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+# INFORMATION BOTTLENECK: EXACT ANALYSIS OF (QUANTIZED) NEURAL NETWORKS
+
+Stephan S. Lorenzen, Christian Igel & Mads Nielsen
+
+Department of Computer Science
+
+University of Copenhagen
+
+{lorenzen,igel,madsn}@di.ku.dk
+
+# ABSTRACT
+
+The information bottleneck (IB) principle has been suggested as a way to analyze deep neural networks. The learning dynamics are studied by inspecting the mutual information (MI) between the hidden layers and the input and output. Notably, separate fitting and compression phases during training have been reported. This led to some controversy including claims that the observations are not reproducible and strongly dependent on the type of activation function used as well as on the way the MI is estimated. Our study confirms that different ways of binning when computing the MI lead to qualitatively different results, either supporting or refusing IB conjectures. To resolve the controversy, we study the IB principle in settings where MI is non-trivial and can be computed exactly. We monitor the dynamics of quantized neural networks, that is, we discretize the whole deep learning system so that no approximation is required when computing the MI. This allows us to quantify the information flow without measurement errors. In this setting, we observed a fitting phase for all layers and a compression phase for the output layer in all experiments; the compression in the hidden layers was dependent on the type of activation function. Our study shows that the initial IB results were not artifacts of binning when computing the MI. However, the critical claim that the compression phase may not be observed for some networks also holds true.
+
+# 1 INTRODUCTION
+
+Improving our theoretical understanding of why over-parameterized deep neural networks generalize well is arguably one of main problems in current machine learning research (Poggio et al., 2020). Tishby & Zaslavsky (2015) suggested to analyze deep neural networks based on their Information Bottleneck (IB) concept, which is built on measurements of mutual information (MI) between the activations of hidden layers and the input and target (Tishby et al., 1999), for an overview see Geiger (2020). Shwartz-Ziv & Tishby (2017) empirically studied the IB principle applied to neural networks and made several qualitative observations about the training process; especially, they observed a fitting phase and a compression phase. The latter information-theoretic compression is conjectured to be a reason for good generalization performance and has widely been considered in the literature (Abrol & Tanner, 2020; Balda et al., 2018; 2019; Chelombiev et al., 2019; Cheng et al., 2019; Darlow & Storkey, 2020; Elad et al., 2019; Fang et al., 2018; Gabrie et al., 2019; Goldfeld et al., 2019; Jónsson et al., 2020; Kirsch et al., 2020; Tang Nguyen & Choi, 2019; Noshad et al., 2019; Schiemer & Ye, 2020; Shwartz-Ziv & Alemi, 2020; Wickström et al., 2019; Yu et al., 2020). The work and conclusions by Shwartz-Ziv & Tishby (2017) received a lot of critique, with the generality of their claims being doubted; especially Saxe et al. (2018) argued that the results by Shwartz-Ziv & Tishby do not generalize to networks using a different activation function. Their critique was again refuted by the original authors with counter-claims about incorrect estimation of the MI, highlighting an issue with the approximation of MI in both studies.
+
+Our goal is to verify the claims by Shwartz-Ziv & Tishby and the critique by Saxe et al. in a setting where the MI can be computed exactly. These studies consider neural networks as theoretical entities working with infinite precision, which makes computation of the information theoretic quantities problematic (for a detailed discussion we refer to Geiger, 2020, see also Section 3). Assuming
+
+continuous input distributions, a deterministic network using any of the standard activation functions (e.g., RELU, TANH) can be shown to have infinite MI (Amjad & Geiger, 2019). If an empirical input distribution defined by a data set  $\mathcal{D}$  is considered (as it is the case in many of the previous studies), then randomly-initialized deterministic neural networks with invertible activation functions will most likely result in trivial measurements of MI in the sense that the MI is finite but always maximal, that is, equal to  $\log |\mathcal{D}|$  (Goldfeld et al., 2019; Amjad & Geiger, 2019). In order to obtain non-trivial measurements of MI, real-valued activations are usually discretized by binning the values, throwing away information in the process. The resulting estimated MI can be shown to be highly dependent on this binning, we refer to Geiger (2020) for a detailed discussion. Instead of approximating the MI in this fashion, we take advantage of the fact that modern computers – and thus neural networks – are discrete in the sense that a floating point value can typically take at most  $2^{32}$  different values. Because 32-bit precision networks may still be too precise to observe compression (i.e., information loss), we apply quantization to the neural network system to an extent that we can compute informative quantities, that is, we amplify the effect of the information loss due to the discrete computations in the neural network. One may argue that we just moved the place where the discretization is applied. This is true, but leads to a fundamental difference: previous studies applying the discretization post-hoc rely on the in general false assumption that the binned MI approximates the continuous MI well – and thus introduce measurement errors, which may occlude certain phenomena and/or lead to artifactual observations. In contrast, our computations reflect the true information flow in a network during training. Our study confirms that estimation of MI by binning may lead to strong artifacts in IB analyses and shows that:
+
+- Both fitting and compression phases occur in the output SOFTMAX layer.  
+- For the hidden layers, the fitting phase occurs for both TANH and RELU activations.  
+- When using TANH in the hidden layers, compression is only observed in the last hidden layer.  
+- When using RELU, we did not observe compression in the hidden layers.  
+- Even when applying low precision quantization, more complex networks with many neurons in each layer are observed to be too expressive to exhibit compression, as no information is lost.  
+- Our setting excludes that the MI approximation is the reason for these different IB dynamics.
+
+The next section introduces the IB concept with a focus on its application to neural networks including the critique and controversy as well as related work. Section 3 discusses issues relating to the estimation of MI, and the idea behind our contribution. Section 4 presents our experiments, results and discussion before we conclude in Section 5.
+
+# 2 THE INFORMATION BOTTLENECK
+
+Preliminaries. Given a continuous random variable (r.v.)  $X$  with density function  $p(x)$  and support  $\mathcal{X}$ , the continuous entropy  $\mathrm{H}(X)$  of  $X$  is a measure of the uncertainty associated with  $X$  and is given by  $\mathrm{H}(X) = -\int_{\mathcal{X}} p(x) \log p(x) \, \mathrm{d}x$ . Given two r.v.s  $X$  and  $Y$  with density functions  $p(x)$  and  $q(y)$  and supports  $\mathcal{X}$  and  $\mathcal{Y}$ , the mutual information  $\mathrm{I}(X; Y)$  of  $X$  and  $Y$  is a measure of the mutual "knowledge" between the two variables. The symmetric  $\mathrm{I}(X; Y)$  is given by  $\mathrm{I}(X; Y) = \int_{\mathcal{Y}} \int_{\mathcal{X}} p(x, y) \log \frac{p(x, y)}{p(x)p(y)} \, \mathrm{d}x \, \mathrm{d}y$ . In many cases it is impossible to compute the continuous entropy and MI for continuous r.v.s exactly, due to limited samples or computational limits, or because it may not be finite (Geiger, 2020). Instead, we often estimate the quantities by their discrete counterparts. When  $X$  is a discrete r.v., we consider the Shannon entropy  $\mathrm{H}(X) = -\sum P(x) \log P(x)$ . Correspondingly, the mutual information  $\mathrm{I}(X; Y)$  of two discrete r.v.s  $X, Y$ , is given by  $\mathrm{I}(X; Y) = \sum_{x, y} P(x, y) \log \frac{P(x, y)}{P(x)P(y)}$ . We have the following useful identity for both the continuous and discrete MI:
+
+$$
+\mathrm {I} (X; Y) = \mathrm {H} (X) - \mathrm {H} (X | Y), \tag {1}
+$$
+
+where  $\mathrm{H}(X|Y)$  is the conditional entropy of  $X$  given  $Y$ .
+
+IB Definition. The IB method was proposed by Tishby et al. (1999). It is an information theoretic framework for extracting relevant components of an input r.v.  $X$  with respect to an output r.v.  $Y$ . These relevant components are found by "squeezing" the information from  $X$  through a bottleneck, in the form of an r.v.  $T$ . In other words,  $T$  is a compression of  $X$ . The idea generalizes rate distortion theory, in which we wish to compress  $X$ , obtaining  $T$ , such that  $I(X;T)$  is maximized subject to a constraint on the expected distortion  $d(x,t)$  wrt. the joint distribution  $p(x,t)$  (Tishby et al., 1999). In the IB framework, the distortion measure  $d$  is replaced by the negative loss in MI between  $T$  and the output  $Y$ ,  $I(T;Y)$ . Both IB and rate distortion are lossy compression schemes. $^{1}$
+
+The data processing inequality (DPI)  $\operatorname{I}(Y;X) \geq \operatorname{I}(Y;T)$  holds for the IB; that is, the bottleneck r.v. cannot contain more information about the label than the input.
+
+One drawback of the information bottleneck method is the dependence on the joint distribution,  $p(x,y)$ , which is generally not known. Shamir et al. (2010) addressed this issue and showed that the MI, the main ingredient in the method, can be estimated reliably with fewer samples than required for estimating the true joint distribution. As common in the IB literature, whenever we discuss the MI computed on a finite data set  $\mathcal{D}$ , we assume that  $p(x,y)$  corresponds to the empirical distribution defined by  $\mathcal{D}$ , which is true for the experiments in Section 4.1. In practice, the assumption has to be relaxed to the data being drawn i.i.d. from  $p(x,y)$ . However, any uncertainty resulting from the finite sample estimation in the latter case is not considered in our discussions.
+
+IB In Deep Learning. Tishby & Zaslavsky (2015) applied the IB concept to neural networks. They view the layers of a deep neural network (DNN) as consecutive compressions of the input. They consider the Markov chain
+
+$$
+Y \rightarrow X \rightarrow T _ {1} \rightarrow T _ {2} \rightarrow \ldots \rightarrow T _ {L} = \hat {Y},
+$$
+
+where  $T_{i}$  denotes the  $i$ 'th hidden layer of the  $L$ -layer network and  $T_{L} = \hat{Y}$  denotes the output of the network. Again, the bottleneck must satisfy the DPI:
+
+$$
+\mathrm {I} (Y; X) \geq \mathrm {I} (Y; T _ {1}) \geq \mathrm {I} (Y; T _ {2}) \geq \dots \geq \mathrm {I} (Y; \hat {Y}), \tag {2}
+$$
+
+$$
+I (X; X) \geq I (X; T _ {1}) \geq I (X; T _ {2}) \geq \dots \geq I (X; \hat {Y}). \tag {3}
+$$
+
+Estimating the MI of continuous variables is difficult (Alemi et al., 2017), as evident from the many different methods proposed (Kriskov et al., 2004; Kolchinsky & Tracey, 2017; Noshad et al., 2019). In the discrete case,  $\operatorname{I}(X;T)$  and  $\operatorname{I}(T;Y)$  can be computed as
+
+$$
+\mathrm {I} (X; T) = \mathrm {H} (T) - \mathrm {H} (T | X) = \mathrm {H} (T), \tag {4}
+$$
+
+$$
+\mathrm {I} (T; Y) = \mathrm {I} (Y; T) = \mathrm {H} (T) - \mathrm {H} (T | Y), \tag {5}
+$$
+
+following from (1) and using in (4) the assumption that  $T$  is a deterministic function of  $X$ . However, for deterministic neural networks the continuous entropies may not be finite (Goldfeld et al., 2019; Saxe et al., 2018; Amjad & Geiger, 2019).
+
+Shwartz-Ziv & Tishby (2017) estimate the MI via (4) and (5) by discretizing  $T$  and then computing the discrete entropy. They trained a network (shown in Figure 1a) on a balanced synthetic data set consisting of 12-bit binary inputs and binary labels. The network was trained for a fixed number of epochs while training and test error were observed. For every epoch and every layer  $T$ , The discretization is done by binning of  $T$ : Given upper and lower bounds  $b_{u}, b_{l}$ , and  $m \in \mathbb{N}$ , we let  $B: \mathbb{R} \to [m]$  denote the binning operation, that maps  $x \in [b_{l}, b_{u}]$  to the index of the corresponding bin from the set of  $m$  uniformly distributed bins in  $[b_{l}, b_{u}]$ . Overloading the notation, we apply  $B$  directly to a vector in  $\mathbb{R}^d$  in order to obtain the resulting vector in  $[m]^d$  of bin indices. Using discretized  $T' = B(T)$ ,  $\operatorname{I}(X; T')$  and  $\operatorname{I}(T'; Y)$  are then computed directly by (4) and (5), using estimates of  $P(T')$ ,  $P(Y)$  and  $P(T'|Y)$  over all samples  $\mathcal{D}$  of  $X$ .
+
+Shwartz-Ziv & Tishby used the TANH activation function for the hidden layers, with  $b_{l} = -1$ ,  $b_{u} = 1$  ( $b_{l} = 0$  for the output SOFTMAX layer) and  $m = 30$  bins. The estimated I ( $X;T$ ) and I ( $T;Y$ ) are plotted in the information plane, providing a visual representation of the information flow in
+
+![](images/7f7fc31de4f0bc1127080cfb21fa03f011d6a439eec27da2a1c3d4741e78ed19.jpg)  
+(a)
+
+![](images/b2c055536c3c4821d92ffea1417264935d74ba391eaf0c947120a2c6342e389b.jpg)  
+(b)  
+Figure 1: (a) The network used in the standard setting, with the number of neurons in each fully connected layer. The output layer activation function is SOFTMAX, while either TANH or RELU is used for the hidden layers. (b) An example of an information plane for a 6 layer network, with the observed phases marked. Only the last three layers are distinguishable.
+
+the network during training (see example in Figure 1b). Based on the obtained results, Shwartz-Ziv & Tishby (2017) make several observations; one notable observation is the occurrence of two phases: an empirical risk minimization phase and a compression phase. The first phase, also referred to as fitting phase, is characterized by increasing  $\mathrm{I}(T;Y)$  related to a decreasing loss. The subsequent compression phase is characterized by decreasing  $\mathrm{I}(X;T)$ , and it has been argued that this compression leads to better generalization.
+
+Critique and Controversy. The work by Tishby & Zaslavsky (2015) and Shwartz-Ziv & Tishby (2017) has jump-started an increasing interest in the IB method for deep learning, with several papers investigating or extending their contributions, see the review by Geiger (2020). However, as mentioned, their work also received criticism. In particular, the compression phase as a general phenomenon has been called into question. Saxe et al. (2018) published a paper refuting several of the claims made by Shwartz-Ziv & Tishby (2017). They based their criticism on a replication of the experiment done by Shwartz-Ziv & Tishby (2017) where they replaced the bounded TANH activation function with the unbounded RELU activation function. When discretizing, they used the maximum activation observed across all epochs for the upper binning bound  $b_{u}$ . The article claims that the two phases observed by Shwartz-Ziv & Tishby occurred because the activations computed using TANH saturate close to the boundaries  $-1$  and  $1$ . The claim is supported by experiments using RELU activations and  $m = 100$  bins, in which the two phases are not observed.
+
+The critique paper by Saxe et al. was published at ICLR 2018, but started a discussion already during the review process the previous year, when Shwartz-Ziv & Tishby defended their paper in the online discussion forum OpenReview.net², posting a response titled “Data falsifying the claims of this ICLR submission all together” (Saxe et al., 2017). The response specifically states that “The authors don’t know how to estimate mutual information correctly”, referring to Saxe et al. (2018), and goes on to provide an example with a network using RELU activations, which does indeed exhibit the two phases. In response, Saxe et al. performed further experiments using different estimators for MI: a state-of-the-art non-parametric KDE approach (Kolchinsky & Tracey, 2017) and a  $k$ -NN based estimator (Kriskov et al., 2004). The authors still did not observe the two phases claimed by Shwartz-Ziv & Tishby.
+
+Following the discussion on OpenReview.net, several other papers have also commented on the controversy surrounding the information bottleneck. Noshad et al. (2019) presented a new MI estimator EDGE, based on dependency graphs, and tested it on the specific counter example using
+
+RELU activations as suggested by Saxe et al. (2018), and they observed the two phases. Table I in the review by Geiger (2020) provides a nice overview over empirical IB studies and if the compression phase was observed (Darlow & Storkey, 2020; Jónsson et al., 2020; Kirsch et al., 2020; Noshad et al., 2019; Raj et al., 2020; Shwartz-Ziv & Tishby, 2017) or not (Abrol & Tanner, 2020; Balda et al., 2018; 2019; Tang Nguyen & Choi, 2019; Shwartz-Ziv & Alemi, 2020; Yu et al., 2020) or the results were mixed (Chelombiev et al., 2019; Cheng et al., 2019; Elad et al., 2019; Fang et al., 2018; Gabrié et al., 2019; Goldfeld et al., 2019; Saxe et al., 2018; Schiemer & Ye, 2020; Wickstrøm et al., 2019). In conclusion, an important part of the controversy surrounding the IB hinges on the estimation of the information-theoretic quantities – this issue has to be solved before researching the information flow.
+
+Related Work. The effect of estimating MI by binning has been investigated before, we again refer to Geiger (2020) for a good overview and discussion. Shwartz-Ziv & Alemi (2020) consider infinite ensembles of infinitely-wide networks, which renders MI computation feasible, but do not observe a compression phase. Chelombiev et al. (2019) applies adaptive binning, which, while less prone to issues caused by having the "wrong" number of bins, is still an estimation and thus also suffers from the same problems. Goldfeld et al. (2019) explores IB analysis by use of stochastic neural networks which allow them to show that the compression phase in these noisy networks occurs due to clustering of the hidden representations. While theoretically interesting, the stochastic neural networks are still qualitatively different from deterministic ones and thus not directly applicable in practice. Raj et al. (2020) conducted an IB analysis of binary networks, where both the activations and weights can only be  $\pm 1$ , which allows for exact computation of MI. The binary networks are significantly different from the networks used in the original studies, whereas applying the IB analysis to quantized versions of networks from these studies allows for a more direct comparison. At a first glance, the work by Raj et al. (2020) could be viewed as taking our approach to the extreme. However, the computations and training dynamics of the binary networks are qualitatively different from our study and the original IB work. For example, the binary networks require modifications for gradient estimation (e.g., Raj et al. consider straight-through-estimator, approximate sign, and swish sign).
+
+# 3 ESTIMATING ENTROPY AND MUTUAL INFORMATION IN NEURAL NETWORKS
+
+The IB principle is defined for continuous and discrete input and output, however continuous MI in neural networks is in general infeasible to compute, difficult to estimate, and conceptually problematic.
+
+# 3.1 Binning and LOW COMPRESSION IN PRECISE NETWORKS
+
+Consider for a moment a deterministic neural network on a computer with infinite precision. Assuming a constant number of training and test patterns, a constant number of update steps, invertible activation functions in the hidden layers and pairwise different initial weights (e.g., randomly initialized). Given any two different input patterns, two neurons will have different activations with probability one (Goldfeld et al., 2019; Geiger, 2020). Thus, on any input, a hidden layer  $T$  will be unique, meaning that, when analysing data set  $\mathcal{D}$ , all observed values of  $T$  will be equally likely, and the entropy  $\mathrm{H}(T)$  will be trivial in the sense that it is maximum, that is,  $\log |\mathcal{D}|$ .
+
+However, when applying the binning  $T' = B(T)$ , information is dropped, as  $B$  is surjective and several different continuous states may map to the same discrete state; for  $m$  bins and  $d_{T}$  neurons in  $T$  the number of different states of  $T'$  is  $m^{d_{T}}$ , and thus more inputs from  $\mathcal{D}$  will map to the same state. As a consequence, the estimated entropy is decreased. Furthermore, the rate of decrease follows the number of bins; the smaller the number of bins  $m$ , the smaller the estimated  $\mathrm{H}(T)$  (as more values will map to the same state). The entropy  $\mathrm{H}(T')$ , and therefore  $\mathrm{I}(X; T')$  in deterministic networks, is upper bounded by  $d_{T} \log m$ . Thus, changing the number of bins used in the estimation changes the results. We also refer to Geiger (2020) for a more detailed discussion of this phenomenon<sup>3</sup>.
+
+Computations in digital computers are not infinitely precise, a 32-bit floating point variable, for instance, can take at most  $2^{32}$  different values. In theory, this means that we can actually consider the states of a neural network discrete and compute the exact MI directly. In practice, we expect the precision to be high enough, that - when using invertible activation functions - the entropy of the
+
+layers will be largely uninformative, as almost no information loss/compression will occur. However, using low precision quantized neural networks, we can compute the exact MI in the network without  $\mathrm{H}(T)$  being trivial – what we regard as the ideal setting to study IB conjectures.
+
+# 3.2 EXACT MUTUAL INFORMATION IN QUANTIZED NETWORKS
+
+Quantization of neural networks was originally proposed as a means to create more efficient networks, space-wise and in terms of inference running time, for instance to make them available on mobile devices (Jacob et al., 2018; Hubara et al., 2017). The idea is to use low precision  $k$ -bit weights (typically excluding biases) and activations, in order to reduce storage and running time. For our use case, we are mainly interested in quantized activations, as they contain the information that is moving through the network.
+
+The naive approach is to quantize after training (post-training quantization). In terms of estimating the entropy/MI, this corresponds to applying a binning with  $2^{k}$  bins to the non-quantized activations as done in the previous studies (Shwartz-Ziv & Tishby, 2017; Saxe et al., 2018), as information available to the network at the time of training is dropped, and thus we expect it to show similar artifacts. Instead, we investigate networks trained using quantization aware training (QAT), in which activations are quantized on the fly during training (Jacob et al., 2018). Specifically, forward passes are quantized while backward passes remain non-quantized. However, the IB and all MI computations in the quantized networks only refer to quantized quantities (i.e., forward pass computations) and thus measure the true information in the network at the considered training step. The actual quantization happens after each layer (after application of the activation function). During training, the algorithm keeps track of the minimum  $a_{l}^{(T)}$  and maximum  $a_{u}^{(T)}$  activations in each layer  $T$  across the epoch, and the activation of each neuron in  $T$  is mapped to one of  $2^{k}$  uniformly distributed values between  $a_{l}^{(T)}$  and  $a_{u}^{(T)}$ , before being passed to the next layer (Jacob et al., 2018). Thus for each epoch, the neurons in a layer  $T$  take at most  $2^{k}$  different values, meaning at most  $2^{kd_{T}}$  different states for  $T$  (because of the SOFTMAX activation function the effective number of states is  $2^{k(d_{Y}-1)}$  for the final layer  $\hat{Y}$ ).
+
+# 4 EXPERIMENTS
+
+We conducted the exact bottleneck analysis in two different setups:
+
+- The setting of Shwartz-Ziv & Tishby (2017) and Saxe et al. (2018) using the network shown in Figure 1a with either TANH or RELU activations, fitted on the same synthetic data set consisting of  $|\mathcal{D}| = 2^{12}$  12-bit binary input patterns with balanced binary output (Section 4.1).  
+- Learning a two-dimensional representation for the MNIST handwritten digits, where a fully connected network similar to the one in the previous setting using a bottleneck architecture with RELU activations for all hidden layers is trained (Section 4.2).
+
+For each setting, we applied  $k = 8$  bit quantization aware training to the given network. As in previous studies (Shwartz-Ziv & Tishby, 2017; Saxe et al., 2018), we used an  $80\% / 20\%$  training/test split to monitor fitting, while computing the MI based on the entire data set (training and test data). The networks were trained using mini batches of size 256 and the Adam optimizer with a learning rate of  $10^{-4}$ . Weights of layer  $T$  were initialized using a truncated normal with mean 0 and standard deviation  $1 / \sqrt{d_T}$ .
+
+In addition, we also replicated the experiments of Shwartz-Ziv & Tishby (2017) and Saxe et al. (2018) using varying number of bins, and we investigated 32- and 4-bit quantization. We also considered IB analysis of a few other architectures for the MNIST task, including architectures with wider bottlenecks, without a bottleneck, and using convolutions. These experiments are described in Supplementary Material A, C, and E.
+
+Our implementation is based on Tensorflow (Abadi et al., 2016). Experiments were run on an Intel Core i9-9900 CPU with 8 3.10GHz cores and a NVIDIA Quadro P2200 GPU.4.
+
+![](images/62fc9f4dd5b15a5c0cfc1e9ec687ee5c2087fc221fa2fcae3960260dd888419f.jpg)  
+Figure 2: The quantized TANH network, trained in the standard setting with the discrete MI computed exactly. The full information plane (left) and the upper right area (right) are shown. Plotted MI values are the means over the 50 repetitions.
+
+![](images/99cddfbcbc6e2595c57b0f98c803267716a49bb0d46eeec0ee8090dd39244eef.jpg)
+
+![](images/186d41e69119898784519c236a64a90f080c3ee022fdc4f674ff4243300f4ab6.jpg)  
+Figure 3: The quantized RELU network, trained in the standard setting with the discrete MI computed exactly. The full information plane (left) and the upper right area (right) are shown. Plotted MI values are the means over the 50 repetitions.
+
+![](images/d009202614aff9b51d898f5512c6e25a53a201888711393aeb4b453a3f5abcc9.jpg)
+
+# 4.1 EXACT MUTUAL INFORMATION FOR SYNTHETIC DATA
+
+The 8-bit quantized network from Figure 1a was fitted for 8000 epochs to the synthetic data set considered in Shwartz-Ziv & Tishby (2017), using either TANH or RELU for the hidden layers. The experiment was repeated 50 times and the means of obtained measurements are reported.
