globe, hair product, hanger, juice, keyboard, lime, lotion bottle, magazine, milk, milkshake, minced meat, mouse, mug, notes, oven tray, pancake, paper, pear, pie, pillow, plate, plum, poundcake, pudding, radio, remote control, salad, salmon, slippers, sports ball, sundae, teddybear, toilet paper, toothbrush, toothpaste, towel, towel rack, toy, washing sponge, water glass, whipped cream, wine, wineglass   
+• Rooms: bedroom, bathroom, living room, kitchen.
+
+# A.2 Tasks
+
+We use the object rearrangement tasks for evaluation. The task is to search for one or more objects in the house and move them to the desired positions. We use natural language as the interface to specify the tasks. Thus, the agent should take as input the natural language instruction and observations, and then output actions.
+
+The tasks are randomly sampled from different distributions. We define various types of object rearrangement tasks for evaluation:
+
+• Simple: this task is to move one object in the house to the desired location. The combination of the object and desired location has appeared in the training dataset.   
+• Novel Simple: this task is to move one object in the house to the desired location. The combination of the object and desired location hasnot appeared in the training dataset.   
+• Comp.: this task is composed of 2 Simple tasks, moving more than one object in the house to their desired location. This kind of task has a longer planning horizon as it requires moving multiple objects to complete. The combinations of Simple tasks have appeared in the training dataset.   
+• Novel Comp. (2): this task is composed of 2 Simple tasks, moving more than one object in the house to their desired location. The combinations of Simple tasks have not appeared in the training dataset.   
+• Novel Comp. (3): this task is composed of 3 Simple tasks, moving more than one object in the house to their desired location. This kind of task has the longest planning horizon. The combinations of Simple tasks have not appeared in the training dataset.
+
+We also have different household environments:
+
+• Seen Apartment: the map of the apartment is shown in Figure 3. These household environments are the same as the ones in the training set, while the object positions are randomly initialized according to a pre-defined commonsense distribution in VirtualHome [24]. • Unseen Apartment: the map of the apartment is shown in Figure 4. These household environments are not the same as the ones in the training set. The object positions are also sampled from a different pre-defined commonsense distribution in VirtualHome [24].
+
+![](images/98287ae96eaee6175debc4f12c11110d0bb713b95b41500cced14e5ef7d2db9a.jpg)  
+Figure 3: The map of the seen apartments in our setting. These household environments are the same as the ones in the training set, while the object positions are randomly initialized according to a commonsense distribution.
+
+![](images/f9cc3c7756485d19e82943f6d52f5d61abf5fe7be6b6f203af086ce54e4f8e2f.jpg)  
+Figure 4: The map of the unseen apartments in our setting. These household environments are not the same as the ones in the training set. The object positions are also sampled from a different commonsense distribution.
+
+# A.3 Goal specification
+
+Similar to prior works [22], we define the goal in the VirtualHome system by a set of predicates. For instance, a goal can be defined by Inside(apple, fridge):2; Inside(plate, dishwasher):1, meaning “put two apples inside the fridge and put one plate inside the dishwasher.” For Simple and Novel Simple tasks, it only requires moving one object, while Comp. and Novel Comp. have more than one object to move.
+
+# A.4 Actions
+
+In VirtualHome, the agent is able to navigate in the environment, grab an object, put an object inside the containers (e.g., fridge) or on the surfaces (e.g., table), open and close the container, etc. The actions in VirtualHome are grounded to moveable objects, containers, or rooms in the environment. For example, Open(5) is to open an object with index (5). The list of available actions in our setting are listed below:
+
+• Walk(<item>): walk to the <item>. The <item> can be a moveable object, a container, or a room. The precondition of this action is that the <item> is visible. The effect of this action is that the agent is close to the <item> if the <item> is an object or inside the <item> if the <item> is a room. The action is translated into the sentence “walk to the <name of item>” when feeding into LLMs.   
+Open(<item>): open the <item>. The <item> can be a moveable object or a container. The precondition of this action is that the agent should be close to <item>. The effect of this action is that the <item> is opened. The action is translated into the sentence “open the <name of item>” when feeding into LLMs. Close(<item>): close the <item>. The <item> can be a moveable object or a container. The precondition of this action is that the agent should be close to <item>. The effect of this action is that the <item> is closed. The action is translated into the sentence “close the <name of item>” when feeding into LLMs.   
+• Grab(<item>): grab the <item>. The <item> should be a moveable object. The precondition of this action is that the agent should be close to the <item>, and the agent is not holding any objects. The effect of this action is that the agent will hold the <item>. The action is translated into the sentence “grab the <name of item>” when feeding into LLMs.   
+• PutIn(<item1>, <item2>): put the moveable object <item1> inside the container <item2>. The precondition of this action is that the agent should be close to the <item2> and holding <item1>. The effect of this action is that the agent is not holding any objects, and the <item1 $>$ is inside the <item2>. The action is translated into the sentence “put the <name of item1> inside the <name of item2>” when feeding into LLMs.   
+PutBack(<item1>, <item2>): put the moveable object <item1> on the surface <item2>. The precondition of this action is that the agent should be close to the <item2> and holding <item1>. The effect of this action is that the agent is not holding any objects, and the <item1> is on the <item2>. The action is translated into the sentence “put the <name of item1> on the <name of item2>” when feeding into LLMs.
+
+# A.5 Observations
+
+We use the same representation as [22] for partial observation. The observation is a list of visible objects and relationships between those objects. Each object or container has a state: open or close. The fine-tuned GPT2 policy [22] also uses the 3d coordinates of the object. We also use relationships to connect different objects, such as Inside(apple, fridge). Those relationships are translated to natural language descriptions when feeding into LLMs, such as “an apple is inside the fridge.”
+
+# B Data gathering
+
+Similar to prior works [25, 22], we collect expert trajectories in VirtualHome using regression planning with handcrafted heuristics3. The expert has full observation of the environment. Given the goal predicates and full observation, the agent will use the handcrafted heuristics for each task to effectively search for the solutions. The expert also has a handcrafted mechanism for compositional tasks to decompose one task into subtasks and finish them progressively. For each trajectory, we include the goal predicates (used by the VirtualHome system and the expert agent), the goal instruction (used by the agent), the partial observation for each time step (not used by the expert agent, the expert agent uses full observation), and the expert actions.
+
+# C Implementation details of belief in LLM-MCTS
+
+This section introduces our implementation details for the belief of states in GPT3.5-MCTS. The source code will be released at https://llm-mcts.github.io before the publication.
+
+# C.1 State representation
+
+We represent the states by a list of objects and their relationships. Each object has a unique name and id in the simulator, as well as the state of the object. We use the same unique name and id in our state representation. The relationships connect different objects, containers, surfaces, and rooms. The VirtualHome contains 59 different types of relationships, including Inside, On, Close, Facing, etc. We use the same type of relationships in our state representation.
+
+# C.2 Belief
+
+The belief of the state also contains a list of objects and their relationships. However, we parameterize the relationships by a vector, representing the probability that the relationship is true. This vector is affiliated with the object representation. For simplicity, we only include the relationships Inside, On in our belief, as we only query LLM about the object positions to build up the commonsense belief of the state.
+
+When building up a state’s belief, we query LLM to predict the position of each moveable object, container, and surface. The position of a moveable object is specified by the relationships (i.e., Inside or $\mathtt { O n }$ ) between itself and a container or surface. The position of a container or a surface is specified by its relationship (i.e., Inside) to the room. We use sampling to approximate the distribution of the position. The moveable objects’ belief of position is represented by a vector whose dimension is the same as the total number of containers and surfaces in the house. Each vector entry denotes the probability that whether the object is inside a specific container or on a specific surface is true. When asking LLM to predict the object positions, we asked LLM for $M$ times and received multiple responses from LLM. We then count each entry’s total number of predictions and normalize them to become a probability distribution. We initialize the value of other unsampled entries in the vector by a lower bound of the probability $1 \times 1 0 ^ { - 3 }$ to ensure that the model will not eliminate other possibilities when the commonsense model is wrong.
+
+The agent will receive new observations to update their belief when interacting with the environment. We will first predict the next state of the agent by the transition function and then update the belief of the object positions by new observations. Suppose the object is inside the current observation. In that case, the other entry of the relations between objects will be masked out by zero, and the entry of the relationships in observation will be replaced by the value of one. However, if a relationship is not inside the observation, the value of the corresponding entry will be replaced by zero, and the vector will be normalized again.
+
+# D Visualized examples
+
+We provide a set of successful (shown in Figure 5) and failed trajectories (shown in Figure 6) to give a better understanding of the tasks and our method. Policy, model, and translation errors are the primary causes of failures. Among these, policy errors are responsible for the majority of the failures. Often time, the policy produces unreasonable behaviors that mislead the search procedure. For example, it usually outputs inadmissible actions, such as “walk to the cutlery fork” where the “cutlery fork” is not in the observation (shown in Figure 6 (a)). It also produces back-and-forth behaviors, resulting in an unreasonable heuristic and slowing the search procedure. For example, when putting objects inside the microwave, it is sometimes struck by repeatedly opening and closing the microwave. For model error, the predicted positions of objects are not always correct. Since a random rollout policy is employed, incorrect object states can result in higher $Q$ -values than correct states, leading to misguided exploration (shown in Figure 6 (b)). The wrong translation also compromises the performance as we translate the response from LLM to admissible action or object names to ensure executability. This is caused in part by the VirtualHome environments, as the policy might not understand the underlying logic of the actions in VirtualHome, such as you have to walk close to interact with the object. Thus, if the LLM outputs “open fridge” but is not close enough to the fridge, the action will be translated to other admissible actions (“open fridge” is not inside the admissible actions for this case as it is invalid due to the setting of VirtualHome).
+
+Instruction: Put one plate inside the   
+kitchen cabinet and one chicken   
+inside microwave.   
+1: Walk to kitchen   
+2: Walk to plate   
+3: Grab plate   
+4: Walk to kitchen cabinet   
+5: Open kitchen cabinet   
+6: Put plate inside kitchen cabinet   
+7: Move fridge   
+8: Open fridge   
+9: Grab chicken   
+10: Walk to microwave   
+11: Open microwave   
+12: Put chicken inside microwave
+
+Instruction: Put one apple on the kitchen table and one toothbrush inside the bathroom cabinet.
+
+![](images/7e889755628e7fc0f820dc40ce53c62b36928d42c21199d1cc22ece2c70822c6.jpg)![](images/32503abb3807addb506d0049e114bb1a427c843162d0931f8743ae69e5296361.jpg)  
+Figure 5: Successful examples
+
+1: Walk to fridge   
+2: Open fridge   
+3: Walk to apple   
+4: Grab apple   
+5: Walk to kitchen table   
+6: Put apple on kitchen table   
+7: Walk to bathroom   
+8: Walk to toothbrush   
+9: Grab toothbrush   
+10: Open bathroom cabinet   
+11: Put toothbrush inside bathroom   
+cabinet   
+1: Walk to kitchen   
+2: Walk to kitchen cabinet   
+3: Open kitchen cabinet   
+4: Walk to dishwasher   
+5: Open dishwasher   
+6: Grab cutlery fork   
+7: Walk to kitchen table   
+8: Put cutlery fork on kitchen table   
+9: Walk to apple   
+10: Grab apple   
+11: Walk fridge   
+12: Open fridge   
+13: Put apple inside fridge   
+14: Close fridge   
+15: Walk to kitchen cabinet   
+16: Open kitchen cabinet (Failed)   
+17: Grab chips   
+18: Walk to living room   
+19: Walk to coffee table   
+20: Put chips on coffee table
+
+Instruction: Put one cutlery fork on   
+the coffee table   
+1: Walk to kitchen   
+2: Walk to cutlery knife   
+3: Walk to cutlery knife   
+4: Walk to cutlery knife   
+Instruction: Put one apple inside the   
+fridge   
+1: Walk to living room   
+2: Walk to coffee table   
+3: Walk to bedroom   
+4: Walk to nightstand   
+5: Walk to kitchen   
+6: Walk to kitchen cabinet   
+7: Open kitchen cabinet
+
+![](images/c8cca9d5d4e0d507619f15269eda7be86f0cddfe8ea390954ada91288b22b58d.jpg)  
+Figure 6: Failed examples. (a) Policy error and translation error. LLM outputs walk to the cutlery fork, but the cutlery fork is not in observation. We use embeddings to evaluate the most similar valid actions. Therefore it translates the action to one similar action “walk to cutlery knife.” The action has an incorrect semantic meaning and causes failure. (b) model error. The LLM predicts the apple is on the nightstand in the bedroom and on the coffee table in the living room. As we are using random rollout to get the estimation of the reward, there will be situations when the incorrect actions result in a higher estimated $Q$ value, thereby misleading the exploration.
+
+Table 3: Runtime performance (second $\pm$ standard error)   
+
+<table><tr><td></td><td>GPT3.5-MCTS</td><td>GPT3.5 Policy</td><td>UCT</td><td>GPT2 Policy</td></tr><tr><td>Runtime</td><td>67.83 ± 18.92</td><td>1.19 ± 0.81</td><td>120.0 ± 0.0</td><td>0.33 ± 0.07</td></tr></table>
+
+# E Runtime performance
+
+The runtime performance to make one-step decisions for simple object-rearrangement tasks is reported in Fig E. The experimental setup is the same as our main experiments in VirtualHome. The simple tasks are the tasks that only require the rearrangement of one item generated from the same distribution as the training dataset. We used 100 times simulation during the tree search for GPT3.5-MCTS in our experiments. For UCT, we bound the runtime by 120 seconds. The details of our hardware for experiments are enclosed below:
+
+• CPU: Intel(R) Xeon(R) Gold 6240 CPU $@$ 2.60GHz (72 cores) • GPU: NVIDIA GeForce RTX 2080 Ti
+
+# F Prompts
+
+One example prompt for the LLM policy and the exact prompt we used for building up the commonsense belief is shown below.
+
+# Listing 1: Example prompt for the heuristic policy
+
+You need to generate a high-level plan for completing a household task using the allowed actions and visible objects. Allowed actions: walk to <object>, walk to <room>, walk to <container>, walk to <surface>, grab <object>, open <container>, close <container>, put <object> on <surface>, put <object> inside <container>.   
+3 Rooms in the house: bedroom, bathroom, living room, kitchen   
+4 You need to strictly follow the format in the following examples:   
+5 Goal: Put one apple inside the fridge   
+6 Completed actions: walk to the kitchen, walk to the apple Current Observation: a kitchen table is inside the kitchen, a kitchen counter is inside the kitchen, an apple is on the kitchen counter, a plate is on the kitchen table, a banana is on the kitchen counter, a fridge is inside the kitchen and fridge is closed, a kitchen cabinet is inside the kitchen and kitchen cabinet is closed, a cutlery knife is on the kitchen table, a microwave is inside the kitchen and microwave is closed, a dishwasher is inside the kitchen and dishwasher is closed.   
+8 Next actions: grab the apple, walk to the fridge, open the fridge, put the apple inside the fridge, close the fridge, done.   
+9 Now, finish the next following task.   
+10   
+11 Goal: Put one apple on the kitchen table   
+12 Completed actions: walk to the kitchen   
+13 Current observation: a kitchen table is inside the kitchen, an apple is on the kitchen table, a kitchen counter is inside the kitchen, an apple is on the kitchen counter, a cutlery knife is on the kitchen counter, a fridge is inside the kitchen and fridge is closed, a kitchen cabinet is inside the kitchen and kitchen cabinet is closed, a kitchen table is inside the kitchen, a plate is on the kitchen table, a pounding cake is on the kitchen table, a microwave is inside the kitchen and microwave is closed, a dishwasher is inside the kitchen and dishwasher is closed. Ney
+
+# Listing 2: Example prompt for the commonsense world model
+
+You need to predict the positions of the moveable objects, containers, and surfaces in the apartment according to the commonsense. Rooms in the apartment: bedroom, bathroom, living room, kitchen. 3 Containers in the apartment: bathroom cabinet, kitchen cabinet, bathroom counter, fridge, oven, dishwasher, microwave, stove, bathroom cabinet. 4 Surfaces in the apartment: bed, bookshelf, cabinet, coffee table, cutting board, floor, fryingpan, kitchen counter, kitchen table, nightstand, sofa, stove. 5 You need to strictly follow the format in the following examples: 6 Question: what are the possible positions of strawberry? Answer: Inside fridge, On kitchen table. 8 Question: what are the possible positions of soap? 9 Answer: On bathroom counter. 10 Question: what are the possible positions of water cup? 11 Answer: On kitchen table, Inside dishwasher. 12 Now, answer the next following question. 13 Question: what are the possible positions of apple?
+
+# Listing 3: Example prompt for the natural language instruction interpretation
+
+You need to interpret the natural language goal into a formal goal representation 2 For example, 3 Goal: put 1 toothbrush inside the bathroomcabinet. 4 Formal goal: (INSIDE, toothbrush, bathroomcabinet, 1) 5 Goal: put 1 toothbrush inside the bathroomcabinet, put 1 apple on the kitchentable. 6 Formal goal: (INSIDE, toothbrush, bathroomcabinet, 1)-(ON, apple, kitchentable, 1) Goal: put 1 toothbrush inside the bathroomcabinet, put 1 apple on the kitchentable, put 1 chicken inside the fridge. 8 Formal goal: (INSIDE, toothbrush, bathroomcabinet, 1)-(ON, apple, kitchentable, 1)-(INSIDE, chicken, fridge, 1) 9 Now, interpret the next following goals:

parse/dev/kroqZZb-6s/kroqZZb-6s.md

@@ -0,0 +1,294 @@
+# CLUSTER-BASED FEATURE IMPORTANCE LEARNING FOR ELECTRONIC HEALTH RECORD TIME-SERIES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The recent availability of Electronic Health Records (EHR) has allowed for the development of algorithms predicting inpatient risk of deterioration and trajectory evolution. However, prediction of disease progression with EHR is challenging since these data are sparse, heterogeneous, multi-dimensional, and multi-modal time-series. As such, clustering is used to identify similar groups within the patient cohort to improve prediction. Current models have shown some success in obtaining cluster representation of patient trajectories, however, they i) fail to obtain clinical interpretability for each cluster, and ii) struggle to learn meaningful cluster numbers in the context of the imbalanced distribution of disease outcomes. We propose a supervised deep learning model to cluster EHR data based on the identification of clinically understandable phenotypes with regard to both outcome prediction and patient trajectory. We introduce novel loss functions to address the problems of class imbalance and cluster collapse, and furthermore propose a feature-time attention mechanism to identify cluster-based phenotype importance across time and feature dimensions. We tested our model in over 100,000 unique trajectories from hospitalised patients with Type-II respiratory failure to predict four different outcomes. Our model yielded added interpretability to cluster formation and outperformed benchmarks by at least $5 \%$ in mean AUROC.
+
+# 1 INTRODUCTION
+
+Chronic conditions such as Chronic Obstructive Pulmonary Disease (COPD) and Cardiovascular Disease (CVD) describe a broad spectrum of medical ailments, and affect a significant percentage of the overall population (Adeloye et al., 2015). Such diseases are characterized by the existence of multiple distinct patient subgroups, largely distinguished by differences in pathology and in the response to different treatments and medical interventions Turner et al. (2015); Vogelmeier et al. (2018). Exacerbation of COPD, a condition of respiratory failure, can result in emergency hospital admission and mortality if it is not well treated and managed. Early identification of COPD patients’ subgroups is therefore of high medical importance and relevance. EHR time-series data are typically used to determine clinically relevant COPD inpatient subgroups, and have been applied to detect risk of deterioration (Pikoula et al., 2019).
+
+However, modelling disease progression and risk prediction is challenging due to the extreme data heterogeneity nature of EHRs. Firstly, EHR data contains a mixture of demographic or static variables (i.e. time independent such as age and sex), and multi-dimensional time-series (e.g Heart Rate, HR, and laboratory measurements, such as blood tests). Secondly, EHR time-series are multi-modal as different features are collected from different devices, representing distinct clinical properties of relevance. Similarly, time-series features are sampled at different times and have low and distinct sampling rates, as well as different missing value properties. Furthermore, each feature is associated with different noise and evolution patterns.
+
+Recent advances in deep learning (DL) approaches have shown promising results in EHR modelling due to their capacity to handle complex data (Rajkomar et al., 2018). Nonetheless, DL approaches lack relevant interpretability frameworks to be scaled and applied in hospital settings. Several such models have since been proposed to tackle this issue (Mayhew et al., 2018), however, most of them focus on a subset of EHR features (usually vital signs only) and fail to provide a clinically-focused phenotypic analysis of learnt patient sub-groups (via clustering).
+
+This work builds on previous research in literature to introduce a cluster-based feature-time attention mechanism to predict patient outcomes based on EHR data. Our method also leverages phenotypic information to aid in clinical interpretability, not only making use of demographics and vital-signs information but also of relevant laboratory measurements (all present in the EHR) to provide a more complete patient physiological status. Our contributions include the following:
+
+• An end-to-end DL supervised model to cluster EHR patient data based on the identification of clinically understandable cluster phenotypes with regard to both outcome prediction and patient trajectory in a multi-class setting;   
+• A weighted loss to address data imbalance for both tasks of clustering and prediction, a common issue in the medical domain;   
+• The incorporation of a novel loss mechanism in the model, to address the issue of cluster collapse and promote sample assignment to all available clusters;   
+• Finally, the inclusion of a novel interpretability framework, derived from a cluster-based feature-time attention layer, aiming to identify relevant timestamps and feature variables to represent the patient physiology, cluster assignment and, ultimately, outcome prediction.
+
+This paper is structured as follows. In Section 2, we describe previous research in EHR time-series modelling, clustering and attention methods. Section 3 introduces the dataset used for analysis and provides description of the proposed model. The experimental setup and results of our analysis are presented in Section 4 and discussion takes place in Section 5. Finally, concluding remarks and future work are available in Section 6.
+
+# 2 RELATED WORK
+
+EHR data comprise complex time-series data, being high-dimensional, multi-modal and heterogeneous, and thus presenting challenges when used in machine learning models (Keogh & Kasetty (2003); Rani & Sikka (2012)). An important goal in a medical setting is to identify phenotypically separable clusters with distinct phenotypic profiles (which we denote as phenotypic clustering hereafter). For the purpose of this work, cluster phenotypes result from the combination of two distinct components: a) the evolution profile of patient trajectories’ within the cluster, and b) the characterisation of the cluster with regards to clinical variables of interest. The latter may include features not used for clustering and may provide information about the underlying or future health status. Traditional clustering models such as K-Means or hierarchical clustering have been shown to fail to capture the existing time-dependent feature relationships. As such, variants have been proposed to mitigate this problem. A temporal version of the K-Means algorithm, Time-Series K-Means (TSKM, Tavenard et al. (2020)), models the distance between time-series of different datapoints, using the Euclidean distance (which is equivalent to considering all temporal observations as an independent feature value for the corresponding patient admission), or time-series alignment strategies such as Dynamic-Time Warping (DTW, Berndt & Clifford (1994)) and soft-DTW (Cuturi & Blondel (2017)).
+
+Recent DL architectures, ranging from Auto-Encoders (AE, Ma et al. (2019)), Convolutional Neural Networks (CNN, Munir et al. (2018)) and others, have shown great promise when applied to timeseries data across a variety of domains. Fortuin et al. (2019) proposed a Self-Organising Map - Variational Auto-Encoder (SOM-VAE), is a state-of-the-art, unsupervised, DL clustering algorithm which extends a variational auto-encoder architecture (Kingma & Welling, 2013) for observation learning and representation, through the addition of a Markov model (Gagniuc, 2017), to infer temporal evolution within the latent space. Clustering is performed in the low dimensional latent space through the use of self-organising maps (Kohonen, 1982) to obtain a discrete, topologically-interpretable latent representation of the learnt clusters. In a supervised setting, AC-TPC (Lee & Van Der Schaar, 2020) serves as the current state-of-the-art for identifying phenotypically separable clusters in patient trajectories in EHR data. AC-TPC maps EHR data into a latent space via an encoder, and uses an actor-critic network (Konda & Tsitsiklis, 2000) which leverages clinical outcomes to aid in cluster formation and obtaining cluster phenotypes. Neither SOM-VAE and AC-TPC provide clinically meaningful interpretation of feature-time importance or outcome of interest.
+
+Attention mechanisms have recently been proposed to provide greater interpretability to Recurrent Neural Networks (RNN) and to aid in dealing with long-term dependencies (Vaswani et al., 2017; Xu et al., 2015), and have also been used in modelling EHR time-series (Schwab et al., 2017; Shashikumar et al., 2018). RETAIN (Choi et al., 2016) proposes a two-level reverse attention mechanism to mimic physician’s decision process and predict a future diagnosis. In other recent works, attention mechanisms based on bi-directional RNN and CNN outperformed standard classification models in predicting high risk vascular diseases with the addition of medication information as input data (Kim et al., 2017). A drawback of such attention mechanisms is the focus on temporal interpretability only, and inability to look at individual features, which is key in a medical setting. To solve this issue, Shamout et al. (2019) considered independent RNN per feature, with a concatenation of the resulting latent vectors. However, the latter does not allow the joint modelling across both feature and time dimensions. Alternatively, Kaji et al. (2019); Gandin et al. (2021) proposed learning attention weights directly on the original inputs, prior to being transformed by a RNN, which does not allow modelling of the resulting latent representations. To the best of our knowledge, no existing models have been proposed that jointly leverage both feature and time dimensions (feature-time) to determine clinical observation relevance on clustered EHR data.
+
+# 3 METHODS
+
+# 3.1 DATASET AND PRE-PROCESSING
+
+Our dataset was retrieved from a retrospective database of routinely collected observations from concluded hospital admissions between March 2014 and March 2018 (the HAVEN project, REC reference: 16/SC/0264 and Confidential Advisory Group reference 08/02/1394). The database includes EHR measurements of adult patients admitted to four hospitals from the Oxford University Hospitals NHS Foundation Trust. Note that the HAVEN dataset does not include data from Intensive Care Units (ICU), and we have excluded observations taken in the Emergency Department. Key characteristics of HAVEN cohort data include a) heterogeneity, b) multi-modality, and difference in: c) noise distributions, d) sampling rates, e) missing values, etc. Such properties are common across EHR settings, and are challenging with respect to learning useful representations and predictions.
+
+We used the protocol defined in Pimentel et al. (2019) to subset the cohort to those patients at risk of developing Type-II Respiratory Failure (T2RF) in hospital (a diagram of the data selection steps can be found in Figure A.1 in the Appendix). Four patient outcomes were considered in our analysis: i) no event during hospital stay, leading to successful discharge from the hospital, or the first instance of one of three possible events, ii) unplanned entry to ICU, iii) cardiac arrest (also named ”Cardiac” hereafter) and iv) death. Outcome groups are not clearly separable (see Tables A.2, A.3 in the Appendix), so patient clusters will naturally contain a mix of different admission outcomes. In this setting, the clinically relevant component of a cluster phenotype (henceforth denoted as cluster outcome propensity or cluster outcome) is represented as a categorical distribution indicating the corresponding propensity for cluster-assigned patients to each corresponding outcome.
+
+For each admission, observations were grouped according to mean window observation value into 4 hour blocks based on the time to outcome (discharge in the case of no event during stay) - only observations within 24 and 72 hours before the outcome were considered. This time window was selected based on those traditionally used for validating Early Warning Score (EWS) systems (baseline models used by UK NHS staff to track inpatient physiology, (Royal College of Physicians, 2017) and clinical input, such that the target phenotype represents the patient status in the subsequent 24 hours. Features were transformed according to min-max normalization due to skewness and heterogeneity in their distributions. Patient admissions were randomly split into train, validation and test sets. Missing values were imputed based on the previously observed time block - all remaining missing observations were imputed according to the feature median from the aggregated validation and test data (see Section 4 for description of train-test data split). Imputed values were flagged in a three-dimensional mask matrix.
+
+After processing, input data contained over 100,000 unique patient trajectories corresponding to 4,266 unique patient admissions (only last patients admissions were considered in our analysis). Original trajectories for the patient cohort are shown in the Appendix in Figures A.4, A.5, A.6 for different variables/features. A lack of clear outcome group separability can be observed across temporal and static variables. Furthermore, we note the high degree of imbalance in the data - admissions with no event account for over $8 6 . 8 \%$ of the total number of admissions, while event classes correspond to $1 0 . 3 \%$ Death, $1 . 8 \%$ ICU and $1 . 1 \%$ Cardiac.
+
+# 3.2 PROPOSED MODEL
+
+We propose a novel model, which we denote by Cluster-bAsed iMportancE Learning fOr Timeseries (CAMELOT). Our proposed methodology is displayed in Figure $1 ^ { 1 }$ . Our model builds on previous literature on 3 key items: a) a modified loss function to target the multi-class imbalance, b) a novel loss function to ensure cluster assignment and phenotype are representative, and c) a novel feature-time attention-level framework to boost representation and introduce feature-time interpretability for cluster assignment.
+
+![](images/5738edb9246a09ec91147e913fda26d231ec701bd4e53ddb8fc11d9be6192f35.jpg)  
+Figure 1: Diagram of proposed model. MLP - Multilayer Perceptron neural network blocks; RNN - Recurrent Neural Network.
+
+Let $N$ denote the number of patients and $D _ { f }$ the number of input features. Input data consists of a set of patient trajectorobservations for patient s , $\mathbb { X } = \{ \{ \mathbf { x } _ { n , t } \} _ { t = 1 } ^ { T _ { n } } \} _ { n = 1 } ^ { N }$ , where tcomes $T _ { n }$ mum number of temporal. Input trajectory data for $n$ $\mathbb { Y } = \left\{ \mathbf { y } _ { n } \right\} _ { n = 1 } ^ { N }$   
+the $n$ -th patient is represented as $\mathbf X _ { n } = [ \mathbf x _ { n , 1 } , . . . , \mathbf x _ { n , T _ { n } } ]$ , where each $\mathbf { x } _ { n , t } \in \mathbb { R } ^ { D _ { f } }$ is referred to as an observation (vector), with a maximum of observed $D _ { f }$ feature values. The corresponding patient outcome is a one-hot encoded vector $\mathbf { y } _ { n } \in \mathbb { R } ^ { 4 }$ (more generally, the dimension of ${ \bf y } _ { n }$ equals to the number of possible outcomes).
+
+Our DL model can be decomposed into 3 neural network blocks: an Encoder, Identifier and Predictor. We refer the action of each network respectively as E, I and $\mathrm { \bf P }$ (for example, $I ( \mathbf { x } )$ denotes the output of the Identifier given some input vector $\mathbf { x }$ ). Both the Identifier and Predictor are Multilayer Perceptrons (MLP), networks of stacked feed-forward dense layers. On the other hand, the Encoder block can be further sub-divided into a) a stack of RNN layers and b) our proposed custom attention layer (see Section 3.3 for further details). Separately, we also consider a set of trainable cluster representation vectors, $\mathcal { C } = \{ \mathbf { c } _ { 1 } , . . . , \mathbf { c } _ { K } \}$ . We assign the outcome for cluster $i$ as $P ( \pmb { c } _ { i } )$ .