+
+Figure 2 and Figure 3 show the information planes for the quantized TANH and RELU networks respectively; additional plots showing the variance of the repetitions are included in Supplementary Material B. Both quantized networks obtained good fits with similar accuracies to non-quantized networks; see Figure F.14 and Figure F.15 in the supplementary material.
+
+For the quantized TANH network, the information plane is very similar to the plane obtained in the standard setting using  $2^{8} = 256$  bins. We clearly observe a fitting and compression phase in the output layer, while I  $(X;T)$  and I  $(T;Y)$  in general are large for the input and hidden layers. Looking at the hidden layer before the output layer, we do observe a fitting and a compression phase, but
+
+![](images/d86acc97267342784e51b8ce105665c2e950d9499f769bec5773fb37ad00ea85.jpg)  
+Figure 4: The information plane for the bottleneck structured network fitted on MNIST. Plotted MI values are the means over the 20 repetitions. The compression phase in the 2 neuron layer is highlighted.
+
+with the fitting phase interrupted by a small compression phase. For the remaining layers, the MI measurements with input and label remained almost constant throughout the training process.
+
+Likewise, the information plane for the quantized RELU network looks very similar to the plane obtained with binning. The MI is slightly higher, but the most notably difference is the occurrence of a slight increase in  $\mathrm{I}(X;T)$  after approximately 1800 epochs, observed only for the hidden layer before the output layer. For the remaining hidden layers, only the fitting phase is observed, with no or very little compression; an observation mirrored in the binning experiments by Saxe et al. (2018).
+
+# 4.2 EXACT MUTUAL INFORMATION FOR LEARNING AN MNIST REPRESENTATION
+
+We considered the task of leaning a two-dimensional representation of the MNIST handwritten digits (Bottou et al., 1994; Deng, 2012). We fitted a network with four hidden layers with RELU activations and 16, 8, 4 and 2 neurons, respectively, where the last two neurons feed into a 10 neuron SOFTMAX output layer. The network was trained for 3000 epochs, and the experiment was repeated 20 times.
+
+The bottleneck architecture for learning a two-dimensional representation leads to a non-trivial IB analysis. Using more expressive networks (e.g., without or with a wider bottleneck) resulted in trivial, constant maximum MI measurements (see Supplementary Material E).
+
+Figure 4 shows the information plane obtained for the network; additional plots showing the variance between repetitions are included in Supplementary Material B. The network exhibits sub-par performance due to the limiting bottleneck structure, obtaining a maximum accuracy of around  $90\%$  (see Figure F.16 in the supplementary material). However, the architecture does admit a non-trivial analysis with a clear fitting phase in the output layer and the three last hidden layers. Small compression phases are also observed in the RELU layers around 1500 epochs, similar to the last hidden layer of the network trained on synthetic data. The compression indicates that the learnt representations for digits from the same class become more similar.
+
+# 4.3 DISCUSSION
+
+Our experiments largely confirm the findings by Saxe et al. (2018): we observe no compression phase in the quantized RELU network for the synthetic data. While the quantized networks do not directly correspond to the networks in the original studies (Shwartz-Ziv & Tishby, 2017; Saxe et al., 2018), they amplify the effect of the networks being executed on a digital computer. Note that using quantization (as often done for mobile and edge computing as well as when working on encrypted data) does not violate any assumption in the IB theory – on the contrary, as argued before, it prevents the information theoretic measures from becoming trivial.
+
+The MI generally being higher in our experiments confirms our discussion from Section 3; information is well-preserved through the network. There is no longer a loss of information from the
+
+estimation method. All information loss – and hence the dynamics observed – is a result of the neural network learning. Performing the same experiments with the full 32-bit non-quantized networks, this is even more pronounced, as almost no loss of information is observed between layers. This setting approaches the continuous case, in which we would get  $H(T) = \log |\mathcal{D}|$  for invertible activation functions. We include information planes for 32-bit networks with exact MI computation in Supplementary Material C, Figure C.11. Conversely, when training networks with even lower precision, we see a higher information loss between layers. This can be expected from  $H(T) \leq 2^{d_Tk}$  and the small, decreasing number  $d_{T}$  of neurons in layer  $T$  (see Figure 1b). Using  $k = 4$  bit precision, more layers are clearly distinguishable, with results similar to those obtained by estimating MI by binning; however, the 4-bit quantized networks also provide a worse fit – indicating that when estimating MI by binning, information vital to the network is neglected in the analysis. We provide information planes for 4-bit quantized networks in Supplementary Material C, Figure C.10.
+
+A similar observation could be made for the network fitted on MNIST. Using the low complexity bottleneck architecture, we obtained non-trivial information measurements. Having only two neurons in the smallest layer limits the network in terms of performance, yielding a sub-optimal accuracy around  $90\%$ . Using more complex networks, for instance, with a wider bottleneck or without a bottleneck, we obtained accuracies over  $95\%$  (see Figure F.16), but, even when using 8-bit quantization, the information planes were trivial, with only the last layer discernible (see Supplementary Material E). We observed similar trivial information planes for a convolutional architecture, further supporting that only very little information is actually lost in the complex networks (even with quantization), making the information bottleneck, when computed exactly, a limited tool for analysing deep learning.
+
+# 5 CONCLUSIONS AND LIMITATIONS
+
+The experiments and discussion of Shwartz-Ziv & Tishby (2017) and Saxe et al. (2018), as well as several follow-up studies, indicate that estimating MI by binning may lead to strong artifacts. We empirically confirmed these issues (see Supplementary Material A): varying the number of bins in the discretization procedure, we observed large differences in the qualitative analysis of the same network, confirming that the results of previous studies highly depend on the estimation method.
+
+To alleviate these issues, we computed the exact MI in quantized neural networks. In our experiments using 8-bit quantized networks, the fitting phase was observed for TANH and RELU network architectures. The compression phase was observed in the output SOFTMAX layer in both cases. For the TANH network, the MI was only non-trivial for the last two layers, but not for the other hidden layers. We observed compression in the last hidden layer when using TANH activations; similar to the phase observed by Shwartz-Ziv & Tishby (2017). For the RELU network, the MI was below its upper bound in all layers; however, no compression was observed in any of the hidden layers using RELU activations. The latter observation confirms the statements by Saxe et al. (2018) about the IB in RELU networks.
+
+Applying the analysis to a network trained on MNIST, we found that a bottleneck architecture learning a low-dimensional representation showed a compression phase. The compression indicates that the learnt representations for digits from the same class become more similar, as intended for representation learning. When applied to networks without a bottleneck (e.g., a simple convolutional architecture), the IB analysis produced only trivial information planes, which shows the limitations of the IB analysis (even with exact computations) for deep learning in practice.
+
+The main contribution of our work is that we eliminated measurements artifacts from experiments studying the IB in neural networks. By doing so, future research on the IB concept for deep learning can focus on discussing the utility of the IB principle for understanding learning dynamics – instead of arguing about measurement errors.
+
+The proposed quantization as such is no conceptual limitation, because for invertible activation functions studying a (hypothetical) truly continuous system would lead to trivial results. The biggest limitation of our study is obviously that we could not (yet) resolve the controversy about the compression phase (which was, admittedly, our goal), because our experiments confirmed that it depends on the network architecture whether the phase is observed or not. However, the proposed framework now allows for accurate future studies of this and other IB phenomena.
+
+# REPRODUCIBILITY STATEMENT
+
+Source code and instructions for reproducing our experiments are available online: https://github.com/StephanLorenzen/ExactIBAnalysisInQNNs
+
+# ACKNOWLEDGEMENTS
+
+The authors would like to thank the anonymous reviewers for their constructive comments. SSL acknowledges funding by the Danish Ministry of Education and Science, Digital Pilot Hub and Skylab Digital. MN acknowledges support through grant NNF20SA0063138 from Novo Nordisk Fonden. CI acknowledges support by the Villum Foundation through the project Deep Learning and Remote Sensing for Unlocking Global Ecosystem Resource Dynamics (DeReEco). The authors acknowledge support by the Pioneer Centre for AI (The Danish National Research Foundation, grant no. P1).
+
+# REFERENCES
+
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+Vinayak Abrol and Jared Tanner. Information-bottleneck under mean field initialization. In International Conference on Machine Learning (ICML): Workshop on Uncertainty and Robustness in Deep Learning, 2020.  
+Alexander Amir Alemi, Ian S. Fischer, Joshua V. Dillon, and Kevin Murphy. Deep variational information bottleneck. arXiv, 1612.00410, 2017.  
+Rana Ali Amjad and Bernhard C. Geiger. Learning representations for neural network-based classification using the information bottleneck principle. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(9):2225-2239, 2019.  
+Emilio Rafael Balda, Arash Behboodi, and Rudolf Mathar. An information theoretic view on learning of artificial neural networks. In International Conference on Signal Processing and Communication Systems (ICSPCS), pp. 1-8. IEEE, 2018.  
+Emilio Rafael Balda, Arash Behboodi, and Rudolf Mathar. On the trajectory of stochastic gradient descent in the information plane, 2019. URL https://openreview.net/forum?id=SkMON20ctX.  
+Léon Bottou, Corinna Cortes, John S. Denker, Harris Drucker, Isabelle Guyon, L.D. Jackel, Yann LeCun, Urs A. Müller, Eduard Säckinger, Patrice Simard, and Vladimir Vapnik. Comparison of classifier methods: a case study in handwritten digit recognition. In International Conference on Pattern Recognition (ICPR), pp. 77-82, 1994.  
+Ivan Chelombiev, Conor Houghton, and Cian O'Donnell. Adaptive estimators show information compression in deep neural networks. In International Conference on Learning Representations (ICLR), 2019.  
+Hao Cheng, Dongze Lian, Shenghua Gao, and Yanlin Geng. Utilizing information bottleneck to evaluate the capability of deep neural networks for image classification. Entropy, 21(5):456, 2019.  
+Luke Nicholas Darlow and Amos Storkey. What information does a resnet compress? arXiv, 2003.06254, 2020.  
+Li Deng. The MNIST database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine, 29(6):141-142, 2012.  
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+
+Hui Fang, Victoria Wang, and Motonori Yamaguchi. Dissecting deep learning networks—visualizing mutual information. Entropy, 20(11):823, 2018.  
+Marylou Gabrie, Andre Manoel, Clément Luneau, Jean Barbier, Nicolas Macris, Florent Krzakala, and Lenka Zdeborova. Entropy and mutual information in models of deep neural networks. Journal of Statistical Mechanics: Theory and Experiment, 2019(12):124014, 2019.  
+Bernhard C. Geiger. On information plane analyses of neural network classifiers - a review. arXiv, 2003.09671, 2020.  
+Ziv Goldfeld, Ewout Van Den Berg, Kristjan Greenewald, Igor Melnyk, Nam Nguyen, Brian Kingsbury, and Yury Polyanskiy. Estimating information flow in deep neural networks. In International Conference on Machine Learning (ICML), pp. 2299-2308, 2019.  
+Itay Hubara, Matthieu Courbariaux, Daniel Soudry, Ran El-Yaniv, and Yoshua Bengio. Quantized neural networks: Training neural networks with low precision weights and activations. Journal of Machine Learning Research, 18(1):6869-6898, 2017.  
+Benoit Jacob, Skirmantas Kligys, Bo Chen, Menglong Zhu, Matthew Tang, Andrew Howard, Hartwig Adam, and Dmitry Kalenichenko. Quantization and training of neural networks for efficient integer-arithmetic-only inference. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 2704-2713, 2018.  
+Hlynur Jónsson, Giovanni Cherubini, and Evangelos Eleftheriou. Convergence behavior of dnns with mutual-information-based regularization. Entropy, 22(7):727, 2020.  
+Andreas Kirsch, Clare Lyle, and Yarin Gal. Scalable training with information bottleneck objectives. In International Conference on Machine Learning (ICML): Workshop on Uncertainty and Robustness in Deep Learning, 2020.  
+Artemy Kolchinsky and Brendan Tracey. Estimating mixture entropy with pairwise distances. Entropy, 19, 2017. doi: 10.3390/e19070361.  
+Alexander Kraskov, Harald Stögbauer, and Peter Grassberger. Estimating mutual information. Physical Review E, 69:066138, 2004.  
+Morteza Noshad, Yu Zeng, and Alfred O Hero. Scalable mutual information estimation using dependence graphs. In IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 2962-2966. IEEE, 2019.  
+Tomaso Poggio, Andrzej Banburski, and Qianli Liao. Theoretical issues in deep networks. Proceedings of the National Academy of Sciences, 117(48):30039-30045, 2020.  
+Vishnu Raj, Nancy Nayak, and Sheetal Kalyani. Understanding learning dynamics of binary neural networks via information bottleneck. arXiv, 2006.07522, 2020.  
+Andrew M. Saxe, Yamini Bansal, Joel Dapello, Madhu Advani, Artemy Kolchinsky, and Brendan D. Tracey. Discussion on openreview.net. https://openreview.net/forum?id=ry_WPG-A-, 2017. Accessed January 2017.  
+Andrew M. Saxe, Yamini Bansal, Joel Dapello, Madhu Advani, Artemy Kolchinsky, and Brendan D. Tracey. On the information bottleneck theory of deep learning. In International Conference on Learning Representations (ICLR), 2018.  
+Martin Schiemer and Juan Ye. Revisiting the information plane, 2020. URL https://openreview.net/forum?id=Hyljn1SFwr.  
+Ohad Shamir, Sivan Sabato, and Naftali Tishby. Learning and generalization with the information bottleneck. Theoretical Computer Science, 411(29):2696-2711, 2010.  
+Ravid Shwartz-Ziv and Alexander A Alemi. Information in infinite ensembles of infinitely-wide neural networks. In Symposium on Advances in Approximate Bayesian Inference, pp. 1-17, 2020.  
+Ravid Shwartz-Ziv and Naftali Tishby. Opening the black box of deep neural networks via information. arXiv, abs/1703.00810, 2017.
+
+Thanh Tang Nguyen and Jaesik Choi. Markov information bottleneck to improve information flow in stochastic neural networks. Entropy, 21(10):976, 2019.  
+Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. In 2015 IEEE Information Theory Workshop (ITW), pp. 1-5, 2015.  
+Naftali Tishby, Fernando C. Pereira, and William Bialek. The information bottleneck method. In *The 37'th Allerton Conference on Communication, Control and Computing*, 1999.  
+Kristoffer Wickström, Sigurd Løkse, Michael Kampffmeyer, Shujian Yu, Jose Principe, and Robert Jenssen. Information plane analysis of deep neural networks via matrix-based Renyi's entropy and tensor kernels. arXiv, 1909.11396, 2019.  
+Shujian Yu, Kristoffer Wickström, Robert Jenssen, and José C Principe. Understanding convolutional neural networks with information theory: An initial exploration. IEEE Transactions on Neural Networks and Learning Systems, 32:435-442, 2020.
+
+# A ESTIMATING MUTUAL INFORMATION BY BINNING
+
+This section presents a replication of the experiments from Shwartz-Ziv & Tishby (2017) and Saxe et al. (2018), but varying the number of bins for estimating MI. Non-quantized TANH and RELU networks were trained in the standard setting, and MI was estimated using 30, 100 and 256 bins. The number of bins were chosen to match the numbers used in the earlier works (30 used by Shwartz-Ziv & Tishby, 100 used by Saxe et al.) and to match the discretization in the quantized neural networks considered in Section 4.1.
+
+![](images/bbb61ed8f27afbab8b7dd13a5a0d6f2ff58b8a94a52529881c357ca5aadbe50e.jpg)  
+Figure A.5: Information planes for the TANH network with various number of bins. 30 bins corresponds to a replication of the results from Shwartz-Ziv & Tishby (2017). The network is trained in the standard setting, and the average of 50 runs is plotted.
+
+![](images/a289d373f8b1592c2ba9d9e5606a4eed885e5bb73d0ff656779446d0a5b75cb4.jpg)  
+Figure A.6: Information planes for the RELU network with various number of bins. 100 bins replicate the results from Saxe et al. (2018). The network is trained in the standard setting, and the average of 50 runs is plotted.
+
+The resulting information planes are shown for the TANH and RELU networks in Figure A.5 and Figure A.6 respectively.
+
+From Figure A.5, we clearly see the results of Shwartz-Ziv & Tishby (2017) replicated using 30 bins for the TANH network; two phases are clearly visible in each layer. As expected, for every layer  $T$  with binned  $T' = B(T)$ ,  $\mathrm{I}(X;T')$  and  $\mathrm{I}(T';Y)$  increase with the number of bins used with the phases still remaining visible in the distinguishable layers.
+
+For the RELU network with 100 bins, the results of Saxe et al. (2018) were also replicated, that is, the compression phase was not observed. When using only 30 bins for the RELU network, the estimation broke down completely in the sense that the DPI (2) is violated: I  $(T'; Y)$  is non-monotone, a phenomenon occurring because of the estimation, which has also been observed by Geiger (2020). Again, we see for any layer  $T$  that I  $(X; T')$  and I  $(T'; Y)$  increase with  $m$ , although to a lesser degree
+
+![](images/f564b2e2d8d9cfa7c248e21b5303ad4fc1fcfebeb1815162b4449303bdf801d6.jpg)  
+Figure B.7: Mean and variance for  $\operatorname{I}(X;T)$  (left) and  $\operatorname{I}(T;Y)$  (center), and the information plane for the median deviating (L2-distance from the mean) repetition (right), for the TANH network in the standard setting.
+
+![](images/eb3a98582b872c8f5a003ba3f442601dbffafba8fcc17df501baf0b490f5a725.jpg)
+
+![](images/2b66a0e8780cf09079f799a144a3b734d06bc06defa6de544176c7329a1a36b1.jpg)
+
+![](images/c073922d4198ebe606b1f52a5171b448d7b362394bd7c7b72417acf364d8ee4f.jpg)  
+Figure B.8: Mean and variance for  $\mathrm{I}(X;T)$  (left) and  $\mathrm{I}(T;Y)$  (center), and the information plane for the median deviating (L2-distance from the mean) repetition (right), for the RELU network in the standard setting.
+
+![](images/f132ac437d65a23fee270d56a3dfa924fe0a5b2ac28c48545bd7487245274935.jpg)
+
+![](images/8fae22b87fbb08a8eac469c9f3c24686a3e0099c10963ef3211c7dc74f63b8a6.jpg)
+
+than for the TANH network, most likely due to the RELU activation functions actually dropping information (by zeroing negative activations).
+
+In general, the estimation of MI in either network is highly dependable on the number of bins used, indicating that the interpretation of the information plane is not straight forward, when using binning for estimation.
+
+# B MI VARIANCE
+
+In this work, we investigate the effect of MI approximation in previous IB studies. These original studies are based on a qualitative analysis of mean curves. However, the question arises of whether mean curves correctly represent individual learning processes. While we do not investigate this question in detail, the following section provides plots illustrating the variance between experimental trials.
+
+Figures B.7, B.8 and B.9 show the mean and variance of  $\mathrm{I}(X;T)$  (left) and  $\mathrm{I}(T;Y)$  (center), as well as the information plane for a single trial (right), for the TANH and RELU networks in the standard setting, and the bottleneck network applied to MNIST, respectively. For the single trial, we selected the median deviating repetition, ranked by the L2-distance from the mean.
+
+From the plots, we see that when the mean MI is large, the variance is usually low. The variance is similar across the different networks (note the different scaling of the y-axes), with slightly larger variance for the RELU activations. Together with the median deviating information plane, the plots suggest that the mean information plane, which averages out less pronounced phases of MI increase and decrease, represents individual repetitions fairly well.
+
+![](images/4edec9b6f45ae60da0810b191f19315fe6394fe92bd4ff77897a6b430e66a332.jpg)  
+Figure B.9: Mean and variance for  $\mathrm{I}(X;T)$  (left) and  $\mathrm{I}(T;Y)$  (center), and the information plane for the median deviating (L2-distance from the mean) repetition (right), for the Bottleneck network applied to MNIST.
+
+![](images/9bed0ad33e86d346e1b42b8f4dc731afa0a7d504232135b22017440d0f6daa4f.jpg)
+
+![](images/712b0dad6e38819455a19a9b9114f3b1489ec3cca1400a4786c119a9dec64170.jpg)
+
+![](images/8edcd25fba53330d88d9404f8fd8f50d013890a3a6737249a1b6f2f77b4a9ec8.jpg)  
+Figure C.10: Information planes for 4-bit quantized TANH network (left) and RELU network (right). The mean of 30 runs is plotted.
+
+![](images/da469bd34eb22a4e62ae9a015708c73e860ced8b5aa0473328db96a68daf5878.jpg)
+
+Additionally, the compression phase in the TANH network can clearly be seen in Figure B.7 (left), while it is absent in the RELU network (Figure B.8).
+
+# C EFFECT OF THE QUANTIZATION PRECISION
+
+This section presents information planes obtained using 4- and 32-bit quantized neural networks trained in the standard setting, in order to investigate the effect of the precision used in the quantization.
+
+Figure C.10 depicts the resulting information planes for the 4-bit networks. As each neuron can take only  $2^{4} = 16$  different values, the total number of possible states per layer decreases significantly in this setting, and as expected we see lower overall MI measurements. For the TANH network, several more layers become distinguishable. The observed information planes look similar to those observed in the original experiments by Shwartz-Ziv & Tishby (2017) and Saxe et al. (2018). However, the network accuracy has now degraded compared to the non-quantized networks (see Supplementary Material F), which indicates that the binning used in the estimation of the MI in previous experiments has discarded information vital to the network.
+
+Figure C.11 shows the resulting information planes for the 32-bit networks. As expected, we see an overall increase in MI; the information drops only very slowly through the network. Each layer has many possible states and – given the small data set – we get closer to the behavior of a continuous system.
+
+![](images/556727a588b145e80d194e4c8ca203d1f46fbfb9d6ff8b5cf93458d632ace470.jpg)  
+Figure C.11: Information planes for 32-bit quantized TANH network (left) and RELU network (right). The mean of 30 runs is plotted.
+
+![](images/6d8a5fa22b195384258e8a26efc41d14db3e099731bdbe83b8c31b51b62352d6.jpg)
+
+![](images/a545d3c895e7dbac4958dd1445e7db855cdbd00d6ae4004273d9f18e628f7785.jpg)  
+Figure D.12: The information planes for the TANH (left) and RELU (right) networks when prefetching on random labels for 1000 epochs. The means of 20 repetitions are shown. In the plots,  $Y$  always refers to the true labels.
+
+![](images/46c7383270c637765b2e673ffa63d2655c92b7728ad16c003ae8830f89a14b2d.jpg)
+
+# D QUANTIZED NETWORK WITH RANDOMIZED PREFITTING
+
+This section presents results for the standard setting using 8-bit quantized networks, but with weights prefitted to random labels. First, the network is fitted for 1000 epochs to the synthetic data set with the labels shuffled. The network is then trained for 8000 epochs with correct labels (as done in Section 4.1). The experiment is repeated 20 times. The resulting information planes for the TANH and the RELU network are presented in Figure D.12, left and right respectively.
+
+Inspecting the plots, we see that, for all discernible layers,  $\mathrm{I}(X;T)$  appears to increase when fitted to random labels. This is not surprising as the input is not shuffled.
+
+As a consequence of the increase in  $\mathrm{I}(X;T)$ , the hidden layers (most notable in the RELU network) also see an increase in  $\mathrm{I}(T;Y)$ . As the ability to distinguish different inputs from a hidden layer  $T$  increases, so does  $\mathrm{I}(T;Y)$ .
+
+As expected, the network accuracy does not increase when training on random labels, and accordingly there is no lateral movement in the information plane for the output layer.
+
+![](images/0813f7d718c00b6397141ca65dd2906e7499b8d92f3490f33625df0aa6688ddf.jpg)
+
+![](images/f30632f0506eadfbc0403d3dc95774fceaac03b316b993e3d374530e5728fb87.jpg)
+
+![](images/a97f88ca37a3b302b872b7ee79fd4001ae4ff0df5ac18138f6537d3f7ada159c.jpg)  
+Figure E.13: Information planes for the additional networks trained on MNIST. Note for HOURGLASS that the three last hidden layers almost coincide completely.
+
+![](images/7ef57e0c6cc742d2a1e52766a1f63365493c03ddb479be9d0e749f215e6f5a91.jpg)
+
+# E MNIST WITH OTHER NETWORK ARCHITECTURES
+
+This section presents the results of training a number of 8-bit quantized networks with varying architectures on MNIST. We consider the same setup as in Section 4.2, but with the following architectures (all using RELU activations in the hidden layers and a final 10 layer SOFTMAX output):
+
+- The BOTTLENECK-4 network with 4 hidden layers of widths 16, 12, 8 and 4, i.e. the bottleneck has width 4.  
+- The HOURGLASS network with 6 hidden layers of widths 16, 8, 4, 2, 4, 8. The bottleneck width is still 2, but the network expands again after the bottleneck, creating an hourglass shape.  
+- The  $4 \times 10$  network with 4 hidden layers, each of width 10.  
+- The CONV network, a simple convolutional network with structure: CV-MP-CV-MP-FC, where CV denotes a (3,3) 2-channel convolutional layer, MP denotes a (2,2) max pooling and FC denotes a fully connected RELU layer with 20 neurons.
+
+Below, we denote the network used in Section 4.2 by BOTTLENECK-2. Each experiment is repeated 20 times and the resulting information planes are reported in Figure E.13. As expected, given the increased complexity, all networks exhibit better performance than the bottleneck structured network considered in Section 4.2 (the accuracy of all networks are presented in Figure F.16 in Supplementary
+
+Material F), obtaining accuracies close to or above  $95\%$  for all networks except the HOURGLASS network.
+
+From Figure E.13, we observe the information curves for the BOTTLENECK-4 network to be similar in shape to the BOTTLENECK-2, but with higher MI measurements in general. This is not surprising, as the layers are wider and thus more expressive.
+
+As expected, the HOURGLASS network also exhibits information curves similar to BOTTLENECK-2 (as the architecture is similar for the first few layers). Not surprisingly, the two expanding layers of widths 4 and 8 almost coincide completely with the bottleneck layer; information cannot increase but is also not decreased significantly.
+
+The two more expressive networks without bottlenecks, 4x10 and CONV, have almost completely trivial information curves. Limited fitting for the output layer in 4x10 and compression in the output layers for both networks can be observed. The experiments indicate the limitations of exact IB analysis for complex, large scale networks.
+
+# F NETWORK ACCURACIES
+
+![](images/cfb3f7d5f6c46c3b5d567f143302009660d7ce6869256624f786dde462cc5127.jpg)  
+Figure F.14 and Figure F.15 report the accuracies obtained by the TANH and RELU networks when fitted on the synthetic data, respectively. As can be seen, the networks suffers only slightly when  
+Figure F.14: Accuracies of the quantized TANH networks (three on the left) and the non-quantized TANH network (right). The means and  $95\%$  confidence intervals over 50 repetitions are reported.
+
+![](images/6e70f5433c200187113759b2f3c57432e85f048a58afe367622379b07d3f2763.jpg)  
+Figure F.15: Accuracies of the quantized RELU networks (three on the left) and the non-quantized RELU network (right). The means and  $95\%$  confidence intervals over 50 repetitions are reported.
+
+using only 4 bits in the quantization.
+
+Figure F.16 reports the accuracy of the networks fitted to MNIST. Unsurprisingly, the more complex networks (BOTTLNECK-4, 4x10, CONV) obtain better accuracy with lower variance, compared to the networks with a low-width bottleneck (BOTTLNECK-2, HOURGLASS).
+
+![](images/d9ac575e3754f93a00b08569bc06ea31b8b4c4d3d496cd2323177b4c58c93b7d.jpg)  
+Figure F.16: Accuracies of the quantized networks applied to MNIST. The means and  $95\%$  confidence intervals over 20 repetitions are reported. The decrease in training accuracy for the bottleneck structured network (left) is due to the optimization objective being cross-entropy.

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+# INFORMATION GAIN PROPAGATION: A NEW WAY TO GRAPH ACTIVE LEARNING WITH SOFT LABELS
+
+Wentao Zhang $^{1}$ , Yexin Wang $^{1}$ , Zhenbang You $^{1}$ , Meng Cao $^{2}$ , Ping Huang $^{2}$ , Jiulong Shan $^{2}$ , Zhi Yang $^{1,3}$ , Bin Cui $^{1,3,4}$
+
+<sup>1</sup>School of CS, Peking University <sup>2</sup>Apple  
+$^{3}$  National Engineering Laboratory for Big Data Analysis and Applications  
+$^{4}$ Institute of Computational Social Science, Peking University (Qingdao), China  
+$^{1}$  {wentao.zhang, yexinwang, zhenbangyou, liyang.cs, yangzhi, bin.cui} @oku.edu.cn  
+2{mengcao,Huang.ping,jlshan}  $@$  apple.com
+
+# ABSTRACT
+
+Graph Neural Networks (GNNs) have achieved great success in various tasks, but their performance highly relies on a large number of labeled nodes, which typically requires considerable human effort. GNN-based Active Learning (AL) methods are proposed to improve the labeling efficiency by selecting the most valuable nodes to label. Existing methods assume an oracle can correctly categorize all the selected nodes and thus just focus on the node selection. However, such an exact labeling task is costly, especially when the categorization is out of the domain of individual expert (oracle). The paper goes further, presenting a soft-label approach to AL on GNNs. Our key innovations are: i) relaxed queries where a domain expert (oracle) only judges the correctness of the predicted labels (a binary question) rather than identifying the exact class (a multi-class question), and ii) new criteria of maximizing information gain propagation for active learner with relaxed queries and soft labels. Empirical studies on public datasets demonstrate that our method significantly outperforms the state-of-the-art GNN-based AL methods in terms of both accuracy and labeling cost.
+
+# 1 INTRODUCTION
+
+Graph Neural Networks (GNNs) have recently achieved remarkable success in various graph-based tasks, ranging from traffic networks, biology, to social networks (Zhang et al., 2020b; Wang et al., 2019; Li et al., 2019; Do et al., 2019; Wu et al., 2020b). Despite their effectiveness and popularity, GNNs typically require a large amount of labeled data to achieve satisfactory performance. However, obtaining these labels is a time-consuming, laborious, and costly process. Therefore, how to do this economically and efficiently has attracted great attention both in academia and industry. One of the most popular strategies to tackle this challenge is Active Learning (AL) (Aggarwal et al., 2014). By combining model training and node selection for labeling, AL significantly reduces labeling cost by selecting the most valuable nodes to label.
+
+However, previous AL methods assume the hard label (namely the exact label, that is, the label specifying the exact class of the node) can always be provided by the oracle. As illustrated in Fig. 1, for any selected node, the active learners in the aforementioned methods ask questions like "which category does it exactly belong to?". Such queries and assumptions exceed the capability of oracle in many labeling tasks requiring domain knowledge. For example, the task in ogbn-papers100M is to leverage the citation graph to infer the labels of the arXiv papers into 172 arXiv subject areas (a single-label, 172-class classification problem). In this example, a specialist/expert in the subject areas of machine learning is incapable of labeling query instances with subject areas of finance (such as mathematical finance or computational finance), which is out of his domain knowledge.
+
+In this paper, we propose a novel active learning paradigm for GNNs in which only soft labels are required. Two salient features of our paradigm are as follows:
+
+![](images/116fdd03ba552e8e9abf0d3d62875e510340671d548a0de5f7a30c4eb8fb4e1e.jpg)  
+Figure 1: An example of our new node labeling strategy.
+
+![](images/4b01925e5b711dfbdc9eb69ede29c1ed72b84cf7350fa9431e4fa8d02e8e42b4.jpg)  
+Figure 2: An example of IGP.
+
+First, we propose relaxed queries where domain expert (oracle) only judges the correctness of the predicted labels (by GNN models) rather than the exact categorization. Specifically, we propose to select the class label with the highest predicted probability and then ask the oracle to judge whether the class label is correct rather than directly asking for the exact class label. In this way, the multi-class classification task is relaxed into a binary-class classification task for the oracle. Under such queries, if the model prediction is incorrect (we assume the oracle does not make mistakes), we annotate the node with the soft label by re-normalizing the model predicted softmax outputs over the remaining classes. As illustrated in Fig.1, we allow relaxed queries such as "Does the sample belong to this class?" (e.g., the first class in Fig. 1), since this query is specific to the domain of experts, hence more ready to be answered. As far as we are concerned, no previous AL methods can deal with such relaxed queries.
+
+Second, the node selection criteria in previous AL methods are specially designed for hard-labeling oracles. As one of the most commonly used AL query strategies, uncertainty sampling queries the labels of the nodes which current model is least certain with, ensuring the largest entropy reduction on a single node when given the hard label from oracle. However, under our relaxed queries, we prove that the uncertainty sampling cannot guarantee the largest information gain (i.e., entropy reduction) on a single node. In this paper, we propose a new criterion to explicitly maximize information gain propagation (IGP) for active learners with relaxed queries and soft labels on graph. As shown in Fig. 2, the labeled nodes (with color) can propagate their label information with different influence magnitude and then reduce the uncertainty of adjacent unlabeled nodes (without color). Considering the influence propagation in GNNs, we select the nodes that can maximize expected information gain in total under the relaxed queries, i.e., reducing the aggregated uncertainty of a neighborhood region beyond a single node. To the best of our knowledge, this is the first work to propose the information gain propagation with relaxed queries, and showing that it is highly effective.
+
+In summary, the core contributions of this work are the following: First, we provide a new AL paradigm for GNNs methodology with relaxed queries and soft labels. Second, we propose a novel node selection criterion that explicitly maximizes propagation of information gain under the new AL paradigm on GNNs. Third, experimental results show our paradigm significantly outperforms the compared baselines by a large margin in five open graph datasets.
+
+# 2 PRELIMINARY
+
+# 2.1 PROBLEM FORMULATION
+
+Let  $\mathcal{G} = (\mathcal{V},\mathcal{E})$  with  $|\mathcal{V}| = N$  nodes and  $|\mathcal{E}| = M$  edges be the graph, and  $c$  is the number of classes.  $\mathbf{X} = \{\pmb{x}_1,\pmb{x}_2\dots,\pmb{x}_N\}$  is the node feature matrix, the one-hot vector  $\pmb{y}_i\in \mathbb{R}^c$  and and  $\pmb{x}_i\in \mathbb{R}^d$  are the ground-truth label and node feature for node  $v_{i}\in \mathcal{V}$ , respectively. We consider a general AL setting in this work. Suppose the full node set  $\mathcal{V}$  is partitioned into training set  $\mathcal{V}_{train}$ , validation set  $\mathcal{V}_{val}$ , and test set  $\mathcal{V}_{test}$ , and  $\mathcal{V}_{train}$  can further be partitioned into the labeled set  $\mathcal{V}_l$  and unlabeled set  $\mathcal{V}_u$ , respectively. Given the labeling size  $\mathcal{B}$ , the loss function  $\ell$ , the goal of graph-based AL is to select a subset  $\mathcal{V}_l\subset \mathcal{V}_{train}$  to label, so that the model  $f$  trained with the supervision of  $\mathcal{V}_l$  can get the lowest loss on  $\mathcal{V}_{test}$ :
+
+$$
+\underset {\mathcal {V} _ {l}: \left| \mathcal {V} _ {l} \right| = \mathcal {B}} {\arg \min } \mathbb {E} _ {v _ {i} \in \mathcal {V} _ {t e s t}} \left[ \ell \left(\boldsymbol {y} _ {i}, \hat {\boldsymbol {y}} _ {i}\right) \right], \tag {1}
+$$
+
+where  $\hat{y}_i$  is the softmax outputs of node  $v_i$  given by the model  $f$ .
+
+Different from the previous setting,  $\mathcal{B}$  is the labeling cost (i.e., money) rather than the size in our new scenario. As the experts are hard to annotate the node out of their domain, we assume the conventional
+
+query cost  $c - 1$  times as much as our relaxed query for each selected node. Correspondingly, the objective in our new scenario is
+
+$$
+\underset {\mathcal {V} _ {l}: M (\mathcal {V} _ {l}) = \mathcal {B}} {\arg \min } \mathbb {E} _ {v _ {i} \in \mathcal {V} _ {t e s t}} \left[ \ell \left(\boldsymbol {y} _ {i}, \hat {\boldsymbol {y}} _ {i}\right) \right], \tag {2}
+$$
+
+where  $M(\mathcal{V}_l)$  is the labeling cost for annotating  $\mathcal{V}_l$ , and we focus on  $f$  being GNNs due to their state-of-the-art performance in many semi-supervised node classification tasks.
+
+# 2.2 GRAPH NEURAL NETWORKS
+
+Unlike the images, text or tabular data, where training data are independently and identically distributed, samples are connected by edges in the graph, and each node in a GNN can aggregate the node embedding or feature from its neighbors to enhance their embedding along edges. Let  $f\left( {\mathbf{A},{\mathbf{X}}^{\left( k\right) };{\mathbf{W}}^{\left( k\right) }}\right)$  be any GNN,where  $\mathbf{A}$  is an adjacency matrix,  ${\mathbf{X}}^{\left( k\right) }$  is the feature matrix of  $k$  -th layer,and  ${\mathbf{W}}^{\left( k\right) }$  are the learned weights of  $k$  -th layer.
+
+Taking the widely used Graph Convolution Network (GCN) (Kipf & Welling, 2017a) as an example, each GCN layer can be formulated as:
+
+$$
+\mathbf {X} ^ {(k + 1)} = f \left(\mathbf {A}, \mathbf {X} ^ {(k)}, \mathbf {W} ^ {(k)}\right) = \delta \left(\widetilde {\mathbf {D}} ^ {- 1} \widetilde {\mathbf {A}} \mathbf {X} ^ {(k)} \mathbf {W} ^ {(k)}\right), \tag {3}
+$$
+
+where  $\mathbf{X}$  (and  $\mathbf{X}^{(0)}$ ) is the original node feature,  $\mathbf{X}^{(k)}$  and  $\mathbf{X}^{(k + 1)}$  are the embeddings of layer  $k$  and  $k + 1$  respectively. Besides,  $\widetilde{\mathbf{D}}$  is the diagonal node degree matrix for normalization and  $\widetilde{\mathbf{A}} = \mathbf{A} + \mathbf{I}_N$  is the adjacent matrix with self connection, where  $\mathbf{I}_N$  is the identity matrix. Compared with the DNN layer which formulated as  $\mathbf{X}^{(k + 1)} = \delta (\mathbf{X}^{(k)}\mathbf{W}^{(k)})$ , the feature matrix  $\mathbf{X}^{(k)}$  will be firstly enhanced with the feature propagation operation  $\widetilde{\mathbf{D}}^{-1}\widetilde{\mathbf{A}}\mathbf{X}^{(k)}$ , and then boosts the semi-supervised node classification performance by getting more nodes involved in the model training.
+
+# 2.3 ACTIVE LEARNING
+
+Common AL. AL can improve labeling efficiency by selecting the most valuable samples to label. Considering the informativeness, Uncertainty Sampling (Yang et al., 2015; Zhu et al., 2008) selects the nodes which has the most uncertain model prediction. Based on ensembles (Zhang et al., 2020a), Query-by-Committee (Burbidge et al., 2007; Melville & Mooney, 2004) trains a committee of models, and selects samples according to the extent to which the models disagree. Furthermore, Density-based (Zhu et al., 2008; Tang et al., 2002) and Clustering-based (Du et al., 2015; Nguyen & Smeulders, 2004) and Diversity-based (Wu et al., 2020a; Jiang & Gupta, 2021) methods can effectively select the most representative samples. All the methods above are proposed for binary or multi-class classification, and some methods (Qi et al., 2008; Yan & Huang, 2018) are proposed for the multi-label classification problem. Besides, considering the class imbalance, some methods (Aggarwal et al., 2020; Attenberg & Provost, 2010) are also proposed to tackle this issue with AL.
+
+GNN-based AL. Despite the effectiveness of common AL methods, it is unsuitable to directly apply them to GNNs since the characteristic of influence propagation has not been considered. To tackle this issue, both AGE (Cai et al., 2017) and ANRMAB (Gao et al., 2018) introduce the density of node embedding and PageRank centrality into the node selection criterion. SEAL (Li et al., 2020) devises a novel AL query strategy in an adversarial way, and RIM (Zhang et al., 2021b) considers the noisy oracle in the node labeling process of graph data. Besides, ALG (Zhang et al., 2021a) proposes to maximize the effectiveness of all influenced nodes of GNNs. Recently, Grain (Zhang et al., 2021c) introduces a new diversity influence maximization objective, which takes the diversity of influenced nodes into consideration.
+
+Both AGE (Cai et al., 2017) and ANRMAB (Gao et al., 2018) are designed for AL on GNNs, which adopt the uncertainty, density, and node degree to select nodes. ANRMAB improves AGE by introducing a multi-armed bandit mechanism for adaptive decision-making. Considering the high training time of GNN, ALG (Zhang et al., 2021a) decouples the GNN model and proposes a new node selection metric that maximizes the effective reception field. Grain (Zhang et al., 2021c) further generalizes the reception field to the number of activated nodes in social influence maximization and introduces the diversity influence maximization for node selection. However, all these AL methods
+
+assume the oracle knows the information of all categories, thereby being able to correctly annotate all the selected nodes. This is impractical if the oracle encounters some areas that they are not familiar with, especially when the number of categories is large. IGP is the first work to actively labeling graph data under domain-specific experts.
+
+# 3 PROPOSED METHOD
+
+This section presents IGP, the first graph-based AL framework that considers the relaxed queries and soft labels. IGP can operate on any GNN model  $f$ . We measure the information gain of selecting a single node in Sec. 3.1. To extend the information gain of a single node to the full graph, we firstly estimate the influence magnitude of graph propagation in Sec. 3.2, and then combine it with the information gain and introduce the new criterion: information gain propagation in Sec. 3.3. Last, we propose to select nodes which can maximize the information gain propagation in Sec. 3.4.
+
+# 3.1 INFORMATION GAIN
+
+Different from the previous annotation method, which asks the oracle to directly provide annotated class  $l$ , the oracle in IGP only needs to judge whether the model prediction  $\hat{\pmb{y}}$  is correct. Rather than discard the incorrect prediction, we propose that both the correct and incorrect predictions will provide the information gain and thus enhance the label supervision.
+
+An oracle is required to answer whether the model prediction is correct, i.e.,  $\hat{y}$  belongs to class  $l$ . After the judgement of the oracle, the uncertainty of the node labels will be decreased. Therefore, we define the normalized soft label by combining the model prediction and oracle judgement as follows.
+
+Definition 3.1 (Normalized Label). For node  $v_{i}$ , the soft label (model predicted softmax output) is denoted as  $\hat{y}_i = [a_1, a_2, \dots, a_c]$ . The pseudo label (denoted as  $l$ ) is set as the class with the highest value in  $\hat{y}_i$ . After the oracle judges whether  $v_{i}$  belongs to class  $l$ , each element  $a_t'$  in the normalized soft label  $\hat{y}_i' = [a_1', a_2', \dots, a_c']$  is defined as
+
+$$
+a _ {t} ^ {\prime} = \left\{ \begin{array}{l l} \mathbb {1} (t = l) & v _ {i} \text {b e l o n g s t o c l a s s} l \\ \frac {a _ {t} \cdot \mathbf {1} (t = l)}{\sum_ {j \neq l} a _ {j}} & v _ {i} \text {d o e s n o t b e l o n g t o c l a s s} l \end{array} \right. \tag {4}
+$$
+
+After the query,  $\hat{\pmb{y}}_i$  is updated as the normalized soft label  $\hat{\pmb{y}}_i^{\prime}$  by Eq. 4 (See more details in Appendix B).
+
+The oracle may agree with the pseudo label (denoted as  $v_{i+}$ ), or disagree (denoted as  $v_{i-}$ ). In either cases, the entropy drops.
+
+The expected information gain (IG) is calculated based on these two cases.
+
+Definition 3.2 (Information Gain (IG)). For node  $v_{i}$ , the IG of annotating it is defined as
+
+$$
+I G (v _ {i}) = H (\hat {\mathbf {y}} _ {i}) - P (v _ {i} -) H \left(\hat {\mathbf {y}} _ {i} ^ {\prime}, v _ {i} -\right), \tag {5}
+$$
+
+where  $H$  is the entropy (well-known in the information theory), e.g.,  $H(\hat{\mathbf{y}}_i) = -\sum_{p=1}^k a_p \log a_p$  ( $\log$  always stands for  $\log_2$  in this paper). Therefore,  $H(\hat{\mathbf{y}}_i)$  and  $H(\hat{\mathbf{y}}_i')$  stand for the entropy of  $v_i$  before and after annotation respectively. Also,  $H(\hat{\mathbf{y}}_i', v_i+)$  and  $H(\hat{\mathbf{y}}_i', v_i-)$  are the two cases of  $H(\hat{\mathbf{y}}_i')$  where  $v_i+$  and  $v_i-$  occurs respectively.
+
+IG is exactly the expectation of entropy reduction, as proved in Appendix C.
+
+The confidence of the model in the pseudo label is shown in the soft label. Hence, before the annotation, this is a good measurement for the probability that the oracle will agree with the pseudo label, namely  $P(v_{i} + )$ .
+
+Here following theorem shows the difference between IG and entropy.
+
+Theorem 3.1. The node with the highest entropy is not necessarily the node that brings the highest IG.
+
+The proof of this theorem can be found in Appendix A.
+
+More examples about IG in differing cases, that is, the oracle may agree or disagree with the pseudo label, can be found in Appendix D.
+
+# 3.2 INFLUENCE MAGNITUDE ESTIMATION
+
+Each newly annotated node will propagate its label information to its  $k$ -hop neighbors in a  $k$ -layer GNN and correspondingly influences the label distribution of these nodes. Since the influence of a node on its different neighbors can be various, we measure it by calculating how much change in the input label of node  $v_{i}$  affects the propagated label of node  $v_{j}$  after  $k$ -steps propagation (Wang & Leskovec, 2020; Xu et al., 2018).
+
+Definition 3.3 (Influence Magnitude). The influence magnitude score of node  $v_{i}$  on node  $v_{j}$  after  $k$ -step propagation is the L1-norm of the expected Jacobian matrix  $\hat{I}_{f}(v_{j}, v_{i}, k) = \left\| \mathbb{E}[\partial \mathbf{X}_{j}^{(k)} / \partial \mathbf{X}_{i}^{(0)}] \right\|_{1}$ . Formally, we normalize the influence magnitude score as:
+
+$$
+I _ {f} \left(v _ {j}, v _ {i}, k\right) = \frac {\hat {I _ {f}} \left(v _ {j} , v _ {i} , k\right)}{\sum_ {v _ {w} \in \mathcal {V}} \hat {I _ {f}} \left(v _ {j} , v _ {w} , k\right)}. \tag {6}
+$$
+
+For a  $k$ -layer GNN, the influence magnitude score  $I_{f}(v_{j}, v_{i}, k)$  captures the sum over probabilities of all possible influential paths with length of  $k$  from node  $v_{i}$  to  $v_{j}$ . Intuitively, node  $v_{i}$  will influence  $v_{j}$  by a larger extent and thus  $I_{f}(v_{j}, v_{i}, k)$  is larger if  $v_{i}$  is easier to achieve at  $v_{j}$  after the random walk. In our AL setting, larger  $I_{f}(v_{j}, v_{i}, k)$  means the label distribution of node  $v_{j}$  will be influenced more by node  $v_{i}$  if  $v_{i}$  is labeled.
+
+# 3.3 INFORMATION GAIN PROPAGATION
+
+For a  $k$ -layer GNN, the supervision signal of one node can be propagated to its  $k$ -hop neighborhood nodes and then influences the label distribution of these neighbors. Therefore, it is unsuitable to just consider the information gain of the node itself in the node selection process. So, we extend the information gain of a single node defined in Def. 3.2 to its  $k$ -hop neighbors in the graph. We firstly combine the influence magnitude with the normalized soft label and then measure the information gain propagation of all the influenced nodes. Concretely, the information gain propagation of node  $v_{j}$  from the labeled node  $v_{i}$  is defined as follows.
+
+Definition 3.4 (information gain propagation). Given the model predicted softmax outputs  $\hat{y}_i$  of node  $v_i$ , the influence magnitude  $I_f(v_j, v_i, k)$  of node  $v_i$  on  $v_j$ , the information gain propagation of node  $v_j$  from labeling node  $v_i$  is defined as:
+
+$$
+I G P \left(v _ {j}, v _ {i}, k\right) = H \left(\sum_ {v _ {m} \in \mathcal {V} _ {l}} I _ {f} \left(v _ {j}, v _ {m}, k\right) \hat {\boldsymbol {y}} _ {m} ^ {\prime}\right) - H \left(\sum_ {v _ {m} \in \mathcal {V} _ {l} \cup \{v _ {i} \}} I _ {f} \left(v _ {j}, v _ {m}, k\right) \hat {\boldsymbol {y}} _ {m} ^ {\prime}\right). \tag {7}
+$$
+
+Since the new soft label  $\hat{\pmb{y}}_i^{\prime}$  is unavailable for node  $v_{i}$  before the annotation, the expected information gain propagation (IGP) is calculated by
+
+$$
+I G P \left(v _ {j}, v _ {i}, k\right) = P \left(v _ {i} +\right) I G P \left(v _ {j}, v _ {i}, k, v _ {i} +\right) + P \left(v _ {i} -\right) I G P \left(v _ {j}, v _ {i}, k, v _ {i} -\right), \tag {8}
+$$
+
+where  $IGP(v_{j}, v_{i}, k, v_{i+})$  is the IGP when the oracle agrees with the pseudo label. In this case,  $\hat{\pmb{y}}_{i}^{\prime}$  is a one-hot vector according to E.q. 4.
+
+# 3.4 MAXIMIZING INFORMATION GAIN PROPAGATION
+
+To maximize the uncertainty of all the influenced nodes in the semi-supervised GNN training, we aim to select and annotate a subset  $\mathcal{V}_l$  from  $\mathcal{V}$  so that the maximum information gain propagation can be achieved according to Eq. 9. Specifically, we optimize this problem by defining the information gain propagation maximization objective as below.