+
+A model call is as follows: Given the $n$ -th patient input trajectory data $\mathbf { X } _ { n }$ , the Encoder network returns a latent representation $\mathbf { z } _ { n } : = E ( \mathbf { X } _ { n } ) \in \mathbb { R } ^ { l }$ . Consequently, the Identifier network computes cluster assignment probabprobability assignment of ies, to $\pi _ { n } : = I ( \mathbf { z } _ { n } ) \in \mathbb { R } ^ { K }$ .  of h element of clusters. A c $\pi _ { n }$ , e $\boldsymbol { \pi } _ { n } ^ { i }$ resents theis selected $\mathbf { z } _ { n }$ $i$ $K$ $k _ { \mathrm { s a m p } } ^ { n }$ according to categorical sampling $( k _ { \mathrm { s a m p } } ^ { n } \sim \mathbf { C a t } ( \pi _ { n } ) )$ , and the corresponding cluster representation, $\mathbf { c } _ { \mathrm { s a m p } } ^ { \mathbf { n } } : = \mathbf { c } _ { \mathbf { k } _ { \mathrm { s a m p } } ^ { \mathbf { n } } }$ is then selected from $\mathcal { C }$ . The output of the model is $y _ { \mathrm { p r e d } } : = P ( \mathbf { c } _ { \mathrm { s a m p } } ^ { n } ) \in \mathbb { R } ^ { 4 }$ . We note that $k _ { \mathrm { s a m p } } ^ { n }$ is only sampled during a training phase; at prediction stage, cluster selection follows the equation $k _ { \mathrm { p r e d } } ^ { n } = \underset { i = 1 , . . . , K } { \arg \operatorname* { m a x } } \pi _ { n } ^ { i }$ .
+
+# 3.3 ENCODER NETWORK AND A CUSTOM ATTENTION LAYER
+
+The diagram of our proposed Encoder network is presented in Figure 2. The Encoder contains (i) a Recurrent Neural Network (RNN) block of stacked Long Short-Term Memory (LSTM) layers, and (ii) a customised attention layer, which computes a latent representation by comparing input data with the sequence of output states from the RNN block. We use the same notation as above, and write the sequence of output states of the final LSTM layer as $\mathbf { o } _ { n , 1 } , . . . , \mathbf { o } _ { n , T _ { n } }$ , with ${ \bf o } _ { n , i } \in \mathbb { R } ^ { l }$ . Theoretically, each ${ \bf o } _ { n , t }$ corresponds to a representative summary of input patient information up until time $t$ . We propose to approximate ${ \bf o } _ { n , t }$ as a linear combination of latent representations of each individual feature, thereby allowing the separation of output states into contributions from each feature. Note that it is important feature transformations be time-independent, to avoid overparametrising the model, over-fitting and ensure feature representation maps are similar across time.
+
+![](images/10d1f2dae6d4c161e7b4ed89343af6140dc1df3e400dc01609dabc40230f10c6.jpg)  
+Figure 2: Diagram of the Encoder network composed of an LSTM Encoder and a custom attention layer.
+
+Our attention layer behaves as a set of $D _ { f }$ feed-forward neural network layers, $\mathbf { U } _ { 1 } , . . . , \mathbf { U } _ { D _ { f } }$ , jointly represented by: (i) a matrix of learnable kernel weights $D \in \mathbb { R } _ { l \times D _ { f } }$ . We write $D = [ \mathbf { D } _ { 1 } , . . . , \mathbf { D } _ { D _ { f } } ]$ ; (ii) a matrix of learnable bias vectors $\boldsymbol { B } \in \mathbb { R } _ { l \times D _ { f } }$ . Similarly, we can write $B = [ \mathbf { B } _ { 1 } , . . . , \mathbf { B } _ { D _ { f } } ]$ ; and (iii) an activation function, $\sigma$ which matches the output activation of the RNN block.
+
+Input data for patient $n$ , $\mathbf { X } _ { n }$ is fed as input to the RNN block, which outputs a sequence of latent output states $( { \bf { \bar { o } } } _ { n , t } ) _ { t = 1 } ^ { T _ { n } }$ . For $t = 1 , . . . , T _ { n }$ , we compute $D _ { f }$ feature representations in latent space as:
+
+$$
+R _ { n , t } : = \sigma ( D \odot \mathbf { x } _ { n , t } + B )
+$$
+
+where ${ \cal R } _ { n , t } = [ { \bf R } _ { n , t } ^ { 1 } , . . . , { \bf R } _ { n , t } ^ { D _ { f } } ]$ is our collection of feature representations, $\sigma$ is applied element-wise and $\mathbf { A } : = { \cal D } \odot \mathbf { x } _ { n , t }$ is a matrix satisfying $A _ { i , j } = D _ { i , j } ( \mathbf { x } _ { n , t } ) _ { j }$ . Equivalently, $\mathbf { R } _ { n , t } ^ { i }$ is the output of a dense layer, $\mathbf { U } _ { i }$ with kernel $\mathbf { D } _ { i }$ , bias $\mathbf { B } _ { i }$ , activation $\sigma$ and input $( \mathbf { x _ { n , t } } ) _ { i }$ . We approximate $\mathbf { o } _ { n , t } \approx$ $\begin{array} { r } { \sum _ { i = 1 } ^ { D _ { f } } \alpha _ { t } ^ { i } { \bf R } _ { n , t } ^ { i } \ = \ R _ { n , t } \alpha _ { t } } \end{array}$ . This approximation is minimised following a least squares criterion, which has a well-known solution, $\widehat { \alpha } _ { t }$ , and corresponding optimal approximation $\widehat { \mathbf { o } } _ { n , t }$ .
+
+Given, $\hat { \mathbf { o } } _ { n , t }$ , a similar procedure is used to compute a context vector as $\begin{array} { r } { \mathbf { z } : = \sum _ { t } \beta _ { t } \hat { \mathbf { o } } _ { n , t } } \end{array}$ , where weights $\beta$ are learned to provide a more representative context vector.
+
+# 3.4 ATTENTION MAP VISUALISATION
+
+Given cluster representation vectors, $\mathbf { c } _ { k }$ , we can compute a cluster-wise feature-time attention visualisation map as follows. First, we normalise feature-weights $\widehat { \mathbf { a } } _ { t }$ according to a softmax function, $\mathbf { s } _ { t } = \sigma ( \widehat { \pmb { \alpha } } _ { t } ) \in \mathbb { R } ^ { D _ { f } }$ , where $\sigma$ is the softmax function: $\begin{array} { r } { \sigma ( \mathbf { x } ) = \frac { \exp { | \mathbf { x } | } } { \| \exp { | \mathbf { x } | } \| _ { 1 } } } \end{array}$ . Secondly, we compute tsolved as before. We similarly normalise cluster-wise weights, $\gamma _ { t } ^ { k }$ according to a least-square approximation of $\gamma _ { n , t } ^ { k }$ tto obtain cluster temporal scores, $\begin{array} { r } { \dot { \mathbf { c } } _ { k } \approx \sum _ { t = 1 } ^ { T _ { n } } \widehat { \mathbf { o } } _ { n , t } \gamma _ { n , t } ^ { k } } \end{array}$ $e _ { n , t } ^ { k } = \sigma ( \gamma _ { n , t } ^ { k } )$ , and . Finally, we can compute $K$ scoring matrices, $M _ { n } ^ { 1 } , . . . , M _ { n } ^ { K } \in \mathbb { R } _ { T _ { n } \times D _ { f } }$ :  M kn 
 t,f $\left( M _ { n } ^ { k } \right) _ { t , f } = e _ { n , t } ^ { k } \pmb { s } _ { t } ^ { f }$ . Note that: $\begin{array} { r } { \| M _ { n } ^ { k } \| _ { 1 } = \sum _ { t } e _ { n , t } ^ { k } \sum _ { f } s _ { t } ^ { f } = \sum _ { t } e _ { n , t } ^ { k } = 1 } \end{array}$ . Given that Matrices $M _ { n } ^ { k }$ are normalised, they may be consequently, visualised as a normalised feature-time map for cluster assignment relevance and provide further model interpretability.
+
+The model is optimised through consideration of three distinct loss functions. We introduce a weighted cross-entropy loss function:
+
+$$
+L _ { \mathrm { p r e d } } ( y _ { \mathrm { t r u e } } , y _ { \mathrm { p r e d } } ) = - \sum _ { c = 1 } ^ { C } w _ { c } y _ { \mathrm { t r u e } } ^ { c } \log \left( y _ { \mathrm { p r e d } } ^ { c } \right) = - w _ { c ^ { \prime } } \log ( y _ { \mathrm { p r e d } } ) _ { c ^ { \prime } }
+$$
+
+where $c ^ { \prime }$ is the true outcome for a particular patient. We propose inversely proportional normalised weights: $\textstyle \sum _ { c = 1 } ^ { C } w _ { c } = 1$ and $w _ { c }$ is inversely proportional to the class distribution, i.e., $\begin{array} { r } { w _ { c } \propto \frac { N } { N _ { c } } } \end{array}$ with $N$ being the number of patients and $N _ { c }$ being the number of patients with outcome label $c$ . Class weighting penalises misclassification more heavily on less sampled classes.
+
+We also propose a novel distribution loss function, $\operatorname { L } _ { \mathrm { d i s t } } ( \pi )$ . We define the average cluster probability of assignment as $\begin{array} { r } { \pi _ { C } : = \frac { 1 } { N } \sum _ { n } \pi _ { n } } \end{array}$ . Then we introduce ${ \cal L } _ { \mathrm { d i s t } } ( \pmb { \pi } ) = - { \cal H } ( \pmb { \pi } _ { C } )$ , where $H$ denotes entropy. Note that $L _ { \mathrm { d i s t } }$ is minimised when $\pi _ { C }$ is uniform, ensuring all clusters are ’explored’ and have comparable number of samples. While clusters should not necessarily be assigned the same number of samples, this loss helps to overcome a concern of cluster collapse, where clusters do not separate and samples are assigned to a very small number of non-representative clusters. Finally, to separate cluster representation vectors, we define the cluster separation loss as $\begin{array} { r } { L _ { \mathrm { c l u s } } ( \mathcal { C } ) = \dot { - } \frac { 1 } { K ( K - 1 ) } \sum _ { i , j } \| \mathbf { c } _ { i } - \mathbf { c } _ { j } \| ^ { 2 } } \end{array}$
+
+To optimise our model, iterative gradients are applied according to weighted combinations of the above loss functions with hyper-parameter weights $\alpha , \beta$ :
+
+1. Outcome Predictor is updated according to $L _ { \mathrm { p r e d } }$ ;   
+2. Encoder and Identifier are trained according to $L _ { \mathrm { p r e d } } + \alpha L _ { \mathrm { d i s t } }$ ;   
+3. Cluster representation vectors are updated with regards to $L _ { \mathrm { p r e d } } + \beta L _ { \mathrm { c l u s } }$ .
+
+# 3.6 INITIALISATION
+
+Our proposed model also follows a set of initialisation pre-training procedures. Firstly, the Encoder and Outcome Predictor are pre-trained according to a classification task $( \tilde { \mathbf { y } } = P ( E ( \bar { \mathbf { x } } ) ) )$ , with corresponding loss $L _ { \mathrm { p r e d } }$ . Latent state representations $E ( \mathbf { x } )$ are clustered through a K-means algorithm with $K$ clusters across the whole training set. Cluster representation vectors are initialised as given by the resulting cluster centroids, and finally the Cluster Identifier network is pre-trained to identify clusters as predicted by the K-Means algorithm with categorical cross-entropy loss. Implementation was completed in Python, with TensorFlow 2, scikit-learn and NumPy. All experiments were run with 1 Tesla v100 GPU, and 8 CPUs Intel(R) Xeon(R) Gold 6246 $@$ 3.30GHz.
+
+# 4 RESULTS
+
+For Benchmark purposes, we considered TSKM as a classic clustering benchmark, and SOM-VAE and AC-TPC as state-of-the-art phenotypic clustering methods. AC-TPC considers temporal subsequences of a complete patient set of observations - for comparison purposes, we consider only the model output for the complete patient sequence. For simplicity, we present results with all input features considered.
+
+All models were trained on the same training set ( $60 \%$ of the complete input data) and evaluated against the same test set (remaining $40 \%$ ). For DL models, we further split the training set into a purely training and validation sets. All experiments with varying hyper-parameters were repeated 10 times with a fixed set of 10 distinct seeds, and results are reported according to average metric and standard deviation. A complete list of the hyper-parameters considered for each model is included in Table A7 in the Appendix. In bold, top-performing hyper-parameters are indicated. Optimal integer hyper-parameters $( K , l )$ were selected according to an “Occam’s Razor” approach - for each parameter, we assign it the highest value such that increasing this amount does not lead to a significant increase in performance according to mean AUROC and a Friedman’s hypothesis test. Neural network size parameters were kept consistent across all DL models where applicable. All other optimal hyper-parameters were selected according to highest AUROC performance conditional on the model predicting each class (e.g. not making predictions solely for no event or Death Events).
+
+We evaluated clustering performance through standard clustering metrics, including Silhouette score (SIL, Rousseeuw (1987)), Davies-Bouldin Index (DBI, Davies (1979)), Variance Ratio Criterion (VRI, Calinski & Harabasz (1974)). Results for all clustering models are displayed in Table 3. ´ In Table 4, we evaluated the (multi-class) prediction performance with regards to Area-under-theReceiver-Operating-Curve (AUROC), unweighted mean F1-score, unweighted mean Recall, and Normalised Mutual Information (NMI). For purely unsupervised models (SOM-VAE and TSKM), an outcome predictive pipeline was constructed by assigning patient admissions to clusters, and consequently to the empirical outcome distribution in the corresponding cluster. The prediction task was also benchmarked against traditional classifiers for outcome prediction in EHR data, namely Support Vector Machines (SVM), XGBoost (XGB) and NEWS2 (i.e., the National Early Warning Score used in the UK hospitals). Furthermore, to evaluate other Neural Network models as benchmarks and also justify both proposed mechanisms, we furthermore considered two other benchmarks: a) ENC-PRED: a stacked LSTM Encoder, followed by a MLP network for outcome prediction, and b) ATTEP; a model equivalent to CAMELOT, except the original entropy loss is considered over the clustering dist loss. Supervised performance for all the above models is included in Table4. As noted, convergence was particularly difficult for ATTEP due to cluster collapse.
+
+Table 3: Clustering separability results by the different clustering methodologies given input data with all available features (static, vital-signs, serum and haematological variables). For each metric and model, the average score and standard deviation are returned. The best values for each metric are indicated in bold.   
+
+<table><tr><td rowspan=1 colspan=1>Metric</td><td rowspan=1 colspan=1>1        TSKM</td><td rowspan=1 colspan=1>SOM-VAE</td><td rowspan=1 colspan=1>AC-TPC</td><td rowspan=1 colspan=1>CAMELOT(proposed)</td></tr><tr><td rowspan=1 colspan=1>SIL</td><td rowspan=1 colspan=1>0.35(±0.01)</td><td rowspan=1 colspan=1>0.25(±0.08)</td><td rowspan=1 colspan=1>0.04(±0.01)</td><td rowspan=1 colspan=1>0.11(±0.04)</td></tr><tr><td rowspan=1 colspan=1>DBI</td><td rowspan=1 colspan=1>1.19 (±0.08)</td><td rowspan=1 colspan=1>1.89(±0.63)</td><td rowspan=1 colspan=1>4.34(±0.80)</td><td rowspan=1 colspan=1>3.12(±0.53)</td></tr><tr><td rowspan=1 colspan=1>VRI</td><td rowspan=1 colspan=1>554.6(±2.50)</td><td rowspan=1 colspan=1>12.8(± 9.32)</td><td rowspan=1 colspan=1>66.5(± 18.7)</td><td rowspan=1 colspan=1>216.7(±6.2)</td></tr></table>
+
+Table 4: Outcome prediction scores across all models, displayed with an average and standard deviation of a set of 10 seeds (except NEWS2, which is deterministic). The best values for each metric are indicated in bold. For clustering algorithms, cluster outcome distributions were taken to be the empirically observed distribution in each cluster.   
+
+<table><tr><td rowspan=1 colspan=1>Metric</td><td rowspan=1 colspan=1>AUROC</td><td rowspan=1 colspan=1>F1-score</td><td rowspan=1 colspan=1>Recall</td><td rowspan=1 colspan=1>NMI</td></tr><tr><td rowspan=1 colspan=1>SVM</td><td rowspan=1 colspan=1>0.50(±0.02)</td><td rowspan=1 colspan=1>0.23 (±0.00)</td><td rowspan=1 colspan=1>0.25(±0.00)</td><td rowspan=1 colspan=1>0.01(±0.02)</td></tr><tr><td rowspan=1 colspan=1>XGB</td><td rowspan=1 colspan=1>0.65(±0.01)</td><td rowspan=1 colspan=1>0.23(±0.00)</td><td rowspan=1 colspan=1>0.22(±0.00)</td><td rowspan=1 colspan=1>0.03(±0.04)</td></tr><tr><td rowspan=1 colspan=1>NEWS2</td><td rowspan=1 colspan=1>0.61</td><td rowspan=1 colspan=1>0.29</td><td rowspan=1 colspan=1>0.34</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td rowspan=1 colspan=1>TSKM</td><td rowspan=1 colspan=1>0.55(±0.01)</td><td rowspan=1 colspan=1>0.24(±0.03)</td><td rowspan=1 colspan=1>0.26(±0.02)</td><td rowspan=1 colspan=1>0.01(±0.03)</td></tr><tr><td rowspan=1 colspan=1>SOM-VAE</td><td rowspan=1 colspan=1>0.61(±0.09)</td><td rowspan=1 colspan=1>0.27(±0.05)</td><td rowspan=1 colspan=1>0.27(±0.03)</td><td rowspan=1 colspan=1>0.05(±0.03)</td></tr><tr><td rowspan=1 colspan=1>AC-TPC</td><td rowspan=1 colspan=1>0.68(±0.01)</td><td rowspan=1 colspan=1>0.38(±0.01)</td><td rowspan=1 colspan=1>0.36(±0.01)</td><td rowspan=1 colspan=1>0.17(±0.02)</td></tr><tr><td rowspan=1 colspan=1>ENC-PRED</td><td rowspan=1 colspan=1>0.57(±0.02)</td><td rowspan=1 colspan=1>0.25(±0.02)</td><td rowspan=1 colspan=1>0.26(±0.02)</td><td rowspan=1 colspan=1>0.06(±0.03)</td></tr><tr><td rowspan=1 colspan=1>ATTEP</td><td rowspan=1 colspan=1>0.67(±0.02)</td><td rowspan=1 colspan=1>0.36(±0.02)</td><td rowspan=1 colspan=1>0.36(±0.02)</td><td rowspan=1 colspan=1>0.16(± 0.03)</td></tr><tr><td rowspan=1 colspan=1>CAMELOT(proposed)</td><td rowspan=1 colspan=1>0.73(±0.02)</td><td rowspan=1 colspan=1>0.36(±0.01)</td><td rowspan=1 colspan=1>0.38(±0.02)</td><td rowspan=1 colspan=1>0.20(±0.03)</td></tr></table>
+
+On top of performance evaluation with regards to clustering separability and outcome prediction, we display a comparison between the learnt cluster phenotypes of the proposed model and that of phenotypic clustering benchmark AC-TPC. For each cluster, the corresponding outcome propensity $P ( \mathbf { c } )$ is shown as a bar plot over the 4 possible outcomes with corresponding probability value. We also display cluster outcome propensity plots for both TSKM and SOM-VAE in the Appendix (Figures A.8 and A.9). We note that the cluster outcome propensity distributions learnt by CAMELOT also align with the empirical outcome relevance in the learnt clusters (Table A.12). Lastly, we also display feature-time cluster relevance attention maps in Figure 6. For each cluster, a patient was randomly selected from the set of patients in the corresponding cluster, and a corresponding featuretime attention matrix and visualised as a heatmap.
+
+# 5 DISCUSSION
+
+Our proposed model shows an improvement in clustering performance ( see Table 3) when compared to the current phenotypic clustering benchmark (AC-TPC), and outperforms SOM-VAE according to VRI. Although the cluster separability metrics are superior in the case of TSKM, this is expected given a metric bias towards convex clusters, and the convexity of the K-Means based algorithm. In particular, DL clustering occurs in a latent space, which, unfortunately, is not easily comparable with an algorithm targetting the input space (such as K-Means). Furthermore, we argue clusters learnt by TSKM are less relevant that our model’s as a) TSKM clusters are extremely hard to distinguish with regards to outcome propensity (as evidenced by very low performance on a prediction task (Table 4)), and b) TSKM clusters are less separable with regards to trajectory evolution, as there is less separation of mean HR trajectories, and less cluster separation when data is projected to a two-dimensional domain with t-stochastic neighbour embedding (tSNE) - both figures are in the Appendix, Figures A10 and A11.
+
+![](images/fbc878ea7f8978b76b270f1c8ad192ef506787ef45aae1a8c31f4e7cbfb1dbb5.jpg)  
+Figure 5: Comparison of bar plots of cluster outcome propensity distributions for the proposed model and benchmark AC-TPC. On the left (blue), distributions are displayed for each cluster (out of a total of 6), and each phenotype corresponds to the probability of an outcome. Similar results are shown on the right (yellow) for AC-TPC. The title of each sub-plot indicates the cluster considered, as well as the number of patients assigned to a given cluster.
+
+![](images/f965e66d1d50bc859ff0affce0c21413c73abd79fde8c4c21f623ef58512f29d.jpg)  
+Figure 6: Feature-Time Cluster Relevance Map. Each Heatmap represents a feature-time relevance matrix for a random patient assigned to a given cluster. Vertical Axis indicates time to outcome, in hours, while horizontal axis indicates different input features.
+
+With regards to predictive power, it can be seen in Table 4 that our model outperforms both standard classifiers (at least $8 \%$ ) and benchmarking clustering methods (at least $5 \%$ ) according to mean AUROC. A similar increase can be seen in other classification metrics, with the exception of F1-score, where model performance is slightly below to that of AC-TPC. Our model is able to more accurately determine patterns in the data than the previously proposed models, as EHR data is extremely complex and heterogeneous. It is particularly promising that the model obtains good predictive task results despite a clustering bottleneck (i.e. sample predicted outcomes are done through the assigned cluster, as opposed to tailored to the precise input data). While it is possible that other models could show better performance on the direct task of outcome prediction given EHR input, such models can potentially be associated with lack of robustness or input sensitivity difficulties. Furthermore, it is likely they would struggle with identifying relevant trends and properties of clinical interest. As such, for new admissions, such models could provide a prediction for the overall outcome, but no robust understanding of how this outcome will occur, and how to prevent potential risks of deterioration, let alone the ability to pool data from other similar patients.
+
+Figure 5 shows the advantage of two key aspects of our methodology. Our model identifies clear, separable cluster outcome distribution and provides a useful layer of interpretability to clinicians to understand a potential risk of deterioration. Our model also identifies a more diverse set of cluster outcomes than AC-TPC, which only picks up 3 different cluster outcome distributions, and doesn’t identify the presence of the ”ICU” and ”Cardiac” classes. We also show other cluster-phenotype benchmark results in Appendix A.8 and A.9. With regards to clusters learnt by CAMELOT, clusters 0 and 3 are the clusters with most ill cohort - they are largely representative of death and cardiac events on the subsequent 24 hours. On the other hand, while cluster 2, 5 are healthier, with a smaller chance of adverse events. Clusters 1 and 4 are largely “healthy” cluster, with reduced risks of the most intense adverse events. We note cluster outcome propensity distributions learnt by AC-TPC are unable to provide this level of detailed information. Furthermore, the propensity distribution learnt by our model matches with the empirical number outcome events observed in each cluster (displayed in Appendix Table A12). Note, furthermore, that the model managed to successfully navigate a heavy class-imbalance setting. Representative clusters are able to capture different-sized sub-populations, yet still identify potential risks of deterioration.
+
+On the other hand, learnt cluster attention maps introduced in Figure 6 introduce yet another layer of interpretability to our proposed clustering model. The personalised attention maps highlight the relevant feature-time pairs driving patient cluster assignment, and can be used to identify the most important clinical variables. For instance, analysing Figure 5 suggests clusters with highest propensity for either of Death or Cardiac Arrest events are Cluster 0, 3 (and 2 to a slightly smaller extent). This reflects in the resulting attention maps, where SBP and FIO2 are highlighted as key clinical variables for cluster assignment. This conclusion is further corroborated when considering descriptive statistics of CAMELOT clusters (Table A13), as SBP and FIO2 are some of the few variables with some significant separation across clusters, and when considering trajectory evolutions (Figure A14). Lastly, note that feature-time weights are also relevant if potential deterioration events did not take place - so that we are more confident on a patient’s health status. As an example, attention maps for clusters 1 and 4 (reasonably healthy clusters) indicate $\mathrm { F I O _ { 2 } }$ as very relevant towards the latter stages of the admission - this is likely due to these patients not showing an increase in oxygen intake (as they did not need it). Thus, attention maps can be very versatile.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+In this work, we propose a novel deep learning model for the task of identification of phenotypically separable clusters applied to EHR data for. As part of our proposed model, we propose 2 distinct loss functions and introduce a novel feature-time attention layer to better represent patient data and to introduce a feature-time relevance map for each cluster. Our experiments show promising results with the addition of both methodological tools above, on both cluster separability and outcome prediction performance. The addition of the feature-time layer has the added benefit of introducing key interpretability tools for researchers to understand relevant regions for good patient physiology representation as well as an indication of what can lead to patient deterioration.
+
+There are multiple interesting avenues of investigation building on this work. On the other hand, the current attention layer mechanisms could potentially be improved with the addition of temporal weight smoothness, or, alternatively, weight regularization to encourage exploration of the complete feature-time space. Alternatively, cluster selection through a neural network mechanism introduces high capacity at the potential cost of robustness and cluster collapse. Potentially, more traditional methods incorporated into a similarly complex pipeline can achieve better performing through a clearer identification of cluster regions in latent space. Furthermore, methodological improvements will also benefit from a more extensive testing across other diverse datasets and other potential areas of application.
+
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+
+# A APPENDIX
+
+# A.1 DATA
+
+A description of the complete pipeline of data re-processing, following the protocol defined in Pimentel et al. (2019), is shown in Figure A.1.
+
+![](images/e4d91bfcdb064c7d529fad3995ff440c293e1ec8dcc0985d03521510a371044a.jpg)  
+A.1: HAVEN processing
+
+A total of 26 input features were considered. Firstly, 4-hourly vital-sign sets which included 8 features: Heart Rate (HR), Respiratory Rate (RR), Systolic Blood Pressure (SBP), Diastolic Blood Pressure (DBP), peripheral Oxygen Saturation $\mathrm { ( S p O _ { 2 } ) }$ ), Temperature (TEMP), level of consciousness via the AVPU scale - Alert, Verbal, Pain, Unresponsive - and estimated Fraction of Inspired Oxygen $\mathrm { ( F i O _ { 2 } }$ , available when an oxygen mask is applied to the patient). Each set consisted of a timestamp and the vital-sign numerical values. Secondly, 4 demographic variables were selected (modelled as static variables): age, sex, and admission type (elective or surgical). Thirdly, we included 6 features resulting from biochemistry blood tests, denoted as ’Serum’: Serum levels of urea, albumin, creatinine, sodium, potassium and C-reactive protein. Finally, 8 haematological blood test features were also included: white and haemaglobin cell counts, concentration of eosinophils, basophils, neutrophils, and lymphocytes, as well as eosinophil-to-basophil and neutrophil-to-lymphocyte ratios. These features were selected based on domain knowledge of features related to severity in the prognosis and outcome of T2RF inpatients.
+
+Descriptive statistics for all input variables is described in Table A.2. Median and inter-quartile range (IQR) is displayed for continuous and categorical variables, while binary variables are shown according to number of counts in the dataset and corresponding cohort proportion. Statistics are displayed for the complete data (”All”), but also for each sub-cohort defined by the overall outcome. We can observe that these sub-cohorts are not clearly separable and are hard to identify solely from this information.
+
+A summary of the patient cohort in relation to outcomes and target phenotypes can be seen in Table A.3. Challenges with regards to obtaining phenotypically separable clusters can similarly be observed - there is no clear significant difference between the target outcome sub-cohorts with regards to demographic input variables. With regards to outcome distribution, we also note the high degree of imbalance in the dataset - the large majority of the patients in our dataset suffered from no adverse events (over $86 \%$ ), while only 48 had a Cardiac event, and 76 were re-directed to the ICU.
+
+The lack of outcome sub-cohort separability can further observed in a temporal domain. Figures A.4, A.5, A.6 plot the mean trajectories of different temporal variables sets for each outcome subcohort, respectively, according to vital signs, haematological and serum features. Mean is calculated based on the time to outcome, and missing observations are disregarded and ignored.
+
+<table><tr><td>eeh 67901 1013 92796</td><td>.95.90/9 26-70918 81-91)81 111-1111 9-09) 9 （- 6-56) 56</td><td>% 850850 (% 10&#x27;80)0181 1% 00 91) 19</td><td>31155151 1100-00.0000 7781-4536 0-00 （900-0:000 385-00.0 009 711-75111 115</td><td>353158185 11:5550000 （21.01- 33.01/00 11.010-2.0000.00 2105350 WUo &quot; UC  Ub</td></tr><tr><td>eaa 4 9941 15484</td><td>399.95 099 (001- L 68 (1-81） 61 151000010 (L-85）99 96-166 24-181 IDI</td><td>88-418 (%10.9) (% 89) (% 88D）8</td><td>17 00- 00.0 7855581 076109 (110-00&#x27; E00 (50-0-20:) 00 100-000000 773153935 05555</td><td>15100000 27.07-00.0- 00.0 3.055505.09000 8555555955555 11.1-1150 W &quot;U y U U</td></tr><tr><td>5 % 6 2</td><td>2/90-.90.99 （66- 0 0588 （11-11 71 2112-5555 IDI 61-91)81 6-) 56 （-</td><td>%680 (08- 19) 69 (% 005) 8 (%0D)8</td><td>11.00-050030 1.1-6.90 5511511552 22133555 (010-00&#x27;)0 (50-0-20:) 200 1-55 1-55</td><td>1.2525222535 1.001500000 （27- 000 215001 15-155 </td></tr><tr><td>caprea 48 3 </td><td>39.90.0559 171-10 505 （001-88 811-0015 (L-LS）59 6-06) 56 1-11</td><td>IDI (Z8-69) 9L 3%2388 （%140 (%057D9</td><td>1101-555550 818856 01-51 (00-10 900 90/0-0:0200 8-55 905-9888 1-000</td><td>11.07500.0.0 22.0- 0000 11051515815 5515225555 23.5-3000 </td></tr><tr><td>916011 6119821 A 994</td><td>3833505905.99 2820 111-11 914 76-20 (61-91)81 (9L-09)L9 (L6 - 56) S6 （）1</td><td>%692）11 (% 09 7111 (18-29) % 64)111</td><td>1.11-061111 11111-3553355 (20-00 010 (000-10000 （5-5 05 201-555090 (891-) 911 911-999</td><td>1.055-15.0000000） 3000-00.00 00.00 (00.601 - 0085 00./ 871501125559 （40-09.110 4 44-44 4444</td></tr><tr><td>eu M eu au</td><td>snontmuo</td><td>eoogee egu reug</td><td>Ceonunuog</td><td>Ceonunuog</td></tr><tr><td></td><td>（a)spag ag gggiae Hum 1 8 % ℃</td><td>3</td><td>品 /6v0x /6v0x 7/6v0[x 7/6v0x 7/6v0[x</td><td>Toun D Tou Toun 8 7</td></tr><tr><td>rrerago Trgrongae o ed Bereeeeg</td><td>eneregeeengor eepggepeamg seereprggotsse eerrprrgr tgteer areoaeaea Tare H</td><td>ogsosn eee sspeges sossapereeg gee eret egge</td><td>geg rette opgggoog Bpggrooog rgoegg rgnonngn go mggeaag</td><td>erereeeenee egret rereeer re ger se eer T trttsstttra</td></tr><tr><td>N o n</td><td>ZOdS JII ndAA R R 8 </td><td>repueg a eotgms 品</td><td>5 5 8 3 </td><td>A 5R R 8 </td></tr><tr><td></td><td>sus</td><td>ttt umes</td><td>oeege</td><td></td></tr></table>
+
+A.2: Descriptive statistics and information of all input data features. Variables are displayed with type, description, units and average statistics. We separate all features according to medical literature, including vital-sign, static, serum and haematological variables, and we also display statistics per outcome sub-cohort, defined as a cohort with those patients assigned to a given outcome.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>No Event</td><td rowspan=1 colspan=1>Death</td><td rowspan=1 colspan=1>ICU</td><td rowspan=1 colspan=1>Cardiac</td></tr><tr><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>3701</td><td rowspan=1 colspan=1>441</td><td rowspan=1 colspan=1>76</td><td rowspan=1 colspan=1>48</td></tr><tr><td rowspan=1 colspan=1>Age (IQR)</td><td rowspan=1 colspan=1>71(61-80)</td><td rowspan=1 colspan=1>81(74-88)</td><td rowspan=1 colspan=1>69 (61-74)</td><td rowspan=1 colspan=1>76 (69-82)</td></tr><tr><td rowspan=1 colspan=1>Gender,M</td><td rowspan=1 colspan=1>1810 (48.9%)</td><td rowspan=1 colspan=1>247 (56.0%)</td><td rowspan=1 colspan=1>38 (50.0%)</td><td rowspan=1 colspan=1>28 (58.33%)</td></tr><tr><td rowspan=1 colspan=1>CCI (IQR)</td><td rowspan=1 colspan=1>4(3-13)</td><td rowspan=1 colspan=1>14(4- 21)</td><td rowspan=1 colspan=1>7(4- 17)</td><td rowspan=1 colspan=1>15(4- 23)</td></tr><tr><td rowspan=1 colspan=1>Elective</td><td rowspan=1 colspan=1>1126 (30.4%)</td><td rowspan=1 colspan=1>3 (0.7%)</td><td rowspan=1 colspan=1>8 (10.5%)</td><td rowspan=1 colspan=1>2(4.2%)</td></tr><tr><td rowspan=1 colspan=1>Surgical</td><td rowspan=1 colspan=1>1054 (28.5%)</td><td rowspan=1 colspan=1>48 (10.1%)</td><td rowspan=1 colspan=1>22 (29.0%)</td><td rowspan=1 colspan=1>6 (12.5%)</td></tr></table>
+
+A.3: Descriptive demographic variable information for each outcome sub-cohort.