+
+Objective. Specifically, we propose to maximize the following objective:
+
+$$
+\max  _ {\mathcal {V} _ {l}} F (\mathcal {V} _ {l}) = \sum_ {v _ {i} \in \mathcal {V} _ {l}} \sum_ {v _ {j} \in R F (v _ {i})} I G P (v _ {j}, v _ {i}, k), \mathbf {s}. \mathbf {t}. \mathcal {V} _ {l} \subseteq \mathcal {V}, | \mathcal {V} _ {l} | = \mathcal {B}. \tag {9}
+$$
+
+where  $RF(v_{i})$  is the receptive field of node  $v_{i}$ , i.e., the node itself with its  $k$ -hop neighbors in a  $k$ -layer GNN. Considering the influence propagation, the proposed objective can be used to find a subset  $\mathcal{V}_{l}$  that can maximize the information gain of all influenced nodes as more as possible.
+
+![](images/598f3a4fd948c812a77430215b3b94ee6af35b7c18adc7cbccb4acee92a4f1e8.jpg)  
+Figure 3: An overview of the proposed IGP framework.
+
+Algorithm 1: Working pipeline of IGP  
+Input: Initial labeled set  $\mathcal{V}_0$  query batch size  $b$  Output: Labeled set  $\nu_{l}$    
+1 Stage 1: Node Selection   
+2  $\mathcal{V}_l = \mathcal{V}_0,\mathcal{V}_b = \emptyset$  .   
+3 for  $t = 1,2,\ldots ,b$  do Select the most valuable node:  $v_{i} = \arg \max_{v\in \mathcal{V}_{train}\backslash \mathcal{V}_{l}}F(\mathcal{V}_{l}\cup \{v\})$  .   
+5  $\mathcal{V}_l = \mathcal{V}_l\cup \{v_i\} ,\mathcal{V}_b = \mathcal{V}_b\cup \{v_i\} ,\mathcal{V}_{train} = \mathcal{V}_{train}\setminus \{v_i\} ;$    
+6 Stage 2: Node Labeling   
+7 for  $v_{i}\in \mathcal{V}_{b}$  do Judge the correctness of the predicted pseudo label  $\bar{y}_i$  if  $\bar{y}_i$  is correct (agreed by the oracle) then. Labeling  $v_{i}$  with  $\hat{\pmb{y}}_i^{\prime}$  according to Eq.4; else. Labeling  $v_{i}$  with  $\hat{\pmb{y}}_i^{\prime}$  according to Eq.4;  $\mathcal{V}_l = \mathcal{V}_l\setminus \{v_i\} ,\mathcal{V}_{train} = \mathcal{V}_{train}\cup \{v_i\} ;$    
+14 Return V
+
+# 4 IGP FRAMEWORK
+
+# 4.1 FRAMEWORK OVERVIEW
+
+As shown in Fig. 3, for each batch of node selection, IGP first trains the GNN model with the soft labels. With the model prediction, IGP measures the expected information gain of labeling each node. Considering both the influence magnitude and information gain, IGP further measures the information gain propagation of each unlabeled node and selects a batch of nodes that can maximize the information gain propagation on the graph. After that, with the relaxed queries, the oracle only judges the correctness of the model prediction, and the label distribution will be normalized correspondingly. At last, we re-train the GNN model with update the model prediction. The above process is repeated until the labeling cost  $\mathcal{B}$  runs out.
+
+# 4.2 WORKING PIPELINE
+
+Node Selection. Without losing generality, consider a batch setting where  $b$  nodes are selected in each iteration. Algorithm 1 provides a sketch of our greedy selection method for GNNs. Given the training set  $\mathcal{V}_{train}$ , and query batch size  $b$ , we first select the node  $v_{i}$ , which generates the maximum marginal gain (Line 3). We then update the labeled set  $\mathcal{V}_l$ , the oracle annotated set  $\mathcal{V}_b$ , and the training set  $\mathcal{V}_{train}$  (Line 4). This process is repeated until a batch of  $b$  nodes have been selected. Note that the normalized soft label  $\hat{\pmb{y}}_i'$  is unavailable before the node labeling process. For better efficiency, we assume  $\hat{\pmb{y}}_i'$  is the model predicted soft label  $\hat{\pmb{y}}_i$  in the node selection process. If  $\hat{\pmb{y}}_i$  is not confident (i.e., [0.4,0.3,0.3] for a three-classification problem), it will be of little influence on the label distribution of its neighbors. On the contrary, if  $\hat{\pmb{y}}_i$  is closer to  $\hat{\pmb{y}}_i'$  if it is confident. Consequently, it is reasonable to make such an assumption.
+
+Node Labeling After getting a batch of the selected node-set  $\nu_{b}$ , we acquire the label from an oracle. As introduced before, different from previous works which require the oracle to annotate the hard
+
+label directly, the oracle in our framework only needs to judge the correctness of the pseudo label  $\bar{y}_i$  (Line 8). If  $\bar{y}_i$  is correct, we update its soft label  $\hat{\pmb{y}}_i'$  according to Eq. 4 (Line 10). Note that we can still get information gain even if  $\bar{y}_i$  is incorrect. For any node  $v_i$  with an incorrect pseudo label, we firstly update its soft label  $\hat{\pmb{y}}_i'$  (Line 10). After that, we remove it from  $\mathcal{V}_l$  and add it to  $\mathcal{V}_{train}$  (Line 13). In this way, this node has the chance to be re-selected in the next batch. Note that the model will produce a different label given the fact that the oracle rejects the previously predicted label.
+
+Model Training Intuitively, a node with smaller entropy contains contributes more to the training process. In other words, a one-hot label contributes most, while other normalized soft labels also provide weak supervision. For GNN model training, we use the weighted loss function as follows
+
+$$
+\mathcal {L} = - \sum_ {v _ {i} \in \mathcal {V} _ {l}, H (v _ {i}) = 0} \hat {\mathbf {y}} _ {i} ^ {\prime} \log \hat {\mathbf {y}} _ {i} + \alpha \sum_ {v _ {i} \in \mathcal {V} _ {l}, H (v _ {i}) \neq 0} \hat {\mathbf {y}} _ {i} ^ {\prime} \log \frac {\hat {\mathbf {y}} _ {i} ^ {\prime}}{\tilde {\mathbf {y}} _ {i}}, \tag {10}
+$$
+
+where  $\alpha$  is a hyper-parameter which controls the importance of the incorrectly predicted nodes by the model. Note that the labeled set has two types of labels: 1) one-hot hard labels (i.e.,  $H(v_{i}) = 0$ ), which is correctly predicted by the model; 2) soft labels (i.e.,  $H(v_{i}) \neq 0$ ), which has incorrect model prediction but also provides weak supervision. We adopt the KL divergence as the loss function to learn from the weak supervised soft label in both cases (cross-entropy is just a special case of KL divergence). Note that KL divergence measures the distance between the ground truth and the current prediction, and it is especially appropriate for methods with a basis of information theory.
+
+# 4.3 EFFICIENCY OPTIMIZATION
+
+By leveraging some recent works on scalable and parallelizable social influence maximization (Aral & Dhillon, 2018), we could enable IGP to efficiently deal with large-scale graphs. The key idea is to identify and dismiss uninfluential nodes to dramatically reduce the amount of computation for evaluating the information gain propagation. For example, we can use the node degree or the distribution of random walkers throughout the nodes (Kim et al., 2017) to filter out a vast number of uninfluential nodes.
+
+# 5 EXPERIMENTS
+
+# 5.1 EXPERIMENTAL SETTINGS
+
+Datasets. We evaluate IGP in both inductive and transductive settings (Hamilton et al., 2017): three citation networks (i.e., Citeseer, Cora, and PubMed) (Kipf & Welling, 2017a), one large social network (Reddit), and one OGB dataset (ogbn-arxiv).
+
+GNN Models. We conduct experiments using the widely used GCN model and demonstrate the generalization of IGP on other GNNs such as SGC (Wu et al., 2019), APPNP (Klicpera et al., 2019) and MVGRL (Hassani & Khasahmadi, 2020).
+
+Baselines. We compare IGP with the following baselines: Random, AGE (Cai et al., 2017), ANRMAB (Gao et al., 2018), GPA (Hu et al., 2020), SEAL (Li et al., 2020), ALG (Zhang et al., 2021a), and GRAIN (Zhang et al., 2021c)
+
+Settings. For each method, we use the hyper-parameter tuning toolkit (Li et al., 2021a;b) or follow the original papers to find the optimal hyper-parameters. To eliminate randomness, we repeat each method ten times and report the mean performance. Specifically, for each method, we choose a small set of labeled nodes as an initial pool - two nodes are randomly selected for each class.
+
+# 5.2 PERFORMANCE COMPARISON
+
+End-to-end Comparison. We choose the labeling budget as 20 labels per class to show the end-to-end accuracy. Table 1 shows that GRAIN, ALG, AGE, and ANRMAB outperform Random in all the datasets, as they are specially designed for GNNs. GRAIN and ALG perform better than other baselines except for IGP because they consider the RF. However, IGP further boosts the accuracy by a significant margin. Remarkably, IGP improves the test accuracy of the best baseline, i.e., GRAIN, by  $1.6 - 2.2\%$  on the three citation networks,  $0.9\%$  on Reddit, and  $0.6\%$  on ogn-arxiv.
+
+Table 1: The test accuracy (%) on different datasets with the same labeling budget.  
+
+<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>PubMed</td><td>Reddit</td><td>ogbn-arxiv</td></tr><tr><td>Random</td><td>78.8(±0.8)</td><td>70.8(±0.9)</td><td>78.9(±0.6)</td><td>91.1(±0.5)</td><td>68.2(±0.4)</td></tr><tr><td>AGE</td><td>82.5(±0.6)</td><td>71.4(±0.6)</td><td>79.4(±0.4)</td><td>91.6(±0.3)</td><td>68.9(±0.3)</td></tr><tr><td>ANRMAB</td><td>82.4(±0.5)</td><td>70.6(±0.6)</td><td>78.2(±0.3)</td><td>91.5(±0.3)</td><td>68.7(±0.2)</td></tr><tr><td>GPA</td><td>82.8(±0.4)</td><td>71.6(±0.4)</td><td>79.9(±0.5)</td><td>91.8(±0.2)</td><td>69.2(±0.3)</td></tr><tr><td>SEAL</td><td>83.2(±0.5)</td><td>72.1(±0.4)</td><td>80.3(±0.4)</td><td>92.1(±0.3)</td><td>69.5(±0.1)</td></tr><tr><td>ALG</td><td>83.6(±0.6)</td><td>73.6(±0.5)</td><td>80.9(±0.3)</td><td>92.4(±0.3)</td><td>70.1(±0.2)</td></tr><tr><td>GRAIN</td><td>84.2(±0.3)</td><td>74.2(±0.3)</td><td>81.8(±0.2)</td><td>92.5(±0.1)</td><td>70.3(±0.2)</td></tr><tr><td>IGP</td><td>86.4(±0.6)</td><td>75.8(±0.3)</td><td>83.6(±0.5)</td><td>93.4(±0.2)</td><td>70.9(±0.3)</td></tr></table>
+
+![](images/83cc33b271d71546e0690db94b9b08175ee4837411e63f0f751083a0313772a9.jpg)  
+(a) Cora
+
+![](images/fb895407113e17a5dda43a9c09c44326205661f40e93fd95d1c374c6143b04e6.jpg)  
+(b) Citeseer
+
+![](images/7284191aca0e79d7190565d89d79ccbc3bd50cfc6c42908f640aab4b95245a3b.jpg)  
+(c) PubMed  
+Figure 4: The test accuracy across different labeling budgets for model training.
+
+Accuracy under Different Labeling Budgets. To show the influence of the labeling budget, we test the accuracy of different AL methods under different labeling budgets on three citation datasets. Concretely, we range the budgets of oracle labels from  $2C$  to  $20C$  ( $C$  is the number of classes) and show the test accuracy of GCN in Figure 4. The experimental results demonstrate that with the increase of labeling budget, the accuracy of IGP grows fairly fast, thereby outperforming the baselines with an even greater margin.
+
+Generalization. In addition to GCN, IGP can also be applied to a large variety of GNN variants. GCN, SGC, MVGRL, and APPNP are four representative GNNs (Chen et al., 2021) which adopt different message passings. Unlike the coupled GCN, both SGC and APPNP are decoupled, while their orderings when doing feature propagation and transformation are different. Besides, MVGRL is a classic self-supervised GNN. We test the generalization ability of IGP by evaluating the aforementioned four types of GNNs on  $20C$  ( $C$  is the number of classes) nodes selected by IGP and other baselines in the AL scenario, and the corresponding results are shown in Table 2. The results suggest that IGP consistently outperforms the other baselines, regardless of whether the GCC is decoupled. Moreover, the result also shows our proposed method IGP can significantly outperform the compared baselines on the top of more sophisticated self-supervised GNN such as MVGRL (Hassani & Khasahmadi, 2020). As shown in the table, the test accuracy of IGP could outperform ALG and GRAIN by more than  $1.8\%$  on PubMed. Therefore, we draw a conclusion that IGP can generalize to different types of GNNs well.
+
+Ablation study. IGP combines informative selecting, informative training, and information quantity in the method. To verify the necessity of each component, we evaluate IGP on GCN by disabling one component at a time. We evaluate IGP: (i) without the node labeled with soft label when training (called “No Informative Training (IT)”; (ii) without the information gain propagation when selecting nodes (called “No Informative Selection (IS)”; (iii) without the influence magnitude and regarding the information among all the nodes as the same (called “No Information Quantity (IM)”);(iv) without the normalized soft label and set the soft label uniformly (called “No Normalized Label (NL)”); Table 3 displays the results of these three settings, and from the table we can learn the following information. 1) If informative training is ignored, the test accuracy will decrease in all three datasets, e.g., the accuracy gap is as large as  $2.9\%$  on PubMed. This is because informative training help utilizing the information of the nodes labeled by probability. 2) Lacking information selecting will lead to worse accuracy, since the information gain of the nodes can not only benefit itself but also its neighbors in the graph. It is more important than other components since removing the information selecting will lead to a more substantial accuracy gap. For example, the gap on Cora is  $2.5\%$ , which is higher than the other gap  $(2.3\%)$ . 3) Information quantity also significantly impacts model accuracy on all datasets, because it will lead to a wrong preference on the nodes with larger degree, although some of them have little influence on other nodes. 4) If we remove normalized label and set the soft
+
+Table 2: Test accuracy of different models on PubMed.  
+
+<table><tr><td>Method</td><td>SGC</td><td>APPNP</td><td>GCN</td><td>MVGRL</td></tr><tr><td>Random</td><td>77.6(±0.8)</td><td>79.2(±0.6)</td><td>78.9(±0.6)</td><td>79.3(±0.5)</td></tr><tr><td>AGE</td><td>78.8(±0.5)</td><td>79.9(±0.5)</td><td>79.4(±0.4)</td><td>79.9(±0.4)</td></tr><tr><td>ANRMAB</td><td>77.8(±0.4)</td><td>78.7(±0.5)</td><td>78.2(±0.3)</td><td>78.9(±0.3)</td></tr><tr><td>GPA</td><td>79.5(±0.6)</td><td>80.2(±0.4)</td><td>79.9(±0.5)</td><td>80.4(±0.3)</td></tr><tr><td>SEAL</td><td>79.8(±0.5)</td><td>80.5(±0.5)</td><td>80.3(±0.4)</td><td>80.7(±0.3)</td></tr><tr><td>ALG</td><td>80.5(±0.4)</td><td>81.2(±0.5)</td><td>80.9(±0.3)</td><td>81.3(±0.2)</td></tr><tr><td>GRAIN</td><td>81.1(±0.3)</td><td>82.0(±0.4)</td><td>81.8(±0.2)</td><td>82.1(±0.1)</td></tr><tr><td>IGP</td><td>83.2(±0.6)</td><td>83.7(±0.5)</td><td>83.6(±0.5)</td><td>83.9(±0.4)</td></tr></table>
+
+Table 3: Influence of different components in test accuracy(%)  
+
+<table><tr><td>Method</td><td>Cora</td><td>Δ</td><td>Citeseer</td><td>Δ</td><td>PubMed</td><td>Δ</td></tr><tr><td>No IT</td><td>84.1(±0.7)</td><td>-2.3</td><td>73.2(±0.3)</td><td>-2.6</td><td>80.7(±0.5)</td><td>-2.9</td></tr><tr><td>No IS</td><td>83.9(±0.5)</td><td>-2.5</td><td>73.1(±0.4)</td><td>-2.7</td><td>80.4(±0.4)</td><td>-3.2</td></tr><tr><td>No IM</td><td>84.5(±0.6)</td><td>-1.9</td><td>73.6(±0.5)</td><td>-2.2</td><td>81.5(±0.6)</td><td>-2.1</td></tr><tr><td>No NL</td><td>84.4(±0.5)</td><td>-2.0</td><td>73.4(±0.3)</td><td>-2.4</td><td>81.3(±0.4)</td><td>-2.3</td></tr><tr><td>IGP</td><td>86.4(±0.6)</td><td>-</td><td>75.8(±0.3)</td><td>-</td><td>83.6(±0.5)</td><td>-</td></tr></table>
+
+![](images/15eccf121227579ee2d204a8c2999d577d9dd0360fe7d41ffe682eee1604e929.jpg)  
+Figure 5: The node distribution with different labels of different methods on the Cora dataset.
+
+![](images/f673c04456a6c82b97777409d30cf14a9ad5be484ca76a7176bc9945822a367e.jpg)
+
+label uniformly, the test accuracy will decrease apparently. Although the performance degradation is smaller than no IT and no IS, the results still show the necessity of normalized label.
+
+Interpretability To show the insight of IGP, we evaluate and draw the distribution of the nodes with soft labels (in red color) and hard labels (in blue color), also those without labels (in gray color) for two methods: AGE and IGP when the labeling budget is  $6C$  for Cora in GCN, the more information entropy the nodes have, the smaller its radium (one-hot labeled node has the largest radium). The result in Figure 5 shows that compared to AGE, IGP has enough labeled nodes with a balanced distribution, both in probability and one-hot, which could explain the better performance of NC-ALG in node classification. We also show the influence on information entropy in Appendix.
+
+# 6 CONCLUSION
+
+This paper presents Informationoin Gain Propagation (IGP), the first GNN-based AL method which explicitly considers AL with soft labels. Concretely, we provide a new way for active learners to exhaustively label nodes with a limited budget, using a combination of domain experts and GNN model. IGP allows us to use relaxed queries where domain expert (oracle) only judges the correctness of the predicted labels rather than the exact categorization, and ii) new criteria of maximizing information gain propagation for the active learner with relaxed queries and soft labels. Empirical studies on real-world graphs show that our approach outperforms competitive baselines by a large margin, especially when the number of classes is large.
+
+# 7 REPRODUCIBILITY
+
+The source code of IGP can be found in Github (https://github.com/zwt233/IGP). To ensure reproducibility, we have provided the overview of datasets in Section F.1 and Table 4 in Appendix F. The detailed hyperparameter settings for our IGP can be found in Appendix F.2. Our experimental environment is presented in Appendix F.2, and please refer to "README.md" in the Github repository for more details.
+
+# ACKNOWLEDGMENTS
+
+This work is supported by NSFC (No. 61832001, 61972004), Beijing Academy of Artificial Intelligence (BAAI), and PKU-Tencent Joint Research Lab. Bin Cui is the corresponding author.
+
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+
+# A PROOFS
+
+Theorem A.1. The node with highest entropy (in terms of the soft label given by the model) is not necessarily the node that brings highest expected information gain.
+
+Let us consider a three classification problem. Assume there is a node  $s$  with soft label  $[1 - 2a, a, a]$ , where  $0 \leq a \leq \frac{1}{3}$  so that  $1 - 2a \geq a$ . Therefore, the entropy of  $s$  is given by
+
+$$
+H (s) = - (1 - 2 a) \log (1 - 2 a) - a \log a - a \log a = - (1 - 2 a) \log (1 - 2 a) - 2 a \log a \tag {11}
+$$
+
+When the oracle gives a negative answer, the soft label will become  $[0, 0.5, 0.5]$ , and thus the entropy will become  $-0.5\log 0.5 - 0.5\log 0.5 = \log 2 = 1$ . Consequently, the expected information gain of  $s$  is given by
+
+$$
+I G (s) = (1 - 2 a) (H (s) - 0) + 2 a (H (s) - 1) = H (s) - 2 a \tag {12}
+$$
+
+Take the derivatives of the above two variables, we have
+
+$$
+\begin{array}{l} \frac {\mathrm {d} H (s)}{\mathrm {d} a} = \left[ - (1 - 2 a) \log (1 - 2 a) - 2 a \log a \right] ^ {\prime} \\ = - \left\{\left[ (1 - 2 a) \log (1 - 2 a) \right] ^ {\prime} + 2 (a \log a) ^ {\prime} \right\} \\ = - \left\{\left[ \log (1 - 2 a) + \frac {1}{\ln 2} \right] * (- 2) + 2 \left(\log a + \frac {1}{\ln 2}\right) \right\} \tag {13} \\ = 2 \left[ \log (1 - 2 a) - \log a \right] \\ = 2 \log \frac {1 - 2 a}{a} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \frac {\mathrm {d} I G (s)}{\mathrm {d} a} = \frac {\mathrm {d} (H (s) - 2 a)}{\mathrm {d} a} \\ = \frac {\mathrm {d} H (s)}{\mathrm {d} a} - 2 \tag {14} \\ = 2 \log {\frac {1 - 2 a}{a}} - 2 \\ = 2 (\log \frac {1 - 2 a}{a} - 1) \\ \end{array}
+$$
+
+Since  $0 \leq a \leq \frac{1}{3}$ ,  $\frac{\mathrm{d}H(s)}{\mathrm{d}a}$  is always greater than or equal to zero. In other words,  $H(s)$  increases with  $a$ .
+
+However,  $\frac{\mathrm{d}IG(s)}{\mathrm{d}a}$  is greater than or equal to zero when  $0 \leq a \leq \frac{1}{4}$ , and less than zero when  $\frac{1}{4} < a \leq \frac{1}{3}$ . That being said, when  $a$  is less than  $\frac{1}{4}$ ,  $IG(s)$  increases with  $a$ ; nonetheless, when  $a$  is greater than  $\frac{1}{4}$ ,  $IG(s)$  decreases with  $a$ , although  $H(s)$  keeps increasing in this case.
+
+To give a specific counterexample, let  $a_1 = 0.25$  and  $a_2 = 0.3$  and denote these nodes as  $s_1$  and  $s_2$  respectively. Thus, the soft labels of  $s_1$  and  $s_2$  are  $[0.5, 0.25, 0.25]$  and  $[0.4, 0.3, 0.3]$ , and we have  $H(s_1) = 1.5$ ,  $H(s_2) \approx 1.571$ ,  $IG(s_1) = 1$  and  $IG(s_2) \approx 0.971$ . Evidently,  $H(s_1) < H(s_2)$  and  $IG(s_1) > IG(s_2)$ .
+
+Therefore, the theorem follows.
+
+# B NORMALIZED LABEL
+
+For a certain node, if the soft label given by the model is  $[a_1, a_2, \dots, a_c]$ , and the oracle indicates that the node does not belong to class  $l$ , then the probability that the node belongs to class  $i$  ( $i \neq l$ ) is given by
+
+$$
+\frac {a _ {i}}{\sum_ {j \neq l} a _ {j}} \tag {15}
+$$
+
+In essence, this is a conditional probability. Formally, let  $j$  be the ground truth label of the node, and then our target can be expressed as
+
+$$
+P (j = i \mid j \neq l) \tag {16}
+$$
+
+With the definition of conditional probability, the above can be transformed as follows
+
+$$
+P (j = i \mid j \neq l) = \frac {P (j = i , j \neq l)}{P (j \neq l)} = \frac {P (j = i)}{P (j \neq l)} \tag {17}
+$$
+
+According to the known condition,  $P(j = i) = a_{i}$  and  $P(j \neq l) = \sum_{j \neq l} a_{j}$ . Therefore, the theorem follows. The theorem suggests that, when the oracle gives a negative answer, the normalized soft label should be set with accordance to the soft label given by the model, rather than the uniform distribution.
+
+Table 4: Overview of the Five Datasets  
+
+<table><tr><td>Dataset</td><td>#Nodes</td><td>#Features</td><td>#Edges</td><td>#Classes</td><td>#Train/Val/Test</td><td>Task type</td><td>Description</td></tr><tr><td>Cora</td><td>2,708</td><td>1,433</td><td>5,429</td><td>7</td><td>1,208/500/1,000</td><td>Transductive</td><td>citation network</td></tr><tr><td>Citeseer</td><td>3,327</td><td>3,703</td><td>4,732</td><td>6</td><td>1,827/500/1,000</td><td>Transductive</td><td>citation network</td></tr><tr><td>Pubmed</td><td>19,717</td><td>500</td><td>44,338</td><td>3</td><td>18,217/500/1,000</td><td>Transductive</td><td>citation network</td></tr><tr><td>ogbn-arxiv</td><td>169,343</td><td>128</td><td>1,166,243</td><td>40</td><td>90,941/29,799/48,603</td><td>Transductive</td><td>citation network</td></tr><tr><td>Reddit</td><td>232,965</td><td>602</td><td>11,606,919</td><td>41</td><td>155,310/23,297/54,358</td><td>Inductive</td><td>social network</td></tr></table>
+
+# C INFORMATION GAIN (IG)
+
+For node  $v_{i}$ , the IG of annotating it is defined as
+
+$$
+I G (v _ {i}) = H \left(\hat {\mathbf {y}} _ {i}\right) - P (v _ {i} -) H \left(\hat {\mathbf {y}} _ {i} ^ {\prime}, v _ {i} -\right), \tag {18}
+$$
+
+Starting from the expectation of entropy reduction, we show how it is transformed into IG as follows. Note that if the oracle agrees with the pseudo label, i.e., event  $v_{i} +$ , the hard label will be obtained, thus the entropy dropping to 0.
+
+$$
+\begin{array}{l} \operatorname {E} \left[ H \left(\hat {\mathbf {y}} _ {i}\right) - H \left(\hat {\mathbf {y}} _ {i} ^ {\prime}\right) \right] = P \left(v _ {i} +\right) \left(H \left(\hat {\mathbf {y}} _ {i}\right) - H \left(\hat {\mathbf {y}} _ {i} ^ {\prime}, v _ {i} +\right)\right) + P \left(v _ {i} -\right) \left(H \left(\hat {\mathbf {y}} _ {i}\right) - H \left(\hat {\mathbf {y}} _ {i} ^ {\prime}, v _ {i} -\right)\right) \\ = P \left(v _ {i} +\right) \left(H \left(\hat {\boldsymbol {y}} _ {i}\right) - 0\right) + P \left(v _ {i} -\right) \left(H \left(\hat {\boldsymbol {y}} _ {i}\right) - H \left(\hat {\boldsymbol {y}} _ {i} ^ {\prime}, v _ {i} -\right)\right) \tag {19} \\ = \left(P \left(v _ {i} +\right) + P \left(v _ {i} -\right)\right) H \left(\hat {\mathbf {y}} _ {i}\right) - P \left(v _ {i} -\right) H \left(\hat {\mathbf {y}} _ {i} ^ {\prime}, v _ {i} -\right) \\ = I G (v _ {i}) \\ \end{array}
+$$
+
+# D EXAMPLE OF EXPECTED INFORMATION GAIN
+
+- Case 1:  $v_{i}$  belongs to the oracle annotated class  $l$ . The prediction is correct in this case, and the information changes from the uncertain soft label to the one-hot label. Suppose the model predicted softmax outputs  $\hat{y}_{i}$  is [0.5, 0.3, 0.2], the label is changed into [1, 0, 0] and the corresponding information gain is defined as  $H([0.5, 0.3, 0.2]) - H([1, 0, 0])$ , where  $H(\cdot)$  is the entropy function.  
+- Case 2:  $v_{i}$  doesn't belong to the oracle annotated class  $l$ . In this case, the model prediction is incorrect, but the uncertainty of the soft label is also decreased because we have already known that the true label may belong to the remaining classes except for the class  $l$ . Similar to case 1, suppose the predicted label  $\hat{y}_{i}$  is [0.5, 0.3, 0.2], the soft label is changed into [0, 0.6, 0.4] and the corresponding information gain is defined as  $H([0.5, 0.3, 0.2]) - H([0, 0.6, 0.4])$ .
+
+# E TIME COMPLEXITY ANALYSIS
+
+Given a  $K$ -layer GNN, the propagation matrix  $(N\times N$ , with  $M$  nonzero values) and label matrix  $(N\times c)$  should be multiplied  $K$  times in IGP. Therefore, in the stage of node selection (line 3-5 in Algorithm 1), selecting the node that maximizes the propagation of information gain  $F(\mathcal{V}_l\cup \{v\})$  from unlabeled nodes requires  $\mathcal{O}(KMc)$ , where  $K$  is the number of layers of GNNs,  $M$  is the number of edges and  $c$  is the number of classes. All in all, the cost of selecting a batch of b nodes is  $\mathcal{O}(bKMc)$ . Suppose we filter out all but  $n$  nodes with a degree larger than a threshold and get the corresponding propagation matrix  $(n\times N$ , with  $m$  nonzero values), we can reduce the overall time complexity to  $\mathcal{O}(bKmc)$ .
+
+# F EXPERIMENTAL DETAILS
+
+# F.1 DATASET DESCRIPTION
+
+Cora, Citeseer, and Pubmed are three popular citation network data-sets, and we follow the public training/validation/test split in GCN Kipf & Welling (2017b). In these three networks, papers from
+
+![](images/727dd881db237ea246358205cb9fe15012340b1b6b4acd996cccdeb1b6d29d23.jpg)  
+Figure 6: the change of information entropy for the whole dataset when labeling budget grows.
+
+different topics are considered as nodes, and the edges are citations among the papers. The node attributes are binary word vectors, and class labels are the topics papers belong to.
+
+Reddit is a social network dataset derived from the community structure of numerous Reddit posts. It is a well-known inductive training dataset, and the training/validation/test split in our experiment is the same as that in GraphSAGE Hamilton et al. (2017). The public version provided by GraphSAINT<sup>2</sup> Zeng et al. (2020) is used in our paper.
+
+ogbn-arxiv is a directed graph, representing the citation network among all Computer Science (CS) arXiv papers indexed by MAG. The training/validation/test split in our experiment is the same as the public version. The public version provided by  $\mathrm{OGB}^3$  is used in our paper.
+
+For more specifications about the five aforementioned datasets, see Table 4.
+
+# F.2 IMPLEMENTATION DETAILS
+
+For GPA Hu et al. (2020), to achieve the highest accuracy, we directly use the pre-trained model by its authors on Github. Specifically, for Cora, we choose the model pre-trained on PubMed and Citeseer; for PubMed, we choose the model pre-trained on Cora and Citeseer; for Citeseer and Reddit, we choose the model pre-trained on Cora and PubMed; for ogbn-arxiv, we choose the model pre-trained on PubMed and Reddit. Other hyper-parameters are all identical to the open-source code.
+
+For AGE Cai et al. (2017) and ANRMAB Gao et al. (2018), to procure well-trained models and guarantee that their model-based selection criteria work well, GCN is trained for 200 epochs in each node selection iteration. AGE is implemented with its open-source version and ANRMAB in accordance with its original paper.
+
+For ALG Zhang et al. (2021a) and GRAIN Zhang et al. (2021c), we follow the public code released by the original paper.
+
+For the hyperparameter  $\alpha$ , we set it as 1 in CiteSeer, PubMed and ogbn-arxiv, and as 2 for Cora and Reddit. Besides, the degree for dismissing uninfluential nodes in IGP is 8 for Cora, CiteSeer, PubMed, and 15 for Reddit and ogbn-arxiv.
+
+We carry out all the experiments on an Ubuntu 16.04 system with Intel(R) Xeon(R) CPU E5-2650 v4 @ 2.20GHz, 4 NVIDIA GeForce GTX 1080 Ti GPUs and 256 GB DRAM. All the experiments are implemented in Python 3.6 with Pytorch 1.7.1 Paszke et al. (2019) on CUDA 10.1.
+
+# G INTERPRETABILITY ANALYSIS
+
+We test the change of information entropy for Citeseer dataset when the labeling budget grows from  $2c$  to  $20c$  ( $c$  is the number of classes) and report the result in Figure 6. Compare to other methods that bring fixed information entropy decrease, IGP has much less information entropy and the gap becomes larger when the labeling budget grow, which explains the better performance of our method.
+
+Table 5: Performance along with running time on the Reddit dataset.  
+
+<table><tr><td>Method</td><td>ANRMAB</td><td>GPA</td><td>AGE</td><td>SEAL</td><td>ALG</td><td>IGP</td><td>IG</td><td>GRAIN</td></tr><tr><td>Relative Running Time</td><td>340.00</td><td>331.25</td><td>8.25</td><td>4.00</td><td>1.63</td><td>1.38</td><td>1.25</td><td>1.00</td></tr><tr><td>Test Accuracy(%)</td><td>91.5(±0.3)</td><td>91.8(±0.2)</td><td>91.6(±0.3)</td><td>92.1(±0.3)</td><td>92.4(±0.3)</td><td>93.4(±0.2)</td><td>91.7(±0.3)</td><td>92.5(±0.1)</td></tr></table>
+
+Table 6: The test accuracy (%) on different hyperparameters on the PubMed dataset  
+
+<table><tr><td>parameters</td><td>α=0</td><td>α=0.5</td><td>α=1</td><td>α=2</td></tr><tr><td>SGC</td><td>79.6(±0.7)</td><td>82.8(±0.6)</td><td>83.2(±0.6)</td><td>82.9(±0.5)</td></tr><tr><td>APPNP</td><td>80.4(±0.6)</td><td>83.4(±0.4)</td><td>83.7(±0.5)</td><td>83.6(±0.5)</td></tr><tr><td>GCN</td><td>80.7(±0.4)</td><td>83.5(±0.5)</td><td>83.6(±0.5)</td><td>83.4(±0.6)</td></tr><tr><td>MVGRL</td><td>80.5(±0.6)</td><td>83.2(±0.5)</td><td>83.9(±0.4)</td><td>83.5(±0.4)</td></tr></table>
+
+# H EFFICIENCY ANALYSIS
+
+To test the efficiency of IGP, we also compare its running time per AL iteration with other baselines in a batch of node selection in the Reddit dataset. Besides, to measure the influence of the extra computational cost in maximizing the propagation of information gain in the local neighborhood, we remove the propagation part in IGP and name this baseline IG.
+
+We set the training time of GRAIN as the baseline, the relative training time and the test accuracy of each method are provided in Table 5. As shown in Table 5, our method IGP can get comparable efficiency as GRAIN while achieving the accuracy improvement of  $0.9\%$ . Besides, removing the propagation part in IGP will make the AL process a little faster, but it also leads to large performance degradation of  $1.7\%$  compared with IGP.
+
+# I EXPLORATION OF HYPERPARAMETERS
+
+Our proposed IGP has only one hyperparameter  $\alpha$ , which controls the importance of the weak label. To investigate its impact on IGP, we set  $\alpha$  to different values and then report the corresponding test accuracy with different base GNN models on different datasets. The experimental results in Table 6 show that removing the weak label will lead to large performance degradation in different models, which verifies the effectiveness of our proposed weak label. Besides, our method IGP is robust to the choice of  $\alpha$  since its performance is stable on different models when we increase  $\alpha$  from 0.5 to 2.
+
+# J INFLUENCE OF THE NUMBER OF CLASSES
+
+To clarify the influence of the number of classes, we examine the model accuracies as a function of the number of classes on the ogbn-arxiv dataset. Specifically, we gradually increase the number of classes and the corresponding number of nodes of the ogbn-arxiv dataset, and report the mean test accuracy of 10 runs with and without the soft-label. As shown in the newly added Figure 7(a), although more classes increase the complexity of a classification problem (thus a decreasing model accuracy), we see that the performance gain of adding soft labels decreases slightly as the number of classes increases.
+
+The reason is that although the soft-max is taken over the remaining classes, the distribution of the output of soft-max (i.e., soft label) is expected to be highly skewed, which can help to alleviate the effect of increased class number on the quality of the soft label. To demonstrate this, we add a new Figure 7(b) showing the percentage of the node whose top-K accuracy is larger than  $80\%$ . As the percentages of top K class increases (i.e., K/total class number), more than  $90\%$  nodes only need the predicted top  $20\%$  classes to guarantee the top-K accuracy larger than  $80\%$ , which means few classes predominate the softmax distribution.
+
+![](images/5ea1eab79fb0f059347595d51b00baf94fa8df2852da077ec6ff0149e081e20b.jpg)  
+(a) The influence of soft label.
+
+![](images/5cf0ffc61df2c80c2d03a997ebef675d36352e153986a4e9c6a8049dbf56a97a.jpg)  
+(b) The analysis of class distribution.  
+Figure 7: (Left) The test accuracy of the different number of classes with and without the soft label. (Right) The relationship of probability and percentage of top-K class in all of the classes.
+
+Table 7: The test accuracy (%) with different filtering degrees on the Reddit dataset.  
+
+<table><tr><td>Filtering Degree</td><td>0</td><td>5</td><td>10</td><td>15</td></tr><tr><td>Test Accuracy (%)</td><td>93.6(±0.3)</td><td>93.5(±0.3)</td><td>93.4(±0.2)</td><td>93.4(±0.2)</td></tr></table>
+
+# K ANALYSIS OF SAMPLE COMPLEXITY
+
+With a non-perfect oracle, our method could reduce sample complexity compared with other AL strategies because we improve the utilization of queries and samples. Traditional AL algorithms have to discard the data samples of queries that have not obtained hard labels from non-perfect Oracle. By contrast, we could still leverage these data samples in a soft-label manner, thus reducing the error quickly than other AL algorithms.
+
+Although some AL algorithms enjoy theoretical guarantees on the traditional linear machine learning models (e.g., linear regression), analyzing the sample complexity of AL for deep learning is quite difficult due to the non-linear nature of neural networks. Most sample complexity is analyzed under the passive learning for neural networks in the literature. In addition to the non-linear nature of neural networks, the graph neural networks (GNNs) are even more complicated as graph data is no longer independent and identically distributed (iid), making the theoretical analysis of sample complexity extremely difficult for GNN-based AL.
+
+# L ANALYSIS OF THE FILTERING DEGREE
+
+To clarify how much accuracy would drop when we filter the small-degree nodes, we increase the filtering degree from 0 to 15 and then report the corresponding mean test accuracy of 10 runs with GCN in Table 7. Note that the threshold we used in the previous experiments is 15 in Reddit.
+
+As shown in Table 7, the test accuracy only drops slightly by  $0.2\%$  when we increase the threshold from 0 to 15, meaning that our efficiency optimization will not introduce a large negative influence on the prediction ability of the GCN model. Compared with the uninfluential nodes with small node degrees, labeled nodes will larger degrees will propagate their label information to more adjacent nodes and thus make larger entropy reduction for the whole dataset. As a result, nodes with larger degrees have a larger probability of being selected in IGP even if the efficiency optimization is not introduced (i.e., setting the filtering degree to 0).

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+# INFORMATION PRIORITIZATION THROUGH EMPOWERMENT IN VISUAL MODEL-BASED RL
+
+Homanga Bharadhwaj *
+
+Carnegie Mellon University
+
+Mohammad Babaeizadeh
+
+Google Research, Brain Team
+
+Dumitru Erhan
+
+Google Research, Brain Team
+
+Sergey Levine
+
+Google Research, Brain Team
+
+University of California Berkeley
+
+# ABSTRACT
+
+Model-based reinforcement learning (RL) algorithms designed for handling complex visual observations typically learn some sort of latent state representation, either explicitly or implicitly. Standard methods of this sort do not distinguish between functionally relevant aspects of the state and irrelevant distractors, instead aiming to represent all available information equally. We propose a modified objective for model-based RL that, in combination with mutual information maximization, allows us to learn representations and dynamics for visual model-based RL without reconstruction in a way that explicitly prioritizes functionally relevant factors. The key principle behind our design is to integrate a term inspired by variational empowerment into a state-space model based on mutual information. This term prioritizes information that is correlated with action, thus ensuring that functionally relevant factors are captured first. Furthermore, the same empowerment term also promotes faster exploration during the RL process, especially for sparse-reward tasks where the reward signal is insufficient to drive exploration in the early stages of learning. We evaluate the approach on a suite of vision-based robot control tasks with natural video backgrounds, and show that the proposed prioritized information objective outperforms state-of-the-art model based RL approaches with higher sample efficiency and episodic returns.
+
+# 1 INTRODUCTION
+
+Model-based reinforcement learning (RL) provides a promising approach to accelerating skill learning: by acquiring a predictive model that represents how the world works, an agent can quickly derive effective strategies, either by planning or by simulating synthetic experience under the model. However, in complex environments with high-dimensional observations (e.g., images), modeling the full observation space can present major challenges. While large neural network models have made progress on this problem (Finn & Levine, 2017; Ha & Schmidhuber, 2018; Hafner et al., 2019a; Watter et al., 2015; Babaeizadeh et al., 2017), effective learning in visually complex environments necessitates some mechanism for learning representations that prioritize functionally relevant factors for the current task. This needs to be done without wasting effort and capacity on irrelevant distractors, and without detailed reconstruction. Several recent works have proposed contrastive objectives that maximize mutual information between observations and latent states (Hjelm et al., 2018; Ma et al., 2020; Oord et al., 2018; Srinivas et al., 2020). While such objectives avoid reconstruction, they still do not distinguish between relevant and irrelevant factors of variation. We thus pose the question: can we devise non-reconstructive representation learning methods that explicitly prioritize information that is most likely to be functionally relevant to the agent?
+
+In this work, we derive a model-based RL algorithm from a combination of representation learning via mutual information maximization (Poole et al., 2019) and empowerment (Mohamed & Rezende, 2015). The latter serves to drive both the representation and the policy toward exploring and representing functionally relevant factors of variation. By integrating an empowerment-based term into a
+
+![](images/47cfd3d1de38b16dd19e19014b10e30cc9720118faf188736beadb1803ea6c37.jpg)  
+Figure 1: Overview of InfoPower.  $I(\mathcal{O}_t;Z_t)$  is the contrastive learning objective for learning an encoder to map from image  $\mathcal{O}$  to latent  $Z$ .  $I(A_{t - 1};Z_t|Z_{t - 1})$  is the empowerment objective that prioritizes encoding controllable representations in  $Z$ .  $-I(i_{t + 1};Z_{t + 1}|Z_t,A_t)$  helps learn a latent forward dynamics model so that future  $Z_{t + k}$  can be predicted from current  $Z_{t}$ .  $I(R_{t};Z_{t})$  helps learn a reward prediction model, such that the agent can learn a plan  $A_{t},..A_{t + k},..$  through latent rollouts. Together, this combination of terms produces a latent state space model for MBRL that captures all necessary information at convergence, while prioritizing the most functionally relevant factors via the empowerment term.
+
+mutual information framework for learning state representations, we effectively prioritize information that is most likely to have functional relevance, which mitigates distractions due to irrelevant factors of variation in the observations. By integrating this same term into policy learning, we further improve exploration, particularly in the early stages of learning in sparse-reward environments, where the reward signal provides comparatively little guidance.
+
+Our main contribution is InfoPower, a model-based RL algorithm for control from image observations that integrates empowerment into a mutual information based, non-reconstructive framework for learning state space models. Our approach explicitly prioritizes information that is most likely to be functionally relevant, which significantly improves performance in the presence of time-correlated distractors (e.g., background videos), and also accelerates exploration in environments when the reward signal is weak. We evaluate the proposed objectives on a suite of simulated robotic control tasks with explicit video distractors, and demonstrate up to  $20\%$  better performance in terms of cumulative rewards at 1M environment interactions with  $30\%$  higher sample efficiency at 100k interactions.
+
+# 2 PROBLEM STATEMENT AND NOTATION
+
+A partially observed Markov decision process (POMDP) is a tuple  $(S, \mathcal{A}, T, R, \mathcal{O})$  that consists of states  $s \in S$ , actions  $a \in \mathcal{A}$ , rewards  $r \in R$ , observations  $o \in \mathcal{O}$ , and a state-transition distribution  $T(s'|s, a)$ . In most practical settings, the agent interacting with the environment doesn't have access to the actual states in  $S$ , but to some partial information in the form of observations  $\mathcal{O}$ . The underlying state-transition distribution  $T$  and reward distribution  $R$  are also unknown to the agent.
+
+In this paper, we consider the observations  $o \in \mathcal{O}$  to be high-dimensional images, and so the agent should learn a compact representation space  $Z$  for the latent state-space model. The problem statement is to learn effective representations from observations  $\mathcal{O}$  when there are visual distractors present in the scene, and plan using the learned representations to maximize the cumulative sum of discounted rewards,  $J = \mathbb{E}[\sum_{t} \gamma^{t-1} r_t]$ . The value of a state  $V(Z_t)$  is defined as the expected cumulative sum of discounted rewards starting at state  $Z_t$ .
+
+We use  $q(\cdot)$  to denote parameterized variational approximations to learned distributions. We denote random variables with capital letters and use lowercase letters to denote particular realizations (e.g.,  $z_{t}$  denotes the value of  $Z_{t}$ ). Since the underlying distributions are unknown, we evaluate all expectations through Monte-Carlo sampling with observed state-transition tuples  $(o_{t}, a_{t-1}, o_{t-1}, z_{t}, z_{t-1}, r_{t})$ .
+
+# 3 INFORMATION PRIORITIZATION FOR THE LATENT STATE-SPACE MODEL
+
+Our goal is to learn a latent state-space model with a representation  $Z$  that prioritizes capturing functionally relevant parts of observations  $\mathcal{O}$ , and devise a planning objective that explores with the learned representation. To achieve this, our key insight is integration of empowerment in the visual model-based RL pipeline. For representation learning we maximize  $\mathrm{MI}\max_ZI(\mathcal{O},Z)$  subject to a prioritization of the empowerment objective  $\max_{Z}I(A_{t - 1};Z_{t}|Z_{t - 1})$ . For planning, we maximize the empowerment objective along with reward-based value with respect to the policy  $\max_{A}I(A_{t - 1};Z_{t}|Z_{t - 1}) + I(R_{t};Z_{t})$ .
+
+In the subsequent sections, we elaborate on our approach, InfoPower, and describe lower bounds to MI that yield a tractable algorithm.
+
+# 3.1 LEARNING CONTROLLABLE FACTORS AND PLANNING THROUGH EMPOWERMENT
+
+Controllable representations are features of the observation that correspond to entities which the agent can influence through its actions. For example, in quadrupedal locomotion, this could include the joint positions, velocities, motor torques, and the configurations of any object in the environment that the robot can interact with. For robotic manipulation, it could include the joint actuators of the robot arm, and the configurations of objects in the scene that it can interact with. Such representations are denoted by  $S^{+}$  in Fig. 2, which we can formally define through conditional independence as the smallest subspace of  $S$ ,  $S^{+} \leq S$ , such that  $I(A_{t-1}; S_t | S_t^{+}) = 0$ . This conditional independence relation can be seen in Fig. 2. We explicitly priori
+
+![](images/51409ce436f7dba7b939ad06300c85cde07f6f8a6ea44d90a0cc1956846f5af9.jpg)  
+Figure 2: PGM showing decomposition of state  $S$  into controllable parts  $S^{+}$  (directly influenced by actions  $A$ ), parts not influenced by actions that still influence the reward,  $\tilde{S}^{-}$ , and distractors  $DS^{-}$ .  $\tilde{S}_{t + 1}^{-}$  may be influenced by  $\tilde{S}_t^-$  (arrow not shown to reduce clutter) but not by  $\tilde{S}_t^+$ .
+
+tize the learning of such representations in the latent space by drawing inspiration from variational empowerment (Mohamed & Rezende, 2015).
+
+The empowerment objective can be cast as maximizing a conditional information term  $I(A_{t-1}; Z_t | Z_{t-1}) = \mathcal{H}(A_{t-1} | Z_{t-1}) - \mathcal{H}(A_{t-1} | Z_t, Z_{t-1})$ . The first term  $\mathcal{H}(A_{t-1} | Z_{t-1})$  encourages the chosen actions to be as diverse as possible, while the second term  $-\mathcal{H}(A_{t-1} | Z_t, Z_{t-1})$  encourages the representations  $Z_t$  and  $Z_{t+1}$  to be such that the action  $A_t$  for transition is predictable. While prior approaches have used empowerment in the model-free setting to learn policies by exploration through intrinsic motivation (Mohamed & Rezende, 2015), we specifically use this objective in combination with MI maximization for prioritizing the learning of controllable representations from distracting images in the latent state-space model.
+
+We include the same empowerment objective in both representation learning and policy learning. For this, we augment the maximization of the latent value function that is standard for policy learning in visual model-based RL (Sutton, 1991), with  $\max_A I(A_{t-1}; Z_t | Z_{t-1})$ . This objectives complements value-based-learning and further improves exploration by seeking controllable states. We empirically analyze the benefits of this in sections 4.3 and 4.5.
+
+In Appendix A.1 we next describe two theorems regarding learning controllable representations. We observe that the  $\max \sum_{t} I(A_{t-1}; Z_t | Z_{t-1})$  objective for learning latent representations  $Z$ , when used along with the planning objective, provably recovers controllable parts of the observation  $\mathcal{O}$ , namely  $S^{+}$ . This result in Theorem 1 is important because in practice, we may not be able to represent every possible factor of variation in a complex environment. In this situation, we would expect that when  $|Z| \ll |\mathcal{O}|$ , learning  $Z$  under the objective  $\max \sum_{t} I(A_{t-1}; Z_t | Z_{t-1})$  would encode  $S^{+}$ .
+
+We further show through Theorem 2 that the inverse information objective alone can be used to train a latent-state space model and a policy through an alternating optimization algorithm that converges to a local minimum of the objective  $\max \sum_{t} I(A_{t-1}; Z_t | Z_{t-1})$  at a rate inversely proportional to
+
+the number of iterations. In Section 4.3 we empirically show how this objective helps achieve higher sample efficiency compared to pure value-based policy learning.
+
+# 3.2 MUTUAL INFORMATION MAXIMIZATION FOR REPRESENTATION LEARNING
+
+For visual model-based RL, we need to learn a representation space  $Z$ , such that a forward dynamics model defining the probability of the next state in terms of the current state and the current action can be learned. The objective for this is  $\sum_{t} - I(i_{t};Z_{t}|Z_{t - 1},A_{t - 1})$ . Here,  $i_t$  denotes the dataset indices that determine the observations  $p(o_{t}|i_{t}) = \delta (o_{t} - o_{t^{\prime}})$ . In addition to the forward dynamics model, we need to learn a reward predictor by maximizing  $\sum_{t}I(R_{t};Z_{t})$ , such that the agent can plan ahead in the future by rolling forward latent states, without having to execute actions and observe rewards in the real environment.