+
+![](images/74381f1f28f3fbe43252cc669d8791edb9470703e6404d9eaf70b1bfb815d80e.jpg)  
+A.4: Plot of mean vital-sign trajectories (median with respect to $\mathrm { S p O _ { 2 } }$ ) in solid line as given by the 4 outcome groups: admissions with a) Cardiac-, b) Death- , or c) ICU-, and d) No-events. The respective standard errors are represented by the dashed lines. We visualised trajectories from up to 7 days prior to an outcome event or discharge - the black lines represent the time window (72 - 24 hours prior to an event or discharge) considered for input to all models.
+
+# A.2 MODEL TRAINING
+
+A list indicating the grid-search range of hyper-parameters considered in our experiments are indicated in Table A.7. For simplicity, we define $\mathbb { P } : = \{ 0 . 0 0 1 , 0 . 0 1 , 0 . 1 , 1 , 1 0 \}$ , $\mathbb { L } : = \overline { { \{ 3 2 , 6 4 , 1 2 8 , 2 5 6 \} } }$ and $\mathbb { K } : = \{ 3 , . . . , 2 0 \}$ . In bold, top-performing hyper-parameters according to target metrics defined in Section 4 are highlighted.
+
+# A.3 RESULTS COMPARISON
+
+In Figures A.8 and Figures A.9 we display cluster outcome propensity distributions for some of our experiments with benchmark clustering models SOM-VAE and TSKM, respectively. Both models do not naturally associate clusters with a distribution - we estimate the cluster outcome as the empirical outcome distribution for the patient cohort assigned to the corresponding cluster.
+
+![](images/7cbbf11689c46b245b146e330e6341b475e20e0bfd241a6057f254c1cb9c118a.jpg)
+
+A.5: Plot of mean haematological trajectories in solid line as given by the 4 outcome groups: admissions with a) Cardiac-, b) Death- , or c) ICU-, and d) No-events. The respective standard errors are represented by the dashed lines. We visualised trajectories from up to 7 days prior to an outcome event or discharge - the black lines represent the time window (72 - 24 hours prior to an event or discharge) considered for input to all models.
+
+![](images/6857f5028848658d1c77a80478710fe7dde4adbca510a0e3cec145e48bade1a8.jpg)
+
+A.6: Plot of mean serum trajectories in solid line as given by the 4 outcome groups: admissions with a) Cardiac, b) Death- , or c) ICU-, and d) No-events. The respective standard errors are represented by the dashed lines. We visualised trajectories from up to 7 days prior to an outcome event or discharge - the black lines represent the time window (72 - 24 hours prior to an event or discharge) considered for input to all models.
+
+We note that clusters learnt by both clustering benchmark models have identical outcomes, which provides no useful clinical interpretability to the cluster-defined populations, as well as likely not assisting models to learn relevant cluster representations.
+
+![](images/13ed1700a4683e286da2aa988e865f24c992993ec189f75196e62cb691a50882.jpg)
+
+A.8: Bar plots of learnt cluster phenotypes for SOM-VAE with optimal hyper-parameters. Each plot represents a cluster - its phenotype is the corresponding empirical outcome distribution in its cluster-assigned patient cohort.
+
+We go further in comparing clusters learnt by TSKM and by our proposed model. We argue clusters learnt by CAMELOT are much more relevant towards our goal. We show this through two distinct plots. Firstly, in Figure A10, we display a scatter plot of patients in each cluster (CAMELOT on the right and TSKM clusters on the left) after projection to two dimensions. Projection was completed through a principal component analysis reduction to 50 dimensions, followed by t-stochastic neighbour embedding dimensionality projection to two.
+
+Furthermore, we also demonstrated that TSKM does not learn as separable cluster trajectory evolution profiles as CAMELOT. This is shown in Figure A11, where Heart-Rate mean trajectories for each cluster (i.e., average HR observations aligned to the same time until end of observations for patients in the clusters) are displayed. It is clear that CAMELOT cluster trajectories are easier to separate.
+
+A complete description of the number of patient admissions with a given outcome per learnt cluster in our proposed model can be seen in Table A.12.
+
+<table><tr><td rowspan=1 colspan=1>Parameter</td><td rowspan=1 colspan=1>TSKM</td><td rowspan=1 colspan=1>SOM-VAE</td><td rowspan=1 colspan=1>AC-TPC</td><td rowspan=1 colspan=1>CAMELOT</td><td rowspan=1 colspan=1>SVM</td><td rowspan=1 colspan=1>XGB</td></tr><tr><td rowspan=1 colspan=1>seeds</td><td rowspan=1 colspan=6>{1001，1012,1134,2475,6138,7415,1663，7205,9253，1782}</td></tr><tr><td rowspan=1 colspan=1>α</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>P(0.1)</td><td rowspan=1 colspan=1>P(0.01)</td><td rowspan=1 colspan=1>P(0.01)</td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>β</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>P(0.1)</td><td rowspan=1 colspan=1>P(0.01)</td><td rowspan=1 colspan=1>P(0.001)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>Y</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>{0.1,0.2,0.6}</td></tr><tr><td rowspan=1 colspan=1>latent dim</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>L(64)</td><td rowspan=1 colspan=1>L(128)</td><td rowspan=1 colspan=1>L(128)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>SOM dim</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>L²(4,4)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>K</td><td rowspan=1 colspan=1>K(7)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>K(6)</td><td rowspan=1 colspan=1>K(6)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>kernel</td><td rowspan=1 colspan=1>{&#x27;DTW&#x27;,&#x27;euclidean&#x27;}</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>{&#x27;polynomial&#x27;,&#x27;rbf&#x27;}</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>C</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>P(10)</td><td rowspan=1 colspan=1>·</td></tr><tr><td rowspan=1 colspan=1>n-estimators</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>{100,200,300}</td></tr><tr><td rowspan=1 colspan=1>depth</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>{1,3,5,10}</td></tr><tr><td rowspan=1 colspan=1>min-child-weight</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>{1,2,3,5}</td></tr></table>
+
+A.7: Parameter range used for Grid-search hyper-parameter optimisation. For each model, the list of parameter values tested is indicated. In bold, the optimum set of hyper-parameters is indicated for each model.
+
+![](images/a6991e898e692311abc357a9c435ca942fbae30bc7e5d5871c2e5be19ca4c51c.jpg)
+
+A.9: Bar plots of learnt cluster phenotypes for TSKM with $K = 6$ . Each plot represents a cluster - its phenotype is the corresponding empirical outcome distribution in its cluster-assigned patient cohort.
+
+![](images/3dcd345e38b4554d4d584bab086c69a6024e86ecdd3680fd243fb3d55ef2e7f0.jpg)  
+A.10: Scatter plot of cluster patient data after projection to 2 dimensions.
+
+<table><tr><td rowspan=1 colspan=1>Outcome</td><td rowspan=1 colspan=1>Healthy</td><td rowspan=1 colspan=1>Death</td><td rowspan=1 colspan=1>ICU</td><td rowspan=1 colspan=1>Cardiac</td></tr><tr><td rowspan=1 colspan=1>Cluster 0</td><td rowspan=1 colspan=1>149</td><td rowspan=1 colspan=1>44</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>7</td></tr><tr><td rowspan=1 colspan=1>Cluster 1</td><td rowspan=1 colspan=1>739</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>Cluster 2</td><td rowspan=1 colspan=1>579</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Cluster 3</td><td rowspan=1 colspan=1>93</td><td rowspan=1 colspan=1>92</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>Cluster 4</td><td rowspan=1 colspan=1>373</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Cluster 5</td><td rowspan=1 colspan=1>288</td><td rowspan=1 colspan=1>84</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>10</td></tr></table>
+
+A.12: Table with empirical number of outcome admissions observed for each cluster learnt by the proposed model.
+
+![](images/b3fb0a392cc615f242cba3e700b96546397a7311be21c745c43047a1a0cdbf91.jpg)  
+A.11: Plot of mean Heart-Rate (HR) trajectory in solid line as given by the TSKM learnt clusters (top) and CAMELOT (bottom). The respective standard errors are represented by the dashed lines. We visualised trajectories from up to 7 days prior to an outcome event or discharge - the black lines represent the time window (72 - 24 hours prior to an event or discharge) considered for input to all models.
+
+We also computed summary statistics for the learnt CAMELOT clusters. For each of the resulting clusters, median, and quartile values were computed and plotted, except on the case of binary variables, where only the number of positive occurrences (and the corresponding proportion in the cluster) are shown.
+
+Lastly, we plot the mean cluster trajectory evolution for SBP and FIO2 to present supportive evidence for the personalised attention maps in Figure 6. These two features were selected from attention map analysis.
+
+<table><tr><td>2 3</td><td>1007-000080 17811-05.010 07/26-04550 （/51-07）0/79 6566-.155 33.55-5.1.11 39353.7599 01-0D01</td><td>.18-0.1950 (%55240981 (%908) 0010 (%8S9D0s9</td><td>1711-1015 18.51-58551 (5:1-0.9)78 (0-00)r0 (r0-0&#x27;0)00 25-5555 (51-10)0</td><td>(0/11-0.95.51 3383555577 1.15-5.1-5.830 1-811 3.20-.0.00 14-4 (71-8) 6.9</td></tr><tr><td>tssh 3</td><td>(081-0&#x27;91021 1.14-.101-110 -700 (0&#x27;SL-0&#x27;09)0&#x27;L9 (0/6-0/6) 0.96 2.11-.0 63955755599</td><td>(0.81-085000 (01-0D01 (% 66&#x27;TS) 0&#x27;961 3%72050 697966</td><td>(1-8.11 (711-8 1111 (0-10)70 (r0-0&#x27;0)r0 91-554 (66-1)7 111-5.00</td><td>1701530300 2.21-8557 (0/8-0.95).1 25537075 18-5 124-96505 （89-)11</td></tr><tr><td>0 2</td><td>13.115-3.515.101 305157585/5 107381060 (0&#x27;8L-0&#x27;09)0&#x27;89 (0,96-0,16)0.06 2.7-5.115.71</td><td>6373-535599 (01-0D0T 0&#x27;58-0.69)59 (%/9) (%96&#x27;9)08 3655551</td><td>(11-1 1715-88751 (r0-0&#x27;0)00 (r0-00)00 (5-0080 （9:41-9.9)504 (1-50 6.0</td><td>11715357315.350 23/87-5.600 1011-0 （1721-7157711 12724455 3555555 1711-71511</td></tr><tr><td>0</td><td>1.1135.7503000 006-00)00 （081-090021 7-0850.59 0/6-0.60.56 2.-..0 33933.799</td><td>(0&#x27;08-085)0&#x27;1 1-0D01 (%00.0)0.100 (% 156781 60005070</td><td>（711-16111 （11-1156 (0-00)0 (r0-0&#x27;0)00 5-6057 (16-84)6.9 17.1-6.070</td><td>171531015010 220-500.00 17.7-5/1500 18110-811114 336-5551 1214-9555 18-7471</td></tr><tr><td>E</td><td>13.753.15.711） .16-00.08 (281-0.90081 （181-029)00 0/6-0.6.6 2.-.1.1 8395553595</td><td>1718-3.59)5.11 3%7555955 %0/227000 162515050 1-0D01</td><td>1.11-11110 (6.71-8/0.0 (20-00) 10 (r0-00)00 9-800 (501-55)55 （7:1-8:00</td><td>1711537015110 2371-3.20 (&#x27;16-08)0&#x27;1 5715-1115.94 901-859 24594 38-71)85</td></tr><tr><td>so 20</td><td>1.114-7111.110 0.6-0.0.00 (061-5900&#x27;81 515-0/215159 /6-596 2.11-.1.10 33355.55791</td><td>:18-769086 (% 5069)051 3567045.00 1-0D01 %50600</td><td>(0/01-88) 6.6 6551-5155 (801-5)9 (70-00)r0 (r0-00)00 86-8087 (1-90)6.0</td><td>2:730700 117153005.000 （161-09876 3.87-3800 132-3550 17/7-501 511 (91-128</td></tr><tr><td>n 里 </td><td>snonurjuog garnga s HUu SHUI % % ℃</td><td>（)eoae </td><td>Cenurnuon</td><td>Cenunuon</td></tr><tr><td>rsrerg</td><td></td><td></td><td>T/6v0[X T/6v0[X 品 T/6v0[X T/6v0[X</td><td>品 T/ouu T/owu 1</td></tr><tr><td>einaied</td><td>errenge reeeaet sessreogooerse eeseorldeeee ereroaeear allta aaeeeea 5 0 P R R </td><td>ereeeeee rsseeieas sostperagis etaigee argegatr 2</td><td>oenllgo eeode rg gonrgooo gpgr gonroog rrregrgiiteg rgegrnrgoong pgonrgeoo geggge 8 8 8 R </td><td>reeeeeeee egege re l i are glratt rge gia eaeeeereeeereeern R AB R 0 50 B</td></tr><tr><td>is gea</td><td></td><td>ee T srns</td><td></td><td>rrleeeah</td></tr></table>
+
+A.13: Descriptive statistics and information of all input data features. Variables are displayed with type, description, units and average statistics. We separate all features according to medical literature, including vitalsign, static, serum and haematological variables. Statistics are shown for each cohort as learnt by our model.
+
+![](images/06221e576edfe0b1abc5ba347060d4ddf5caedd3233369f4cf81a8aa6a3dfd37.jpg)  
+A.14: Plot of mean Systolic Blood Pressure (SBP) trajectories in solid line as given by the clusters learnt by our model (top). In the bottom, mean FIO2 trajectories are displayed. The respective standard errors are represented by the dashed lines. We visualised trajectories from up to 7 days prior to an outcome event or discharge - the black lines represent the time window $7 2 - 2 4$ hours prior to an event or discharge) considered for input to all models.

parse/dev/kroqZZb-6s/kroqZZb-6s_content_list.json

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+    {
+        "type": "text",
+        "text": "CLUSTER-BASED FEATURE IMPORTANCE LEARNING FOR ELECTRONIC HEALTH RECORD TIME-SERIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
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+        "page_idx": 0
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+    {
+        "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": "The recent availability of Electronic Health Records (EHR) has allowed for the development of algorithms predicting inpatient risk of deterioration and trajectory evolution. However, prediction of disease progression with EHR is challenging since these data are sparse, heterogeneous, multi-dimensional, and multi-modal time-series. As such, clustering is used to identify similar groups within the patient cohort to improve prediction. Current models have shown some success in obtaining cluster representation of patient trajectories, however, they i) fail to obtain clinical interpretability for each cluster, and ii) struggle to learn meaningful cluster numbers in the context of the imbalanced distribution of disease outcomes. We propose a supervised deep learning model to cluster EHR data based on the identification of clinically understandable phenotypes with regard to both outcome prediction and patient trajectory. We introduce novel loss functions to address the problems of class imbalance and cluster collapse, and furthermore propose a feature-time attention mechanism to identify cluster-based phenotype importance across time and feature dimensions. We tested our model in over 100,000 unique trajectories from hospitalised patients with Type-II respiratory failure to predict four different outcomes. Our model yielded added interpretability to cluster formation and outperformed benchmarks by at least $5 \\%$ in mean AUROC. ",
+        "bbox": [
+            233,
+            265,
+            764,
+            517
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            544,
+            336,
+            560
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chronic conditions such as Chronic Obstructive Pulmonary Disease (COPD) and Cardiovascular Disease (CVD) describe a broad spectrum of medical ailments, and affect a significant percentage of the overall population (Adeloye et al., 2015). Such diseases are characterized by the existence of multiple distinct patient subgroups, largely distinguished by differences in pathology and in the response to different treatments and medical interventions Turner et al. (2015); Vogelmeier et al. (2018). Exacerbation of COPD, a condition of respiratory failure, can result in emergency hospital admission and mortality if it is not well treated and managed. Early identification of COPD patients’ subgroups is therefore of high medical importance and relevance. EHR time-series data are typically used to determine clinically relevant COPD inpatient subgroups, and have been applied to detect risk of deterioration (Pikoula et al., 2019). ",
+        "bbox": [
+            174,
+            575,
+            825,
+            714
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, modelling disease progression and risk prediction is challenging due to the extreme data heterogeneity nature of EHRs. Firstly, EHR data contains a mixture of demographic or static variables (i.e. time independent such as age and sex), and multi-dimensional time-series (e.g Heart Rate, HR, and laboratory measurements, such as blood tests). Secondly, EHR time-series are multi-modal as different features are collected from different devices, representing distinct clinical properties of relevance. Similarly, time-series features are sampled at different times and have low and distinct sampling rates, as well as different missing value properties. Furthermore, each feature is associated with different noise and evolution patterns. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent advances in deep learning (DL) approaches have shown promising results in EHR modelling due to their capacity to handle complex data (Rajkomar et al., 2018). Nonetheless, DL approaches lack relevant interpretability frameworks to be scaled and applied in hospital settings. Several such models have since been proposed to tackle this issue (Mayhew et al., 2018), however, most of them focus on a subset of EHR features (usually vital signs only) and fail to provide a clinically-focused phenotypic analysis of learnt patient sub-groups (via clustering). ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This work builds on previous research in literature to introduce a cluster-based feature-time attention mechanism to predict patient outcomes based on EHR data. Our method also leverages phenotypic information to aid in clinical interpretability, not only making use of demographics and vital-signs information but also of relevant laboratory measurements (all present in the EHR) to provide a more complete patient physiological status. Our contributions include the following: ",
+        "bbox": [
+            176,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• An end-to-end DL supervised model to cluster EHR patient data based on the identification of clinically understandable cluster phenotypes with regard to both outcome prediction and patient trajectory in a multi-class setting;   \n• A weighted loss to address data imbalance for both tasks of clustering and prediction, a common issue in the medical domain;   \n• The incorporation of a novel loss mechanism in the model, to address the issue of cluster collapse and promote sample assignment to all available clusters;   \n• Finally, the inclusion of a novel interpretability framework, derived from a cluster-based feature-time attention layer, aiming to identify relevant timestamps and feature variables to represent the patient physiology, cluster assignment and, ultimately, outcome prediction. ",
+        "bbox": [
+            217,
+            186,
+            825,
+            353
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This paper is structured as follows. In Section 2, we describe previous research in EHR time-series modelling, clustering and attention methods. Section 3 introduces the dataset used for analysis and provides description of the proposed model. The experimental setup and results of our analysis are presented in Section 4 and discussion takes place in Section 5. Finally, concluding remarks and future work are available in Section 6. ",
+        "bbox": [
+            174,
+            367,
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+            436
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            462,
+            344,
+            478
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "EHR data comprise complex time-series data, being high-dimensional, multi-modal and heterogeneous, and thus presenting challenges when used in machine learning models (Keogh & Kasetty (2003); Rani & Sikka (2012)). An important goal in a medical setting is to identify phenotypically separable clusters with distinct phenotypic profiles (which we denote as phenotypic clustering hereafter). For the purpose of this work, cluster phenotypes result from the combination of two distinct components: a) the evolution profile of patient trajectories’ within the cluster, and b) the characterisation of the cluster with regards to clinical variables of interest. The latter may include features not used for clustering and may provide information about the underlying or future health status. Traditional clustering models such as K-Means or hierarchical clustering have been shown to fail to capture the existing time-dependent feature relationships. As such, variants have been proposed to mitigate this problem. A temporal version of the K-Means algorithm, Time-Series K-Means (TSKM, Tavenard et al. (2020)), models the distance between time-series of different datapoints, using the Euclidean distance (which is equivalent to considering all temporal observations as an independent feature value for the corresponding patient admission), or time-series alignment strategies such as Dynamic-Time Warping (DTW, Berndt & Clifford (1994)) and soft-DTW (Cuturi & Blondel (2017)). ",
+        "bbox": [
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+        "type": "text",
+        "text": "Recent DL architectures, ranging from Auto-Encoders (AE, Ma et al. (2019)), Convolutional Neural Networks (CNN, Munir et al. (2018)) and others, have shown great promise when applied to timeseries data across a variety of domains. Fortuin et al. (2019) proposed a Self-Organising Map - Variational Auto-Encoder (SOM-VAE), is a state-of-the-art, unsupervised, DL clustering algorithm which extends a variational auto-encoder architecture (Kingma & Welling, 2013) for observation learning and representation, through the addition of a Markov model (Gagniuc, 2017), to infer temporal evolution within the latent space. Clustering is performed in the low dimensional latent space through the use of self-organising maps (Kohonen, 1982) to obtain a discrete, topologically-interpretable latent representation of the learnt clusters. In a supervised setting, AC-TPC (Lee & Van Der Schaar, 2020) serves as the current state-of-the-art for identifying phenotypically separable clusters in patient trajectories in EHR data. AC-TPC maps EHR data into a latent space via an encoder, and uses an actor-critic network (Konda & Tsitsiklis, 2000) which leverages clinical outcomes to aid in cluster formation and obtaining cluster phenotypes. Neither SOM-VAE and AC-TPC provide clinically meaningful interpretation of feature-time importance or outcome of interest. ",
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+        "type": "text",
+        "text": "Attention mechanisms have recently been proposed to provide greater interpretability to Recurrent Neural Networks (RNN) and to aid in dealing with long-term dependencies (Vaswani et al., 2017; Xu et al., 2015), and have also been used in modelling EHR time-series (Schwab et al., 2017; Shashikumar et al., 2018). RETAIN (Choi et al., 2016) proposes a two-level reverse attention mechanism to mimic physician’s decision process and predict a future diagnosis. In other recent works, attention mechanisms based on bi-directional RNN and CNN outperformed standard classification models in predicting high risk vascular diseases with the addition of medication information as input data (Kim et al., 2017). A drawback of such attention mechanisms is the focus on temporal interpretability only, and inability to look at individual features, which is key in a medical setting. To solve this issue, Shamout et al. (2019) considered independent RNN per feature, with a concatenation of the resulting latent vectors. However, the latter does not allow the joint modelling across both feature and time dimensions. Alternatively, Kaji et al. (2019); Gandin et al. (2021) proposed learning attention weights directly on the original inputs, prior to being transformed by a RNN, which does not allow modelling of the resulting latent representations. To the best of our knowledge, no existing models have been proposed that jointly leverage both feature and time dimensions (feature-time) to determine clinical observation relevance on clustered EHR data. ",
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+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 DATASET AND PRE-PROCESSING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our dataset was retrieved from a retrospective database of routinely collected observations from concluded hospital admissions between March 2014 and March 2018 (the HAVEN project, REC reference: 16/SC/0264 and Confidential Advisory Group reference 08/02/1394). The database includes EHR measurements of adult patients admitted to four hospitals from the Oxford University Hospitals NHS Foundation Trust. Note that the HAVEN dataset does not include data from Intensive Care Units (ICU), and we have excluded observations taken in the Emergency Department. Key characteristics of HAVEN cohort data include a) heterogeneity, b) multi-modality, and difference in: c) noise distributions, d) sampling rates, e) missing values, etc. Such properties are common across EHR settings, and are challenging with respect to learning useful representations and predictions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We used the protocol defined in Pimentel et al. (2019) to subset the cohort to those patients at risk of developing Type-II Respiratory Failure (T2RF) in hospital (a diagram of the data selection steps can be found in Figure A.1 in the Appendix). Four patient outcomes were considered in our analysis: i) no event during hospital stay, leading to successful discharge from the hospital, or the first instance of one of three possible events, ii) unplanned entry to ICU, iii) cardiac arrest (also named ”Cardiac” hereafter) and iv) death. Outcome groups are not clearly separable (see Tables A.2, A.3 in the Appendix), so patient clusters will naturally contain a mix of different admission outcomes. In this setting, the clinically relevant component of a cluster phenotype (henceforth denoted as cluster outcome propensity or cluster outcome) is represented as a categorical distribution indicating the corresponding propensity for cluster-assigned patients to each corresponding outcome. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For each admission, observations were grouped according to mean window observation value into 4 hour blocks based on the time to outcome (discharge in the case of no event during stay) - only observations within 24 and 72 hours before the outcome were considered. This time window was selected based on those traditionally used for validating Early Warning Score (EWS) systems (baseline models used by UK NHS staff to track inpatient physiology, (Royal College of Physicians, 2017) and clinical input, such that the target phenotype represents the patient status in the subsequent 24 hours. Features were transformed according to min-max normalization due to skewness and heterogeneity in their distributions. Patient admissions were randomly split into train, validation and test sets. Missing values were imputed based on the previously observed time block - all remaining missing observations were imputed according to the feature median from the aggregated validation and test data (see Section 4 for description of train-test data split). Imputed values were flagged in a three-dimensional mask matrix. ",
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+    {
+        "type": "text",
+        "text": "After processing, input data contained over 100,000 unique patient trajectories corresponding to 4,266 unique patient admissions (only last patients admissions were considered in our analysis). Original trajectories for the patient cohort are shown in the Appendix in Figures A.4, A.5, A.6 for different variables/features. A lack of clear outcome group separability can be observed across temporal and static variables. Furthermore, we note the high degree of imbalance in the data - admissions with no event account for over $8 6 . 8 \\%$ of the total number of admissions, while event classes correspond to $1 0 . 3 \\%$ Death, $1 . 8 \\%$ ICU and $1 . 1 \\%$ Cardiac. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.2 PROPOSED MODEL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We propose a novel model, which we denote by Cluster-bAsed iMportancE Learning fOr Timeseries (CAMELOT). Our proposed methodology is displayed in Figure $1 ^ { 1 }$ . Our model builds on previous literature on 3 key items: a) a modified loss function to target the multi-class imbalance, b) a novel loss function to ensure cluster assignment and phenotype are representative, and c) a novel feature-time attention-level framework to boost representation and introduce feature-time interpretability for cluster assignment. ",
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+    {
+        "type": "image",
+        "img_path": "images/5738edb9246a09ec91147e913fda26d231ec701bd4e53ddb8fc11d9be6192f35.jpg",
+        "image_caption": [
+            "Figure 1: Diagram of proposed model. MLP - Multilayer Perceptron neural network blocks; RNN - Recurrent Neural Network. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Let $N$ denote the number of patients and $D _ { f }$ the number of input features. Input data consists of a set of patient trajectorobservations for patient s , $\\mathbb { X } = \\{ \\{ \\mathbf { x } _ { n , t } \\} _ { t = 1 } ^ { T _ { n } } \\} _ { n = 1 } ^ { N }$ , where tcomes $T _ { n }$ mum number of temporal. Input trajectory data for $n$ $\\mathbb { Y } = \\left\\{ \\mathbf { y } _ { n } \\right\\} _ { n = 1 } ^ { N }$   \nthe $n$ -th patient is represented as $\\mathbf X _ { n } = [ \\mathbf x _ { n , 1 } , . . . , \\mathbf x _ { n , T _ { n } } ]$ , where each $\\mathbf { x } _ { n , t } \\in \\mathbb { R } ^ { D _ { f } }$ is referred to as an observation (vector), with a maximum of observed $D _ { f }$ feature values. The corresponding patient outcome is a one-hot encoded vector $\\mathbf { y } _ { n } \\in \\mathbb { R } ^ { 4 }$ (more generally, the dimension of ${ \\bf y } _ { n }$ equals to the number of possible outcomes). ",
+        "bbox": [
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+        "type": "text",
+        "text": "Our DL model can be decomposed into 3 neural network blocks: an Encoder, Identifier and Predictor. We refer the action of each network respectively as E, I and $\\mathrm { \\bf P }$ (for example, $I ( \\mathbf { x } )$ denotes the output of the Identifier given some input vector $\\mathbf { x }$ ). Both the Identifier and Predictor are Multilayer Perceptrons (MLP), networks of stacked feed-forward dense layers. On the other hand, the Encoder block can be further sub-divided into a) a stack of RNN layers and b) our proposed custom attention layer (see Section 3.3 for further details). Separately, we also consider a set of trainable cluster representation vectors, $\\mathcal { C } = \\{ \\mathbf { c } _ { 1 } , . . . , \\mathbf { c } _ { K } \\}$ . We assign the outcome for cluster $i$ as $P ( \\pmb { c } _ { i } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A model call is as follows: Given the $n$ -th patient input trajectory data $\\mathbf { X } _ { n }$ , the Encoder network returns a latent representation $\\mathbf { z } _ { n } : = E ( \\mathbf { X } _ { n } ) \\in \\mathbb { R } ^ { l }$ . Consequently, the Identifier network computes cluster assignment probabprobability assignment of ies, to $\\pi _ { n } : = I ( \\mathbf { z } _ { n } ) \\in \\mathbb { R } ^ { K }$ .  of h element of clusters. A c $\\pi _ { n }$ , e $\\boldsymbol { \\pi } _ { n } ^ { i }$ resents theis selected $\\mathbf { z } _ { n }$ $i$ $K$ $k _ { \\mathrm { s a m p } } ^ { n }$ according to categorical sampling $( k _ { \\mathrm { s a m p } } ^ { n } \\sim \\mathbf { C a t } ( \\pi _ { n } ) )$ , and the corresponding cluster representation, $\\mathbf { c } _ { \\mathrm { s a m p } } ^ { \\mathbf { n } } : = \\mathbf { c } _ { \\mathbf { k } _ { \\mathrm { s a m p } } ^ { \\mathbf { n } } }$ is then selected from $\\mathcal { C }$ . The output of the model is $y _ { \\mathrm { p r e d } } : = P ( \\mathbf { c } _ { \\mathrm { s a m p } } ^ { n } ) \\in \\mathbb { R } ^ { 4 }$ . We note that $k _ { \\mathrm { s a m p } } ^ { n }$ is only sampled during a training phase; at prediction stage, cluster selection follows the equation $k _ { \\mathrm { p r e d } } ^ { n } = \\underset { i = 1 , . . . , K } { \\arg \\operatorname* { m a x } } \\pi _ { n } ^ { i }$ . ",
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+    {
+        "type": "text",