+
+Finally, we need to learn an encoder for encoding observations  $\mathcal{O}$  to latents  $Z$ . Most successful prior works have used reconstruction-loss as a natural objective for learning this encoder (Babaeizadeh et al., 2017; Hafner et al., 2019b;a). A reconstruction-loss can be motivated by considering the objective  $I(\mathcal{O}, Z)$  and computing its BA lower
+
+bound (Agakov, 2004).  $I(o_{t};z_{t}) \geq \mathbb{E}_{p(o_{t},z_{t})}[\log q_{\phi^{\prime}}(o_{t}|z_{t})] + \mathcal{H}(p(o_{t}))$ . The first term here is the reconstruction objective, with  $q_{\phi^{\prime}}(o_t|z_t)$  being the decoder, and the second term can be ignored as it doesn't depend on  $Z$ . However, this reconstruction objective explicitly encourages encoding the information from every pixel in the latent space (such that reconstructing the image is possible) and hence is prone to not ignoring distractors.
+
+In contrast, if we consider other lower bounds to  $I(\mathcal{O}, Z)$ , we can obtain tractable objectives that do not involve reconstructing high-dimensional images. We can obtain an  $NCE$ -based lower bound (Hjelm et al., 2018):  $I(o_{t}; z_{t}) \geq \mathbb{E}_{q_{\phi}(z_{t} | o_{t}) p(o_{t})}[\log f_{\theta}(z_{t}, o_{t}) - \log \sum_{t' \neq t} f_{\theta}(z_{t}, o_{t'})]$ , where  $q_{\phi}(z_{t} | o_{t})$  is the learned encoder,  $o_{t}$  is the observation at timestep  $t$  (positive sample), and all observations in the replay buffer  $o_{t'}$  are negative samples.  $f_{\theta}(z_{t}, o_{t'}) = \exp(z_{t}^{T} W_{\theta} z_{t'})$ . The lower-bound is a form of contrastive learning as it maximizes compatibility of  $z_{t}$  with the corresponding observation  $o_{t}$  while minimizing compatibility with all other observations across time and batch.
+
+Although prior work has explored NCE-based bounds for contrastive learning in RL (Srinivas et al., 2020), to the best of our knowledge, prior work has not used this in conjunction with empowerment for prioritizing information in visual model-based RL. Similarly, the Nguyen-Wainwright-Jordan (NWJ) bound (Nguyen et al., 2010), which to the best our knowledge has not been used by prior works in visual model-based RL, can be obtained as,
+
+$$
+I (o _ {t}; z _ {t}) \geq \mathbb {E} _ {q _ {\phi} (z _ {t} | o _ {t}) p (o _ {t})} [ f _ {\theta} (z _ {t}, o _ {t}) ] - e ^ {- 1} \mathbb {E} _ {q _ {\phi} (z _ {t} | o _ {t}) p (o _ {t})} e ^ {f _ {\theta} (z _ {t}, o _ {t})},
+$$
+
+where  $f_{\theta}$  is a critic. There exists an optimal critic function for which the bound is tightest and equality holds.
+
+We refer to the InfoNCE and NWJ lower bound based objectives as contrastive learning, in order to distinguish them from a reconstruction-loss based objective, though both are bounds on mutual
+
+Algorithm 1: Information Prioritization in Visual Model-based RL (InfoPower)
+
+Initialize dataset  $\mathcal{D}$  with random episodes.
+
+Initialize model parameters  $\phi, \chi, \psi, \eta$ .
+
+Initialize dual variable  $\lambda$
+
+while not converged do
+
+for update step  $c = 1..C$  do
+
+// Model learning
+
+Sample data  $\{(a_t, o_t, r_t)\}_{t = k}^{k + L} \sim \mathcal{D}$ .
+
+Compute latents  $z_{t} \sim p_{\phi}(z_{t}|z_{t - 1},a_{t - 1},o_{t})$
+
+Calculate  $\mathcal{L}$  based on section 3.4.
+
+$$
+\left(\phi , \chi , \psi , \eta\right) \leftarrow \left(\phi , \chi , \psi , \eta\right) + \nabla_ {\phi , \chi , \psi , \eta} \mathcal {L}
+$$
+
+$$
+\lambda \leftarrow \lambda - \nabla_ {\lambda} \mathcal {L}
+$$
+
+// Behavior learning
+
+Rollout latent plan,  $S\gets S\cup \{z_t,a_t,r_t\}$
+
+$$
+V (z _ {t}) \approx \mathbb {E} _ {\pi} \left[ \ln q _ {\eta} \left(r _ {t} \mid z _ {t}\right) + \ln q _ {\psi} \left(a _ {t - 1} \mid z _ {t}, z _ {t - 1}\right) \right]
+$$
+
+Update policy  $\pi$  and value model
+
+# end
+
+// Environment interaction
+
+for time step  $t = 0..T - 1$  do
+
+$$
+z _ {t} \sim p _ {\phi} \left(z _ {t} \mid z _ {t - 1}, a _ {t - 1}, o _ {t}\right); a _ {t} \sim \pi \left(a _ {t} \mid z _ {t}\right)
+$$
+
+$$
+r _ {t}, o _ {t + 1} \leftarrow \text {e n v . s t e p} \left(a _ {t}\right).
+$$
+
+# end
+
+Add data  $\mathcal{D}\gets \mathcal{D}\cup \{(o_t,a_t,r_t)_{t = 1}^T\}$
+
+# end
+
+information. We denote a lower bound to MI by  $\underline{I}(o_t, z_t)$ . We empirically find the NWJ-bound to perform slightly better than the NCE-bound for our approach, explained in section 4.5.
+
+# 3.3 OVERALL OBJECTIVE
+
+We now motivate the overall objective, which consists of maximizing mutual information while prioritizing the learning of controllable representations through empowerment in a latent state-space model. Based on the discussions in Sections 3.1 and 3.2, we define the overall objective for representation learning as
+
+$$
+\max  _ {Z _ {0: H - 1}} \sum_ {t = 0} ^ {H - 1} I (\mathcal {O} _ {t}; Z _ {t}) \text {s . t .} \sum_ {t = 0} ^ {H - 1} \overbrace {\left(- I \left(i _ {t} ; Z _ {t} | Z _ {t - 1} , A _ {t - 1}\right) + I \left(A _ {t - 1} ; Z _ {t} | Z _ {t - 1}\right) + I \left(R _ {t} ; Z _ {t}\right)\right)} ^ {\mathcal {C} _ {t}} \geq c _ {0}.
+$$
+
+The objective is to maximize a MI term  $I(\mathcal{O}_t;Z_t)$  through contrastive learning such that a constraint on  $\mathcal{C}_t$  holds for prioritizing the encoding of forward-predictive, reward-predictive and controllable representations. We define the overall planning objective as
+
+$$
+\max  _ {A _ {0: H - 1}} \sum_ {t = 0} ^ {H - 1} I (A _ {t - 1}; Z _ {t} | Z _ {t - 1}) + V (Z _ {t}) \quad ; A _ {t} = \pi (Z _ {t}) \quad ; V (Z _ {t}) \approx \sum_ {t} R _ {t}.
+$$
+
+The planning objective is to learn a policy as a function of the latent state  $Z$  such that the empowerment term and a reward-based value term are maximized over the horizon  $H$ .
+
+We can perform the constrained optimization for representation learning through the method of Lagrange Multipliers, by the primal and dual updates shown in Section A.2. In order to analyze this objective, let  $|\mathcal{O}| = n$  and  $|Z| = d$ . Since,  $\mathcal{O}$  corresponds to images and  $Z$  is a bottlenecked latent representation,  $d \ll n$ .
+
+$I(\mathcal{O},Z)$  is maximized when  $Z$  contains all the information present in  $\mathcal{O}$  such that  $Z$  is a sufficient statistic of  $\mathcal{O}$ . However, in practice, this is not possible because  $|Z|\ll |\mathcal{O}|$ . When  $c_{0}$  is sufficiently large, and the constraint  $\sum_{t}\mathcal{C}_{t}\geq c_{0}$  is satisfied,  $Z = [S^{+},\tilde{S}^{-}]$ . Hence, the objective  $\max \sum_{t}I(\mathcal{O}_{t},Z_{t})$  s.t.  $\sum_{t}\mathcal{C}_{t}\geq c_{0}$  cannot encode anything else in  $Z$ , in particular it cannot encode distractors  $DS^{-}$ .
+
+To understand the importance of prioritization through  $\mathcal{C}_t\geq c_0$ , consider  $\max \sum_{t}I(\mathcal{O}_{t},Z_{t})$  without the constraint. This objective would try to make  $Z$  a sufficient statistic of  $\mathcal{O}$ , but since  $|Z|\ll |\mathcal{O}|$ , there are no guarantees about which parts of  $\mathcal{O}$  are getting encoded in  $Z$ . This is because both distractors  $S^{-}$  and non-distractors  $S^{+}$ ,  $D\tilde{S}^{-}$  are equally important with respect to  $I(\mathcal{O},Z)$ . Hence, the constraint helps in prioritizing the type of information to be encoded in  $Z$ .
+
+# 3.4 PRACTICAL ALGORITHM AND IMPLEMENTATION DETAILS
+
+To arrive at a practical algorithm, we optimize the overall objective in section 3.3 through lower bounds on each of the MI terms. For  $I(\mathcal{O}, Z)$  we consider two variants, corresponding to the NCE and NWJ lower bounds described in Section 3.2. We can obtain a variational lower bound on each of the terms in  $\mathcal{C}_t$  as follows, with detailed derivations in Appendix A.3:
+
+$$
+\begin{array}{l} - I \left(i _ {t}; z _ {t} \mid a _ {t - 1}\right) \geq - \sum_ {t} \mathbb {E} \left[ D _ {\mathrm {K L}} \left(p \left(z _ {t} \mid z _ {t - 1}, a _ {t - 1}, o _ {t}\right) \mid \mid q _ {\chi} \left(z _ {t} \mid z _ {t - 1}, a _ {t - 1}\right)\right) \right] \\ I \left(r _ {t}; z _ {t}\right) \geq \mathbb {E} _ {p \left(r _ {t} \mid o _ {t}\right)} \left[ \log q _ {\eta} \left(r _ {t} \mid z _ {t}\right) \right] + \mathcal {H} \left(p \left(r _ {t}\right)\right) \\ I \left(a _ {t - 1}; z _ {t} \mid z _ {t - 1}\right) \geq \mathbb {E} _ {p \left(o _ {t} \mid z _ {t - 1}, a _ {t - 1}\right) q _ {\phi} \left(z _ {t} \mid o _ {t}\right)} \left[ \log q _ {\psi} \left(a _ {t - 1} \mid z _ {t}, z _ {t - 1}\right) \right] + \mathbb {E} [ \mathcal {H} (\pi \left(a _ {t - 1} \mid z _ {t - 1}\right)) ] \\ \end{array}
+$$
+
+Here,  $q_{\phi}(z_t|o_t)$  is the observation encoder,  $q_{\psi}(a_{t-1}|z_t,z_{t-1})$  is the inverse dynamics model,  $q_{\eta}(r_t|z_t)$  is the reward decoder, and  $q_{\chi}(z_t|z_{t-1},a_{t-1})$  is the forward dynamics model. The inverse model helps in learning representations such that the chosen action is predictable from successive latent states. The forward dynamics model helps in predicting the next latent state given the current latent state and the current action, without having the next observation. The reward decoder predicts the reward (a scalar) at a time-step given the corresponding latent state. We use dense networks for all the models, with details in Appendix A.7. We denote by  $\underline{C}_t$ , the lower bound to  $C_t$  based on the sum of the terms above. We construct a Lagrangian  $\mathcal{L} = \sum_{t}\underline{I}(o_t;z_t) + \lambda (\underline{C}_t - c_0)$  and optimize it by primal-dual gradient descent. An outline of this is shown in Algorithm 1.
+
+Planning Objective. For planning to choose actions at every time-step, we learn a policy  $\pi (a|z)$  through value estimates of task reward and the empowerment objective. We learn value estimates with  $V(z_{t})\approx \mathbb{E}_{\pi}[\ln q_{\eta}(r_{t}|z_{t}) + \ln q_{\psi}(a_{t - 1}|z_{t},z_{t - 1})]$ . We estimate  $V(z_{t})$  similar to Equation 6 of (Hafner et al., 2019a). The empowerment term  $q_{\psi}(a_{t - 1}|z_t,z_{t - 1})$  in policy learning incentivizes choosing actions  $a_{t - 1}$  such that they can be predicted from consecutive latent states  $z_{t - 1},z_{t}$ . This biases the policy to explore controllable regions of the state-space. The policy is trained to maximize the estimate of the value, while the value model is trained to fit the estimate of the value that changes as the policy is updated.
+
+Finally, we note that the difference in value function of the underlying MDP  $Q^{\pi}(o,a)$  and the latent MDP  $\hat{Q}^{\pi}(z,a)$ , where  $z\sim q_{\phi}(z|o)$  is bounded, under some regularity assumptions. We provide this result in Theorem 3 of the Appendix Section A.4. The overall procedure for model learning, planning, and interacting with the environment is outlined in Algorithm 1.
+
+# 4 EXPERIMENTS
+
+Through our experiments, we aim to understand the following research questions:
+
+1. How does InfoPower compare with the baselines in terms of episodic returns in environments with explicit background video distractors?  
+2. How sample efficient is InfoPower when the reward signal is weak ( $< 100k$  env steps when the learned policy typically doesn't achieve very high rewards)?  
+3. How does InfoPower compare with baselines in terms of behavioral similarity of latent states?
+
+Please refer to the website for a summary and qualitative visualization results https://sites.google.com/view/information-empowerment
+
+# 4.1 SETUP
+
+We perform experiments with modified DeepMind Control Suite environments (Tassa et al., 2018), with natural video distractors from ILSVRC dataset (Russakovsky et al., 2015) in the background. The agent receives only image observations at each time-step and does not receive the ground-truth simulator state. This is a very challenging setting, because the agents must learn to ignore the distractors and abstract out representations necessary for control. While natural videos that are unrelated to the task might be easy to ignore, realistic scenes might have other elements that resemble the controllable elements, but are not actually controllable (e.g., other cars in a driving scenario). To emulate this, we also add distractors that resemble other potentially controllable robots, as shown for example in Fig. 3 (1st and 6th), but are not actually influenced by actions.
+
+In addition to this setting, we also perform experiments with gray-scale distractors based on videos from the Kinetics dataset (Kay et al., 2017). This setting is adapted exactly based on prior works (Fu et al., 2021; Zhang et al., 2020) and we compare directly to the published results.
+
+We compare InfoPower with state-of-the-art baselines that also learn world models for control: Dreamer (Hafner et al., 2019a), C-Dreamer that is a contrastive version of Dreamer similar to Ma et al. (2020), TIA (Fu et al., 2021) that learns explicit reconstructions for both the distracting background and agent separately, DBC (Zhang et al., 2020), and DeepMDP (Gelada et al., 2019). We also compare variants of InfoPower with different MI bounds (NCE and NWJ), and ablations of it that remove the empowerment objective from policy learning.
+
+![](images/5a8f45a700831c2d67581b4c17249705e3700ee8afa7babefec8269aafa4f92e.jpg)
+
+![](images/a025c1f4591c9d2b8195f77e0011fa77874488869e719e24ccea05362605b4cf.jpg)
+
+![](images/0856bc80532ce84446dd65fbf12013459142a7b26dafb1d27cfa8b7115807179.jpg)
+
+![](images/426b5c8b25c94fcea448644080584ec978e52333d7fd6c919c3f4e6a87c246d0.jpg)  
+Figure 3: Some of the natural video background distractors in our experiments. The videos change every 50 time-steps. Some backgrounds (for e.g. the top left and the bottom right) have more complex distractors in the form of agent-behind-agent i.e. the background has pre-recorded motion of a similar agent that is being controlled.
+
+![](images/3fdfa4ad3c6f624b88032febc47153214db112d06f1222f086dccac85448f72c.jpg)
+
+![](images/c01ea0fedd0de3d578fa9c032de2003de56e2aca95e8264bafb485e31b20019f.jpg)
+
+# 4.2 A BEHAVIORAL SIMILARITY METRIC BASED ON GRAPH KERNELS
+
+In order to measure how good the learned latent representations are at capturing the functionally relevant elements in the scene, we introduce a behavioral similarity metric with details in A.5. Given
+
+![](images/dd58329efe83620ed0a5972516319ef7e540f894efe7e4bfaa6d9fba43fd88d4.jpg)
+
+![](images/cd17b4aa89c9b48a15741c4fd16f6d9b896c69080d0d402c4348fdc8906e1b21.jpg)
+
+![](images/5beff254eb8b996e37606220ac5eaaf58a34515f72873030ff83817a2b147a70.jpg)
+
+![](images/1394ad68af65c5afe8d9b5162c3e065aaf798231d71a475fc85679b3f7f5e64f.jpg)
+
+Figure 4: Evaluation of InfoPower and baselines in a suite of DeepMind Control tasks with natural video distractors in the background. The x-axis denotes the number of environment interactions and the y-axis shows the episodic returns. The S.D is over 4 random seeds. Higher is better.  
+![](images/7e9dca1593e635563daaa496d48a262f5ffbf10e81c66ba68b8832048a221c6c.jpg)  
+InfoPower dreamer cdreamer tia dbc deepmpdp
+
+![](images/52867dcb113a5012f62bd6357b48549ff88ae9d5fee434ab606165f1c78a9479.jpg)
+
+![](images/64cd26384a88e6979419088e2be9b24b2452db17c6bb018a258ec87e26db29a5.jpg)
+
+![](images/5f6695ddd06657678a498c0974502083ff6835041f53d88d07f05b8cddbcd5a9.jpg)
+
+the sets  $S_{z} = \{z^{i}\}_{i=1}^{n}$  and  $S_{z_{\mathrm{gt}}} = \{z_{\mathrm{gt}}^{i}\}_{i=1}^{n}$ , we construct complete graphs by connecting every vertex with every other vertex through an edge. The weight of an edge is the Euclidean distance between the respective pairs of vertices. The label of each vertex is an unique integer and corresponding vertices in the two graphs are labelled similarly.
+
+Let the resulting graphs be denoted as  $G_{z} = (V_{z},E_{z})$  and  $G_{z_{\mathrm{gt}}} = (V_{z_{\mathrm{gt}}},E_{z_{\mathrm{gt}}})$  respectively. Note that both these graphs are shortest path graphs, by construction. Let,  $e_i = \{u_i,v_i\}$  and  $e_j = \{u_j,v_j\}$ . We define a kernel  $\hat{k}$  that measures similarity between the edges  $e_i,e_j$  and the labels on respective vertices.  $\hat{k} (e_i,e_j) =$
+
+$$
+k _ {v} (l (u _ {i}), l (u _ {j})) k _ {e} (l (e _ {i}), l (e _ {j})) k _ {v} (l (v _ {i}), l (v _ {j})) + k _ {v} (l (u _ {i}), l (v _ {j})) k _ {e} (l (e _ {i}), l (e _ {j})) k _ {v} (l (v _ {i}), l (u _ {j}))
+$$
+
+Here, the function  $l(\cdot)$  denotes the labels of vertices and the weights of edges.  $k_{v}$  is a Dirac delta kernel i.e.  $k_{e}(x,y) = 1 - \delta (x,y)$  (Note that  $\delta (x,y) = 1$  iff  $x = y$ , else  $\delta (x,y) = 0$ ) and  $k_{e}$  is a Brownian ridge kernel,  $k_{e}(x,y) = \frac{1}{c}\max (0,c - |x - y|)$ , where  $c$  is a large number. We define the shortest path kernel to measure similarity between the two graphs, as,  $k(G_{z},G_{z_{\mathrm{gt}}}) = \frac{1}{|E_z|}\sum_{e_i\in E_z}\sum_{e_j\in E_{z_{\mathrm{gt}}}}\hat{k} (e_i,e_j)$ . The value of  $k(G_{z},G_{z_{\mathrm{gt}}})$  is low when a large number of pairs of corresponding vertices in both the graphs have the same edge length. We expect methods that recover latent state representations which better reflect the true underlying simulator state (i.e., the positions of the joints) to have higher values according to this metric.
+
+# 4.3 SAMPLE EFFICIENCY ANALYSIS
+
+In Fig. 4, we compare InfoPower and the baselines in terms of episodic returns. This version of InfoPower corresponds to an NWJ bound on MI which we find works slightly better than the NCE bound variant analyzed in section 4.5. It is evident that InfoPower achieves higher returns before 1M steps of training quickly compared to the baselines, indicating higher sample efficiency. This suggests the effectiveness of the empowerment model in helping capture controllable representations early on during training, when the agent doesn't take on actions that yield very high rewards.
+
+# 4.4 BEHAVIORAL SIMILARITY OF LATENT STATES
+
+In this section, we analyze how similar are the learned latent representations with respect to the underlying simulator states. The intuition for this comparison is that the proprioceptive features in the simulator state abstract out distractors in the image, and so we want the latent states to be behaviorally similar to the simulator states.
+
+Quantitative results with the defined metric. In Table 1, we show results for behavioral similarity of latent states (Sim), based on the metric in section 4.2. We see that the value of Sim for InfoPower is around  $20\%$  higher than the most competitive baseline, indicating high behavioral similarity of the latent states with respect to the corresponding ground-truth simulator states.
+
+Qualitative visualizations with t-sne. Fig. 5 shows a t-SNE plot of latent states  $z \sim q_{\phi}(z|o)$  for InfoPower and the baselines with visualizations of 3 nearest neighbors for two randomly chosen latent states. We see that the state of the agent is similar in each group for InfoPower, although the
+
+![](images/a1512658a6efb1ea8e31be94c54bfc960fc23a87673d26e19b3722f6ae3d9415.jpg)  
+Figure 5: t-SNE plot of latent states  $z \sim q_{\phi}(z|o)$  with visualizations of three nearest neighbors for two randomly sampled points (in red frame). We see that the state of the agent is similar in each set for InfoPower, whereas for Dreamer, and the most competitive baseline C-Dreamer, the nearest neighbor frames have significantly different agent configurations.
+
+![](images/ddb966c78e28053068585a1d5e6d0dc1d2db9ab7a6cae55f9ed1c55ed47d777f.jpg)
+
+![](images/ff0e4e5ac64da3ca201f95ef2a763fe8b684b2c663f6b32412091357dfcaf77b.jpg)
+
+![](images/d93d7c3c48f36144263bcc85207e4dfdf35ac20b39e177755bb0d7f60d8afecf.jpg)
+
+![](images/89e3ef090dcb8b779e912ac5629afb45a7cc8377e7e85b54962b328bf3ad375e.jpg)
+
+Figure 6: Evaluation of InfoPower and ablated variants in a suite of DeepMind Control tasks with natural video distractors in the background. The x-axis denotes the number of environment interactions and the y-axis shows the episodic returns. InfoPower-NWJ and InfoPower-NCE are full versions of our method differing only in the MI lower bound. The versions with - Policy do not include the empowerment objective in policy learning, but only use it from representation learning. The S.D is over 4 random seeds. Higher is better.  
+![](images/2fe927f788cefb6e644f1969e0ab27115b0da097e3bf1ec12fd09fbcc45f204d.jpg)  
+InfoPower NWJ InfoPower NCE InfoPower NWJ - Policy InfoPower NCE - Policy
+
+![](images/52df03954942b7826829d6ced844e1a4990394eecf5437e8e43a730862b5e924.jpg)
+
+![](images/fae4ea36de116ee1bbcc704c002cd1b5a2bf7ef84d014bc39159dea26c60a487.jpg)
+
+![](images/3ff72f94341bb03bb7212c55392b127cb8b134fde376814ab257ec57554aec63.jpg)
+
+background scenes are significantly different. However, for the baselines, the nearest neighbor states are significantly different in terms of the pose of the agent, indicating that the latent representations encode significant background information.
+
+# 4.5 ABLATION STUDIES
+
+In Fig. 6, we compare different ablations of InfoPower. Keeping everything else the same, and changing only the MI lower bound to NCE, we see that the performance is almost similar or slightly worse. However, when we remove the empowerment objective from policy optimization (the versions with '-Policy' in the plot), we see that performance drops. The drop is significant in the regions  $< 200k$  environment interactions, particularly in the sparse reward environments - cartpole balance and ball in a cup, indicating the necessity of the empowerment objective in exploration for learning controllable representations, when the reward signal is weak.
+
+# 5 RELATED WORKS
+
+Visual model-based RL. Recent developments in video prediction and contrastive learning have enabled learning of world-models from images (Watter et al., 2015; Babaeizadeh et al., 2017; Finn & Levine, 2017; Hafner et al., 2019a; Ha & Schmidhuber, 2018; Hafner et al., 2019b; Xie et al., 2020). All of these approaches learn latent representations through reconstruction objectives that are amenable for planning. Other approaches have used similar reconstruction based objectives for control, but not for MBRL (Lee et al., 2019; Gregor et al., 2019).