+        "text": "3.3 ENCODER NETWORK AND A CUSTOM ATTENTION LAYER ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The diagram of our proposed Encoder network is presented in Figure 2. The Encoder contains (i) a Recurrent Neural Network (RNN) block of stacked Long Short-Term Memory (LSTM) layers, and (ii) a customised attention layer, which computes a latent representation by comparing input data with the sequence of output states from the RNN block. We use the same notation as above, and write the sequence of output states of the final LSTM layer as $\\mathbf { o } _ { n , 1 } , . . . , \\mathbf { o } _ { n , T _ { n } }$ , with ${ \\bf o } _ { n , i } \\in \\mathbb { R } ^ { l }$ . Theoretically, each ${ \\bf o } _ { n , t }$ corresponds to a representative summary of input patient information up until time $t$ . We propose to approximate ${ \\bf o } _ { n , t }$ as a linear combination of latent representations of each individual feature, thereby allowing the separation of output states into contributions from each feature. Note that it is important feature transformations be time-independent, to avoid overparametrising the model, over-fitting and ensure feature representation maps are similar across time. ",
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+        "img_path": "images/10d1f2dae6d4c161e7b4ed89343af6140dc1df3e400dc01609dabc40230f10c6.jpg",
+        "image_caption": [
+            "Figure 2: Diagram of the Encoder network composed of an LSTM Encoder and a custom attention layer. "
+        ],
+        "image_footnote": [],
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+        "text": "Our attention layer behaves as a set of $D _ { f }$ feed-forward neural network layers, $\\mathbf { U } _ { 1 } , . . . , \\mathbf { U } _ { D _ { f } }$ , jointly represented by: (i) a matrix of learnable kernel weights $D \\in \\mathbb { R } _ { l \\times D _ { f } }$ . We write $D = [ \\mathbf { D } _ { 1 } , . . . , \\mathbf { D } _ { D _ { f } } ]$ ; (ii) a matrix of learnable bias vectors $\\boldsymbol { B } \\in \\mathbb { R } _ { l \\times D _ { f } }$ . Similarly, we can write $B = [ \\mathbf { B } _ { 1 } , . . . , \\mathbf { B } _ { D _ { f } } ]$ ; and (iii) an activation function, $\\sigma$ which matches the output activation of the RNN block. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Input data for patient $n$ , $\\mathbf { X } _ { n }$ is fed as input to the RNN block, which outputs a sequence of latent output states $( { \\bf { \\bar { o } } } _ { n , t } ) _ { t = 1 } ^ { T _ { n } }$ . For $t = 1 , . . . , T _ { n }$ , we compute $D _ { f }$ feature representations in latent space as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2eb53b52877207b1efb34a30ec9354b56381e8b62cec5f3a98fc8fe995eb321b.jpg",
+        "text": "$$\nR _ { n , t } : = \\sigma ( D \\odot \\mathbf { x } _ { n , t } + B )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where ${ \\cal R } _ { n , t } = [ { \\bf R } _ { n , t } ^ { 1 } , . . . , { \\bf R } _ { n , t } ^ { D _ { f } } ]$ is our collection of feature representations, $\\sigma$ is applied element-wise and $\\mathbf { A } : = { \\cal D } \\odot \\mathbf { x } _ { n , t }$ is a matrix satisfying $A _ { i , j } = D _ { i , j } ( \\mathbf { x } _ { n , t } ) _ { j }$ . Equivalently, $\\mathbf { R } _ { n , t } ^ { i }$ is the output of a dense layer, $\\mathbf { U } _ { i }$ with kernel $\\mathbf { D } _ { i }$ , bias $\\mathbf { B } _ { i }$ , activation $\\sigma$ and input $( \\mathbf { x _ { n , t } } ) _ { i }$ . We approximate $\\mathbf { o } _ { n , t } \\approx$ $\\begin{array} { r } { \\sum _ { i = 1 } ^ { D _ { f } } \\alpha _ { t } ^ { i } { \\bf R } _ { n , t } ^ { i } \\ = \\ R _ { n , t } \\alpha _ { t } } \\end{array}$ . This approximation is minimised following a least squares criterion, which has a well-known solution, $\\widehat { \\alpha } _ { t }$ , and corresponding optimal approximation $\\widehat { \\mathbf { o } } _ { n , t }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given, $\\hat { \\mathbf { o } } _ { n , t }$ , a similar procedure is used to compute a context vector as $\\begin{array} { r } { \\mathbf { z } : = \\sum _ { t } \\beta _ { t } \\hat { \\mathbf { o } } _ { n , t } } \\end{array}$ , where weights $\\beta$ are learned to provide a more representative context vector. ",
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+    {
+        "type": "text",
+        "text": "3.4 ATTENTION MAP VISUALISATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Given cluster representation vectors, $\\mathbf { c } _ { k }$ , we can compute a cluster-wise feature-time attention visualisation map as follows. First, we normalise feature-weights $\\widehat { \\mathbf { a } } _ { t }$ according to a softmax function, $\\mathbf { s } _ { t } = \\sigma ( \\widehat { \\pmb { \\alpha } } _ { t } ) \\in \\mathbb { R } ^ { D _ { f } }$ , where $\\sigma$ is the softmax function: $\\begin{array} { r } { \\sigma ( \\mathbf { x } ) = \\frac { \\exp { | \\mathbf { x } | } } { \\| \\exp { | \\mathbf { x } | } \\| _ { 1 } } } \\end{array}$ . Secondly, we compute tsolved as before. We similarly normalise cluster-wise weights, $\\gamma _ { t } ^ { k }$ according to a least-square approximation of $\\gamma _ { n , t } ^ { k }$ tto obtain cluster temporal scores, $\\begin{array} { r } { \\dot { \\mathbf { c } } _ { k } \\approx \\sum _ { t = 1 } ^ { T _ { n } } \\widehat { \\mathbf { o } } _ { n , t } \\gamma _ { n , t } ^ { k } } \\end{array}$ $e _ { n , t } ^ { k } = \\sigma ( \\gamma _ { n , t } ^ { k } )$ , and . Finally, we can compute $K$ scoring matrices, $M _ { n } ^ { 1 } , . . . , M _ { n } ^ { K } \\in \\mathbb { R } _ { T _ { n } \\times D _ { f } }$ : \u0000 M kn \u0001 t,f $\\left( M _ { n } ^ { k } \\right) _ { t , f } = e _ { n , t } ^ { k } \\pmb { s } _ { t } ^ { f }$ . Note that: $\\begin{array} { r } { \\| M _ { n } ^ { k } \\| _ { 1 } = \\sum _ { t } e _ { n , t } ^ { k } \\sum _ { f } s _ { t } ^ { f } = \\sum _ { t } e _ { n , t } ^ { k } = 1 } \\end{array}$ . Given that Matrices $M _ { n } ^ { k }$ are normalised, they may be consequently, visualised as a normalised feature-time map for cluster assignment relevance and provide further model interpretability. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The model is optimised through consideration of three distinct loss functions. We introduce a weighted cross-entropy loss function: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/af0e5e3e122f54c28bb452bd9c37a60531518a63d527559c4b9ca38f92541485.jpg",
+        "text": "$$\nL _ { \\mathrm { p r e d } } ( y _ { \\mathrm { t r u e } } , y _ { \\mathrm { p r e d } } ) = - \\sum _ { c = 1 } ^ { C } w _ { c } y _ { \\mathrm { t r u e } } ^ { c } \\log \\left( y _ { \\mathrm { p r e d } } ^ { c } \\right) = - w _ { c ^ { \\prime } } \\log ( y _ { \\mathrm { p r e d } } ) _ { c ^ { \\prime } }\n$$",
+        "text_format": "latex",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $c ^ { \\prime }$ is the true outcome for a particular patient. We propose inversely proportional normalised weights: $\\textstyle \\sum _ { c = 1 } ^ { C } w _ { c } = 1$ and $w _ { c }$ is inversely proportional to the class distribution, i.e., $\\begin{array} { r } { w _ { c } \\propto \\frac { N } { N _ { c } } } \\end{array}$ with $N$ being the number of patients and $N _ { c }$ being the number of patients with outcome label $c$ . Class weighting penalises misclassification more heavily on less sampled classes. ",
+        "bbox": [
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+        "type": "text",
+        "text": "We also propose a novel distribution loss function, $\\operatorname { L } _ { \\mathrm { d i s t } } ( \\pi )$ . We define the average cluster probability of assignment as $\\begin{array} { r } { \\pi _ { C } : = \\frac { 1 } { N } \\sum _ { n } \\pi _ { n } } \\end{array}$ . Then we introduce ${ \\cal L } _ { \\mathrm { d i s t } } ( \\pmb { \\pi } ) = - { \\cal H } ( \\pmb { \\pi } _ { C } )$ , where $H$ denotes entropy. Note that $L _ { \\mathrm { d i s t } }$ is minimised when $\\pi _ { C }$ is uniform, ensuring all clusters are ’explored’ and have comparable number of samples. While clusters should not necessarily be assigned the same number of samples, this loss helps to overcome a concern of cluster collapse, where clusters do not separate and samples are assigned to a very small number of non-representative clusters. Finally, to separate cluster representation vectors, we define the cluster separation loss as $\\begin{array} { r } { L _ { \\mathrm { c l u s } } ( \\mathcal { C } ) = \\dot { - } \\frac { 1 } { K ( K - 1 ) } \\sum _ { i , j } \\| \\mathbf { c } _ { i } - \\mathbf { c } _ { j } \\| ^ { 2 } } \\end{array}$ ",
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+    {
+        "type": "text",
+        "text": "To optimise our model, iterative gradients are applied according to weighted combinations of the above loss functions with hyper-parameter weights $\\alpha , \\beta$ : ",
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+        "type": "text",
+        "text": "1. Outcome Predictor is updated according to $L _ { \\mathrm { p r e d } }$ ;   \n2. Encoder and Identifier are trained according to $L _ { \\mathrm { p r e d } } + \\alpha L _ { \\mathrm { d i s t } }$ ;   \n3. Cluster representation vectors are updated with regards to $L _ { \\mathrm { p r e d } } + \\beta L _ { \\mathrm { c l u s } }$ . ",
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+    {
+        "type": "text",
+        "text": "3.6 INITIALISATION ",
+        "text_level": 1,
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+        "text": "Our proposed model also follows a set of initialisation pre-training procedures. Firstly, the Encoder and Outcome Predictor are pre-trained according to a classification task $( \\tilde { \\mathbf { y } } = P ( E ( \\bar { \\mathbf { x } } ) ) )$ , with corresponding loss $L _ { \\mathrm { p r e d } }$ . Latent state representations $E ( \\mathbf { x } )$ are clustered through a K-means algorithm with $K$ clusters across the whole training set. Cluster representation vectors are initialised as given by the resulting cluster centroids, and finally the Cluster Identifier network is pre-trained to identify clusters as predicted by the K-Means algorithm with categorical cross-entropy loss. Implementation was completed in Python, with TensorFlow 2, scikit-learn and NumPy. All experiments were run with 1 Tesla v100 GPU, and 8 CPUs Intel(R) Xeon(R) Gold 6246 $@$ 3.30GHz. ",
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+        "text": "4 RESULTS ",
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+        "type": "text",
+        "text": "For Benchmark purposes, we considered TSKM as a classic clustering benchmark, and SOM-VAE and AC-TPC as state-of-the-art phenotypic clustering methods. AC-TPC considers temporal subsequences of a complete patient set of observations - for comparison purposes, we consider only the model output for the complete patient sequence. For simplicity, we present results with all input features considered. ",
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+        "text": "All models were trained on the same training set ( $60 \\%$ of the complete input data) and evaluated against the same test set (remaining $40 \\%$ ). For DL models, we further split the training set into a purely training and validation sets. All experiments with varying hyper-parameters were repeated 10 times with a fixed set of 10 distinct seeds, and results are reported according to average metric and standard deviation. A complete list of the hyper-parameters considered for each model is included in Table A7 in the Appendix. In bold, top-performing hyper-parameters are indicated. Optimal integer hyper-parameters $( K , l )$ were selected according to an “Occam’s Razor” approach - for each parameter, we assign it the highest value such that increasing this amount does not lead to a significant increase in performance according to mean AUROC and a Friedman’s hypothesis test. Neural network size parameters were kept consistent across all DL models where applicable. All other optimal hyper-parameters were selected according to highest AUROC performance conditional on the model predicting each class (e.g. not making predictions solely for no event or Death Events). ",
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+        "type": "text",
+        "text": "We evaluated clustering performance through standard clustering metrics, including Silhouette score (SIL, Rousseeuw (1987)), Davies-Bouldin Index (DBI, Davies (1979)), Variance Ratio Criterion (VRI, Calinski & Harabasz (1974)). Results for all clustering models are displayed in Table 3. ´ In Table 4, we evaluated the (multi-class) prediction performance with regards to Area-under-theReceiver-Operating-Curve (AUROC), unweighted mean F1-score, unweighted mean Recall, and Normalised Mutual Information (NMI). For purely unsupervised models (SOM-VAE and TSKM), an outcome predictive pipeline was constructed by assigning patient admissions to clusters, and consequently to the empirical outcome distribution in the corresponding cluster. The prediction task was also benchmarked against traditional classifiers for outcome prediction in EHR data, namely Support Vector Machines (SVM), XGBoost (XGB) and NEWS2 (i.e., the National Early Warning Score used in the UK hospitals). Furthermore, to evaluate other Neural Network models as benchmarks and also justify both proposed mechanisms, we furthermore considered two other benchmarks: a) ENC-PRED: a stacked LSTM Encoder, followed by a MLP network for outcome prediction, and b) ATTEP; a model equivalent to CAMELOT, except the original entropy loss is considered over the clustering dist loss. Supervised performance for all the above models is included in Table4. As noted, convergence was particularly difficult for ATTEP due to cluster collapse. ",
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+        "type": "table",
+        "img_path": "images/9fe070f0407550852ea9698500bd467a470058ebae3344612c6d5984db986bf2.jpg",
+        "table_caption": [
+            "Table 3: Clustering separability results by the different clustering methodologies given input data with all available features (static, vital-signs, serum and haematological variables). For each metric and model, the average score and standard deviation are returned. The best values for each metric are indicated in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Metric</td><td rowspan=1 colspan=1>1        TSKM</td><td rowspan=1 colspan=1>SOM-VAE</td><td rowspan=1 colspan=1>AC-TPC</td><td rowspan=1 colspan=1>CAMELOT(proposed)</td></tr><tr><td rowspan=1 colspan=1>SIL</td><td rowspan=1 colspan=1>0.35(±0.01)</td><td rowspan=1 colspan=1>0.25(±0.08)</td><td rowspan=1 colspan=1>0.04(±0.01)</td><td rowspan=1 colspan=1>0.11(±0.04)</td></tr><tr><td rowspan=1 colspan=1>DBI</td><td rowspan=1 colspan=1>1.19 (±0.08)</td><td rowspan=1 colspan=1>1.89(±0.63)</td><td rowspan=1 colspan=1>4.34(±0.80)</td><td rowspan=1 colspan=1>3.12(±0.53)</td></tr><tr><td rowspan=1 colspan=1>VRI</td><td rowspan=1 colspan=1>554.6(±2.50)</td><td rowspan=1 colspan=1>12.8(± 9.32)</td><td rowspan=1 colspan=1>66.5(± 18.7)</td><td rowspan=1 colspan=1>216.7(±6.2)</td></tr></table>",
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+        "type": "table",
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+        "table_caption": [
+            "Table 4: Outcome prediction scores across all models, displayed with an average and standard deviation of a set of 10 seeds (except NEWS2, which is deterministic). The best values for each metric are indicated in bold. For clustering algorithms, cluster outcome distributions were taken to be the empirically observed distribution in each cluster. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Metric</td><td rowspan=1 colspan=1>AUROC</td><td rowspan=1 colspan=1>F1-score</td><td rowspan=1 colspan=1>Recall</td><td rowspan=1 colspan=1>NMI</td></tr><tr><td rowspan=1 colspan=1>SVM</td><td rowspan=1 colspan=1>0.50(±0.02)</td><td rowspan=1 colspan=1>0.23 (±0.00)</td><td rowspan=1 colspan=1>0.25(±0.00)</td><td rowspan=1 colspan=1>0.01(±0.02)</td></tr><tr><td rowspan=1 colspan=1>XGB</td><td rowspan=1 colspan=1>0.65(±0.01)</td><td rowspan=1 colspan=1>0.23(±0.00)</td><td rowspan=1 colspan=1>0.22(±0.00)</td><td rowspan=1 colspan=1>0.03(±0.04)</td></tr><tr><td rowspan=1 colspan=1>NEWS2</td><td rowspan=1 colspan=1>0.61</td><td rowspan=1 colspan=1>0.29</td><td rowspan=1 colspan=1>0.34</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td rowspan=1 colspan=1>TSKM</td><td rowspan=1 colspan=1>0.55(±0.01)</td><td rowspan=1 colspan=1>0.24(±0.03)</td><td rowspan=1 colspan=1>0.26(±0.02)</td><td rowspan=1 colspan=1>0.01(±0.03)</td></tr><tr><td rowspan=1 colspan=1>SOM-VAE</td><td rowspan=1 colspan=1>0.61(±0.09)</td><td rowspan=1 colspan=1>0.27(±0.05)</td><td rowspan=1 colspan=1>0.27(±0.03)</td><td rowspan=1 colspan=1>0.05(±0.03)</td></tr><tr><td rowspan=1 colspan=1>AC-TPC</td><td rowspan=1 colspan=1>0.68(±0.01)</td><td rowspan=1 colspan=1>0.38(±0.01)</td><td rowspan=1 colspan=1>0.36(±0.01)</td><td rowspan=1 colspan=1>0.17(±0.02)</td></tr><tr><td rowspan=1 colspan=1>ENC-PRED</td><td rowspan=1 colspan=1>0.57(±0.02)</td><td rowspan=1 colspan=1>0.25(±0.02)</td><td rowspan=1 colspan=1>0.26(±0.02)</td><td rowspan=1 colspan=1>0.06(±0.03)</td></tr><tr><td rowspan=1 colspan=1>ATTEP</td><td rowspan=1 colspan=1>0.67(±0.02)</td><td rowspan=1 colspan=1>0.36(±0.02)</td><td rowspan=1 colspan=1>0.36(±0.02)</td><td rowspan=1 colspan=1>0.16(± 0.03)</td></tr><tr><td rowspan=1 colspan=1>CAMELOT(proposed)</td><td rowspan=1 colspan=1>0.73(±0.02)</td><td rowspan=1 colspan=1>0.36(±0.01)</td><td rowspan=1 colspan=1>0.38(±0.02)</td><td rowspan=1 colspan=1>0.20(±0.03)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "On top of performance evaluation with regards to clustering separability and outcome prediction, we display a comparison between the learnt cluster phenotypes of the proposed model and that of phenotypic clustering benchmark AC-TPC. For each cluster, the corresponding outcome propensity $P ( \\mathbf { c } )$ is shown as a bar plot over the 4 possible outcomes with corresponding probability value. We also display cluster outcome propensity plots for both TSKM and SOM-VAE in the Appendix (Figures A.8 and A.9). We note that the cluster outcome propensity distributions learnt by CAMELOT also align with the empirical outcome relevance in the learnt clusters (Table A.12). Lastly, we also display feature-time cluster relevance attention maps in Figure 6. For each cluster, a patient was randomly selected from the set of patients in the corresponding cluster, and a corresponding featuretime attention matrix and visualised as a heatmap. ",
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+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our proposed model shows an improvement in clustering performance ( see Table 3) when compared to the current phenotypic clustering benchmark (AC-TPC), and outperforms SOM-VAE according to VRI. Although the cluster separability metrics are superior in the case of TSKM, this is expected given a metric bias towards convex clusters, and the convexity of the K-Means based algorithm. In particular, DL clustering occurs in a latent space, which, unfortunately, is not easily comparable with an algorithm targetting the input space (such as K-Means). Furthermore, we argue clusters learnt by TSKM are less relevant that our model’s as a) TSKM clusters are extremely hard to distinguish with regards to outcome propensity (as evidenced by very low performance on a prediction task (Table 4)), and b) TSKM clusters are less separable with regards to trajectory evolution, as there is less separation of mean HR trajectories, and less cluster separation when data is projected to a two-dimensional domain with t-stochastic neighbour embedding (tSNE) - both figures are in the Appendix, Figures A10 and A11. ",
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+        "image_caption": [
+            "Figure 5: Comparison of bar plots of cluster outcome propensity distributions for the proposed model and benchmark AC-TPC. On the left (blue), distributions are displayed for each cluster (out of a total of 6), and each phenotype corresponds to the probability of an outcome. Similar results are shown on the right (yellow) for AC-TPC. The title of each sub-plot indicates the cluster considered, as well as the number of patients assigned to a given cluster. "
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+        "image_caption": [
+            "Figure 6: Feature-Time Cluster Relevance Map. Each Heatmap represents a feature-time relevance matrix for a random patient assigned to a given cluster. Vertical Axis indicates time to outcome, in hours, while horizontal axis indicates different input features. "
+        ],
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+    {
+        "type": "text",
+        "text": "With regards to predictive power, it can be seen in Table 4 that our model outperforms both standard classifiers (at least $8 \\%$ ) and benchmarking clustering methods (at least $5 \\%$ ) according to mean AUROC. A similar increase can be seen in other classification metrics, with the exception of F1-score, where model performance is slightly below to that of AC-TPC. Our model is able to more accurately determine patterns in the data than the previously proposed models, as EHR data is extremely complex and heterogeneous. It is particularly promising that the model obtains good predictive task results despite a clustering bottleneck (i.e. sample predicted outcomes are done through the assigned cluster, as opposed to tailored to the precise input data). While it is possible that other models could show better performance on the direct task of outcome prediction given EHR input, such models can potentially be associated with lack of robustness or input sensitivity difficulties. Furthermore, it is likely they would struggle with identifying relevant trends and properties of clinical interest. As such, for new admissions, such models could provide a prediction for the overall outcome, but no robust understanding of how this outcome will occur, and how to prevent potential risks of deterioration, let alone the ability to pool data from other similar patients. ",
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+    {
+        "type": "text",
+        "text": "Figure 5 shows the advantage of two key aspects of our methodology. Our model identifies clear, separable cluster outcome distribution and provides a useful layer of interpretability to clinicians to understand a potential risk of deterioration. Our model also identifies a more diverse set of cluster outcomes than AC-TPC, which only picks up 3 different cluster outcome distributions, and doesn’t identify the presence of the ”ICU” and ”Cardiac” classes. We also show other cluster-phenotype benchmark results in Appendix A.8 and A.9. With regards to clusters learnt by CAMELOT, clusters 0 and 3 are the clusters with most ill cohort - they are largely representative of death and cardiac events on the subsequent 24 hours. On the other hand, while cluster 2, 5 are healthier, with a smaller chance of adverse events. Clusters 1 and 4 are largely “healthy” cluster, with reduced risks of the most intense adverse events. We note cluster outcome propensity distributions learnt by AC-TPC are unable to provide this level of detailed information. Furthermore, the propensity distribution learnt by our model matches with the empirical number outcome events observed in each cluster (displayed in Appendix Table A12). Note, furthermore, that the model managed to successfully navigate a heavy class-imbalance setting. Representative clusters are able to capture different-sized sub-populations, yet still identify potential risks of deterioration. ",
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+    {
+        "type": "text",
+        "text": "On the other hand, learnt cluster attention maps introduced in Figure 6 introduce yet another layer of interpretability to our proposed clustering model. The personalised attention maps highlight the relevant feature-time pairs driving patient cluster assignment, and can be used to identify the most important clinical variables. For instance, analysing Figure 5 suggests clusters with highest propensity for either of Death or Cardiac Arrest events are Cluster 0, 3 (and 2 to a slightly smaller extent). This reflects in the resulting attention maps, where SBP and FIO2 are highlighted as key clinical variables for cluster assignment. This conclusion is further corroborated when considering descriptive statistics of CAMELOT clusters (Table A13), as SBP and FIO2 are some of the few variables with some significant separation across clusters, and when considering trajectory evolutions (Figure A14). Lastly, note that feature-time weights are also relevant if potential deterioration events did not take place - so that we are more confident on a patient’s health status. As an example, attention maps for clusters 1 and 4 (reasonably healthy clusters) indicate $\\mathrm { F I O _ { 2 } }$ as very relevant towards the latter stages of the admission - this is likely due to these patients not showing an increase in oxygen intake (as they did not need it). Thus, attention maps can be very versatile. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this work, we propose a novel deep learning model for the task of identification of phenotypically separable clusters applied to EHR data for. As part of our proposed model, we propose 2 distinct loss functions and introduce a novel feature-time attention layer to better represent patient data and to introduce a feature-time relevance map for each cluster. Our experiments show promising results with the addition of both methodological tools above, on both cluster separability and outcome prediction performance. The addition of the feature-time layer has the added benefit of introducing key interpretability tools for researchers to understand relevant regions for good patient physiology representation as well as an indication of what can lead to patient deterioration. ",
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+    {
+        "type": "text",
+        "text": "There are multiple interesting avenues of investigation building on this work. On the other hand, the current attention layer mechanisms could potentially be improved with the addition of temporal weight smoothness, or, alternatively, weight regularization to encourage exploration of the complete feature-time space. Alternatively, cluster selection through a neural network mechanism introduces high capacity at the potential cost of robustness and cluster collapse. Potentially, more traditional methods incorporated into a similarly complex pipeline can achieve better performing through a clearer identification of cluster regions in latent space. Furthermore, methodological improvements will also benefit from a more extensive testing across other diverse datasets and other potential areas of application. ",
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+    {
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            195
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 DATA ",
+        "text_level": 1,
+        "bbox": [
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+            212,
+            256,
+            227
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A description of the complete pipeline of data re-processing, following the protocol defined in Pimentel et al. (2019), is shown in Figure A.1. ",
+        "bbox": [
+            176,
+            241,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/e4d91bfcdb064c7d529fad3995ff440c293e1ec8dcc0985d03521510a371044a.jpg",
+        "image_caption": [
+            "A.1: HAVEN processing "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A total of 26 input features were considered. Firstly, 4-hourly vital-sign sets which included 8 features: Heart Rate (HR), Respiratory Rate (RR), Systolic Blood Pressure (SBP), Diastolic Blood Pressure (DBP), peripheral Oxygen Saturation $\\mathrm { ( S p O _ { 2 } ) }$ ), Temperature (TEMP), level of consciousness via the AVPU scale - Alert, Verbal, Pain, Unresponsive - and estimated Fraction of Inspired Oxygen $\\mathrm { ( F i O _ { 2 } }$ , available when an oxygen mask is applied to the patient). Each set consisted of a timestamp and the vital-sign numerical values. Secondly, 4 demographic variables were selected (modelled as static variables): age, sex, and admission type (elective or surgical). Thirdly, we included 6 features resulting from biochemistry blood tests, denoted as ’Serum’: Serum levels of urea, albumin, creatinine, sodium, potassium and C-reactive protein. Finally, 8 haematological blood test features were also included: white and haemaglobin cell counts, concentration of eosinophils, basophils, neutrophils, and lymphocytes, as well as eosinophil-to-basophil and neutrophil-to-lymphocyte ratios. These features were selected based on domain knowledge of features related to severity in the prognosis and outcome of T2RF inpatients. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Descriptive statistics for all input variables is described in Table A.2. Median and inter-quartile range (IQR) is displayed for continuous and categorical variables, while binary variables are shown according to number of counts in the dataset and corresponding cohort proportion. Statistics are displayed for the complete data (”All”), but also for each sub-cohort defined by the overall outcome. We can observe that these sub-cohorts are not clearly separable and are hard to identify solely from this information. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A summary of the patient cohort in relation to outcomes and target phenotypes can be seen in Table A.3. Challenges with regards to obtaining phenotypically separable clusters can similarly be observed - there is no clear significant difference between the target outcome sub-cohorts with regards to demographic input variables. With regards to outcome distribution, we also note the high degree of imbalance in the dataset - the large majority of the patients in our dataset suffered from no adverse events (over $86 \\%$ ), while only 48 had a Cardiac event, and 76 were re-directed to the ICU. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The lack of outcome sub-cohort separability can further observed in a temporal domain. Figures A.4, A.5, A.6 plot the mean trajectories of different temporal variables sets for each outcome subcohort, respectively, according to vital signs, haematological and serum features. Mean is calculated based on the time to outcome, and missing observations are disregarded and ignored. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/1d1d99e397e35021064c39e6b47e96059e8e339499744548c8e13e6c9daea4bc.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>eeh 67901 1013 92796</td><td>.95.90/9 26-70918 81-91)81 111-1111 9-09) 9 （- 6-56) 56</td><td>% 850850 (% 10&#x27;80)0181 1% 00 91) 19</td><td>31155151 1100-00.0000 7781-4536 0-00 （900-0:000 385-00.0 009 711-75111 115</td><td>353158185 11:5550000 （21.01- 33.01/00 11.010-2.0000.00 2105350 WUo &quot; UC  Ub</td></tr><tr><td>eaa 4 9941 15484</td><td>399.95 099 (001- L 68 (1-81） 61 151000010 (L-85）99 96-166 24-181 IDI</td><td>88-418 (%10.9) (% 89) (% 88D）8</td><td>17 00- 00.0 7855581 076109 (110-00&#x27; E00 (50-0-20:) 00 100-000000 773153935 05555</td><td>15100000 27.07-00.0- 00.0 3.055505.09000 8555555955555 11.1-1150 W &quot;U y U U</td></tr><tr><td>5 % 6 2</td><td>2/90-.90.99 （66- 0 0588 （11-11 71 2112-5555 IDI 61-91)81 6-) 56 （-</td><td>%680 (08- 19) 69 (% 005) 8 (%0D)8</td><td>11.00-050030 1.1-6.90 5511511552 22133555 (010-00&#x27;)0 (50-0-20:) 200 1-55 1-55</td><td>1.2525222535 1.001500000 （27- 000 215001 15-155 </td></tr><tr><td>caprea 48 3 </td><td>39.90.0559 171-10 505 （001-88 811-0015 (L-LS）59 6-06) 56 1-11</td><td>IDI (Z8-69) 9L 3%2388 （%140 (%057D9</td><td>1101-555550 818856 01-51 (00-10 900 90/0-0:0200 8-55 905-9888 1-000</td><td>11.07500.0.0 22.0- 0000 11051515815 5515225555 23.5-3000 </td></tr><tr><td>916011 6119821 A 994</td><td>3833505905.99 2820 111-11 914 76-20 (61-91)81 (9L-09)L9 (L6 - 56) S6 （）1</td><td>%692）11 (% 09 7111 (18-29) % 64)111</td><td>1.11-061111 11111-3553355 (20-00 010 (000-10000 （5-5 05 201-555090 (891-) 911 911-999</td><td>1.055-15.0000000） 3000-00.00 00.00 (00.601 - 0085 00./ 871501125559 （40-09.110 4 44-44 4444</td></tr><tr><td>eu M eu au</td><td>snontmuo</td><td>eoogee egu reug</td><td>Ceonunuog</td><td>Ceonunuog</td></tr><tr><td></td><td>（a)spag ag gggiae Hum 1 8 % ℃</td><td>3</td><td>品 /6v0x /6v0x 7/6v0[x 7/6v0x 7/6v0[x</td><td>Toun D Tou Toun 8 7</td></tr><tr><td>rrerago Trgrongae o ed Bereeeeg</td><td>eneregeeengor eepggepeamg seereprggotsse eerrprrgr tgteer areoaeaea Tare H</td><td>ogsosn eee sspeges sossapereeg gee eret egge</td><td>geg rette opgggoog Bpggrooog rgoegg rgnonngn go mggeaag</td><td>erereeeenee egret rereeer re ger se eer T trttsstttra</td></tr><tr><td>N o n</td><td>ZOdS JII ndAA R R 8 </td><td>repueg a eotgms 品</td><td>5 5 8 3 </td><td>A 5R R 8 </td></tr><tr><td></td><td>sus</td><td>ttt umes</td><td>oeege</td><td></td></tr></table>",
+        "bbox": [
+            330,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2: Descriptive statistics and information of all input data features. Variables are displayed with type, description, units and average statistics. We separate all features according to medical literature, including vital-sign, static, serum and haematological variables, and we also display statistics per outcome sub-cohort, defined as a cohort with those patients assigned to a given outcome. ",
+        "bbox": [
+            173,
+            824,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/39a469bfaaa9e7fea1e3879d639423cb8d4044f7a96eeadcdb646fcaefe1a9b0.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>No Event</td><td rowspan=1 colspan=1>Death</td><td rowspan=1 colspan=1>ICU</td><td rowspan=1 colspan=1>Cardiac</td></tr><tr><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>3701</td><td rowspan=1 colspan=1>441</td><td rowspan=1 colspan=1>76</td><td rowspan=1 colspan=1>48</td></tr><tr><td rowspan=1 colspan=1>Age (IQR)</td><td rowspan=1 colspan=1>71(61-80)</td><td rowspan=1 colspan=1>81(74-88)</td><td rowspan=1 colspan=1>69 (61-74)</td><td rowspan=1 colspan=1>76 (69-82)</td></tr><tr><td rowspan=1 colspan=1>Gender,M</td><td rowspan=1 colspan=1>1810 (48.9%)</td><td rowspan=1 colspan=1>247 (56.0%)</td><td rowspan=1 colspan=1>38 (50.0%)</td><td rowspan=1 colspan=1>28 (58.33%)</td></tr><tr><td rowspan=1 colspan=1>CCI (IQR)</td><td rowspan=1 colspan=1>4(3-13)</td><td rowspan=1 colspan=1>14(4- 21)</td><td rowspan=1 colspan=1>7(4- 17)</td><td rowspan=1 colspan=1>15(4- 23)</td></tr><tr><td rowspan=1 colspan=1>Elective</td><td rowspan=1 colspan=1>1126 (30.4%)</td><td rowspan=1 colspan=1>3 (0.7%)</td><td rowspan=1 colspan=1>8 (10.5%)</td><td rowspan=1 colspan=1>2(4.2%)</td></tr><tr><td rowspan=1 colspan=1>Surgical</td><td rowspan=1 colspan=1>1054 (28.5%)</td><td rowspan=1 colspan=1>48 (10.1%)</td><td rowspan=1 colspan=1>22 (29.0%)</td><td rowspan=1 colspan=1>6 (12.5%)</td></tr></table>",
+        "bbox": [
+            199,
+            101,
+            799,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3: Descriptive demographic variable information for each outcome sub-cohort. ",
+        "bbox": [
+            258,
+            194,
+            736,
+            208
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/74381f1f28f3fbe43252cc669d8791edb9470703e6404d9eaf70b1bfb815d80e.jpg",
+        "image_caption": [
+            "A.4: Plot of mean vital-sign trajectories (median with respect to $\\mathrm { S p O _ { 2 } }$ ) in solid line as given by the 4 outcome groups: admissions with a) Cardiac-, b) Death- , or c) ICU-, and d) No-events. The respective standard errors are represented by the dashed lines. We visualised trajectories from up to 7 days prior to an outcome event or discharge - the black lines represent the time window (72 - 24 hours prior to an event or discharge) considered for input to all models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            264,
+            250,
+            725,
+            609
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.2 MODEL TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            738,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A list indicating the grid-search range of hyper-parameters considered in our experiments are indicated in Table A.7. For simplicity, we define $\\mathbb { P } : = \\{ 0 . 0 0 1 , 0 . 0 1 , 0 . 1 , 1 , 1 0 \\}$ , $\\mathbb { L } : = \\overline { { \\{ 3 2 , 6 4 , 1 2 8 , 2 5 6 \\} } }$ and $\\mathbb { K } : = \\{ 3 , . . . , 2 0 \\}$ . In bold, top-performing hyper-parameters according to target metrics defined in Section 4 are highlighted. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3 RESULTS COMPARISON ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            840,
+            380,
+            854
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Figures A.8 and Figures A.9 we display cluster outcome propensity distributions for some of our experiments with benchmark clustering models SOM-VAE and TSKM, respectively. Both models do not naturally associate clusters with a distribution - we estimate the cluster outcome as the empirical outcome distribution for the patient cohort assigned to the corresponding cluster. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/7cbbf11689c46b245b146e330e6341b475e20e0bfd241a6057f254c1cb9c118a.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            261,
+            118,
+            723,
+            371
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.5: Plot of mean haematological trajectories in solid line as given by the 4 outcome groups: admissions with a) Cardiac-, b) Death- , or c) ICU-, and d) No-events. The respective standard errors are represented by the dashed lines. We visualised trajectories from up to 7 days prior to an outcome event or discharge - the black lines represent the time window (72 - 24 hours prior to an event or discharge) considered for input to all models. ",
+        "bbox": [
+            173,
+            407,
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+            459
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/6857f5028848658d1c77a80478710fe7dde4adbca510a0e3cec145e48bade1a8.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            272,
+            506,
+            725,
+            752
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.6: Plot of mean serum trajectories in solid line as given by the 4 outcome groups: admissions with a) Cardiac, b) Death- , or c) ICU-, and d) No-events. The respective standard errors are represented by the dashed lines. We visualised trajectories from up to 7 days prior to an outcome event or discharge - the black lines represent the time window (72 - 24 hours prior to an event or discharge) considered for input to all models. ",
+        "bbox": [
+            174,
+            791,
+            825,
+            843
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We note that clusters learnt by both clustering benchmark models have identical outcomes, which provides no useful clinical interpretability to the cluster-defined populations, as well as likely not assisting models to learn relevant cluster representations. ",
+        "bbox": [
+            176,
+            882,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/13ed1700a4683e286da2aa988e865f24c992993ec189f75196e62cb691a50882.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            227,
+            122,
+            686,
+            321
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.8: Bar plots of learnt cluster phenotypes for SOM-VAE with optimal hyper-parameters. Each plot represents a cluster - its phenotype is the corresponding empirical outcome distribution in its cluster-assigned patient cohort. ",
+        "bbox": [
+            173,
+            393,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We go further in comparing clusters learnt by TSKM and by our proposed model. We argue clusters learnt by CAMELOT are much more relevant towards our goal. We show this through two distinct plots. Firstly, in Figure A10, we display a scatter plot of patients in each cluster (CAMELOT on the right and TSKM clusters on the left) after projection to two dimensions. Projection was completed through a principal component analysis reduction to 50 dimensions, followed by t-stochastic neighbour embedding dimensionality projection to two. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we also demonstrated that TSKM does not learn as separable cluster trajectory evolution profiles as CAMELOT. This is shown in Figure A11, where Heart-Rate mean trajectories for each cluster (i.e., average HR observations aligned to the same time until end of observations for patients in the clusters) are displayed. It is clear that CAMELOT cluster trajectories are easier to separate. ",
+        "bbox": [
+            174,
+            588,
+            825,
+            666
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A complete description of the number of patient admissions with a given outcome per learnt cluster in our proposed model can be seen in Table A.12. ",
+        "bbox": [
+            173,
+            671,
+            823,
+            700
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/4bf31b362c41494646ca16a1ef206b8db6c352eadf7be16a0f68de1053890a0b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Parameter</td><td rowspan=1 colspan=1>TSKM</td><td rowspan=1 colspan=1>SOM-VAE</td><td rowspan=1 colspan=1>AC-TPC</td><td rowspan=1 colspan=1>CAMELOT</td><td rowspan=1 colspan=1>SVM</td><td rowspan=1 colspan=1>XGB</td></tr><tr><td rowspan=1 colspan=1>seeds</td><td rowspan=1 colspan=6>{1001，1012,1134,2475,6138,7415,1663，7205,9253，1782}</td></tr><tr><td rowspan=1 colspan=1>α</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>P(0.1)</td><td rowspan=1 colspan=1>P(0.01)</td><td rowspan=1 colspan=1>P(0.01)</td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>β</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>P(0.1)</td><td rowspan=1 colspan=1>P(0.01)</td><td rowspan=1 colspan=1>P(0.001)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>Y</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>=</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>{0.1,0.2,0.6}</td></tr><tr><td rowspan=1 colspan=1>latent dim</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>L(64)</td><td rowspan=1 colspan=1>L(128)</td><td rowspan=1 colspan=1>L(128)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>SOM dim</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>L²(4,4)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>K</td><td rowspan=1 colspan=1>K(7)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>K(6)</td><td rowspan=1 colspan=1>K(6)</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>kernel</td><td rowspan=1 colspan=1>{&#x27;DTW&#x27;,&#x27;euclidean&#x27;}</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>{&#x27;polynomial&#x27;,&#x27;rbf&#x27;}</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>C</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>P(10)</td><td rowspan=1 colspan=1>·</td></tr><tr><td rowspan=1 colspan=1>n-estimators</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>{100,200,300}</td></tr><tr><td rowspan=1 colspan=1>depth</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>{1,3,5,10}</td></tr><tr><td rowspan=1 colspan=1>min-child-weight</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>{1,2,3,5}</td></tr></table>",
+        "bbox": [
+            174,
+            752,
+            823,
+            882
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.7: Parameter range used for Grid-search hyper-parameter optimisation. For each model, the list of parameter values tested is indicated. In bold, the optimum set of hyper-parameters is indicated for each model. ",
+        "bbox": [
+            169,
+            893,
+            825,
+            921
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/a6991e898e692311abc357a9c435ca942fbae30bc7e5d5871c2e5be19ca4c51c.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            246,
+            137,
+            730,
+            338
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A.9: Bar plots of learnt cluster phenotypes for TSKM with $K = 6$ . Each plot represents a cluster - its phenotype is the corresponding empirical outcome distribution in its cluster-assigned patient cohort. ",
+        "bbox": [
+            171,
+            393,
+            825,
+            421
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/3dcd345e38b4554d4d584bab086c69a6024e86ecdd3680fd243fb3d55ef2e7f0.jpg",
+        "image_caption": [
+            "A.10: Scatter plot of cluster patient data after projection to 2 dimensions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            258,
+            479,
+            733,
+            684
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/1568fffe443b8e4880208e9d61492c2ae751eeaef8a6bb9fa982800d76a56b54.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Outcome</td><td rowspan=1 colspan=1>Healthy</td><td rowspan=1 colspan=1>Death</td><td rowspan=1 colspan=1>ICU</td><td rowspan=1 colspan=1>Cardiac</td></tr><tr><td rowspan=1 colspan=1>Cluster 0</td><td rowspan=1 colspan=1>149</td><td rowspan=1 colspan=1>44</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>7</td></tr><tr><td rowspan=1 colspan=1>Cluster 1</td><td rowspan=1 colspan=1>739</td><td rowspan=1 colspan=1>28</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>Cluster 2</td><td rowspan=1 colspan=1>579</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Cluster 3</td><td rowspan=1 colspan=1>93</td><td rowspan=1 colspan=1>92</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>Cluster 4</td><td rowspan=1 colspan=1>373</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Cluster 5</td><td rowspan=1 colspan=1>288</td><td rowspan=1 colspan=1>84</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>10</td></tr></table>",
+        "bbox": [
+            194,
+            773,
+            803,
+            882
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A.12: Table with empirical number of outcome admissions observed for each cluster learnt by the proposed model. ",
+        "bbox": [
+            171,
+            893,
+            825,
+            920
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/b3fb0a392cc615f242cba3e700b96546397a7311be21c745c43047a1a0cdbf91.jpg",
+        "image_caption": [
+            "A.11: Plot of mean Heart-Rate (HR) trajectory in solid line as given by the TSKM learnt clusters (top) and CAMELOT (bottom). The respective standard errors are represented by the dashed lines. We visualised trajectories from up to 7 days prior to an outcome event or discharge - the black lines represent the time window (72 - 24 hours prior to an event or discharge) considered for input to all models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
+            137,
+            761,
+            354
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We also computed summary statistics for the learnt CAMELOT clusters. For each of the resulting clusters, median, and quartile values were computed and plotted, except on the case of binary variables, where only the number of positive occurrences (and the corresponding proportion in the cluster) are shown. ",
+        "bbox": [
+            174,
+            477,
+            825,
+            539
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Lastly, we plot the mean cluster trajectory evolution for SBP and FIO2 to present supportive evidence for the personalised attention maps in Figure 6. These two features were selected from attention map analysis. ",
+        "bbox": [
+            174,
+            546,
+            825,
+            593
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/e1465e3821a0185fd3c68e7d5c97a784f22575616d59b83454a9b06a2a650a6d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>2 3</td><td>1007-000080 17811-05.010 07/26-04550 （/51-07）0/79 6566-.155 33.55-5.1.11 39353.7599 01-0D01</td><td>.18-0.1950 (%55240981 (%908) 0010 (%8S9D0s9</td><td>1711-1015 18.51-58551 (5:1-0.9)78 (0-00)r0 (r0-0&#x27;0)00 25-5555 (51-10)0</td><td>(0/11-0.95.51 3383555577 1.15-5.1-5.830 1-811 3.20-.0.00 14-4 (71-8) 6.9</td></tr><tr><td>tssh 3</td><td>(081-0&#x27;91021 1.14-.101-110 -700 (0&#x27;SL-0&#x27;09)0&#x27;L9 (0/6-0/6) 0.96 2.11-.0 63955755599</td><td>(0.81-085000 (01-0D01 (% 66&#x27;TS) 0&#x27;961 3%72050 697966</td><td>(1-8.11 (711-8 1111 (0-10)70 (r0-0&#x27;0)r0 91-554 (66-1)7 111-5.00</td><td>1701530300 2.21-8557 (0/8-0.95).1 25537075 18-5 124-96505 （89-)11</td></tr><tr><td>0 2</td><td>13.115-3.515.101 305157585/5 107381060 (0&#x27;8L-0&#x27;09)0&#x27;89 (0,96-0,16)0.06 2.7-5.115.71</td><td>6373-535599 (01-0D0T 0&#x27;58-0.69)59 (%/9) (%96&#x27;9)08 3655551</td><td>(11-1 1715-88751 (r0-0&#x27;0)00 (r0-00)00 (5-0080 （9:41-9.9)504 (1-50 6.0</td><td>11715357315.350 23/87-5.600 1011-0 （1721-7157711 12724455 3555555 1711-71511</td></tr><tr><td>0</td><td>1.1135.7503000 006-00)00 （081-090021 7-0850.59 0/6-0.60.56 2.-..0 33933.799</td><td>(0&#x27;08-085)0&#x27;1 1-0D01 (%00.0)0.100 (% 156781 60005070</td><td>（711-16111 （11-1156 (0-00)0 (r0-0&#x27;0)00 5-6057 (16-84)6.9 17.1-6.070</td><td>171531015010 220-500.00 17.7-5/1500 18110-811114 336-5551 1214-9555 18-7471</td></tr><tr><td>E</td><td>13.753.15.711） .16-00.08 (281-0.90081 （181-029)00 0/6-0.6.6 2.-.1.1 8395553595</td><td>1718-3.59)5.11 3%7555955 %0/227000 162515050 1-0D01</td><td>1.11-11110 (6.71-8/0.0 (20-00) 10 (r0-00)00 9-800 (501-55)55 （7:1-8:00</td><td>1711537015110 2371-3.20 (&#x27;16-08)0&#x27;1 5715-1115.94 901-859 24594 38-71)85</td></tr><tr><td>so 20</td><td>1.114-7111.110 0.6-0.0.00 (061-5900&#x27;81 515-0/215159 /6-596 2.11-.1.10 33355.55791</td><td>:18-769086 (% 5069)051 3567045.00 1-0D01 %50600</td><td>(0/01-88) 6.6 6551-5155 (801-5)9 (70-00)r0 (r0-00)00 86-8087 (1-90)6.0</td><td>2:730700 117153005.000 （161-09876 3.87-3800 132-3550 17/7-501 511 (91-128</td></tr><tr><td>n 里 </td><td>snonurjuog garnga s HUu SHUI % % ℃</td><td>（)eoae </td><td>Cenurnuon</td><td>Cenunuon</td></tr><tr><td>rsrerg</td><td></td><td></td><td>T/6v0[X T/6v0[X 品 T/6v0[X T/6v0[X</td><td>品 T/ouu T/owu 1</td></tr><tr><td>einaied</td><td>errenge reeeaet sessreogooerse eeseorldeeee ereroaeear allta aaeeeea 5 0 P R R </td><td>ereeeeee rsseeieas sostperagis etaigee argegatr 2</td><td>oenllgo eeode rg gonrgooo gpgr gonroog rrregrgiiteg rgegrnrgoong pgonrgeoo geggge 8 8 8 R </td><td>reeeeeeee egege re l i are glratt rge gia eaeeeereeeereeern R AB R 0 50 B</td></tr><tr><td>is gea</td><td></td><td>ee T srns</td><td></td><td>rrleeeah</td></tr></table>",
+        "bbox": [
+            250,
+            122,
+            745,
+            839
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "A.13: Descriptive statistics and information of all input data features. Variables are displayed with type, description, units and average statistics. We separate all features according to medical literature, including vitalsign, static, serum and haematological variables. Statistics are shown for each cohort as learnt by our model. ",
+        "bbox": [
+            173,
+            853,
+            823,
+            892
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/06221e576edfe0b1abc5ba347060d4ddf5caedd3233369f4cf81a8aa6a3dfd37.jpg",
+        "image_caption": [
+            "A.14: Plot of mean Systolic Blood Pressure (SBP) trajectories in solid line as given by the clusters learnt by our model (top). In the bottom, mean FIO2 trajectories are displayed. The respective standard errors are represented by the dashed lines. We visualised trajectories from up to 7 days prior to an outcome event or discharge - the black lines represent the time window $7 2 - 2 4$ hours prior to an event or discharge) considered for input to all models. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            305,
+            174,
+            700,
+            773
+        ],
+        "page_idx": 19
+    }
+]

parse/dev/lL3lnMbR4WU/lL3lnMbR4WU.md

@@ -0,0 +1,449 @@
+# OPEN-VOCABULARY OBJECT DETECTION VIA VISION AND LANGUAGE KNOWLEDGE DISTILLATION
+
+Xiuye $\mathbf { G u } ^ { 1 }$ , Tsung-Yi Lin2, Weicheng ${ \bf K u o } ^ { 1 }$ , Yin Cui1
+
+1Google Research, 2Nvidia∗ {xiuyegu, weicheng, yincui}@google.com tsungyil@nvidia.com
+
+# ABSTRACT
+
+We aim at advancing open-vocabulary object detection, which detects objects described by arbitrary text inputs. The fundamental challenge is the availability of training data. It is costly to further scale up the number of classes contained in existing object detection datasets. To overcome this challenge, we propose ViLD, a training method via Vision and Language knowledge Distillation. Our method distills the knowledge from a pretrained open-vocabulary image classification model (teacher) into a two-stage detector (student). Specifically, we use the teacher model to encode category texts and image regions of object proposals. Then we train a student detector, whose region embeddings of detected boxes are aligned with the text and image embeddings inferred by the teacher. We benchmark on LVIS by holding out all rare categories as novel categories that are not seen during training. ViLD obtains 16.1 mask $\mathsf { A P } _ { r }$ with a ResNet-50 backbone, even outperforming the supervised counterpart by 3.8. When trained with a stronger teacher model ALIGN, ViLD achieves 26.3 $\mathsf { A P } _ { r }$ . The model can directly transfer to other datasets without finetuning, achieving $7 2 . 2 \mathrm { \ A P 5 0 }$ on PASCAL VOC, 36.6 AP on COCO and 11.8 AP on Objects365. On COCO, ViLD outperforms the previous state-of-theart (Zareian et al., 2021) by 4.8 on novel AP and 11.4 on overall AP. Code and demo are open-sourced at https://github.com/tensorflow/tpu/ tree/master/models/official/detection/projects/vild.
+
+# 1 INTRODUCTION
+
+Consider Fig. 1, can we design object detectors beyond recognizing only base categories (e.g., toy) present in training labels and expand the vocabulary to detect novel categories (e.g., toy elephant)? In this paper, we aim to train an open-vocabulary object detector that detects objects in any novel categories described by text inputs, using only detection annotations in base categories.
+
+Existing object detection algorithms often learn to detect only the categories present in detection datasets. A common approach to increase the detection vocabulary is by collecting images with more labeled categories. The research community has recently collected new object detection datasets with large vocabularies (Gupta et al., 2019; Kuznetsova et al., 2020). LVIS (Gupta et al., 2019) is a milestone of these efforts by building a dataset with 1,203 categories. With such a rich vocabulary, it becomes quite challenging to collect enough training examples for all categories. By Zipf’s law, object categories naturally follow a long-tailed distribution. To find sufficient training examples for rare categories, significantly more data is needed (Gupta et al., 2019), which makes it expensive to scale up detection vocabularies.
+
+On the other hand, paired image-text data are abundant on the Internet. Recently, Radford et al. (2021) train a joint vision and language model using 400 million image-text pairs and demonstrate impressive results on directly transferring to over 30 datasets. The pretrained text encoder is the key to the zero-shot transfer ability to arbitrary text categories. Despite the great success on learning image-level representations, learning object-level representations for open-vocabulary detection is still challenging. In this work, we consider borrowing the knowledge from a pretrained openvocabulary classification model to enable open-vocabulary detection.
+
+![](images/12a2540acaad4fac22c4a93650975728dddff1e773f1715a7bdfa0fff24e710c.jpg)  
+Figure 1: An example of our open-vocabulary detector with arbitrary texts. After training on base categories (purple), we can detect novel categories (pink) that are not present in the training data.
+
+We begin with an R-CNN (Girshick et al., 2014) style approach. We turn open-vocabulary detection into two sub-problems: 1) generalized object proposal and 2) open-vocabulary image classification. We train a region proposal model using examples from the base categories. Then we use the pretrained open-vocabulary image classification model to classify cropped object proposals, which can contain both base and novel categories. We benchmark on LVIS (Gupta et al., 2019) by holding out all rare categories as novel categories and treat others as base categories. To our surprise, the performance on the novel categories already surpasses its supervised counterpart. However, this approach is very slow for inference, because it feeds object proposals one-by-one into the classification model.
+
+To address the above issue, we propose ViLD (Vision and Language knowledge Distillation) for training two-stage open-vocabulary detectors. ViLD consists of two components: learning with text embeddings (ViLD-text) and image embeddings (ViLD-image) inferred by an open-vocabulary image classification model, e.g., CLIP. In ViLD-text, we obtain the text embeddings by feeding category names into the pretrained text encoder. Then the inferred text embeddings are used to classify detected regions. Similar approaches have been used in prior detection works (Bansal et al., 2018; Rahman et al., 2018; Zareian et al., 2021). We find text embeddings learned jointly with visual data can better encode the visual similarity between concepts, compared to text embeddings learned from a language corpus, e.g., GloVe (Pennington et al., 2014). Using CLIP text embeddings achieves 10.1 $\mathsf { A P } _ { r }$ (AP of novel categories) on LVIS, significantly outperforming the $3 . \mathbf { 0 } \ \mathrm { A P } _ { r }$ of using GloVe. In ViLD-image, we obtain the image embeddings by feeding the object proposals into the pretrained image encoder. Then we train a Mask R-CNN whose region embeddings of detected boxes are aligned with these image embeddings. In contrast to ViLD-text, ViLD-image distills knowledge from both base and novel categories since the proposal network may detect regions containing novel objects, while ViLD-text only learns from base categories. Distillation enables ViLD to be general in choosing teacher and student architectures. ViLD is also energy-efficient as it works with off-the-shelf open-vocabulary image classifiers. We experiment with the CLIP and ALIGN (Jia et al., 2021) teacher models with different architectures (ViT and EfficientNet).
+
+We show that ViLD achieves 16.1 AP for novel categories on LVIS, surpassing the supervised counterpart by 3.8. We further use ALIGN as a stronger teacher model to push the performance to 26.3 novel AP, which is close (only 3.7 worse) to the 2020 LVIS Challenge winner (Tan et al., 2020) that is fully-supervised. We directly transfer ViLD trained on LVIS to other detection datasets without finetuning, and obtain strong performance of $7 2 . 2 \mathrm { A P } _ { 5 0 }$ on PASCAL VOC, 36.6 AP on COCO and 11.8 AP on Objects365. We also outperform the previous state-of-the-art open-vocabulary detector on COCO (Zareian et al., 2021) by 4.8 novel AP and 11.4 overall AP.
+
+# 2 RELATED WORK
+
+Increasing vocabulary in visual recognition: Recognizing objects using a large vocabulary is a long-standing research problem in computer vision. One focus is zero-shot recognition, aiming at recognizing categories not present in the training set. Early works (Farhadi et al., 2009; Rohrbach et al., 2011; Jayaraman & Grauman, 2014) use visual attributes to create a binary codebook representing categories, which is used to transfer learned knowledge to unseen categories. In this direction, researchers have also explored class hierarchy, class similarity, and object parts as discriminative features to aid the knowledge transfer (Rohrbach et al., 2011; Akata et al., 2016; Zhao et al., 2017; Elhoseiny et al., 2017; Ji et al., 2018; Cacheux et al., 2019; Xie et al., 2020). Another focus is learning to align latent image-text embeddings, which allows to classify images using arbitrary texts. Frome et al. (2013) and Norouzi et al. (2014) are pioneering works that learn a visual-semantic embedding space using deep learning. Wang et al. (2018) distills information from both word embeddings and knowledge graphs. Recent work CLIP (Radford et al., 2021) and ALIGN (Jia et al., 2021) push the limit by collecting million-scale image-text pairs and then training joint image-text models using contrastive learning. These models can directly transfer to a suite of classification datasets and achieve impressive performances. While these work focus on image-level open-vocabulary recognition, we focus on detecting objects using arbitrary text inputs.
+
+![](images/4cdf33b89496acb3d10d8d249574bccf34cd94d73bb4bc63bb612f5742fc93d4.jpg)  
+Figure 2: An overview of using ViLD for open-vocabulary object detection. ViLD distills the knowledge from a pretrained open-vocabulary image classification model. First, the category text embeddings and the image embeddings of cropped object proposals are computed, using the text and image encoders in the pretrained classification model. Then, ViLD employs the text embeddings as the region classifier (ViLD-text) and minimizes the distance between the region embedding and the image embedding for each proposal (ViLD-image). During inference, text embeddings of novel categories are used to enable open-vocabulary detection.
+
+Increasing vocabulary in object detection: It’s expensive to scale up the data collection for large vocabulary object detection. Zhao et al. (2020) and Zhou et al. (2021) unify the label space from multiple datasets. Joseph et al. (2021) incrementally learn identified unknown categories. Zero-shot detection (ZSD) offers another direction. Most ZSD methods align region features to pretrained text embeddings in base categories (Bansal et al., 2018; Demirel et al., 2018; Rahman et al., 2019; Hayat et al., 2020; Zheng et al., 2020). However, there is a large performance gap to supervised counterparts. To address this issue, Zareian et al. (2021) pretrain the backbone model using image captions and finetune the pretrained model with detection datasets. In contrast, we use an image-text pretrained model as a teacher model to supervise student object detectors. All previous methods are only evaluated on tens of categories, while we are the first to evaluate on more than 1,000 categories.
+
+# 3 METHOD
+
+Notations: We divide categories in a detection dataset into the base and novel subsets, and denote them by $C _ { B }$ and $C _ { N }$ . Only annotations in $C _ { B }$ are used for training. We use $\tau ( \cdot )$ to denote the text encoder and $\mathcal { V } ( \cdot )$ to denote the image encoder in the pretrained open-vocabulary image classifier.
+
+# 3.1 LOCALIZATION FOR NOVEL CATEGORIES
+
+The first challenge for open-vocabulary detection is to localize novel objects. We modify a standard two-stage object detector, e.g., Mask R-CNN (He et al., 2017), for this purpose. We replace its classspecific localization modules, i.e., the second-stage bounding box regression and mask prediction layers, with class-agnostic modules for general object proposals. For each region of interest, these modules only predict a single bounding box and a single mask for all categories, instead of one prediction per category. The class-agnostic modules can generalize to novel objects.
+
+# 3.2 OPEN-VOCABULARY DETECTION WITH CROPPED REGIONS
+
+Once object candidates are localized, we propose to reuse a pretrained open-vocabulary image classifier to classify each region for detection.
+
+![](images/2d4f5932293dd4e577ca7d53f30d5a2d564631c7405a1f159060ec49fc0e9fc4.jpg)  
+Figure 3: Model architecture and training objectives. (a) The classification head of a vanilla two-stage detector, e.g., Mask R-CNN. (b) ViLD-text replaces the classifier with fixed text embeddings and a learnable background embedding. The projection layer is introduced to adjust the dimension of region embeddings to be compatible with the text embeddings. (c) ViLD-image distills from the precomputed image embeddings of proposals with an $\mathcal { L } _ { 1 }$ loss. (d) ViLD combines ViLD-text and ViLD-image.
+
+Image embeddings: We train a proposal network on base categories $C _ { B }$ and extract the region proposals $\tilde { r } \in \tilde { P }$ offline. We crop and resize the proposals, and feed them into the pretrained image encoder $\nu$ to compute image embeddings $\mathcal { V } ( \mathrm { c r o p } ( I , \tilde { r } ) )$ , where $I$ is the image.
+
+We ensemble the image embeddings from $1 \times$ and $1 . 5 \times$ crops, as the $1 . 5 \times$ crop provides more context cues. The ensembled embedding is then renormalized to unit norm:
+
+$$
+\mathcal { V } ( \mathrm { c r o p } ( I , \tilde { r } _ { \{ 1 \times , 1 . 5 \times \} } ) ) = \frac { \mathbf { v } } { \lVert \mathbf { v } \rVert } , \mathrm { ~ w h e r e ~ } \mathbf { v } = \mathcal { V } ( \mathrm { c r o p } ( I , \tilde { r } _ { 1 \times } ) ) + \mathcal { V } ( \mathrm { c r o p } ( I , \tilde { r } _ { 1 . 5 \times } ) ) .
+$$
+
+Text embeddings: We generate the text embeddings offline by feeding the category texts with prompt templates, e.g., “a photo of $\{ { \mathrm { c a t e g o r y } } \}$ in the scene”, into the text encoder $\tau$ . We ensemble multiple prompt templates and the synonyms if provided.
+
+Then, we compute cosine similarities between the image and text embeddings. A softmax activation is applied, followed by a per-class NMS to obtain final detections. The inference is slow since every cropped region is fed into $\nu$ .
+
+# 3.3 VILD: VISION AND LANGUAGE KNOWLEDGE DISTILLATION.