+
+MI for representation learning. Mutual Information measures the dependence between two random variables. The task of learning latent representations  $Z$  from images  $\mathcal{O}$  for downstream applications, has been very successful with MI objectives of the form  $\max_{f_1,f_2} I(f_1(\mathcal{O}), f_2(Z))$  (Hjelm & Bachman, 2020; Tian et al., 2020; Oord et al., 2018; Tschannen et al., 2019; Nachum & Yang, 2021). Since calculating MI exactly is intractable optimizing MI based objectives, it is important to construct appropriate MI estimators that lower-bound the true MI objective (Hjelm et al., 2018; Nguyen et al., 2010; Belghazi et al., 2018; Agakov, 2004). The choice of the estimator is crucial, as shown by recent works (Poole et al., 2019), and different estimators yield very different behaviors of the algorithm. We incorporate MI maximization through the NCE (Hjelm et al., 2018) and NWJ (Nguyen et al., 2010) lower bounds, such that typical reconstruction objectives for representation learning which do not perform well with visual distractors, can be avoided.
+
+Inverse models and empowerment. Prior approaches have used inverse dynamics models for regularization in representation learning (Agrawal et al., 2016; Zhang et al., 2018) and as bonuses for improving policy gradient updates in RL (Shelhamer et al., 2016; Pathak et al., 2017). The importance of information theoretic approaches for learning representations that maximize predictive power has been discussed in prior work (Still, 2009) and more recently in Lee et al. (2020). Empowerment (Mohamed & Rezende, 2015) has been used as exploration bonuses for policy learning (Leibfried et al., 2019; Klyubin et al., 2008), and for learning skills in RL (Gregor et al., 2016; Sharma et al., 2019; Eysenbach et al., 2018). In contrast to prior work, we incorporate empowerment both for state space representation learning and policy learning, in a visual model-based RL framework, with the aim of prioritizing the most functionally relevant information in the scene.
+
+RL with environment distractors. Some recent RL frameworks have studied the problem of abstracting out only the task relevant information from the environment when there are explicit distractors (Hansen & Wang, 2020; Zhang et al., 2020; Fu et al., 2021; Ma et al., 2020). Zhang et al. (2020) constrain the latent states by enforcing a bisimulation metric, without a reconstruction objective. Our approach is different from this primarily because of the empowerment objective that provides useful signal for dealiasing controllable vs. uncontrollable representations independent of how strong is the observed reward signal. Fu et al. (2021) model both the relevant and irrelevant aspects of the environment separately, and differ from our approach that prioritizes learning only the relevant aspects. (Shu et al., 2020) was and earlier approach that used contrastive representations for control. More recently, (Nguyen et al., 2021) used temporal predictive coding and (Ma et al., 2020) used InfoNCE based contrastive loss for learning representations while ignoring distractors in visual MBRL. Incorporating data augmentations for improving robustness with respect to environment variations is an orthogonal line of work (Laskin et al., 2020; Hansen & Wang, 2020; Raileanu et al., 2020; Srinivas et al., 2020; Kostrikov et al., 2020), complementary to our approach.
+
+# 6 CONCLUSION
+
+In this paper we derived an approach for visual model-based RL such that an agent can learn a latent state-space model and a policy by explicitly prioritizing the encoding of functionally relevant factors. Our prioritized information objective integrates a term inspired by variational empowerment into a non-reconstructive objective for learning state space models. We evaluate our approach on a suite of vision-based robot control tasks with two sets of challenging video distractor backgrounds. In comparison to state-of-the-art visual model-based RL methods, we observe higher sample efficiency and episodic returns across different environments and distractor settings.
+
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+
+# A APPENDIX
+
+# A.1 THEORETICAL RESULTS ON CONTROLLABLE REPRESENTATIONS
+
+Let  $\Phi(\cdot)$  denote the encoder such that  $Z = \Phi(\mathcal{O})$ .  $\mathcal{O}$  is the observation seen by the agent and  $S$  is the underlying state.  $S^{+}$  is defined as that part of underlying state  $S$  which is directly influenced by actions  $A$  i.e.  $S^{+} \leq S$  s.t.  $I(A_{t-1}; S_t | S_t^{+}) = 0$ .
+
+We next describe two theorems regarding learning controllable representations, with proofs in the Appendix. We observe that the max  $\sum_{t}I(A_{t - 1};\bar{Z}_t|Z_{t - 1})$  objective alone for learning latent representations  $Z$ , along with the planning objective provably recovers controllable parts of the observation  $\mathcal{O}$ , namely  $S^{+}$ .
+
+Theorem 1. The objective max  $\sum_{t}I(A_{t - 1};Z_{t}|Z_{t - 1})$  provably recovers controllable parts  $S^{+}$  of the observation  $\mathcal{O}$ .  $S^{+}$  is defined as that part of underlying state  $S$  which is directly influenced by actions  $A$  i.e.  $S^{+}\subset S$  s.t.  $I(S_{t};A_{t - 1}|S_{t}^{+}) = 0$ .
+
+Proof. Based on Fig. 2 and Fig. 7, we have the following
+
+$$
+\begin{array}{l} \max  I \left(A _ {t - 1}; Z _ {t}\right) \leq \max  I \left(A _ {t - 1}; \mathcal {O} _ {t}\right) \quad \text {D a t a - P r o c e s s i n g I n e q u a l i t y} \\ \leq I \left(A _ {t - 1}; S _ {t}\right) \quad \text {D a t a - P r o c e s s i n g I n e q u a l i t y} \\ \leq I \left(A _ {t - 1}; S _ {t}, S _ {t} ^ {+}\right) \\ \leq I \left(A _ {t - 1}; S _ {t} \mid S _ {t} ^ {+}\right) + I \left(A _ {t - 1}; S _ {t} ^ {+}\right) \quad \text {C h a i n R u l e o f M I} \\ \leq I \left(A _ {t - 1}; S _ {t} ^ {+}\right) \quad \text {C o n d i t i o n a l} \\ \end{array}
+$$
+
+So, the maximum above can be obtained when the encoder  $\Phi$  is an identity function over  $S^{+}$  and a zero function over  $[\tilde{S}^{-}, S^{-}]$ . So,  $Z_{t} = S_{t}^{+}$ . Hence, given  $Z_{t-1}$ ,  $I(A_{t-1}; Z_{t}|Z_{t-1})$  is maximized when  $\Phi$  is an identity function over  $S^{+}$ . So,  $\max \sum_{t} I(A_{t-1}; Z_{t}|Z_{t-1})$  learns the encoder  $\Phi$  such that controllable representations are recovered.
+
+![](images/ac331eca0a7d3a7464f3021d01bbfc8892bfa203d45d35b66e9e8e202099fd75.jpg)
+
+This result is important because in practice, we may not be able to represent every possible factor of variation in a complex environment. In this situation, we would expect that when  $|Z| \ll |\mathcal{O}|$ , learning  $Z$  under the objective  $\max \sum_{t} I(A_{t-1}; Z_t | Z_{t-1})$  would encode  $S^+$ . We next show that the inverse information objective alone can be used to train a latent-state space model and a policy through an alternating optimization algorithm that converges to a local minimum of the objective  $\max \sum_{t} I(A_{t-1}; Z_t | Z_{t-1})$  at a rate inversely proportional to the number of iterations.
+
+Theorem 2.  $\max_{\pi, \psi} \sum_t I(A_{t-1}; Z_t | Z_{t-1}) = \sum_t \mathbb{E}_{\pi(a_{t-1} | z_{t-1}) p(z_t | z_{t-1}, a_{t-1}, o_t)} \log \frac{q_{\psi}(a_{t-1} | z_t, z_{t-1})}{\pi(a_{t-1} | z_{t-1})}$  can be optimized through an alternating optimization scheme that has a convergence rate of  $\mathcal{O}(1/N)$  to a local minima of the objective, where  $N$  is the number of iterations.
+
+Proof. Let  $z \sim q_{\phi}(z|o)$  denote a latent state sampled from the encoder distribution. Learning a world model such that reward prediction and inverse information are maximized can be summarized as:
+
+$$
+\max  _ {\phi , \psi , \eta} \sum_ {t} \mathbb {E} _ {\pi \left(a _ {t} \mid z _ {t}\right), q _ {\phi} \left(z _ {t} \mid o _ {t}\right)} \left[ \mathbb {E} _ {p \left(r _ {t} \mid z _ {t}, a _ {t}\right)} \log q _ {\eta} \left(r _ {t} \mid z _ {t}, a _ {t}\right) + \mathbb {E} _ {p \left(z _ {t + 1} \mid z _ {t}, a _ {t}\right)} \log \frac {q _ {\psi} \left(a _ {t} \mid z _ {t + 1} , z _ {t}\right)}{\pi \left(a _ {t} \mid z _ {t}\right)} \right]
+$$
+
+The policy  $\pi (a_t|z_t)$  is learned by
+
+$$
+\max _ {\pi} \sum_ {t} \mathbb {E} _ {\pi (a _ {t} | z _ {t}), q _ {\phi} (z _ {t} | o _ {t})} \left[ \mathbb {E} _ {p (r _ {t} | z _ {t}, a _ {t})} \log q _ {\eta} (r _ {t} | z _ {t}, a _ {t}) + \mathbb {E} _ {p (z _ {t + 1} | z _ {t}, a _ {t})} \log \frac {q _ {\psi} (a _ {t} | z _ {t + 1} , z _ {t})}{\pi (a _ {t} | z _ {t})} \right]
+$$
+
+Model-based RL in this case involves alternating optimization between policy learning and learning the world model. In the case where there is no reward signal from the environment, i.e.  $p(r_t|z_t)$ , we have the following objective
+
+$$
+\max  _ {\pi} \max  _ {\phi , \psi} \sum_ {t} \mathbb {E} _ {\pi \left(a _ {t} \mid z _ {t}\right), q _ {\phi} \left(z _ {t} \mid s _ {t}\right) p \left(z _ {t + 1} \mid z _ {t}, a _ {t}\right)} \log \frac {q _ {\psi} \left(a _ {t} \mid z _ {t + 1} , z _ {t}\right)}{\pi \left(a _ {t} \mid z _ {t}\right)}
+$$
+
+![](images/484a8b287e82fe4a01bbc0e13905f21b9974a5066add738ad2da70dec9eceb7d.jpg)  
+Figure 7: PGM of the MDP with distractor states. The state observed by the agent  $\mathcal{O}$  consists of three parts  $S^{+}$ ,  $\tilde{S}^{-}$  and  $DS^{-}$ .  $S^{+}$  is the controllable part of the state i.e. it is affected by the actions of the agent, and in turn affects the reward  $R$ ;  $\tilde{S}^{-}$  is not controllable by the agent but affects the reward  $R$  and future  $S^{+}$ ;  $DS^{-}$  is not controllable by the agent and doesn't affect the reward  $R$  and future  $S^{+}$ .
+
+The optimal  $q_{\psi}^{*}$  give a fixed  $\pi$  and  $\phi$  can be derived as in page 335 of Cover & Thomas (2006).
+
+$$
+q _ {\psi} ^ {*} (a _ {t} | z _ {t + 1}, z _ {t}) = \frac {q _ {\phi} (z _ {t} | s _ {t}) p (z _ {t + 1} | z _ {t} , a _ {t}) \pi (a _ {t} | z _ {t})}{\int_ {a} q _ {\phi} (z _ {t} | s _ {t}) p (z _ {t + 1} | z _ {t} , a _ {t}) \pi (a _ {t} | z _ {t})}
+$$
+
+The optimal  $\pi *$  given a fixed  $\psi$  and  $\phi$  can be derived as
+
+$$
+\pi^ {*} (a _ {t} | z _ {t}) = \frac {\exp \left(\mathbb {E} _ {q _ {\phi} (z _ {t} | s _ {t}) p (z _ {t + 1} | z _ {t} , a _ {t})} [ \log q _ {\psi} (a _ {t} | z _ {t + 1} , z _ {t}) ]\right)}{\int_ {a} \exp \left(\mathbb {E} _ {q _ {\phi} (z _ {t} | s _ {t}) p (z _ {t + 1} | z _ {t} , a _ {t}) q} [ \log q _ {\psi} (a _ {t} | z _ {t + 1} , z _ {t}) ]\right)}
+$$
+
+We note that these alternating iterations correspond to an instantiation of the Blahut-Arimoto algorithm (Blahut, 1972) for determining the information theoretic capacity of a channel. Based on (Nakagawa et al., 2021) we see that this iterative optimization procedure between  $q_{\psi}^{*}$  and  $\pi^{*}$  converges, and the worst-case rate of convergence is  $\mathcal{O}(1 / N)$  where  $N$  is the number of iterations.
+
+![](images/97518134861d17a0cb307188149fe16121e44a0392f14144044418ca5c64a194.jpg)
+
+This result is useful because it shows that even in the absence of rewards from the environment  $(r_t = 0\forall t)$ , when planning to minimize regret is not possible, the inverse information objective can be used to train a policy that explores to seek out controllable parts of the state-space. When rewards from the environment are present, we can train the policy with this objective and the value estimates obtained from the cumulative rewards during planning. In Section 4.3 we empirically show how this objective helps achieve higher sample efficiency compared to pure value-based policy learning.
+
+# A.2 PRIMAL DUAL UPDATES
+
+$$
+Z _ {0: H - 1} = Z _ {0: H - 1} + \frac {\partial \sum_ {t = 0} ^ {H - 1} I (\mathcal {O} _ {t} ; Z _ {t}) + \lambda (- I (i _ {t} ; Z _ {t} | Z _ {t - 1} , A _ {t - 1}) + I (A _ {t - 1} ; Z _ {t} | Z _ {t - 1}) + I (R _ {t} ; Z _ {t}) - c _ {0})}{\partial Z _ {0 : H - 1}}
+$$
+
+$$
+\lambda = \lambda - \left(\underbrace {- I \left(i _ {t} ; Z _ {t} \mid Z _ {t - 1} , A _ {t - 1}\right)} _ {\text {f w d d y n a m i c s m o d e l}} + \underbrace {I \left(A _ {t - 1} ; Z _ {t} \mid Z _ {t - 1}\right)} _ {\text {i n v d y n a m i c s m o d e l}} + \underbrace {I \left(R _ {t} ; Z _ {t}\right)} _ {\text {r e w m o d e l}} - c _ {0}\right)
+$$
+
+# A.3 FORWARD, INVERSE, AND REWARD MODELS
+
+$$
+\begin{array}{l} - I \left(i _ {t}; Z _ {t} \mid Z _ {t - 1}, A _ {t - 1}\right) = - \int p \left(z _ {t}, z _ {t - 1}, a _ {t - 1}, i _ {t}, o _ {t}\right) \log \frac {p \left(z _ {t} \mid z _ {t - 1} , a _ {t - 1} , o _ {t}\right)}{p \left(z _ {t} \mid z _ {t - 1} , a _ {t - 1}\right)} \\ = - \int p (z _ {t}, z _ {t - 1}, a _ {t - 1}, o _ {t}) \left(\log \frac {p (z _ {t} | z _ {t - 1} , a _ {t - 1} , o _ {t})}{q _ {\chi} (z _ {t} | z _ {t - 1} , a _ {t - 1})} + \log \frac {q _ {\chi} (z _ {t} | z _ {t - 1} , a _ {t - 1})}{p (z _ {t} | z _ {t - 1} , a _ {t - 1})}\right) \\ \geq - \int p (z _ {t}, z _ {t - 1}, a _ {t - 1}, o _ {t}) \log \frac {p (z _ {t} | z _ {t - 1} , a _ {t - 1} , o _ {t})}{q _ {\chi} (z _ {t} | z _ {t - 1} , a _ {t - 1})} \\ = - D _ {\mathrm {K L}} \left(p \left(z _ {t} \mid z _ {t - 1}, a _ {t - 1}, o _ {t}\right) \mid \mid q _ {\chi} \left(z _ {t} \mid z _ {t - 1}, a _ {t - 1}\right)\right) \\ \end{array}
+$$
+
+We can obtain a variational lower bound on the empowerment
+
+$$
+\begin{array}{l} I \left(a _ {t}; z _ {t + k} \mid z _ {t}\right) = \int p \left(a _ {t}, z _ {t + k}, z _ {t}\right) \log \frac {p \left(a _ {t} \mid z _ {t + k} , z _ {t}\right)}{p \left(a _ {t} \mid z _ {t}\right)} \\ = \int p \left(a _ {t}, z _ {t + k}, z _ {t}\right) \left(\log \frac {q \left(a _ {t} \mid z _ {t + k} , z _ {t}\right)}{p \left(a _ {t} \mid z _ {t}\right)} + \log \frac {p \left(a _ {t} \mid z _ {t + k} , z _ {t}\right)}{q \left(a _ {t} \mid z _ {t + k} , z _ {t}\right)}\right) \\ \geq \int p (a _ {t}, z _ {t + k}, z _ {t}) \log \frac {q (a _ {t} | z _ {t + k} , z _ {t})}{p (a _ {t} | z _ {t})} \\ = \int p \left(a _ {t}, z _ {t + k}, z _ {t}\right) \log q \left(a _ {t} \mid z _ {t + k}, z _ {t}\right) - \int p \left(z _ {t}\right) p \left(z _ {t + k} \mid a _ {t}, z _ {t}\right) p \left(a _ {t} \mid z _ {t}\right) \log p \left(a _ {t} \mid z _ {t}\right) \\ = \mathbb {E} _ {p (a _ {t}, z _ {t + k}, z _ {t})} [ \log q (a _ {t} | z _ {t + k}, z _ {t}) ] + \mathbb {E} _ {p (z _ {t}) p (z _ {t + k} | a _ {t}, z _ {t})} [ \mathcal {H} (p (a _ {t} | z _ {t})) ] \\ \end{array}
+$$
+
+Here,  $q(a_{t}|z_{t + k},z_{t})$  is the inverse dynamics model, and  $\mathcal{H}(p(a_t|z_t))$  is the entropy of the policy.
+
+We can obtain a variational lower bound on the reward model as
+
+$$
+\begin{array}{l} I (r _ {t}; z _ {t}) = \int p (r _ {t}, z _ {t}) \log \frac {p (r _ {t} | z _ {t})}{p (r _ {t})} \\ = \int p \left(r _ {t}, z _ {t}\right) \left(\log \frac {q \left(r _ {t} \mid z _ {t}\right)}{p \left(r _ {t}\right)} + \log \frac {p \left(r _ {t} \mid z _ {t}\right)}{q \left(r _ {t} \mid z _ {t}\right)}\right) \\ = \int p \left(r _ {t}, z _ {t}\right) \log \frac {q \left(r _ {t} \mid z _ {t}\right)}{p \left(r _ {t}\right)} + K L [ p \left(r _ {t}, z _ {t}\right) | | q \left(r _ {t}, z _ {t}\right) ] \\ \geq \int p (r _ {t}, z _ {t}) \log \frac {q (r _ {t} | z _ {t})}{p (r _ {t})} \\ = \int p \left(r _ {t}, z _ {t}\right) \log q \left(r _ {t} \mid z _ {t}\right) - \int p \left(r _ {t}, z _ {t}\right) \log p \left(r _ {t}\right) \\ = \mathbb {E} _ {p (r _ {t}, z _ {t})} [ \log q (r _ {t} | z _ {t}) ] + C o n s t a n t \\ \end{array}
+$$
+
+Here,  $q(r_t|z_t)$  is the reward decoder. Since the reward is a scalar, reconstructing it is computationally simpler compared to reconstructing high dimensional observations as in the BA bound for the observation model.
+
+# A.4 VALUE DIFFERENCE RESULT
+
+We show that the difference in value function of the underlying MDP  $Q^{\pi}(o,a)$  and the latent MDP  $\hat{Q}^{\pi}(z,a)$ , where  $z\sim q_{\phi}(z|o)$  is bounded, under some regularity assumptions. Similar to the assumption in (Gelada et al., 2019), let the policy  $\pi$  have bounded semi-norm value functions under the total variation distance  $D_{\mathrm{TV}}$  i.e.  $|\hat{V}^{\pi}(z)|_{D_{\mathrm{TV}}}\leq K$  and  $|\mathbb{E}_{z_1\sim P}V(z_1) - \mathbb{E}_{z_2\sim Q}V(z_2)|\leq KD_{\mathrm{TV}}(P,Q)$ .
+
+Define the forward dynamics learning objective as  $\min_{\chi}L_{T} = \min_{\chi}D_{\mathrm{KL}}(p(o'|o,a)||q_{\chi}(z'|z,a))$  and the reward model objective as  $\min_{\eta}L_R = \min_{\eta}D_{\mathrm{KL}}(p(r|o)||q_{\eta}(r|z))$
+
+Theorem 3. The difference between the  $Q$ -function in the latent MDP and that in the original MDP is bounded by the loss in reward predictor and forward dynamics model.
+
+$$
+\mathbb {E} _ {(o, a) \sim d ^ {\pi} (o, a), z \sim q _ {\phi} (z | o)} [ Q ^ {\pi} (o, a) - \hat {Q} ^ {\pi} (z, a) ] \leq \frac {\sqrt {L _ {R}} + \gamma K \sqrt {L _ {T}}}{1 - \gamma}
+$$
+
+Proof.
+
+$$
+\begin{array}{l} \mathbb {E} _ {(o, a) \sim d ^ {\pi} (o, a), z \sim q _ {\phi} (z | o)} [ Q ^ {\pi} (o, a) - \hat {Q} ^ {\pi} (z, a) ] \\ \leq \mathbb {E} _ {(o, a) \sim d ^ {\pi} (o, a), z \sim q _ {\phi} (z | o)} \bigl (\left| r (o, a) - \hat {r} (z, a) \right| + \gamma \left| \mathbb {E} _ {o ^ {\prime} \sim p (o ^ {\prime} | o, a)} V (o ^ {\prime}) - \mathbb {E} _ {z ^ {\prime} \sim q _ {\chi} (z ^ {\prime} | z, a)} \hat {V} (z ^ {\prime}) \right| \bigr) \\ \leq \sqrt {D _ {\mathrm {K L}} (p (r | o) | | q _ {\eta} (r | z))} + \mathbb {E} _ {(o, a) \sim d ^ {\pi} (o, a), z \sim q _ {\phi} (z | o)} (\gamma | \mathbb {E} _ {o ^ {\prime} \sim p (o ^ {\prime} | o, a), z ^ {\prime} \sim q _ {\phi} (z ^ {\prime} | o ^ {\prime})} [ V (o ^ {\prime}) - \hat {V} (z ^ {\prime}) ] | \\ + \gamma | \mathbb {E} _ {o ^ {\prime} \sim p (o ^ {\prime} | o, a), z ^ {\prime} \sim q _ {X} (z ^ {\prime} | z, a)} [ V (o ^ {\prime}) - \hat {V} (z ^ {\prime}) ] |) \\ \leq \sqrt {L _ {R}} + \mathbb {E} _ {(o, a) \sim d ^ {\pi} (o, a), z \sim q _ {\phi} (z | o)} (\gamma | \mathbb {E} _ {o ^ {\prime} \sim p (o ^ {\prime} | o, a), z ^ {\prime} \sim q _ {\phi} (z ^ {\prime} | o ^ {\prime})} [ V (o ^ {\prime}) - \hat {V} (z ^ {\prime}) ] |) \\ + \mathbb {E} _ {(o, a) \sim d ^ {\pi} (o, a), z \sim q _ {\phi} (z | o)} (\gamma K D _ {\mathrm {T V}} (p (o ^ {\prime} | o, a) | | q _ {\chi} (z ^ {\prime} | z, a))) \\ \leq \sqrt {L _ {R}} + \gamma \mathbb {E} _ {(o, a) \sim d ^ {\pi} (o, a), z \sim q _ {\phi} (z | o)} \mathbb {E} _ {o ^ {\prime} \sim p (o ^ {\prime} | o, a), z ^ {\prime} \sim q _ {\phi} (z ^ {\prime} | o ^ {\prime})} [ V (o ^ {\prime}) - \hat {V} (z ^ {\prime}) ]) \\ + \gamma K \sqrt {D _ {\mathrm {K L}} (p (o ^ {\prime} | o , a) | | q _ {\chi} (z ^ {\prime} | z , a))} \\ \leq \sqrt {L _ {R}} + \gamma \mathbb {E} _ {(o, a) \sim d ^ {\pi} (o, a), z \sim q _ {\phi} (z | o)} [ V (o) - \hat {V} (z) ]) + \gamma K \sqrt {L _ {T}} \\ \leq \sqrt {L _ {R}} + \gamma \mathbb {E} _ {(o, a) \sim d ^ {\pi} (o, a), z \sim q _ {\phi} (z | o)} [ Q (o, a) - \hat {Q} (z, a) ]) + \gamma K \sqrt {L _ {T}} \\ \end{array}
+$$
+
+So, we have shown that
+
+$$
+\mathbb {E} _ {(o, a) \sim d ^ {\pi} (o, a), z \sim q _ {\phi} (z | o)} [ Q ^ {\pi} (o, a) - \hat {Q} ^ {\pi} (z, a) ] \leq \frac {\sqrt {L _ {R}} + \gamma K \sqrt {L _ {T}}}{1 - \gamma}
+$$
+
+![](images/5075625daaf4b427692506eac396f0616819f0c84bc0fb7962781d20f291134b.jpg)
+
+We can see that, as  $L_{R}, L_{T} \to 0$ , the Q-function in the representation space of the MDP,  $\hat{Q}$ , becomes increasingly closer to that of the original MDP,  $Q$ . Since this result holds for all Q-functions, it also holds for the optimal  $Q^{*}$  and  $\hat{Q}^{*}$ . This result extends sufficiency results in prior works Rakelly et al. (2021) to a setting with stochastic encoders and KL-divergence losses on forward dynamics and reward, as opposed to Wasserstein metrics (Gelada et al., 2019), and bisimulation metrics (Zhang et al., 2020).