+
+We propose ViLD to address the slow inference speed of the above method. ViLD learns region embeddings in a two-stage detector to represent each proposal $r$ . We denote region embeddings by $\mathcal { R } ( \phi ( I ) , r )$ , where $\phi ( \cdot )$ is a backbone model and $\mathcal { R } ( \cdot )$ is a lightweight head that generates region embeddings. Specifically, we take outputs before the classification layer as region embeddings.
+
+Replacing classifier with text embeddings: We first introduce ViLD-text. Our goal is to train the region embeddings such that they can be classified by text embeddings. Fig. 3(b) shows the architecture and training objective. ViLD-text replaces the learnable classifier in Fig. 3(a) with the text embeddings introduced in Sec. 3.2. Only $\mathcal { T } ( C _ { B } )$ , the text embeddings of $C _ { B }$ , are used for training. For the proposals that do not match any groundtruth in $C _ { B }$ , they are assigned to the background category. Since the text “background” does not well represent these unmatched proposals, we allow the background category to learn its own embedding $\mathbf { e } _ { b g }$ . We compute the cosine similarity between each region embedding $\mathcal { R } ( \phi ( I ) , r )$ and all category embeddings, including $\mathcal { T } ( C _ { B } )$ and $\mathbf { e } _ { b g }$ . Then we apply softmax activation with a temperature $\tau$ to compute the cross entropy loss. To train the first-stage region proposal network of the two-stage detector, we extract region proposals $r \in P$ online, and train the detector with ViLD-text from scratch. The loss for ViLD-text can be written as:
+
+$$
+\begin{array} { l }  { \displaystyle { { \bf { e } } _ { r } } = \mathcal { R } ( \phi ( I ) , r ) } \ { \displaystyle { { \bf { e } } _ { r } } } \\ { { \displaystyle { \bf { z } } ( r ) = \left[ s i m ( { \bf { e } } _ { r } , { \bf { e } } _ { b g } ) , \ s i m ( { \bf { e } } _ { r } , { \bf { t } } _ { 1 } ) , \ \cdot \cdot \cdot , \ s i m ( { \bf { e } } _ { r } , { \bf { t } } _ { | C _ { B } | } ) \right] } } \\ { { \displaystyle \mathcal { L } _ { \mathrm { V i L D - t e x t } } = \frac { 1 } { N } \sum _ { r \in P } \mathcal { L } _ { \mathrm { C E } } \Big ( s o f t m a x \big ( { \bf { z } } ( r ) / \tau \big ) , y _ { r } \Big ) } , } \end{array}
+$$
+
+where $s i m ( \mathbf { a } , \mathbf { b } ) = \mathbf { a } ^ { \top } \mathbf { b } / ( \| \mathbf { a } \| \| \mathbf { b } \| )$ , $\mathbf { t } _ { i }$ denotes elements in $\mathcal { T } ( C _ { B } )$ , $y _ { r }$ denotes the class label of region $r$ , $N$ is the number of proposals per image $( | P | )$ , and $\mathcal { L } _ { C E }$ is the cross entropy loss.
+
+During inference, we include novel categories $( C _ { N } )$ and generate $\mathcal { T } ( C _ { B } \cup C _ { N } )$ (sometimes $\mathcal { T } ( C _ { N } )$ only) for open-vocabulary detection (Fig. 2). Our hope is that the model learned from annotations in $C _ { B }$ can generalize to novel categories $C _ { N }$ .
+
+Distilling image embeddings: We then introduce ViLD-image, which aims to distill the knowledge from the teacher image encoder $\nu$ into the student detector. Specifically, we align region embeddings $\mathcal { R } ( \phi ( I ) , \tilde { r } )$ to image embeddings $\mathcal { V } ( \mathrm { c r o p } ( I , \tilde { r } ) )$ introduced in Sec. 3.2.
+
+To make the training more efficient, we extract $M$ proposals $\tilde { r } \in \tilde { P }$ offline for each training image, and precompute the $M$ image embeddings. These proposals can contain objects in both $C _ { B }$ and $C _ { N }$ , as the network can generalize. In contrast, ViLD-text can only learn from $C _ { B }$ . We apply an $\mathcal { L } _ { 1 }$ loss between the region and image embeddings to minimize their distance. The ensembled image embeddings in Sec. 3.2 are used for distillation:
+
+$$
+\mathcal { L } _ { \mathrm { V i L D - i m a g e } } = \frac { 1 } { M } \sum _ { \tilde { \boldsymbol { r } } \in \tilde { P } } \| \mathcal { V } ( \mathrm { c r o p } ( I , \tilde { r } _ { \{ 1 \times , 1 . 5 \times \} } ) ) - \mathcal { R } ( \phi ( I ) , \tilde { r } ) \| _ { 1 } .
+$$
+
+Fig. 3(c) shows the architecture. Zhu et al. (2019) use a similar approach to make Faster R-CNN features mimic R-CNN features, however, the details and goals are different: They reduce redundant context to improve supervised detection; while ViLD-image is to enable open-vocabulary detection on novel categories.
+
+The total training loss of ViLD is simply a weighted sum of both objectives:
+
+$$
+\mathcal { L } _ { \mathrm { V i L D } } = \mathcal { L } _ { \mathrm { V i L D - t e x t } } + w \cdot \mathcal { L } _ { \mathrm { V i L D - i m a g e } } ,
+$$
+
+where $w$ is a hyperparameter weight for distilling the image embeddings. Fig. 3(d) shows the model architecture and training objectives. ViLD-image distillation only happens in training time. During inference, ViLD-image, ViLD-text and ViLD employ the same set of text embeddings as the detection classifier, and use the same architecture for open-vocabulary detection (Fig. 2).
+
+# 3.4 MODEL ENSEMBLING
+
+In this section, we explore model ensembling for the best detection performance over base and novel categories. First, we combine the predictions of a ViLD-text detector with the open-vocabulary image classification model. The intuition is that ViLD-image learns to approximate the predictions of its teacher model, and therefore, we assume using the teacher model directly may improve performance. We use a trained ViLD-text detector to obtain top $k$ candidate regions and their confidence scores. Let $p _ { i }$ ,ViLD-text denote the confidence score of proposal $\tilde { r }$ belonging to category $i$ . We then feed crop $( I , \tilde { r } )$ to the open-vocabulary classification model to obtain the teacher’s confidence score $p _ { i , \mathrm { c l s } }$ . Since we know the two models have different performance on base and novel categories, we introduce a weighted geometric average for the ensemble:
+
+$$
+p _ { i , \mathrm { e n s e m b l e } } = { \left\{ \begin{array} { l l } { p _ { i , { \mathrm { V i L D - t e x t } } } ^ { \lambda } \cdot p _ { i , \mathrm { c l s } } ^ { ( 1 - \lambda ) } , } & { { \mathrm { i f ~ } } i \in C _ { B } } \\ { p _ { i , { \mathrm { V i L D - t e x t } } } ^ { ( 1 - \lambda ) } \cdot p _ { i , \mathrm { c l s } } ^ { \lambda } . } & { { \mathrm { i f ~ } } i \in C _ { N } } \end{array} \right. }
+$$
+
+$\lambda$ is set to $2 / 3$ , which weighs the prediction of ViLD-text more on base categories and vice versa.   
+Note this approach has a similar slow inference speed as the method in Sec. 3.2.
+
+Next, we introduce a different ensembling approach to mitigate the above inference speed issue. Besides, in ViLD, the cross entropy loss of ViLD-text and the $\mathcal { L } _ { 1 }$ distillation loss of ViLD-image is applied to the same set of region embeddings, which may cause contentions. Here, instead, we learn two sets of embeddings for ViLD-text (Eq. 2) and ViLD-image (Eq. 3) respectively, with two separate heads of identical architectures. Text embeddings are applied to these two regions embeddings to obtain confidence scores $p _ { i }$ ,ViLD-text and $p _ { i }$ ,ViLD-image, which are then ensembled in the same way as Eq. 5, with $p _ { i }$ ,ViLD-image replacing $p _ { i , \mathrm { c l s } }$ . We name this approach ViLD-ensemble.
+
+# 4 EXPERIMENTS
+
+Implementation details: We benchmark on the Mask R-CNN (He et al., 2017) with ResNet (He et al., 2016) FPN (Lin et al., 2017) backbone and use the same settings for all models unless explicitly specified. The models use $1 0 2 4 \times 1 0 2 4$ as input image size, large-scale jittering augmentation of range [0.1, 2.0], synchronized batch normalization (Ioffe & Szegedy, 2015; Girshick et al., 2018) of batch size 256, weight decay of 4e-5, and an initial learning rate of 0.32. We train the model from scratch for 180,000 iterations, and divide the learning rate by 10 at $0 . 9 \times$ , $0 . 9 5 \times$ , and $0 . 9 7 5 \times$ of total iterations. We use the publicly available pretrained CLIP model1 as the open-vocabulary classification model, with an input size of $2 2 4 \times 2 2 4$ . The temperature $\tau$ is set to 0.01, and the maximum number of detections per image is 300. We refer the readers to Appendix D for more details.
+
+# 4.1 BENCHMARK SETTINGS
+
+We mainly evaluate on LVIS (Gupta et al., 2019) with our new setting. To compare with previous methods, we also use the setting in Zareian et al. (2021), which is adopted in many zero-shot detection works.
+
+LVIS: We benchmark on LVIS v1. LVIS contains a large and diverse set of vocabulary (1,203 categories) that is more suitable for open-vocabulary detection. We take its 866 frequent and common categories as the base categories $C _ { B }$ , and hold out the 337 rare categories as the novel categories $C _ { N }$ . $\mathsf { A P } _ { r }$ , the AP of rare categories, is the main metric.
+
+COCO: Bansal et al. (2018) divide COCO-2017 (Lin et al., 2014) into 48 base categories and 17 novel categories, removing 15 categories without a synset in the WordNet hierarchy. We follow previous works and do not compute instance masks. We evaluate on the generalized setting.
+
+# 4.2 LEARNING GENERALIZABLE OBJECT PROPOSALS
+
+We first study whether a detector can localize novel categories when only trained on base categories. We evaluate the region proposal networks in Mask R-CNN with a ResNet-50 backbone. Table 1 shows the average recall (AR) (Lin et al., 2014) on novel categories. Training with only base categories performs slightly worse by $\sim 2$ AR at 100, 300, and 1000 proposals, compared to using both base and novel categories. This experiment demonstrates that, without seeing novel categories during training, region proposal networks can generalize to novel categories, only suffering a small performance drop. We believe better proposal networks focusing on unseen category generalization should further improve the performance, and leave this for future research.
+
+Table 1: Training with only base categories achieves comparable average recall (AR) for novel categories on LVIS. We compare RPN trained with base only vs. base $^ +$ novel categories and report the bounding box AR.   
+
+<table><tr><td>Supervision</td><td>ARr@100</td><td>ARr@300</td><td>ARr@1000</td></tr><tr><td>base</td><td>39.3</td><td>48.3</td><td>55.6</td></tr><tr><td>base + novel</td><td>41.1</td><td>50.9</td><td>57.0</td></tr></table>
+
+# 4.3 OPEN-VOCABULARY CLASSIFIER ON CROPPED REGIONS
+
+In Table 2, we evaluate the approach in Sec. 3.2, i.e., using an open-vocabulary classifier to classify cropped region proposals. We use CLIP in this experiment and find it tends to output confidence scores regardless of the localization quality (Appendix B). Given that, we ensemble the CLIP confidence score with a proposal objectness score by geometric mean. Results show it improves both base and novel APs. We compare with supervised baselines trained on base/base+novel categories, as well as Supervised-RFS (Mahajan et al., 2018; Gupta et al., 2019) that uses category frequency for balanced sampling. CLIP on cropped regions already outperforms supervised baselines on $\mathsf { A P } _ { r }$ by a large margin, without accessing detection annotations in novel categories. However, the performances of $\mathsf { A P } _ { c }$ and $\mathsf { A P } _ { f }$ are still trailing behind. This experiment shows that a strong openvocabulary classification model can be a powerful teacher model for detecting novel objects, yet there is still much improvement space for inference speed and overall AP.
+
+Table 2: Using CLIP for open-vocabulary detection achieves high detection performance on novel categories. We apply CLIP to classify cropped region proposals, with or without ensembling objectness scores, and report the mask average precision (AP). The performance on novel categories $( \mathsf { A P } _ { r } )$ is far beyond supervised learning approaches. However, the overall performance is still behind.   
+
+<table><tr><td>Method</td><td>APr</td><td>APc</td><td>APf</td><td>AP</td></tr><tr><td>Supervised (base class only)</td><td>0.0</td><td>22.6</td><td>32.4</td><td>22.5</td></tr><tr><td>CLIP on cropped regions w/o objectness</td><td>13.0</td><td>10.6</td><td>6.0</td><td>9.2</td></tr><tr><td>CLIP on cropped regions</td><td>18.9</td><td>18.8</td><td>16.0</td><td>17.7</td></tr><tr><td>Supervised (base+novel)</td><td>4.1</td><td>23.5</td><td>33.2</td><td>23.9</td></tr><tr><td>Supervised-RFS (base+novel)</td><td>12.3</td><td>24.3</td><td>32.4</td><td>25.4</td></tr></table>
+
+Table 3: Performance of ViLD and its variants. ViLD outperforms the supervised counterpart on novel categories. Using ALIGN as the teacher model achieves the best performance without bells and whistles. All results are mask AP. We average over 3 runs for R50 experiments. †: methods with R-CNN style; runtime is $6 3 0 \times$ of Mask R-CNN style. ‡: for reference, fully-supervised learning with additional tricks.   
+
+<table><tr><td>Backbone</td><td>Method</td><td>APr</td><td>AP</td><td>APf</td><td>AP</td></tr><tr><td rowspan="2">ResNet-50+ViT-B/32</td><td> CLIP on cropped regions+</td><td>18.9</td><td>18.8</td><td>16.0</td><td>17.7</td></tr><tr><td>ViLD-text+CLIP†</td><td>22.6</td><td>24.8</td><td>29.2</td><td>26.1</td></tr><tr><td rowspan="6">ResNet-50</td><td>Supervised-RFS (base+novel)</td><td>12.3</td><td>24.3</td><td>32.4</td><td>25.4</td></tr><tr><td>GloVe baseline</td><td>3.0</td><td>20.1</td><td>30.4</td><td>21.2</td></tr><tr><td>ViLD-text</td><td>10.1</td><td>23.9</td><td>32.5</td><td>24.9</td></tr><tr><td>ViLD-image</td><td>11.2</td><td>11.3</td><td>11.1</td><td>11.2</td></tr><tr><td>ViLD (w=0.5)</td><td>16.1</td><td>20.0</td><td>28.3</td><td>22.5</td></tr><tr><td>ViLD-ensemble (w=0.5)</td><td>16.6</td><td>24.6</td><td>30.3</td><td>25.5</td></tr><tr><td rowspan="2">EfficientNet-b7</td><td>ViLD-ensemble w/ ViT-L/14 (w=1.0)</td><td>21.7</td><td>29.1</td><td>33.6</td><td>29.6</td></tr><tr><td>ViLD-ensemble w/ ALIGN (w=1.0)</td><td>26.3</td><td>27.2</td><td>32.9</td><td>29.3</td></tr><tr><td>ResNeSt269+HTC</td><td>2020 Challenge winner (Tan et al., 2020)</td><td>30.0</td><td>41.9</td><td>46.0</td><td>41.5</td></tr></table>
+
+# 4.4 VISION AND LANGUAGE KNOWLEDGE DISTILLATION
+
+We evaluate the performance of ViLD and its variants (ViLD-text, ViLD-image, and ViLDensemble), which are significantly faster compared to the method in Sec. 4.3. Finally, we use stronger teacher models to demonstrate our best performance. Table 3 summarizes the results.
+
+Text embeddings as classifiers (ViLD-text): We evaluate ViLD-text using text embeddings generated by CLIP, and compare it with GloVe text embeddings (Pennington et al., 2014) pretrained on a large-scale text-only corpus. Table 3 shows ViLD-text achieves $1 0 . 1 \mathrm { \ A P } _ { r }$ , which is significantly better than $3 . 0 \mathrm { \ A P } _ { r }$ using GloVe. This demonstrates the importance of using text embeddings that are jointly trained with images. ViLD-text achieves much higher $\mathsf { A P } _ { c }$ and $\mathsf { A P } _ { f }$ compared to CLIP on cropped regions (Sec. 4.3), because ViLD-text uses annotations in $C _ { B }$ to align region embeddings with text embeddings. The $\mathsf { A P } _ { r }$ is worse, showing that using only 866 base categories in LVIS does not generalize as well as CLIP to novel categories.
+
+Distilling image embeddings (ViLD-image): We evaluate ViLD-image, which distills from the image embeddings of cropped region proposals, inferred by CLIP’s image encoder, with a distillation weight of 1.0. Experiments show that ensembling with objectness scores doesn’t help with other ViLD variants, so we only apply it to ViLD-image. Without training with any object category labels, ViLD-image achieves $1 1 . 2 \mathsf { A P } _ { r }$ and 11.2 overall AP. This demonstrates that visual distillation works for open-vocabulary detection but the performance is not as good as CLIP on cropped regions.
+
+Text+visual embeddings (ViLD): ViLD shows the benefits of combining distillation loss (ViLDimage) with classification loss using text embeddings (ViLD-text). We explore different hyperparameter settings in Appendix Table 7 and observe a consistent trade-off between $\mathsf { A P } _ { r }$ and $\mathsf { A P } _ { c , f }$ , which suggests there is a competition between ViLD-text and ViLD-image. In Table 3, we compare ViLD with other methods. Its $\mathsf { A P } _ { r }$ is 6.0 higher than ViLD-text and 4.9 higher than ViLD-image, indicating combining the two learning objectives boosts the performance on novel categories. ViLD outperforms Supervised-RFS by $3 . 8 \mathrm { A P } _ { r }$ , showing our open-vocabulary detection approach is better than supervised models on rare categories.
+
+Table 4: Performance on COCO dataset compared with existing methods. ViLD outperforms all the other methods in the table trained with various sources by a large margin, on both novel and base categories.   
+
+<table><tr><td>Method</td><td>Training source</td><td>Novel AP</td><td>Base AP</td><td>Overall AP</td></tr><tr><td>Bilen &amp; Vedaldi (2016) Ye et al. (2019)</td><td>image-level labels in CB U CN</td><td>19.7 20.3</td><td>19.6 20.1</td><td>19.6 20.1</td></tr><tr><td>Bansal et al. (2018) Zhu et al. (2020)</td><td>instance-level labels in CB</td><td>0.31 3.41</td><td>29.2 13.8</td><td>24.9 13.0</td></tr><tr><td>Rahman et al. (2020) Zareian et al. (2021)</td><td>image captions in CB U CN</td><td>4.12</td><td>35.9</td><td>27.9</td></tr><tr><td>CLIP on cropped regions</td><td>instance-level labels in CB</td><td>22.8</td><td>46.0</td><td>39.9</td></tr><tr><td>ViLD-text ViLD-image</td><td>image-text pairs from Internet (may contain CB U CN) instance-level labels in CB</td><td>26.3 5.9 24.1</td><td>28.3 61.8</td><td>27.8 47.2</td></tr></table>
+
+Model ensembling: We study methods discussed in Sec. 3.4 to reconcile the conflict of joint training with ViLD-text and ViLD-image. We use two ensembling approaches: 1) ensembling ViLD-text with CLIP (ViLD-text $+ \mathbf { C L I P } )$ ); 2) ensembling ViLD-text and ViLD-image using separate heads (ViLD-ensemble). As shown in Table 3, ViLD-ensemble improves performance over ViLD, mainly on $\mathsf { A P } _ { c }$ and $\mathsf { A P } _ { r }$ . This shows ensembling reduces the competition. ViLD-text+CLIP obtains much higher $\mathsf { A P } _ { r }$ , outperforming ViLD by 6.5, and maintains good $\mathsf { A P } _ { c , f }$ . Note that it is slow and impractical for real world applications. This experiment is designed for showing the potential of using open-vocabulary classification models for open-vocabulary detection.
+
+Stronger teacher model: We use CLIP ViT-L/14 and ALIGN (Jia et al., 2021) to explore the performance gain with a stronger teacher model (details in Appendix D). As shown in Table 3, both models achieve superior results compared with R50 ViLD w/ CLIP. The detector distilled from ALIGN is only trailing to the fully-supervised 2020 Challenge winner (Tan et al., 2020) by $3 . 7 \mathrm { A P } _ { r }$ , which employs two-stage training, self-training, and multi-scale testing etc. The results demonstrate ViLD scales well with the teacher model, and is a promising open-vocabulary detection approach.
+
+# 4.5 PERFORMANCE COMPARISON ON COCO DATASET
+
+Several related works in zero-shot detection and open-vocabulary detection are evaluated on COCO. To compare with them, we train and evaluate ViLD variants following the benchmark setup in Zareian et al. (2021) and report box AP with an IoU threshold of 0.5. We use the ResNet-50 backbone, shorten the training schedule to 45,000 iterations, and keep other settings the same as our experiments on LVIS. Table 4 summarizes the results. ViLD outperforms Zareian et al. (2021) by 4.8 Novel AP and 13.5 Base AP. Different from Zareian et al. (2021), we do not have a pretraining phase tailored for detection. Instead, we use an off-the-shelf classification model. The performance of ViLD-text is low because only 48 base categories are available, which makes generalization to novel categories challenging. In contrast, ViLD-image and ViLD, which can distill image features of novel categories, outperform all existing methods (not apple-to-apple comparison though, given different methods use different settings).
+
+# 4.6 TRANSFER TO OTHER DATASETS
+
+Trained ViLD models can be transferred to other detection datasets, by simply switching the classifier to the category text embeddings of the new datasets. For simplicity, we keep the background embedding trained on LVIS. We evaluate the transferability of ViLD on PASCAL VOC (Everingham et al., 2010), COCO (Lin et al., 2014), and Objects365 (Shao et al., 2019). Since the three datasets have much smaller vocabularies, category overlap is unavoidable and images can be shared among datasets, e.g., COCO and LVIS. As shown in Table 5, ViLD achieves better transfer performance than ViLD-text. In PASCAL and COCO, the gap is large. This improvement should be credited to visual distillation, which better aligns region embeddings with the text classifier. We also compare with supervised learning and finetuning the classification layer. Although across datasets, ViLD has 3-6 AP gaps compared to the finetuning method and larger gaps compared to the supervised method, it is the first time we can directly transfer a trained detector to different datasets using language.
+
+Table 5: Generalization ability of ViLD. We evaluate the LVIS-trained model with ResNet-50 backbone on PASCAL VOC 2007 test set, COCO validation set, and Objects365 v1 validation set. Simply replacing the text embeddings, our approaches are able to transfer to various detection datasets. The supervised baselines of COCO and Objects365 are trained from scratch. †: the supervised baseline of PASCAL VOC is initialized with an ImageNet-pretrained checkpoint. All results are box APs.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">PASCAL VOC+</td><td colspan="3">CoCO</td><td colspan="3">Objects365</td></tr><tr><td>AP50</td><td>AP75</td><td>AP</td><td>AP50</td><td>AP75</td><td>AP</td><td>AP50</td><td>AP75</td></tr><tr><td>ViLD-text</td><td>40.5</td><td>31.6</td><td>28.8</td><td>43.4</td><td>31.4</td><td>10.4</td><td>15.8</td><td>11.1</td></tr><tr><td>ViLD</td><td>72.2</td><td>56.7</td><td>36.6</td><td>55.6</td><td>39.8</td><td>11.8</td><td>18.2</td><td>12.6</td></tr><tr><td>Finetuning</td><td>78.9</td><td>60.3</td><td>39.1</td><td>59.8</td><td>42.4</td><td>15.2</td><td>23.9</td><td>16.2</td></tr><tr><td>Supervised</td><td>78.5</td><td>49.0</td><td>46.5</td><td>67.6</td><td>50.9</td><td>25.6</td><td>38.6</td><td>28.0</td></tr></table>
+
+![](images/7244b96d31b6f6e562fd71b8bca7fb59a44963198d7d6427bc8c5a465606d94d.jpg)  
+Figure 4: Qualitative results on LVIS, COCO, and Objects365. First row: ViLD is able to correctly localize and recognize objects in novel categories. For clarity, we only show the detected novel objects. Second row: The detected objects on base+novel categories. The performance on base categories is not degraded with ViLD. Last two rows: ViLD can directly transfer to COCO and Objects365 without further finetuning.
+
+# 4.7 QUALITATIVE RESULTS
+
+In Fig. 4, we visualize ViLD’s detection results. It illustrates ViLD is able to detect objects of both novel and base categories, with high-quality mask predictions on novel objects, e.g., it well separates banana slices from the crepes (novel category). We also show qualitative results on COCO and Objects365, and find ViLD generalizes well. We show more qualitative results, e.g., interactive detection and systematic expansion, in Appendix A.
+
+# 5 CONCLUSION
+
+We present ViLD, an open-vocabulary object detection method by distilling knowledge from openvocabulary image classification models. ViLD is the first open-vocabulary detection method evaluated on the challenging LVIS dataset. It attains 16.1 AP for novel cateogires on LVIS with a ResNet50 backbone, which surpasses its supervised counterpart at the same inference speed. With a stronger teacher model (ALIGN), the performance can be further improved to 26.3 novel AP. We demonstrate that the detector learned from LVIS can be directly transferred to 3 other detection datasets. We hope that the simple design and strong performance make ViLD a scalable alternative approach for detecting long-tailed categories, instead of collecting expensive detection annotations.
+
+# ETHICS STATEMENT
+
+Our paper studies open-vocabulary object detection, a sub-field in computer vision. Our method is based on knowledge distillation, a machine learning technique that has been extensively used in computer vision, natural language processing, etc. All of our experiments were conducted on public datasets with pretrained models that are either publicly available or introduced in published papers. The method proposed in our paper is a principled method for open-vocabulary object detection that can be used in a wide range of applications. Therefore, the ethical impact of our work would primarily depends on the specific applications. We foresee positive impacts if our method is applied to object detection problems where the data collection is difficult to scale, such as detecting rare objects for self-driving cars. But the method can also be applied to other sensitive applications that could raise ethical concerns, such as video surveillance systems.
+
+# REPRODUCIBILITY STATEMENT
+
+We provide detailed descriptions of the proposed method in Sec. 3. Details about experiment settings, hyper-parameters and implementations are presented in Sec. 4, Appendix C and Appendix D. We release our code and pretrained models at https://github.com/tensorflow/tpu/ tree/master/models/official/detection/projects/vild to facilitate the reproducibility of our work.
+
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+
+# APPENDIX
+
+# A ADDITIONAL QUALITATIVE RESULTS
+
+On-the-fly interactive object detection: We tap the potential of ViLD by using arbitrary text to interactively recognize fine-grained categories and attributes. We extract the region embedding and compute its cosine similarity with a small set of on-the-fly arbitrary texts describing attributes and/or fine-grained categories; we apply softmax with temperature $\tau$ on top of the similarities. To our surprise, though never trained on fine-grained dog breeds (Fig. 5), it correctly distinguishes husky from shiba inu. It also works well on identifying object colors (Fig. 1). The results demonstrate knowledge distillation from an open-vocabulary image classification model helps ViLD to gain understanding of concepts not present in the detection training. Of course, ViLD does not work all the time, e.g., it fails to recognize poses of animals.
+
+![](images/7ef91dbbab058581eab347afce30b5716bc4ad3ce48dccb72abfce67ee51b06c.jpg)  
+Figure 5: On-the-fly interactive object detection. One application of ViLD is using on-the-fly arbitrary texts to further recognize more details of the detected objects, e.g., fine-grained categories and color attributes.
+
+Systematic expansion of dataset vocabulary: In addition, we propose to systematically expand the dataset vocabulary $( \mathbf { v } = \{ v _ { 1 } , . . . , v _ { p } \} )$ with a set of attributes $( \mathbf { a } = \{ a _ { 1 } , . . . , a _ { q } \} )$ as follows:
+
+$$
+{ \begin{array} { r l } & { \operatorname* { P r } ( v _ { i } , a _ { j } \mid \mathbf { e } _ { r } ) = \operatorname* { P r } ( v _ { i } \mid \mathbf { e } _ { r } ) \cdot \operatorname* { P r } ( a _ { j } \mid v _ { i } , \mathbf { e } _ { r } ) } \\ & { \qquad = \operatorname* { P r } ( v _ { i } \mid \mathbf { e } _ { r } ) \cdot \operatorname* { P r } ( a _ { j } \mid \mathbf { e } _ { r } ) , } \end{array} }
+$$
+
+where $\mathbf { e } _ { r }$ denotes the region embedding. We assume $v _ { i }$ ⊥⊥ $a _ { j } \mid \mathbf { e } _ { r }$ , i.e., given $\mathbf { e } _ { r }$ the event the object belongs to category $v _ { i }$ is conditionally independent to the event it has attribute $a _ { j }$ .
+
+Let $\tau$ denote the temperature used for softmax and $\tau$ denote the text encoder as in Eq. 2. Then
+
+$$
+\begin{array} { r } { \operatorname* { P r } ( v _ { i } \mid \mathbf { e } _ { r } ) = s o f t m a x _ { i } ( s i m ( \mathbf { e } _ { r } , { \mathcal { T } } ( \mathbf { v } ) ) / \tau ) , } \\ { \operatorname* { P r } ( a _ { j } \mid \mathbf { e } _ { r } ) = s o f t m a x _ { j } ( s i m ( \mathbf { e } _ { r } , { \mathcal { T } } ( \mathbf { a } ) ) / \tau ) . } \end{array}
+$$
+
+In this way, we are able to expand $p$ vocabularies into a new set of $p \times q$ vocabularies with attributes. The conditional probability approach is similar to YOLO9000 (Redmon $\&$ Farhadi, 2017). We show a qualitative example of this approach in Fig. 6, where we use a color attribute set as a. Our open-vocabulary detector successfully detects fruits with color attributes.
+
+We further expand the detection vocabulary to fine-grained bird categories by using all 200 species from CUB-200-2011 (Wah et al., 2011). Fig. 7 shows successful and failure examples of our openvocabulary fine-grained detection on CUB-200-2011 images. In general, our model is able to detect visually distinctive species, but fails at other ones.
+
+Transfer to PASCAL VOC: In Fig. 8, we show qualitative results of transferring an openvocabulary detector trained on LVIS (Gupta et al., 2019) to PASCAL VOC Detection (2007 test set) (Everingham et al., 2010), without finetuning (Sec. 4.6 in the main paper). Results demonstrate that the transferring works well.
+
+Failure cases: In Fig. 9, we show two failure cases of ViLD. The most common failure cases are the missed detection. A less common mistake is misclassifying the object category.
+
+We show a failure case of mask prediction on PASCAL VOC in Fig. 10. It seems that the mask prediction is sometimes based on low-level appearance rather than semantics.
+
+![](images/fae0671f0f9a2a6d923a34a785fdef4afb021db49abe249c432b8e790d4c8843.jpg)  
+Figure 6: Systematic expansion of dataset vocabulary with colors. We add 11 color attributes (red orange, dark orange, light orange, yellow, green, cyan, blue, purple, black, brown, white) to LVIS categories, which expand the vocabulary size by $1 1 \times$ . Above we show an example of detection results. Our open-vocabulary detector is able to assign the correct color to each fruit. A class-agnostic NMS with threshold 0.9 is applied. Each figure shows top 15 predictions.
+
+![](images/ab0aa787df02645e8929e08eb1cf3086ed07aeeb03ddea83a8a44cba23b3abcc.jpg)  
+Figure 7: Systematic expansion of dataset vocabulary with fine-grained categories. We use the systematic expansion method to detect 200 fine-grained bird species in CUB-200-2011. (a): Our open-vocabulary detector is able to perform fine-grained detection (bottom) using the detector trained on LVIS (top). (b): It fails at recognizing visually non-distinctive species. It incorrectly assigns “Western Gull” to “Horned Puffin” due to visual similarity.
+
+# B ANALYSIS OF CLIP ON CROPPED REGIONS
+
+In this section, we analyze some common failure cases of CLIP on cropped regions and discuss possible ways to mitigate these problems.
+
+Visual similarity: This confusion is common for any classifiers and detectors, especially on large vocabularies. In Fig. 11(a), we show two failure examples due to visual similarity. Since we only use a relatively small ViT-B/32 CLIP model, potentially we can improve the performance with a higher-capacity pretrained model. In Table 6, when replacing this CLIP model with an EfficientNetl2 ALIGN model, we see an increase on AP.
+
+![](images/b76fc928c57b8befef4bf1c3fef38b10ec1356f05e8d1f4fcd3d889285887f6e.jpg)  
+Figure 8: Transfer to PASCAL VOC. ViLD correctly detects objects when transferred to PASCAL VOC, where images usually have lower resolution than LVIS (our training set). In the third picture, our detector is able to find tiny bottles, though it fails to detect the person.
+
+![](images/c954eac2ba14d6d0676fa5b9adfb5a929e0fae112c526102cee9c4c7cb392527.jpg)  
+Figure 9: Failure cases on LVIS novel categories. The red bounding boxes indicate the groundtruths of the failed detections. (a) A common failure type where the novel objects are missing, e.g., the elevator car is not detected. (b) A less common failure where (part of) the novel objects are misclassified, e.g., half of the waffle iron is detected as a calculator due to visual similarity.
+
+Aspect ratio: This issue is introduced by the pre-processing of inputs in CLIP. We use the ViTB/32 CLIP with a fixed input resolution of $2 2 4 \times 2 2 4$ . It resizes the shorter edge of the image to 224, and then uses a center crop. However, since region proposals can have more extreme aspect ratios than the training images for CLIP, and some proposals are tiny, we directly resize the proposals to that resolution, which might cause some issues. For example, the thin structure in Fig. 11(b) right will be highly distorted with the pre-processing. And the oven and fridge can be confusing with the distorted aspect ratio. There might be some simple remedies for this, e.g., pasting the cropped region with original aspect ratio on a black background. We tried this simple approach with both CLIP and ALIGN. Preliminary results show that it works well on the fully convolutional ALIGN, while doesn’t work well on the transformer-based CLIP, probably because CLIP is never trained with black image patches.
+
+Multiple objects in a bounding box: Multiple objects in a region interfere CLIP’s classification results, see Fig. 11(c), where a corner of an aquarium dominates the prediction. This is due to CLIP pretraining, which pairs an entire image with its caption. The caption is usually about salient objects in the image. It’s hard to mitigate this issue at the open-vocabulary classification model’s end. On the other hand, a supervised detector are trained to recognize the object tightly surrounded by the bounding box. So when distilling knowledge from an open-vocabulary image classification model, keeping training a supervised detector on base categories could help, as can be seen from the improvement of ViLD over ViLD-image (Sec. 4.4).
+
+Confidence scores predicted by CLIP do not reflect the localization quality: For example, in Fig. 12(a), CLIP correctly classifies the object, but gives highest scores to partial detection boxes. CLIP is not trained to measure the quality of bounding boxes. Nonetheless, in object detection, it is important for the higher-quality boxes to have higher scores. In Fig. 12(c), we simply re-score by taking the geometric mean of the CLIP confidence score and the objectness score from the proposal model, which yields much better top predictions. In Fig. 12(b), we show top predictions of the Mask
+
+![](images/55e73fc02a3fb642133f91a74000a0d38f2a76059362427fd45a0b0197ee69d5.jpg)  
+Figure 10: An example of ViLD on PASCAL VOC showing a mask of poor quality. The class-agnostic mask prediction head occasionally predicts masks based on low-level appearance rather than semantics, and thus fails to obtain a complete instance mask.
+
+Table 6: ALIGN on cropped regions achieves superior $\mathbf { A P _ { r } }$ , and overall very good performance. It shows a stronger open-vocabulary classification model can improve detection performance by a large margin. We report box APs here.   
+
+<table><tr><td>Method</td><td>APr</td><td>APc</td><td>APf</td><td>AP</td></tr><tr><td>CLIP on cropped regions</td><td>19.5</td><td>19.7</td><td>17.0</td><td>18.6</td></tr><tr><td>ALIGN on cropped regions</td><td>39.6</td><td>32.6</td><td>26.3</td><td>31.4</td></tr></table>
+
+R-CNN model. Its top predictions have good bounding boxes, while the predicted categories are wrong. This experiment shows that it’s important to have both an open-vocabulary classification model for better recognition, as well as supervision from detection dataset for better localization.
+
+# C ADDITIONAL QUANTITATIVE RESULTS
+
+Hyperparameter sweep for visual distillation: Table 7 shows the parameter sweep of different distillation weights using $\mathcal { L } _ { 1 }$ and $\mathcal { L } _ { 2 }$ losses. Compared with no distillation, additionally learning from image embeddings generally yields better performance on novel categories. We find $\mathcal { L } _ { 1 }$ loss can better improve the $\mathsf { A P } _ { r }$ performance with the trade-off against $\mathsf { A P } _ { c }$ and $\mathsf { A P } _ { f }$ . This suggests there is a competition between ViLD-text and ViLD-image.
+
+Table 7: Hyperparameter sweep for visual distillation in ViLD. $\mathcal { L } _ { 1 }$ loss is better than $\mathcal { L } _ { 2 }$ loss. For $\mathcal { L } _ { 1 }$ loss, there is a trend that $\mathsf { A P } _ { r }$ increases as the weight increases, while $\operatorname { A P } _ { f , c }$ decrease. For all parameter combinations, ViLD outperforms ViLD-text on $\operatorname { A P } _ { r }$ . We use ResNet-50 backbone and shorter training iterations (84,375 iters), and report mask AP in this table.   
+
+<table><tr><td>Distill loss</td><td>Distill weight w</td><td>APr</td><td>APc</td><td>APf</td><td>AP</td></tr><tr><td>No distill</td><td>0.0</td><td>10.4</td><td>22.9</td><td>31.3</td><td>24.0</td></tr><tr><td rowspan="3">L2loss</td><td>0.5</td><td>13.7</td><td>21.7</td><td>31.2</td><td>24.0</td></tr><tr><td>1.0</td><td>12.4</td><td>22.7</td><td>31.4</td><td>24.3</td></tr><tr><td>2.0</td><td>13.4</td><td>22.0</td><td>30.9</td><td>24.0</td></tr><tr><td rowspan="4">L1 loss</td><td>0.05</td><td>12.9</td><td>22.4</td><td>31.7</td><td>24.4</td></tr><tr><td>0.1</td><td>14.0</td><td>20.9</td><td>31.2</td><td>23.8</td></tr><tr><td>0.5</td><td>16.3</td><td>19.2</td><td>27.3</td><td>21.9</td></tr><tr><td>1.0</td><td>17.3</td><td>18.2</td><td>25.1</td><td>20.7</td></tr></table>
+
+Box APs and ResNet-152 backbone: Table 8 shows the corresponding box AP of Table 3 in the main paper. In general, box AP is slightly higher than mask AP. In addition, we include the results of ViLD variants with the ResNet-152 backbone. The deeper backbone improves all metrics. The trend/relative performance is consistent for box and mask APs, as well as for different backbones. ViLD-ensemble achieves the best box and mask $\mathsf { A P } _ { r }$ .
+
+Ablation study on prompt engineering: We conduct an ablation study on prompt engineering. We compare the text embeddings ensembled over synonyms and 63 prompt templates (listed in Appendix D) with a non-ensembled version: Using the single prompt template “a photo of {article} {category}”. Table 9 illustrates that ensembling multiple prompts slightly improves the performance by $0 . 4 \mathrm { A P } _ { r }$ .
+
+![](images/2bbd84cf9e2060a48bc4b8fecf84a4a5a226c1aaf7facfb612624f8226168ade.jpg)  
+Figure 11: Typical errors of CLIP on cropped regions. (a): The prediction and the groundtruth have high visual similarity. (b): Directly resizing the cropped regions changes the aspect ratios, which may cause troubles. (c): CLIP’s predictions are sometimes affected by other objects appearing in the region, rather than predicting what the entire bounding box is.
+
+![](images/c41556508228c80c13d828883177c7d71b5d5a89722eaef18da674a02f939962.jpg)  
+Figure 12: The prediction scores of CLIP do not reflect the quality of bounding box localization. (a): Top predictions of CLIP on cropped region. Boxes of poor qualities receive high scores, though the classification is correct. (b): Top predictions of a vanilla Mask R-CNN model. Box qualities are good while the classification is wrong. (c): We take the geometric mean of CLIP classification score and objectiveness score, and use it to rescore (a). In this way, a high-quality box as well as the correct category rank first.
+
+Table 8: Performance of ViLD variants. This table shows additional box APs for models in Table 3 and ResNet-152 results.   
+
+<table><tr><td rowspan="2">Backbone</td><td rowspan="2">Method</td><td colspan="3">Box</td><td rowspan="2"></td><td colspan="4">Mask</td></tr><tr><td>APr</td><td>APc</td><td>APf</td><td>AP</td><td>APr APc</td><td>APf</td><td>AP</td></tr><tr><td>ResNet-50</td><td>CLIP on cropped regions</td><td>19.5</td><td>19.7</td><td>17.0</td><td>18.6</td><td>18.9</td><td>18.8</td><td>16.0</td><td>17.7</td></tr><tr><td>+ViT-B/32</td><td>ViLD-text+CLIP</td><td>23.8</td><td>26.7</td><td>32.8</td><td>28.6</td><td>22.6</td><td>24.8</td><td>29.2</td><td>26.1</td></tr><tr><td rowspan="6">ResNet-50</td><td>Supervised-RFS (base+novel) GloVe baseline</td><td>13.0 3.2</td><td>26.7 22.0</td><td>37.4</td><td>28.5 23.8</td><td>12.3 3.0</td><td>24.3 20.1</td><td>32.4 30.4</td><td>25.4 21.2</td></tr><tr><td></td><td></td><td></td><td>34.9</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ViLD-text</td><td>10.6</td><td>26.1</td><td>37.4</td><td>27.9</td><td>10.1</td><td>23.9</td><td>32.5</td><td>24.9</td></tr><tr><td>ViLD-image</td><td>10.3</td><td>11.5</td><td>11.1</td><td>11.2</td><td>11.2</td><td>11.3</td><td>11.1</td><td>11.2</td></tr><tr><td>ViLD (w=0.5)</td><td>16.3</td><td>21.2</td><td>31.6</td><td>24.4</td><td>16.1</td><td>20.0</td><td>28.3</td><td>22.5</td></tr><tr><td>ViLD-ensemble (w=0.5)</td><td>16.7</td><td>26.5</td><td>34.2</td><td>27.8</td><td>16.6</td><td>24.6</td><td>30.3</td><td>25.5</td></tr><tr><td rowspan="5">ResNet-152</td><td>Supervised-RFS (base+novel)</td><td>16.2</td><td>29.6</td><td>39.7</td><td>31.2</td><td>14.4</td><td>26.8</td><td>34.2</td><td>27.6</td></tr><tr><td>ViLD-text</td><td>12.3</td><td>28.3</td><td>39.7</td><td>30.0</td><td>11.7</td><td>25.8</td><td>34.4</td><td>26.7</td></tr><tr><td>ViLD-image</td><td>12.5</td><td>13.9</td><td>13.4</td><td>13.4</td><td>13.1</td><td>13.4</td><td>13.0</td><td>13.2</td></tr><tr><td>ViLD (w=1.0)</td><td>19.1</td><td>22.4</td><td>31.5</td><td>25.4</td><td>18.7</td><td>21.1</td><td>28.4</td><td>23.6</td></tr><tr><td>ViLD-ensemble (w=2.0)</td><td>19.8</td><td>27.1</td><td>34.5</td><td>28.7</td><td>18.7</td><td>24.9</td><td>30.6</td><td>26.0</td></tr><tr><td rowspan="2">EfficientNet-b7</td><td>ViLD-ensemble w/ ViT-L/14 (w=1.0)</td><td>22.0</td><td>31.5</td><td>38.0</td><td>32.4</td><td>21.7</td><td>29.1</td><td>33.6</td><td>29.6</td></tr><tr><td>ViLD-ensemble w/ ALIGN (w=1.0)</td><td>27.0</td><td>29.4</td><td>36.5</td><td>31.8</td><td>26.3</td><td>27.2</td><td>32.9</td><td>29.3</td></tr></table>
+
+Table 9: Ablation study on prompt engineering. Results indicate ensembling multiple prompt templates slightly improves $\operatorname { A P } _ { r }$ . ViLD w/ multiple prompts is the same ViLD model in Table 3, and ViLD w/ single prompt only changes the text embeddings used as the classifier.   
+
+<table><tr><td>Method</td><td>APr</td><td>APc</td><td>APf</td><td>AP</td></tr><tr><td>ViLD w/ single prompt</td><td>15.7</td><td>19.7</td><td>28.9</td><td>22.6</td></tr><tr><td>ViLD w/ multiple prompts</td><td>16.1</td><td>20.0</td><td>28.3</td><td>22.5</td></tr></table>
+
+# D MORE IMPLEMENTATION DETAILS
+
+ViLD-ensemble architecture: In Fig. 13, we show the detailed architecture and learning objectives for ViLD-ensemble, the ensembling technique introduced in Sec. 3.4.
+
+Model used for qualitative results: For all qualitative results, we use a ViLD model with ResNet152 backbone, whose performance is shown in Table 8.
+
+Details for supervised baselines: For a fair comparison, we train the second stage box/mask prediction heads of Supervised and Supervised-RFS baselines in the class-agnostic manner introduced in Sec. 3.1.
+
+Details for R-CNN style experiments: We provide more details here for the R-CNN style experiments: CLIP on cropped regions in Sec. 4.2 and ViLD-text+CLIP in Sec. 4.3. 1) Generalized object proposal: We use the standard Mask R-CNN R50-FPN model. To report mask AP and compare with other methods, we treat the second-stage refined boxes as proposals and use the corresponding masks. We apply a class-agnostic NMS with 0.9 threshold, and output a maximum of 1000 proposals. The objectness score is one minus the background score. 2) Open-vocabulary classification on cropped regions: After obtaining CLIP confidence scores for the 1000 proposals, we apply a class-specific NMS with a threshold of 0.6, and output the top 300 detections as the final results.
+
+Additional details for ViLD variants: Different from the R-CNN style experiments, for all ViLD variants (Sec. 3.3, Sec. 3.4), we use the standard two-stage Mask R-CNN with the class-agnostic localization modules introduced in Sec. 3.1. Both the $M$ offline proposals and $N$ online proposals are obtained from the first-stage RPN (Ren et al., 2015). In general, the R-CNN style methods and ViLD variants share the same concept of class-agnostic object proposals. We use the second-stage outputs in R-CNN style experiments only because we want to obtain the Mask AP, the main metric, to compare with other methods. For ViLD variants, we remove the unnecessary complexities and show that using a simple one-stage RPN works well.
+
+![](images/63c5146ea253927d7b33b76f425881b87d35cd9cfdf2ae79a0b0c434f4d6533e.jpg)  
+Figure 13: Model architecture and training objectives for ViLD-ensemble. The learning objectives are similar to ViLD. Different from ViLD, we use two separate heads of identical architecture in order to reduce the competition between ViLD-text and ViLD-image objetvies. During inference, the results from the two heads are ensembled as described in Sec. 3.4. Please refer to Fig. 3 for comparison with other ViLD variants.
+
+Architecture for open-vocabulary image classification models: Popular open-vocabulary image classification models (Radford et al., 2021; Jia et al., 2021) perform contrastive pre-training on a large number of image-text pairs. Given a batch of paired images and texts, the model learns to maximize the cosine similarity between the embeddings of the corresponding image and text pairs, while minimizing the cosine similarity between other pairs. Specifically, for CLIP (Radford et al., 2021), we use the version where the image encoder adopts the Vision Transformer (Dosovitskiy et al., 2020) architecture and the text encoder is a Transformer (Vaswani et al., 2017). For ALIGN (Jia et al., 2021), its image encoder is an EfficientNet (Tan & Le, 2019) and its text encoder is a BERT (Devlin et al., 2019).
+
+Details for ViLD with stronger teacher models: In both experiments with CLIP ViT-L/14 and ALIGN, we use EfficientNet-b7 as the backbone and ViLD-ensemble for better performance. We also crop the RoI features from only FPN level $P _ { 3 }$ in the feature pyramid. The large-scale jittering range is reduced to [0.5, 2.0]. For CLIP ViT-L/14, since its image/text embeddings have 768 dimensions, we increase the FC dimension of the Faster R-CNN heads to 1,024, and the FPN dimension to 512. For ViLD w/ ALIGN, we use the ALIGN model with an EfficientNet-l2 image encoder and a BERT-large text encoder as the teacher model. We modify several places in the Mask RCNN architecture to better distill the knowledge from the teacher. We equip the ViLD-image head in ViLD-ensemble with the MBConvBlocks in EfficientNet. Since the MBConvBlocks are fullyconvolutional, we apply a global average pooling to obtain the image embeddings, following the teacher. The ViLD-text head keeps the same Faster R-CNN head architecture as in Mask R-CNN. Since ALIGN image/text embeddings have 1,376 dimensions ( $2 . 7 \times$ CLIP embedding dimension), we increase the number of units in the fully connected layers of the ViLD-text head to 2,048, and the FPN dimension to 1,024.
+
+Text prompts: Since the open-vocabulary classification model is trained on full sentences, we feed the category names into a prompt template first, and use an ensemble of various prompts. Following Radford et al. (2021), we curate a list of 63 prompt templates. We specially include several prompts containing the phrase “in the scene” to better suit object detection, e.g., “There is {article} {category} in the scene”.
+
+Our list of prompt templates is shown below:
+
+’There is {article} {category} in the scene.’ ’There is the {category} in the scene.’ ’a photo of {article} {category} in the scene.’ ’a photo of the {category} in the scene.’ ’a photo of one {category} in the scene.’ ’itap of {article} {category}.’
+
+’itap of my {category}.’   
+’itap of the {category}.’   
+’a photo of {article} {category}.’   
+’a photo of my {category}.’   
+’a photo of the {category}.’   
+’a photo of one {category}.’   
+’a photo of many {category}.’   
+’a good photo of {article} {category}.’   
+’a good photo of the {category}.’   
+’a bad photo of {article} {category}.’   
+’a bad photo of the {category}.’   
+’a photo of a nice {category}.’   
+’a photo of the nice {category}.’   
+’a photo of a cool {category}.’   
+’a photo of the cool {category}.’   
+’a photo of a weird {category}.’   
+’a photo of the weird {category}.’   
+’a photo of a small {category}.’   
+’a photo of the small {category}.’   
+’a photo of a large {category}.’   
+’a photo of the large {category}.’   
+’a photo of a clean {category}.’   
+’a photo of the clean {category}.’   
+’a photo of a dirty {category}.’   
+’a photo of the dirty {category}.’   
+’a bright photo of {article} {category}.’   
+’a bright photo of the {category}.’   
+’a dark photo of {article} {category}.’   
+’a dark photo of the {category}.’   
+’a photo of a hard to see {category}.’   
+’a photo of the hard to see {category}.’   
+’a low resolution photo of {article} {category}.’   
+’a low resolution photo of the {category}.’   
+’a cropped photo of {article} {category}.’   
+’a cropped photo of the {category}.’   
+’a close-up photo of {article} {category}.’   
+’a close-up photo of the {category}.’   
+’a jpeg corrupted photo of {article} {category}.’   
+’a jpeg corrupted photo of the {category}.’   
+’a blurry photo of {article} {category}.’   
+’a blurry photo of the {category}.’   
+’a pixelated photo of {article} {category}.’   
+’a pixelated photo of the {category}.’   
+’a black and white photo of the {category}.’   
+’a black and white photo of {article} {category}.’   
+’a plastic {category}.’   
+’the plastic {category}.’   
+’a toy {category}.’   
+’the toy {category}.’   
+’a plushie {category}.’   
+’the plushie {category}.’   
+’a cartoon {category}.’   
+’the cartoon {category}.’   
+’an embroidered {category}.’   
+’the embroidered {category}.’   
+’a painting of the {category}.’   
+’a painting of a {category}.’

parse/dev/wYqLTy4wkor/wYqLTy4wkor_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "GROUNDING ALEATORIC UNCERTAINTY IN UNSUPERVISED ENVIRONMENT DESIGN ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In reinforcement learning (RL), adaptive curricula have proven highly effective for learning policies that generalize well under a wide variety of changes to the environment. Recently, the framework of Unsupervised Environment Design (UED) generalized notions of curricula for RL in terms of generating entire environments, leading to the development of new methods with robust minimaxregret properties. However, in partially-observable or stochastic settings (those featuring aleatoric uncertainty), optimal policies may depend on the groundtruth distribution over the aleatoric features of the environment. Such settings are potentially problematic for curriculum learning, which necessarily shifts the environment distribution used during training with respect to the fixed groundtruth distribution in the intended deployment environment. We formalize this phenomenon as curriculum-induced covariate shift, and describe how, when the distribution shift occurs over such aleatoric environment parameters, it can lead to learning suboptimal policies. We then propose a method which, given black-box access to a simulator, corrects this resultant bias by aligning the advantage estimates to the ground-truth distribution over aleatoric parameters. This approach leads to a minimax-regret UED method, SAMPLR, with Bayes-optimal guarantees. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Adaptive curricula, which dynamically adjust the distribution of training environments to optimize the performance of the resulting policy, have played a key role in many recent achievements in deep reinforcement learning (RL). Applications have spanned both single-agent RL (Portelas et al., 2020; Wang et al., 2019; Zhong et al., 2020; Justesen et al., 2018), where adaptation occurs over environment variations, and multi-agent RL (MARL), where adaptation can additionally occur over co-players (Silver et al., 2016; Vinyals et al., 2019; Stooke et al., 2021). By presenting the agent with challenges at the threshold of its abilities, such methods demonstrably improve the sample efficiency and the generality of the final policy (Matiisen et al., 2017; Dennis et al., 2020; Jiang et al., 2021b;a). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work introduces a fundamental problem relevant to adaptive curriculum learning methods for RL, which we call curriculum-induced covariate shift (CICS). Analogous to the covariate shift that occurs in supervised learning (SL), CICS refers to a mismatch between the input distribution at training and test time, and in this case, specifically when the distribution shift is caused by the selective sampling performed by an adaptive curriculum. While there may be cases in which CICS impacts model performance in SL, adaptive curricula for SL have generally not been found to be as impactful as in RL (Wu et al., 2021). Therefore, here we focus on addressing this problem specifically as it arises in the RL setting, and leave investigation of its potential impact in SL to future work. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To establish precise language around adaptive curricula, we cast our discussion under the lens of Unsupervised Environment Design (UED, Dennis et al., 2020). UED provides a formal problem description for which curriculum learning is the solution, by defining the Underspecified POMDP (UPOMDP; see Section 2), which expands the classic POMDP with a set of free parameters $\\Theta$ , representing the dimensions along which the environment may vary across episodes. The goal of UED is then to adapt distributions over $\\Theta$ , so to maximize some objective, which could be tied to an RL agent’s performance over this distribution. This allows us to view adaptive curricula as emerging via a multi-agent game between a teacher that proposes environments with parameters $\\theta \\sim \\Theta$ and a student that learns to solve them. In addition to the notational clarity it provides, this formalism lends the analysis of adaptive curricula to useful game theoretic constructs, such as Nash equilibria (NE, Nash et al., 1950). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/80f8a0269d13b24c9d874710957ba3f421688417eb5cf584d9837b0677b95eb0.jpg",
+        "image_caption": [
+            "Figure 1: Adaptive curricula can result in covariate shifts in environment parameters with respect to a fixed ground-truth distribution ${ \\overline { { P } } } ( \\theta )$ (see top path), e.g. whether apple or banana is the correct fruit to choose per level. Here, the policy $\\pi _ { > }$ $( \\pi _ { \\mathcal { J } } )$ always chooses apple (banana). Our method, SAMPLR (bottom path) matches the advantages in sampled levels to the advantages observed if sampling levels from ${ \\overline { { P } } } ( \\theta )$ (blue triangles), thus constraining the optimal policy under the curriculum distribution $P ( \\theta )$ to match that under ${ \\overline { { P } } } ( \\theta )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This game-theoretic view has led to the development of curriculum methods with principled robustness guarantees, such as PAIRED (Dennis et al., 2020) and Prioritized Level Replay (PLR, Jiang et al., 2021a), which showed that curricula optimizing for the student’s regret lead to minimax regret (Savage, 1951) policies at Nash equilibria, implying the agent can solve all solvable environments within the training domain. While other methods can also be cast in this framework, they do not hold the same desirable property at equilibrium. For this reason, we will focus on addressing CICS for regret-maximizing UED, but note that our solution can be used with other UED methods. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To see how the CICS can be problematic, consider the simplified case of training a self-driving car in simulation, so it learns to take the fastest route from home to office. Suppose traffic data shows on $70 \\%$ of the days, Route 1 is faster than Route 2. Moreover, on any given day the self-driving car cannot infer which route is faster ahead of time, so always picking Route 1 is faster in expectation. To support training a policy using an adaptive curriculum, one could build a simulator which sets road conditions per episode based on a random day sampled from the traffic data. However, adaptive curriculum over the traffic settings may oversample days when Route 1 is closed—perhaps because it finds the agent needs more practice on Route 2—shifting the best choice to Route 2 in training. In fact, methods for minimax regret UED like PLR would keep shifting the distribution of fastest route, to maximize the agent’s regret. Their curriculum dynamics would map to the zero-sum game of matching pennies, in which one player wins for guessing whether the other chose heads or tails; the NE corresponds to each player randomly playing each option half the time. Randomly picking the route in this way is suboptimal, because in reality Route 1 is faster in expectation. This example is depicted in Figure 1, where the two routes are replaced by an apple and banana. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "If, on a given day, the faster route could be identified before having to pick one, the agent could choose optimally. Instead, it is an aleatoric parameter, inducing irreducible uncertainty in the limit of infinite experiential data (Der Kiureghian & Ditlevsen, 2009). When CICS occurs over such parameters, with respect to a ground-truth distribution of environments $\\overline { { P } } ( \\boldsymbol { \\theta } )$ , the learned policy can be suboptimal with respect to $\\overline { { P } }$ . It can therefore be useful to ground—that is, to constrain—the aleatoric parameters $\\Theta ^ { \\prime } \\subset \\Theta$ to $\\overline { { P } } ( \\theta ^ { \\prime } )$ when it is known or can be learned, as in simulation or from real-world data. However, grounding all $\\theta$ is undesirable, preventing the curriculum from sampling enough opportunities to learn from useful scenarios with low support under ${ \\overline { { P } } } ( \\theta )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we formalize the problem of CICS in RL, and provide a solution by proposing a UED method to find robustly Bayes-optimal policies where $\\theta ^ { \\prime }$ is grounded to $\\overline { { P } } ( \\theta ^ { \\prime } )$ . Our solution called Sample-Matched PLR (SAMPLR) extends PLR, a state-of-the-art UED algorithm, by constraining the advantage estimates to match those observed if training under $\\overline { { P } } ( \\theta ^ { \\prime } )$ . This advantage correction adapts Off-Belief Learning (Hu et al., 2021) from cooperative MARL, revealing an intriguing connection between curriculum biases observed in single and multi-agent RL. Our experiments in challenging environments based on the NetHack Learning Environment (NLE, Küttler et al., 2020) demonstrate that SAMPLR learns near-optimal policies under CICS, in cases where standard PLR fails. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            155,
+            326,
+            172
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 UNSUPERVISED ENVIRONMENT DESIGN ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The problem of Unsupervised Environment Design (UED, Dennis et al. (2020)) is the problem of automatically generating an adaptive distribution of environments which will lead to policies that successfully transfer within a target domain. The domain of possible environments is represented by an Underspecified POMDP (UPOMDP), which adds a set of free parameters to the standard definition of a POMDP, along which each concrete instantiation, or level, of the UPOMDP. For instance, these free parameters can be the position of obstacles in a maze, or friction coefficients in a physics-based task. Formally a UPOMDP is defined as a tuple $\\mathcal { M } = \\langle A , O , \\Theta , S , T , \\mathcal { I } , \\mathcal { R } , \\gamma \\rangle$ , where $A$ is a set of actions, $O$ is a set of observations, $\\Theta$ is a set of free parameters, $S$ is a set of states, $\\mathcal { T } : S \\times A \\times \\Theta \\to \\Delta ( S )$ is a transition function, $\\mathcal { T } : S  O$ is an observation (or inspection) function, $\\mathcal { R } : S  \\mathbb { R }$ is a reward function, and $\\gamma$ is a discount factor. UED typically approaches the curriculum design problem as training a teacher agent that co-evolves an adversarial curriculum for a student agent, for example, by maximizing the student’s regret. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We will focus on a recent UED algorithm called Robust Prioritized Level Replay $( { \\mathrm { P L R } } ^ { \\perp }$ , Jiang et al., 2021b), which performs environment design via random search. PLR maintains a buffer of the most useful levels for training, according to some learning potential score—typically based on a regret approximation, such as the positive value loss—and with probability $p$ , actively samples the next training level from this level buffer instead of the ground-truth training distribution. This selective-sampling mechanism has been demonstrated to greatly improve sample-efficiency and generalization in several domains, while provably leading to a minimax regret policy for the student at NE. In maximizing regret, PLR curricula naturally avoid unsolvable levels, which have no regret. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 OFF-BELIEF LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            522,
+            379,
+            536
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In cooperative MARL, self-play promotes the formation of cryptic conventions—arbitrary sequences of actions that allow agents to communicate information about the environment state. These conventions are learned jointly among all agents during training, but are arbitrary and hence, indecipherable to independently-trained agents or humans at test time. Crucially, this leads to policies that fail to perform zero-shot coordination (ZSC, Hu et al., 2020), where independentlytrained agents must cooperate successfully without additional learning steps, or ad-hoc team play. Off-Belief Learning (OBL) resolves this problem by forcing agents to assume their co-players act according to a fixed, known policy $\\pi _ { 0 }$ until the current time $t$ , and optimally afterwards, conditioned on this assumption. If $\\pi _ { 0 }$ is playing uniformly random, this removes the possibility of forming arbitrary conventions. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Formally, let $G$ be a decentralized, partially-observable MDP (Dec-POMDP, Bernstein et al., 2002), with state $s$ , joint action $a$ , observation function ${ \\mathcal { T } } ^ { i } ( s )$ for each player $i$ , and transition function ${ \\mathcal { T } } ( s , a )$ . Let the historical trajectory $\\tau = ( s _ { 1 } , a _ { 1 } , . . . a _ { t - 1 } , s _ { t } )$ , and the action-observation history (AOH) for agent $i$ be $\\tau ^ { i } = ( \\mathcal { T } ^ { i } ( s _ { 1 } ) , a _ { 1 } , . . . , a _ { t - 1 } , \\mathcal { T } ^ { i } ( s _ { t } ) )$ . Further, let $\\pi _ { 0 }$ be an arbitrary policy, such as a uniformly random policy, and $B _ { \\pi _ { 0 } } ( \\tau | \\tau ^ { i } ) = P ( \\tau _ { t } | \\tau _ { t } ^ { i } , \\pi _ { 0 } )$ , a belief model predicting the current state, conditioned on the AOH of agent $i$ and the assumption of co-players playing policy $\\pi _ { 0 }$ until the current time $t$ , and optimally according to $\\pi _ { 1 }$ from $t$ and beyond. OBL aims to find the policy $\\pi _ { 1 }$ with the optimal, counter-factual value function, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9031b63b642b7e2f1170ba71f0b36344a1ef7c24dff06273dd49bbfe3db84fba.jpg",
+        "text": "$$\nV ^ { \\pi _ { 0 }  \\pi _ { 1 } } ( \\tau ^ { i } ) = \\mathbb { E } _ { \\tau \\sim \\mathcal { B } _ { \\pi _ { 0 } } ( \\tau ^ { i } ) } [ V ^ { \\pi _ { 1 } } ( \\tau ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            852
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As the agent conditions its policy on the realized $\\operatorname { A O H } \\tau ^ { i }$ , while transition dynamics are based on states sampled from $\\boldsymbol { B }$ , this mechanism is called a fictitious transition. In Section 5, we show how OBL’s fictitious transition can be adapted to the single-agent curriculum learning setting to address CICS, by interpreting the curriculum designer in UED as a co-player. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The mismatch between training and testing distributions of input features is referred to as covariate shift, and has long served as a fundamental problem for the machine learning community. Covariate shifts have been extensively studied in supervised learning (Vapnik & Chervonenkis, 1971; Huang et al., 2006; Bickel et al., 2009; Arjovsky et al., 2019). In RL, prior works have largely focused on covariate shifts due to training on off-policy data (Sutton et al., 2016; Rowland et al., 2020; Espeholt et al., 2018; Hallak & Mannor, 2017; Gelada & Bellemare, 2019; Thomas & Brunskill, 2016) including the important case of learning from demonstrations (Pomerleau, 1988; Ross & Bagnell, 2010). Recent work also aims to learn invariant representations robust to covariate shifts (Zhang et al., 2019; 2021). More generally, CICS can be interpreted as a kind of sample-selection bias (Heckman, 1979). We believe this work to be the first to formalize and provide a solution to the problem of covariate shifts in reinforcement learning due to curriculum learning. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our method fixes a critical flaw that can cause curricula to fail under CICS—an important problem as curricula have been shown to be essential for training RL agents across many of the most challenging domains, including combinatorial gridworlds (Zhong et al., 2020), Go (Silver et al., 2016), StarCraft 2 (Vinyals et al., 2019), and achieving comprehensive task mastery in open-ended environments (Stooke et al., 2021). While this work focuses on PLR, other recent methods include minimax adversarial curricula (Wang et al., 2019; 2020) and curricula based on changes in return (Matiisen et al., 2017; Portelas et al., 2020). Most similar to our work, OFFER (Ciosek & Whiteson, 2017) adapts a curriculum over transition functions and uses importance sampling to correct for biased gradient estimates. Unlike this work, Ciosek & Whiteson (2017) requires whitebox access to the transition function and does not directly study the impact of CICS on the learning dynamics. Curriculum methods have also been studied in goal-conditioned RL (Florensa et al., 2018; Campero et al., 2021; Sukhbaatar et al., 2018; OpenAI et al., 2021), though CICS does not occur here as goals are observed by the agent. Lastly, domain randomization (DR, Sadeghi & Levine, 2017; Peng et al., 2017) can be seen as a degenerate form of UED, though curriculum-based extensions of DR have also been studied (Jakobi, 1997; Tobin et al., 2017). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Prior work has also investigated methods for learning Bayes optimal policies under uncertainty about the task (Zintgraf et al., 2020; Osband et al., 2013), based on the framework of Bayes-adaptive MDPs (BAMDPs) (Bellman, 1956; Duff, 2002). In this setting, the agent can adapt to an unknown MDP over several episodes by acting to reduce its uncertainty about the identity of the MDP. In contrast, SAMPLR learns a robustly Bayes optimal policy for the case of zero-shot transfer. Further unlike these works, our setting assumes the distribution of certain aleatoric parameters are biased during training, which would lead to biased a posteriori uncertainty estimates with respect to the ground-truth distribution when optimizing for the BAMDP objective. Instead, SAMPLR proposes a means to correct for this bias assuming knowledge of the true environment parameters for each level, to which we can safely assume access in curriculum learning. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 CURRICULUM-INDUCED COVARIATE SHIFT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As UED algorithms formulate curriculum learning as a multi-agent game between teacher and student agents, we can formalize when CICS become problematic by considering the equilibrium point of this game: Let $\\theta$ be the environment parameters controlled by UED, $\\overline { { P } } ( \\boldsymbol { \\theta } )$ , the ground truth distribution of $\\theta$ , and $P ( \\theta )$ , the curriculum distribution at equilibrium. Further, let $\\tau _ { t }$ be $\\mathbf { \\bar { \\rho } } ( o _ { 1 } , a _ { 1 } , . . . , a _ { t - 1 } , o _ { t } )$ , the student agent’s action-observation history (AOH) until time $t$ (though we will use simply $\\tau$ when clear from context). The optimal action-value function $Q ^ { * }$ with respect to ${ \\overline { { P } } } ( \\theta )$ can then be expressed as a marginalization over $\\theta$ : ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b045ccc2b13f490516e2e70bf83f08c6301dc528352ac3e3212135734be1f866.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\overline { { \\boldsymbol { Q } } } ^ { * } ( \\boldsymbol { a } _ { t } | \\tau _ { t } ) = \\underbrace { \\mathbb { E } \\tau _ { t + 1 : \\infty } } _ { \\approx \\pi ^ { * } } \\left[ \\displaystyle \\sum _ { l = 0 } ^ { \\infty } \\gamma ^ { l } \\boldsymbol { r } _ { t + l } \\right] = \\displaystyle \\sum _ { \\theta } \\overline { { P } } ( \\theta | \\tau _ { t } ) \\overline { { Q } } _ { \\overline { { P } } } ^ { * } ( \\boldsymbol { a } _ { t } | \\tau _ { t } , \\theta ) } \\\\ & { \\qquad \\propto \\displaystyle \\sum _ { \\theta } \\overline { { P } } ( \\theta ) \\overline { { P } } ( \\tau _ { t } | \\theta ) \\overline { { Q } } _ { \\overline { { P } } } ^ { * } ( \\boldsymbol { a } _ { t } | \\tau _ { t } , \\theta ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "From Equation 2, we see that $\\overline { { Q } } ^ { * } ( a _ { t } | \\tau _ { t } )$ remains optimal under different values of ${ \\overline { { P } } } ( \\theta )$ at each time $t$ as long as it is possible to infer $\\theta$ deterministically from $\\tau _ { t }$ , implying that $\\overline { { P } } ( \\theta ^ { \\prime } | \\tau _ { t } ) = 1$ for some $\\theta$ , in which case the RHS of Equation 1 reduces to the LHS. If $\\overline { { P } } ( \\theta | \\tau _ { t } ) < 1$ for all $\\theta$ , then some subset $\\theta ^ { \\prime } \\subset \\Theta$ results in irreducible uncertainty at time $t$ , thereby constituting aleatoric parameters. Letting $Q ^ { * } ( a _ { t } | \\tau _ { t } )$ be the optimal action-value function when $\\overline { { P } } ( \\overline { { \\theta } } )$ is replaced with $P ( \\theta )$ in Equation 2, we can then state that curriculum-induced covariate shift results in suboptimal policies with respect to the ground truth distribution when ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/41d9cd52cadf7ce416ac03ff9c317a1e153aad4fb7cefe7f7b7526cf377496c1.jpg",
+        "image_caption": [
+            "Figure 2: A standard RL transition (left) and the fictitious transition used by SAMPLR (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/50a7a3951ee257f59089389e48b89ad7e5d26703903bbf2947fbdcc73ca22229.jpg",
+        "text": "$$\n\\arg \\operatorname* { m a x } _ { a } Q ^ { * } ( a | \\tau _ { t } ) \\neq \\arg \\operatorname* { m a x } _ { a } \\overline { { Q } } ^ { * } ( a | \\tau _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Moreover, this formulation highlights how this effect results from the presence of aleatoric parameters. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This description categorizes errors made by a policy trained on $P$ evaluated on levels drawn from ${ \\overline { { P } } } ( \\theta )$ into two categories: The first type of error, which we call a mistake, simply arises when $P ( \\theta ^ { \\prime \\prime } ) = \\dot { 0 }$ for some non-aleatoric $\\theta ^ { \\prime \\prime }$ where $\\overline { { P } } ( \\theta ^ { \\prime \\prime } ) > 0$ , i.e. the policy did not train on trajectories needed to learn to behave optimally under some set of otherwise identifiable level parameters. This is distinct from the second kind of error, which we call a misunderstanding, and which corresponds to the problematic mismatch between training and test time distributions of specifically aleatoric parameters $\\theta ^ { \\prime }$ , so that $P ( \\theta ^ { \\prime } ) \\neq \\overline { { P } } ( \\theta ^ { \\prime } )$ , as we just previously characterized. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This taxonomy also clarifies errors in cooperative MARL, where the co-players shape environment interactions, and thus play a similar role to the UED teacher. Failures in ZSC can then be diagnosed as due to (i) a mistake, because the co-players fail to generate trajectories that occur with test-time co-players; or (ii), a misunderstanding, because the train-time co-players shift the training distribution of aleatoric parameters $P ( \\theta ^ { \\prime } )$ , which impacts the inference of $\\begin{array} { r } { \\overline { { P } } \\dot { ( \\tau ) } = \\sum _ { \\theta ^ { \\prime } } \\overline { { P } } ( \\theta ^ { \\prime } ) \\overline { { P } } ( \\tau | \\theta ^ { \\prime } ) } \\end{array}$ needed for optimal cooperation—for example, through the use of cryptic conventions. This view then connects generalization errors in cooperative MARL to those in single-agent RL, implying that methods like OBL devised to solve one type of error in one of the settings may be adapted for the other. Indeed, we now describe our method, which does exactly this: By adapting OBL to single-agent curriculum learning, we can ground the values of $\\overline { { P } } ( \\theta ^ { \\prime } | \\tau _ { t } )$ by forcing $P (  { \\hat { \\theta } } ^ { \\prime } ) = \\overline { { P } } ( \\theta ^ { \\prime } )$ , thereby ensuring that the distribution of the aleatoric parameters at equilibrium is equivalent to their ground truth distribution. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 SAMPLE-MATCHED PLR (SAMPLR) ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now describe how OBL’s fictitious transition can be adapted for $\\mathrm { \\bf P I R ^ { \\perp } }$ (Jiang et al., 2021a) to address CICS, resulting in Sample-Matched PLR (SAMPLR). To avoid CICS, we must ground $\\theta ^ { \\prime }$ to the ground-truth distribution $\\overline { { P } } ( \\theta ^ { \\prime } )$ , while allowing the remaining parameters in $\\Theta$ to vary under UED, so as to still benefit from a curriculum. To achieve this, we adapt the fictitious transition to single-agent curriculum learning by treating the UED teacher as a co-player—one that performs the single action of choosing the level $\\theta$ at the start of each episode, and subsequently performs no-ops. Under this fictitious transition, we ground the teacher’s choice of $\\theta$ such that the aleatoric parameters $\\theta ^ { \\prime }$ are assumed to be sampled from ${ \\overline { { P } } } ( \\theta )$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Thus, at each time $t$ , the agent takes actions $a _ { t }$ based on its AOH as usual, but estimates the advantage $A ( a _ { t } , s _ { t } )$ using fictitious transitions, which assume subsequent state transitions and rewards occur with $\\Theta ^ { \\prime }$ fixed to $\\theta ^ { \\prime } \\sim \\overline { { P } } ( \\theta ^ { \\prime } )$ , sampled at the start of the episode. More formally, the fictitious transition is performed as $s _ { t } ^ { \\prime } \\sim \\dot { B } ( \\dot { s } _ { t } ^ { \\prime } | \\tau )$ , $\\bar { a } _ { t } \\sim \\pi ( \\cdot | \\tau )$ , $s _ { t + 1 } ^ { \\prime } = \\dot { \\mathcal T } ( s _ { t } ^ { \\prime } , a _ { t } )$ , and $r _ { t } ^ { \\prime } = \\dot { \\mathcal { R } } ( s _ { t + 1 } )$ , where $\\tau$ is the ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/aa9266fe1719456c4c3d06f701b48b3ff56f786fb1c5a323a1dbd3d4dafcc859.jpg",
+        "table_caption": [
+            "Algorithm 1: Sample-Matched PLR (SAMPLR) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Randomly initialize policyπ(Φ),an empty level buffer Λof size K,and belief model B(st|T). while not converged do Sample replay-decision Bernoulli,d~ PD(d)</td></tr><tr><td colspan=\"2\">if d=O or|A|=O then</td></tr><tr><td colspan=\"2\">Sample level θ from level generator</td></tr><tr><td colspan=\"2\">Collect π&#x27;s trajectory T on θ, with a stop-gradient Φ⊥</td></tr><tr><td colspan=\"2\">Use observed ground-truth states to update B</td></tr><tr><td colspan=\"2\">else</td></tr><tr><td colspan=\"2\">Use PLR to sample a replay level from the level store,θ ~ Λ</td></tr><tr><td colspan=\"2\">Collect fictitious trajectory T&#x27; on 0,based on s&#x27;t ~ B</td></tr><tr><td colspan=\"2\">Update π with rewards R(T&#x27;)</td></tr><tr><td colspan=\"2\"></td></tr><tr><td colspan=\"2\">end</td></tr><tr><td colspan=\"2\">Compute PLR score, S = score(T&#x27;,π)</td></tr><tr><td colspan=\"2\">Update Λ with θ using score S end</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "AOH of the student. Figure 2 summarizes this transition mechanism. Here, the belief model $\\boldsymbol { B } ( \\boldsymbol { s } _ { t } | \\tau )$ can be expressed as ",
+        "bbox": [
+            173,
+            340,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/83e950a36bbe418cab1e37e16e37b06999d7aaf26cc573c27f25e4516965c5b5.jpg",
+        "text": "$$\nB ( s _ { t } | \\tau ) = \\sum _ { \\theta ^ { \\prime } } \\overline { { P } } ( s _ { t } | \\tau , \\theta ^ { \\prime } ) \\overline { { P } } ( \\theta ^ { \\prime } | \\tau ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
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+            419
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This shows that, assuming blackbox simulator access, we can generally implement $\\boldsymbol { B }$ as follows: Periwith $\\theta ^ { \\prime } \\sim \\overline { { P } } ( \\theta ^ { \\prime } )$ ring training, . We use these $\\{ ( \\theta _ { k } ^ { \\prime } , \\tau _ { k } ) \\} _ { k = 1 } ^ { N }$ $N$ trajectories, such that each is gepairs to update a posterior model ${ \\overline { { P } } } ( \\theta ^ { \\prime } | \\tau )$ under a level that predicts the underlying $\\theta ^ { \\prime }$ given $\\tau$ . We can then sample from by first sampling $\\theta \\sim \\overline { { P } } ( \\theta ^ { \\prime } | \\tau )$ , followed by stepping forward a parallel simulator that has been initially reset to the current AOH $\\tau$ , with fixed $\\Theta ^ { \\prime } = \\theta ^ { \\prime }$ , thereby yielding a desired sample of $s _ { t }$ according to $\\boldsymbol { B }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In practice, it is often the case that $\\theta ^ { \\prime }$ can be uniquely identified by some revelatory event by time $t$ , so that $\\overline { { { \\cal P } } } ( \\theta ^ { \\prime } | \\tau ) = 1$ for some $\\theta ^ { \\prime }$ and $\\tau$ , and $\\overline { { P } } ( s _ { t } | \\tau , \\mathbf { \\dot { \\theta } } ^ { \\prime } ) = \\overline { { P } } ( s _ { t } | \\check { \\tau } )$ otherwise. In this case, we can implement the fictitious transition by setting $\\theta ^ { \\prime } \\sim \\overline { { P } } ( \\theta ^ { \\prime } )$ at the start of each episode; subsequent transitions will then be consistent with $\\boldsymbol { B }$ . For example, in the fruit choice example, whether apple or banana was the right goal is deterministically revealed by the final reward, and otherwise, does not impact transition dynamics. Additionally, we often only have limited access to $\\overline { { P } } ( \\theta )$ throughout training, for example, if sampling ${ \\overline { { P } } } ( \\theta )$ is costly. In this case, we can learn an estimate $\\hat { P } ( \\theta ^ { \\prime } )$ using the samples we do collect from ${ \\overline { { P } } } ( \\theta )$ , which can occur online. We then use $\\hat { P } ( \\theta ^ { \\prime } )$ in sampling fictitious transitions during UED. We refer to the resulting $\\hat { P } ( \\theta ^ { \\prime } )$ as a learned belief prior. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "SAMPLR, summarized in Algorithm 1, incorporates this fictitious transition by replacing the advantages of trajectories on replay levels sampled by $\\mathrm { P L R ^ { \\perp } }$ with their fictitious counterparts, as only these trajectories are used by $\\mathrm { P L R ^ { \\perp } }$ for training. To reduce the cost of sampling $\\overline { { { \\cal P } } } ( \\theta ^ { \\prime } )$ , we can use the new levels regularly sampled by $\\mathrm { P L R ^ { \\perp } }$ to estimate the learned belief prior $\\overline { { P } } ( \\theta ^ { \\prime } )$ , and use $\\overline { { P } } ( \\theta ^ { \\prime } )$ in sampling fictitious transitions on replay levels. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6 GROUNDED POLICIES ARE ROBUSTLY BAYES OPTIMAL ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We can view OBL-based correction as a method for training a policy to be optimal with respect to the ground-truth value function, with levels sampled from some generating distribution $\\Lambda$ defined as: ",
+        "bbox": [
+            171,
+            790,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/08c72489149afbd5d1d245d76f9c0da45a6cc4a3c369b34ba7b9f313fa78661a.jpg",
+        "text": "$$\n\\overline { { { V } } } ^ { \\Lambda } ( \\pi ) = \\mathbb { E } _ { \\tau \\sim { \\mathcal { M } } ^ { \\Lambda } ( \\pi ) } \\left[ \\overline { { { V } } } ^ { \\pi } ( \\tau ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            824,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that when $\\Lambda = \\overline { { { \\cal P } } } ( \\theta )$ this reduces to the ground-truth value function notated simply as $\\overline { { V } } ( \\pi )$ . First, we will note that, for any UED method, our OBL-based correction will ensure that, in equilibrium, the resulting policy is Bayes-optimal on the ground truth beliefs, on any trajectory sampled from $\\mathcal { M } ^ { \\Lambda } ( \\pi )$ , the distribution of trajectories of $\\pi$ in levels sampled from $\\Lambda$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Remark 1. If π is optimal with respect to the grounded value function $\\overline { { V } } ^ { \\Lambda } ( \\pi )$ then it is Bayes optimal with respect to the ground-truth parameter distribution ${ \\overline { { P } } } ( \\theta )$ on the support of $\\mathcal { M } ^ { \\Lambda } ( \\pi )$ . ",
+        "bbox": [
+            174,
+            102,
+            825,
+            136
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proof. By definition we have that $\\pi \\in \\underset { \\pi \\in \\Pi } { \\arg \\operatorname* { m a x } } \\{ \\overline { { V } } ^ { \\Lambda } ( \\pi ) \\} = \\underset { \\pi \\in \\Pi } { \\arg \\operatorname* { m a x } } \\big \\{ \\mathbb { E } _ { \\tau \\sim \\mathcal { M } ^ { \\Lambda } ( \\pi ) } \\left[ \\overline { { V } } ^ { \\pi } ( \\tau ) \\right] \\big \\}$ . Since $\\pi$ can condition on the initial trajectory $\\tau$ , the action selected after each trajectory can be independently optimized. Thus we have, for all $\\tau \\in \\mathcal { M } ^ { \\Lambda } ( \\pi )$ , $\\pi \\in \\arg \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\{ { \\overline { { V } } } ^ { \\pi } ( \\tau ) \\}$ implying that $\\pi$ is the optimal grounded policy and V π = V ∗. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Thus, assuming the base RL algorithm finds Bayes optimal policies, a UED method that optimizes the grounded value function, as done by SAMPLR, will result in Bayes optimal performance over the ground-truth distribution. When the UED method aims to maximize worst-case regret, we can prove an even stronger property we call robust $\\epsilon$ -Bayes optimality. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let $\\overline { { V } } ^ { \\theta } ( \\pi )$ be the value function for $\\pi$ evaluated on a specific level $\\theta$ . We will say that a policy is robustly $\\epsilon$ -Bayes optimal iff for all $\\theta$ in the domain of ${ \\overline { { P } } } ( \\theta )$ and for all $\\pi ^ { \\prime }$ we have ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f23f569a93cd0becfff39aecb03f07101145c38080f737324027e0fed89c493e.jpg",
+        "text": "$$\n\\overline { { { V } } } ^ { \\theta } ( \\pi ) \\geq \\overline { { { V } } } ^ { \\theta } ( \\pi ^ { \\prime } ) - \\epsilon .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Note how this differs from being only $\\epsilon$ -Bayes optimal, which means for all $\\pi ^ { \\prime }$ , ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9f54e4b5711bc956a0fef77e9d57014a884cef9f3acc727d760c2a9134678594.jpg",
+        "text": "$$\n\\overline { { { V } } } ( \\pi ) \\geq \\overline { { { V } } } ( \\pi ^ { \\prime } ) - \\epsilon\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "With robust $\\epsilon$ -Bayes optimality, we must be $\\epsilon$ -optimal even on levels which are rarely sampled from the ground-truth distribution. We will show that if SAMPLR is in an $\\epsilon$ -Nash Equilibrium, then a policy is robustly $\\epsilon$ -Bayes optimal with respect to the grounded value function $\\overline { { { V } } } ( \\dot { \\pi } )$ rather than only $\\epsilon$ -Bayes optimal as one would expect from training directly on the true distribution of levels. ",
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+    {
+        "type": "text",
+        "text": "Theorem 1. If $\\pi$ is $\\epsilon$ -Bayes optimal by $\\overline { { V } } ^ { \\Lambda } ( \\pi )$ for $\\Lambda$ minimizing worst-case regret as is done in SAMPLR, then it is robustly $\\epsilon$ -Bayes optimal with respect to the grounded value function, $\\overline { { V } } ( \\pi )$ . ",
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+        "type": "text",
+        "text": "Proof. Let $\\pi$ be $\\epsilon$ -optimal with respect to $\\overline { { V } } ^ { \\Lambda } ( \\pi )$ where $\\Lambda$ minimizing worst-case regret with respect to $\\pi$ . Let $\\overline { { \\pi } } ^ { * }$ be an optimal grounded policy, and let $\\theta$ be arbitrary. Then we have: ",
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+    {
+        "type": "equation",
+        "img_path": "images/1201ae7ad9f0207d9806dd4db5308d39dc074990b4e68a598ecd8402013cf44b.jpg",
+        "text": "$$\n\\overline { { { V } } } ^ { \\theta } ( \\overline { { { \\pi } } } ^ { * } ) - \\overline { { { V } } } ^ { \\theta } ( \\pi ) \\le \\overline { { { V } } } ^ { \\Lambda } ( \\overline { { { \\pi } } } ^ { * } ) - \\overline { { { V } } } ^ { \\Lambda } ( \\pi ) \\le \\epsilon\n$$",
+        "text_format": "latex",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Where the first inequality follows from $\\Lambda$ minimizing worst-case regret with respect to $\\pi$ , and the second follows from $\\pi$ being $\\epsilon$ -optimal on $\\Lambda$ . Rearranging terms gives the desired condition. □ ",
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+    {
+        "type": "text",
+        "text": "7 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We investigate the performance of SAMPLR with respect to the the standard PLR and domain randomization in environments based on MiniHack (Samvelyan et al., 2021), a library for creating custom environments based on the runtime of the NetHack Learning Environment (NLE) (Küttler et al., 2020). Acting optimally in our environments requires grounding to the ground-truth distribution. ",
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+        "type": "text",
+        "text": "Our agents are trained using PPO (Schulman et al., 2017), using the best hyperparameters found via grid search, and use the policy architecture in Küttler et al. (2020). We tune the PLR-specific hyperparameters shared among $\\mathrm { \\dot { P L } } \\mathrm { \\mathbf { R } } ^ { \\perp }$ and SAMPLR variants for each environment, based on the performance of $\\mathrm { \\bf P } \\mathrm { \\bf L } { \\bf R } ^ { \\perp }$ . Full details of our environments, agent architecture, and hyperparameters are provided in Appendix A, and our SAMPLR implementation, in Appendix B. We compare both standard SAMPLR and a variant called LP-SAMPLR that learns a belief prior over the true goal, against $\\mathrm { P L R ^ { \\perp } }$ and standard PPO baselines. These baselines allows us to separate the relative changes in performance metrics due to curriculum learning and our proposed correction for CICS. ",
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+        "img_path": "images/5a0ad8437b53763848c849d01d988bfe750885e8df97170c00e1a06794f23227.jpg",
+        "image_caption": [
+            "Figure 4: Episodic returns (left) and number of rooms in solved levels (middle) during training (dotted lines) and test on the ground-truth distribution (solid lines), for $q = 0 . 7$ . Normalized test returns and proportion of apple goals during training for various $q$ are shown on the right. Plots show mean and standard error of 10 runs. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "In our experiments, we first adapt the apple and banana example, depicted in Figure 3 into a fully procedurally-generated RL environment set in the world of NetHack. For each level, the agent can only learn to choose optimally in expectation at test time by grounding to the distribution of correct choices in the deployment domain. Specifically, in each level, the agent must traverse between one to eight randomly generated rooms, and in the final room, the agent must choose to eat the apple or the banana. The correct choice is fixed for each level, but indiscernible to the agent. Thus, the identity of the true goal acts as the aleatoric parameter. Figure 3 shows example levels from this environment. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This environment presents a hard exploration challenge for standard RL algorithms, as it requires learning to both navigate multiple rooms, as well as the NLE-specific skills of kicking doors and eating. The doors opening into adjacent rooms are locked. In order to go from one room to the next, the agent must learn to, potentially repeatedly, kick the locked door until it opens. Likewise, upon reaching a piece of fruit in the final room, the agent must learn to deliberately choose to eat the fruit. If the right choice of fruit were determinable per episode, we expect $\\mathrm { P L R ^ { \\perp } }$ ’s adaptive curriculum to improve learning by selectively sampling levels at the threshold of the agent’s abilities. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let $\\pi _ { A }$ be the policy in which the agent always chooses the apple, and $\\pi _ { B }$ , the banana. If the probability of the goal being apple ${ \\overline { { P } } } ( A ) = q$ , the expected return of the agent is $R _ { A } q$ under $\\pi _ { A }$ and $R _ { B } ( 1 - q )$ under $\\pi _ { B }$ . The optimal policy is then to act according to $\\pi _ { A }$ when $q > R _ { B } / ( R _ { A } \\bar { + } R _ { B } ) \\bar { }$ , and according to $\\pi _ { B }$ otherwise. We expect training under domain randomization, which samples each level completely at random from the ground-truth distribution ${ \\overline { { P } } } ( \\theta )$ , defined by the environment parameterization, would converge to the correct choice of $\\pi _ { A }$ or $\\pi _ { B }$ , assuming the environment is learnable by the choice of RL algorithm. In contrast, PLR evolves an adversarial curriculum, incentivizing PLR to shift the distribution over goals throughout training. For example, PLR is incentivized to flip the curriculum distribution to favor eating apples whenever the agent begins to consistently succeed at eating bananas. We thus expect the PLR curriculum to continually oscillate the preferred choice of goal. ",
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+    {
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+        "img_path": "images/6181ee162b5a26f1daf84e32660b3dcfcc8a720ba04615e65a12a27521f7d96d.jpg",
+        "image_caption": [
+            "Figure 3: Levels from the stochastic fruit-choice environment. Across levels, the correct choice of fruit is distributed according to $\\overline { { P } }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we set $R _ { A } = 3$ , $R _ { B } = 1 0$ , and $q$ in $\\{ 0 . 7 , 0 . 5 , 0 . 3 \\}$ , making it always optimal to follow the banana-eating policy $\\pi _ { B }$ , but with the marginal benefit of doing so varying with $q$ . We report train and test performance of each agent over 200M training steps in Figure 4. We find that DR struggles to learn an effective policy, plateauing at an expected return under 1.0; PLR performs even worse, due to its adversarial curriculum shifting the distribution over the correct goal, leading to rapid oscillations in the optimal choice of fruit under the curriculum distribution, as visible in the high-variance oscillations in the proportion of apple goals selected by PLR for each value of $q$ in 4). This makes it difficult for the agent to settle on the optimal policy with respect to any ground-truth distribution. The proportion of each choice outcome, shown in Figure 5, reveals that both DR and PLR policies fail to eat any fruit most of the time, even after 200M training steps. In contrast, both SAMPLR and LP-SAMPLR realize a marked improvement in test performance by grounding PLR’s otherwise wild shifts in $q$ . Figure 4 shows this improvement is most pronounced when the expected difference in $\\pi _ { A }$ and $\\pi _ { B }$ is smaller, and therefore easier for PLR to flip the agent’s policy. Moreover, we see in Figure 5 that both SAMPLR methods present levels with higher room counts on average, indicating that early in training, it is able to discover easier, few-room levels in which the agent can capture reward signal, and which can then be made more complex to push the agent along its threshold of abilities. ",
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+        "img_path": "images/bb074562b692232a71fe532a4bd2cb72fc0fdf886ff185d9a66a6e10c0f927f5.jpg",
+        "image_caption": [
+            "Figure 5: Left: Proportion of training episodes in which the agent fails to eat any fruit; eats the apple; or eats the banana. Right: Number of rooms in levels during training. Plots show mean and standard error of 10 runs. "
+        ],
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+        "image_caption": [
+            "Figure 6: Left: Levels from FireDungeon. Middle: Test return on the ground-truth distribution of FireDungeon. Right: Proportion of training levels with fireproof armor. Plots show mean and standard error of 10 runs. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We next turn to a more challenging environment that introduces additional NetHack-specific dynamics. The FireDungeon environment (see Figure 6 for example levels), requires the agent to navigate through up to 13 chambers, and ultimately choose between chamber A or B, each containing a valid goal. Reaching this goal ends the episode and provides the agent with a sparse reward. Further, in this penultimate chamber, there is an armor, which if worn grants the agent with fire resistance with probability $q$ . Chamber B is marked by a red gemstone by the door and contains enemy units, whose fire attack will instantly kill the agent. Killing each enemy grants the agent with $+ 1$ final reward, which is only provided upon reaching either goal. The agent thus stands to attain a higher reward by attacking the enemies in Chamber B, before reaching the goal, only if the armor is fireproof. This environment presents an even more difficult exploration problem for the agent, yet we see in Figure 6, both SAMPLR variants begin to learn to solve this environment with significantly greater sample-efficiency than DR, while PLR again struggles to learn. As in the case of the previous stochastic choice environment, we see that PLR rapidly oscillates the key aleatoric parameter, which in this case, determines whether the armor is fireproof. ",
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+    {
+        "type": "text",
+        "text": "8 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Using the formal notions of environment parameterizations in the framework of UED, we defined the problem of curriculum-induced covariate shift in RL. Our definition highlights the issues that can arise when there is persisitent uncertainty over the environment parameters, either because the uncertainty is irreducible or because reducing the uncertainty is costly. We then adapted a fictitious transition mechanism previously used to improve zero-shot coordination in cooperative MARL to correct for this covariate shift. We demonstrated that our resulting algorithm, SAMPLR, avoids the pitfalls of this type of covariate shift, while preserving the benefits of curriculum learning. ",
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+    {
+        "type": "text",
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