+
+# A.5 BEHAVIORAL SIMILARITY METRIC DETAILS
+
+Given the sets  $S_{z} = \{z^{i}\}_{i=1}^{n}$  and  $S_{z_{\mathrm{gt}}} = \{z_{\mathrm{gt}}^{i}\}_{i=1}^{n}$ , we construct complete graphs by connecting every vertex with every other vertex through an edge. The weight of an edge is the Euclidean distance between the respective pairs of vertices. The label of each vertex is an unique integer and corresponding vertices in the two graphs are labelled similarly. Let the resulting graphs be denoted as  $G_{z} = (V_{z}, E_{z})$  and  $G_{z_{\mathrm{gt}}} = (V_{z_{\mathrm{gt}}}, E_{z_{\mathrm{gt}}})$  respectively. Note that both these graphs are shortest path graphs, by construction.
+
+Since the Euclidean distances in both the graphs cannot be directly compared, we scale the distances such that the shortest edge weight among all edge weights are equal in both the graphs. Scaling every edge weight by the same number doesn't change the structure of the graph. Now, we define the shortest path kernel to measure similarity between the two graphs, as follows
+
+$$
+k (G _ {z}, G _ {z _ {\mathrm {g t}}}) = \frac {1}{| E _ {z} |} \sum_ {e _ {i} \in E _ {z}} \sum_ {e _ {j} \in E _ {z _ {\mathrm {g t}}}} \hat {k} (e _ {i}, e _ {j})
+$$
+
+Let,  $e_i = \{u_i, v_i\}$  and  $e_j = \{u_j, v_j\}$ . Here, the kernel  $\hat{k}$  measures similarity between the edges  $e_i, e_j$  and the labels on respective vertices. Let the function  $l(\cdot)$  denote the labels of vertices and the weights of edges.
+
+$$
+\hat {k} (e _ {i}, e _ {j}) = k _ {v} (l (u _ {i}), l (u _ {j})) k _ {e} (l (e _ {i}), l (e _ {j})) k _ {v} (l (v _ {i}), l (v _ {j})) + k _ {v} (l (u _ {i}), l (v _ {j})) k _ {e} (l (e _ {i}), l (e _ {j})) k _ {v} (l (v _ {i}), l (u _ {j}))
+$$
+
+Here,  $k_{v}$  is a Dirac delta kernel i.e.  $k_{e}(x,y) = 1 - \delta (x,y)$  (Note that  $\delta (x,y) = 1$  iff  $x = y$ , else  $\delta (x,y) = 0$ ) and  $k_{e}$  is a Brownian ridge kernel,  $k_{e}(x,y) = \frac{1}{c}\max (0,c - |x - y|)$ , where  $c$  is a large number.
+
+So, in summary, the value of  $\hat{k}(e_i,e_j)$  is low when the edge lengths between corresponding pairs of vertices in both the graphs are close. Hence, the value of  $k(G_z,G_{z_{\mathrm{gt}}})$  is low when a large number of pairs of corresponding vertices in both the graphs have the same edge length.
+
+We expect methods that recover latent state representations which better reflect the true underlying simulator state (i.e., the positions of the joints) to have higher values according to this metric.
+
+# A.6 DESCRIPTION OF THE DISTRACTORS
+
+ILSVRC dataset (Russakovsky et al., 2015) The distractors used for the main experiments are natural video backgrounds with RGB images from the ILSVRC dataset. We use 200 videos during training, and reserve 50 videos for testing. An illustration of this for different environments are shown in Fig. 8. These images are snapshots of what is received by the algorithm (64x64x3 images) and no information about proprioceptive features is provided. While natural videos that are unrelated to the task might be easy to ignore, realistic scenes might have other elements that resemble the controllable elements, but are not actually controllable (e.g., other cars in a driving scenario).
+
+To emulate this, we also add distractors that resemble other potentially controllable robots, as shown for example in Fig. 8 (top left and bottom right frames), but are not actually influenced by the actions of the algorithm. These are particularly challenging because the background video has a prerecorded motion of the same (or similar) agent that is being controlled. We include such challenging agent-behind-agent background videos with a frequency of 1 in very 3 videos.
+
+For the experiments in Table 1, examples of different levels of distractors are shown in Fig. 9. The three different levels have distractor windows of size  $32 \times 32$ ,  $40 \times 40$ , and  $64 \times 64$  respectively.
+
+Kinetics dataset (Kay et al., 2017) For comparing directly with results in prior works DBC (Zhang et al., 2020), and TIA (Fu et al., 2021), we evaluate on the setting where the random videos are grayscale images from the Kinetics dataset driving car class. The settings are same as in the prior works and we compare directly to the results in the respective papers.
+
+# A.7 TRAINING AND NETWORK DETAILS
+
+The agent observations have shape  $64 \times 64 \times 3$ , action dimensions range from 1 to 12, rewards per time-step are scalars between 0 and 1. All episodes have randomized initial states, and the initial distractor video background is chosen randomly.
+
+We implement our approach with TensorFlow 2 and use a single Nvidia V100 GPU and 10 CPU cores for each training run. The training time for 1e6 environment steps is 4 hours on average. In comparison, the average training time for 1e6 steps for Dreamer is 3 hours, for CDreamer is 4 hours, for TIA is 4 hours, for DBC is 4.5 hours, and for DeepMDP is 5 hours.
+
+The encoder consists of 4 convolutional layers with kernel size 4 and channel numbers 32, 65, 128, 256. The forward model consists of deterministic and stochastic parts, as in standard in visual model-based RL (Hafner et al., 2019b;a). The stochastic states have size 30 and deterministic state has size 200. The compatibility function  $f_{\theta}(z_t, o_{t'}) = \exp(z_t^T W_{\theta} z_{t'})$  for contrastive learning is learned with a  $200 \times 200 W_{\theta}$  matrix. Here,  $z_{t'}$  is the encoding of  $o_{t'}$ . All other models are implemented as three dense layers of size 300 with ELU activations.
+
+We use ADAM optimizer, with learning rate of 6e-4 for the latent-state space model, and 8e-5 for the value function and policy optimization. The hyper-parameter  $c_{0}$  for the prioritized information constraint is set to 1000, after doing a grid-search over the range [100, 10000] and observing similar performance for 1000 and 10000. 100 training steps are followed by 1 episode of interacting with the environment with the currently trained policy, for observing real transitions. The dataset is initialized with 7 episodes collected with random actions in the environment. We kept the hyper-parameters of InfoPower same across all environments and distractor types.
+
+For fair comparisons, we kept all the common hyper-parameter values same as Dreamer (Hafner et al., 2019a). This was the protocol followed in prior works TIA (Fu et al., 2021) and Ma et al.
+
+![](images/b26606aa4fbbeec57820845fcda54a811b8c209a02d1a81bb2f1113d268c4d2c.jpg)
+
+![](images/e290ba43b19dca6c6acd29bc3f678c4098ed119a5614f7a3695d2e3bfccab00d.jpg)
+
+![](images/5bad1f36d0d6722e6b5791dd9083a3927cb490ddcb86e09de5da3f609ec7e32d.jpg)
+
+![](images/c0bafdd533c235d4a75387256701bfec2505bca016b3c2c5617981dbb4ce1bad.jpg)  
+Figure 8: Illustration of the natural video background distractors used in our experiments. The videos change after every 50 time-steps. Some video backgrounds (for example the top left and the bottom right) have more complex distractors in the form of agent-behind-agent i.e. an agent of similar morphology moves in the background of the agent being controlled.
+
+![](images/5c8fbe6093ed387f5cdcdcb47f957f5c3f88f9983a102a5f6a520f288e2064dc.jpg)
+
+![](images/da4306e5cdec3cca389069094e76e30f776218e503bace1621bd88712ba64099.jpg)
+
+![](images/c0ccdda148f563974b1eabcd126bae04f961a893ca00ba014795e4e19d790d77.jpg)  
+L1
+
+![](images/c6ca88a7cd1306bdca688b35eb99513a328c5e599c6704a4dbe66114b7e80d4d.jpg)  
+L2
+
+![](images/edbf65a7f6b7af332250c9b34ff6370be98f0a173efde88761ec212398d12aaa.jpg)  
+L3  
+Figure 9: Illustration of the different levels of distractors. L1, L2, and L3 respectively have distractor windows of size  $32 \times 32$ ,  $40 \times 40$ , and  $64 \times 64$ .
+
+(2020). For the baseline TIA, we tuned the environment-specific parameters  $\lambda_{R_{adv}}$  and  $\lambda_{O_s}$  as mentioned in Table 2 of (Fu et al., 2021). For  $\lambda_{R_{adv}}$ , we performed a gridsearch over the range  $10k - 50k$  and for  $\lambda_{O_s}$  we performed a gridsearch over the range  $1 - 3$  to obtain the parameters for best performance, which we used for the plots in Fig. 4 and Fig. 11. Respectively for walker-walk, cheetah-run, finger-spin, quadruped-walk, hopper-stand, ball-in-a-cup-catch, cartpole-balance-sparse, quadruped-run, the values of  $\lambda_{R_{adv}}$  are  $30k,30k,20k,40k,30k,40k,20k,30k$ . The values of  $\lambda_{O_s}$  are  $2,2,1.5,2.5,2.5,2.2,2$ .
+
+For baseline DBC (Zhang et al., 2020), we kept all the parameters same as in Table 2 of the paper, because DBC does not have any environment-specific parameters and the same values were used for all environments in the DBC paper. The DeepMDP agent and its hyperparameters are adapted from the implementation provided in the github repo of DBC (Zhang et al., 2020).
+
+# A.8 CONTRASTIVE LEARNING
+
+Since the distracting backgrounds are changing videos of a certain duration, for contrastive learning, we contrast both across time and across batches. Contrasting across time helps model invariance against temporally contiguous distractors (for example in the same video) while contrasting across batches helps in learning invariance across different videos. Concretely, we sample batches from the dataset  $\left(^{(i)}\right)\{(a_{t},o_{t},r_{t})\}_{t = 1}^{H}$ , where  $i$  denotes a batch index, and for each observation  $o_t$  in batch  $i$ , all observations  $o_{t'}$  both in batch  $i$  and in other batches  $j\neq i$  are considered negative samples.
+
+# A.9 RESULT ON VARYING DISTRACTOR LEVELS
+
+We consider different levels of distractors by varying the size of the window where distractors in the background are active. Fig. 9 in the Appendix illustrates this visually. Table 1 shows results for
+
+Table 1: DM Control Walker Stand with natural video distractors at different levels. We tabulate the rewards (Rew) and behavioral similarity (Sim) at  ${500}\mathrm{k}$  and  $1\mathrm{M}$  steps of training. Higher is better for both Rew and Sim.  
+
+<table><tr><td>Name</td><td>Levels</td><td>Rew@500k</td><td>Rew@1M</td><td>Sim@500k</td><td>Sim@1M</td></tr><tr><td>Dreamer</td><td>L1</td><td>197 ± 31</td><td>240 ± 27</td><td>0.73±0.02</td><td>0.74±0.01</td></tr><tr><td>DBC</td><td>L1</td><td>261 ± 25</td><td>390 ± 34</td><td>0.75±0.03</td><td>0.74±0.02</td></tr><tr><td>C-Dreamer</td><td>L1</td><td>291 ± 34</td><td>590 ± 26</td><td>0.73±0.02</td><td>0.79±0.01</td></tr><tr><td>DeepMDP</td><td>L1</td><td>263 ± 31</td><td>340 ± 22</td><td>0.74±0.02</td><td>0.75±0.03</td></tr><tr><td>InfoPower</td><td>L1</td><td>397 ± 22</td><td>650 ± 100</td><td>0.82 ± 0.01</td><td>0.84 ± 0.03</td></tr><tr><td>Dreamer</td><td>L2</td><td>180 ± 36</td><td>197 ± 20</td><td>0.56±0.03</td><td>0.59±0.02</td></tr><tr><td>DBC</td><td>L2</td><td>221 ± 28</td><td>320 ± 33</td><td>0.65±0.02</td><td>0.66±0.02</td></tr><tr><td>C-Dreamer</td><td>L2</td><td>282 ± 30</td><td>550 ± 66</td><td>0.63±0.01</td><td>0.71±0.02</td></tr><tr><td>DeepMDP</td><td>L2</td><td>213 ± 34</td><td>300 ± 25</td><td>0.59±0.02</td><td>0.61±0.04</td></tr><tr><td>InfoPower</td><td>L2</td><td>394 ± 30</td><td>644 ± 101</td><td>0.77 ± 0.03</td><td>0.77 ± 0.03</td></tr><tr><td>Dreamer</td><td>L3</td><td>140 ± 52</td><td>157 ± 10</td><td>0.32±0.02</td><td>0.33±0.03</td></tr><tr><td>DBC</td><td>L3</td><td>165 ± 20</td><td>221 ± 24</td><td>0.45±0.01</td><td>0.48±0.01</td></tr><tr><td>C-Dreamer</td><td>L3</td><td>231 ± 39</td><td>485 ± 86</td><td>0.58±0.02</td><td>0.64±0.01</td></tr><tr><td>DeepMDP</td><td>L3</td><td>153 ± 23</td><td>212 ± 28</td><td>0.44±0.02</td><td>0.49±0.01</td></tr><tr><td>InfoPower</td><td>L3</td><td>389 ± 20</td><td>624 ± 80</td><td>0.71 ± 0.01</td><td>0.74 ± 0.03</td></tr></table>
+
+Table 2: Comparison of baselines TIA and DBC with InfoPower on environments with video distractors from the Kinetics dataset Driving Car class. The results for TIA and DBC are from the respective papers.  
+
+<table><tr><td></td><td colspan="2">TIA</td><td colspan="2">DBC</td><td colspan="2">InfoPower</td></tr><tr><td></td><td>500k</td><td>800k</td><td>500k</td><td>800k</td><td>500k</td><td>800k</td></tr><tr><td>Walker-Run</td><td>389±45</td><td>602±40</td><td>165±92</td><td>210±32</td><td>417±22</td><td>630±33</td></tr><tr><td>Cheetah-Run</td><td>384±149</td><td>579±172</td><td>261±64</td><td>277±81</td><td>425±97</td><td>572±158</td></tr><tr><td>Hopper-Stand</td><td>338±221</td><td>581±214</td><td>110±92</td><td>207±120</td><td>395±140</td><td>615±176</td></tr><tr><td>Ball-in-a-Cup</td><td>201±197</td><td>115±110</td><td>-</td><td>-</td><td>255±104</td><td>263±116</td></tr><tr><td>Walker-Walk</td><td>878±37</td><td>948±28</td><td>545±57</td><td>630±68</td><td>925±18</td><td>972±26</td></tr><tr><td>Finger-Spin</td><td>371±96</td><td>487±107</td><td>801±10</td><td>832±4</td><td>795±34</td><td>787±58</td></tr><tr><td>Hopper-Hop</td><td>47±44</td><td>72±68</td><td>5±12</td><td>0±0</td><td>85±26</td><td>77±74</td></tr></table>
+
+the different approaches on varying distractor levels. The size of the frame for distractors at levels L1, L2, and L3 respectively are 32x32, 40x40, and 64x64. We observe that the baselines perform worse as the distractor window size increases. While, for InfoPower, the performance decreases with increasing level is minimal. This indicates the effectiveness of InfoPower in filtering out background distractors from the observations while learning latent representations. Note that the distractors in Figs. 4 and 6 are of level 3 i.e. same size as the observations 64x64.
+
+# A.10 RESULTS ON DISTRACTORS FROM THE KINETICS DATASET
+
+In this section, we evaluate InfoPower on the same distractors used in prior works, DBC and TIA. The distractor videos are from the Kinetics dataset (Kay et al., 2017) Driving Car class, and converted to grayscale. From Table 2 we see that InfoPower slightly outperforms the baselines. We believe the performance gap is not as significant as Fig. 4 because the distractors being grayscale images, and not RGB might be easier to ignore by the baselines as well as InfoPower.
+
+# A.11 SETTING WITH ONLY SHIFTED AGENT BEHIND AGENT DISTRACTORS
+
+In this section we evaluate InfoPower with the baselines only for the challenging setting of agent-behind-agent distrators where the background contains the same agent being controlled (a walker or a cheetah). An illustration of this is shown in Fig. 10. From the results in Table 3, we see that InfoPower significantly out
+
+![](images/1ac9ef0cc09220893ef07d8cb6c654cd2266534d9c7d9ed65e333d42d9d65dd5.jpg)  
+Figure 10: Illustration of the distractor setting for results in Table 3.
+
+![](images/2b31a52ae74ecd74fe72fea5c986136983d072f2578565169749c3d6be459b4e.jpg)
+
+performs the competitive baselines both at 500k and 1M environment interactions. This suggests the utility of InfoPower in separating out informative elements of the observation from uninformative ones even in challenging settings.
+
+Table 3: Evaluation of InfoPower and the most competitive baselines in a suite of challenging distractors where the background contains a shifted version of the agent being controlled. Results are averaged over 4 random seeds. Fig. 10 shows a visualization of the distractors for this setting.  
+
+<table><tr><td></td><td colspan="2">Walker-Walk-shifted</td><td colspan="2">Cheetah-Run-shifted</td></tr><tr><td></td><td>500k</td><td>1M</td><td>500k</td><td>1M</td></tr><tr><td>DBC</td><td>35±15</td><td>104±28</td><td>41±10</td><td>78±38</td></tr><tr><td>CDreamer</td><td>108±36</td><td>164±45</td><td>73±44</td><td>110±52</td></tr><tr><td>TIA</td><td>79±42</td><td>152±33</td><td>65±24</td><td>115±28</td></tr><tr><td>InfoPower</td><td>168±48</td><td>278±42</td><td>130±46</td><td>207±35</td></tr></table>
+
+![](images/f270e6992273fb17037be8fbca3945970a906bcfc8c79157ec31cbbddccfdefa.jpg)
+
+![](images/c3ae08e690f3a266d1f8e6d9a289f93b8433d2199fc34a634ae4ababdbfa393a.jpg)
+
+![](images/4e16b52547f684c1e1d03f08fed2f45f72596a92807e281ceb30d578b4771f51.jpg)
+
+![](images/acf1ae2c55710c1d03250cd4f649c502b33e0d9512d8302ef7541706fed8b9af.jpg)
+
+![](images/c40d058f1fc974275a4f80f945605fdc1fcb966a46433263e34125ba613b9b50.jpg)  
+Figure 11: Evaluation of InfoPower and baselines with empowerment in policy learning. The setting corresponds to DeepMind Control tasks with RGB natural video distractors in the background (same as that in Figure 5 of the main paper). The x-axis denotes the number of environment interactions and the y-axis shows the episodic returns.
+
+![](images/255c4c70725c4c33bd76fd486a6f644bd1427c73f69b893b8153e7d0b5ca6c03.jpg)
+
+![](images/e57d2c105ea3b2eda5c5839bfa695fb57b28d2afec0628199add02d53fb282f2.jpg)
+
+![](images/0bacfb4044d04b0a1ed25d514216a2a6bd0a0b4c040358b4f311c41a80242c5e.jpg)
+
+# A.12 COMPARISON OF BASELINES WITH EMPOWERMENT IN POLICY LEARNING
+
+In this section, we compare against modified versions of the baselines, such that we include empowerment in the policy learning of the baselines. This is the only modification we make to the baselines, and keep everything else unchanged. We described in section 3.3 that inclusion of empowerment in the objective for visual model-based RL (by modifying both the representation learning objective and the policy learning objective) is an important contribution of our paper. The aim of this experiment is to disentangle the benefits of empowerment in representation learning, for InfoPower with that in policy learning.
+
+We have plotted the results in Fig. 11. We can see that the performance of the baselines is slightly better than Fig. 4 because of the inclusion of empowerment in the policy learning objective which as we motivated previously in section 3.4 helps with exploring controllable regions of the state-space. Note that by adding empowerment, we have effectively modified the baselines.
+
+# A.13 COMPARISON ON ENVIRONMENTS WITHOUT DISTRACTORS
+
+We compare the performance of InfoPower with a state-of-the-art visual model-based RL method, Dreamer in the default DM Control environments without distractors. We see from Table 4 that InfoPower performs slightly better or similar to Dreamer in all the environments consistently. This result complements our main paper result on environments with distractors and confirms that the benefit of InfoPower in challenging distracting environments does not come at a cost of performance in simpler non-distracting environments.
+
+Table 4: Comparison of InfoPower with state-of-the-art visual model-based RL method, Dreamer in the default DM Control environments without distractors. We tabulate reward at  $500\mathrm{k}$  and 1M environment interactions. Results are averaged over 4 random seeds. Higher is better.  
+
+<table><tr><td></td><td colspan="2">Dreamer</td><td colspan="2">InfoPower</td></tr><tr><td></td><td>Rew @ 500k</td><td>Rew @ 1M</td><td>Rew @ 500k</td><td>1M</td></tr><tr><td>Walker Walk</td><td>890±52</td><td>985±10</td><td>910±41</td><td>980±17</td></tr><tr><td>Cheetah Run</td><td>580±190</td><td>811±95</td><td>586±142</td><td>810±102</td></tr><tr><td>Finger Spin</td><td>570±187</td><td>573±96</td><td>590±172</td><td>598±85</td></tr><tr><td>Quadruped Walk</td><td>592±45</td><td>646±50</td><td>623±65</td><td>666±47</td></tr><tr><td>Hopper Stand</td><td>701±68</td><td>906±33</td><td>715±60</td><td>930±39</td></tr><tr><td>Ball in a Cup</td><td>871±102</td><td>910±48</td><td>913±67</td><td>939±52</td></tr><tr><td>Cart-Pole Sparse</td><td>742±58</td><td>849±88</td><td>776±40</td><td>866±80</td></tr><tr><td>Quadruped Run</td><td>450±47</td><td>515±65</td><td>468±42</td><td>510±61</td></tr></table>
+
+# A.14 ADDITIONAL ABLATIONS
+
+In this section, we consider additional ablations, namely a version of of InfoPower that does not have the empowerment objective, a version of InfoPower that does not have the MI maximization objective. We see from Table 5 that removing either of these terms drops performance.
+
+Table 5: DM Walker Stand with natural video distractors  
+
+<table><tr><td>Name</td><td>Obs bound</td><td>Rew@500k</td><td>Rew@1M</td><td>Sim@500k</td><td>Sim@1M</td></tr><tr><td>Dreamer</td><td>BA</td><td>140 ± 52</td><td>157 ± 10</td><td>0.32±0.02</td><td>0.33±0.03</td></tr><tr><td>Contrastive-Dreamer</td><td>NCE</td><td>231 ± 39</td><td>485 ± 86</td><td>0.58±0.02</td><td>0.64±0.01</td></tr><tr><td>NonGenerative-Dreamer</td><td>NWJ</td><td>280 ± 25</td><td>480 ± 93</td><td>0.59±0.01</td><td>0.62±0.05</td></tr><tr><td>InfoPower</td><td>NCE</td><td>254 ± 27</td><td>481 ± 30</td><td>0.69±0.03</td><td>0.72±0.02</td></tr><tr><td>InfoPower</td><td>NWJ</td><td>389 ± 20</td><td>624 ± 80</td><td>0.71±0.01</td><td>0.74±0.03</td></tr><tr><td>Inverse only</td><td>None</td><td>282 ± 70</td><td>367 ± 24</td><td>0.63±0.04</td><td>0.66±0.03</td></tr></table>

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+# INVERSE ONLINE LEARNING: UNDERSTANDING NON-STATIONARY AND REACTIONARY POLICIES
+
+Alex J. Chan, Alicia Curth, and Mihaela van der Schaar
+
+University of Cambridge
+
+Department of Applied Mathematics and Theoretical Physics
+
+Cambridge, UK
+
+{ajc340, amc253, mv472}@cam.ac.uk
+
+# ABSTRACT
+
+Human decision making is well known to be imperfect and the ability to analyse such processes individually is crucial when attempting to aid or improve a decision-maker's ability to perform a task, e.g. to alert them to potential biases or oversights on their part. To do so, it is necessary to develop interpretable representations of how agents make decisions and how this process changes over time as the agent learns online in reaction to the accrued experience. To then understand the decision-making processes underlying a set of observed trajectories, we cast the policy inference problem as the inverse to this online learning problem. By interpreting actions within a potential outcomes framework, we introduce a meaningful mapping based on agents choosing an action they believe to have the greatest treatment effect. We introduce a practical algorithm for retrospectively estimating such perceived effects, alongside the process through which agents update them, using a novel architecture built upon an expressive family of deep state-space models. Through application to the analysis of UNOS organ donation acceptance decisions, we demonstrate that our approach can bring valuable insights into the factors that govern decision processes and how they change over time.
+
+# 1 INTRODUCTION
+
+Decision modelling is often viewed through the lens of policy learning (Bain & Sammut, 1995; Abbeel & Ng, 2004), where the aim is to learn some imitation policy that captures the actions (and thus decisions) of an agent in some structured way (Hüyük et al., 2021a). Unfortunately, this usually implicitly relies on the assumption that the actions taken are already optimal with respect to some objective, and hence do not change over time. While this might be appropriate to approximate the stationary policy of an autonomous agent, such assumptions often break down when applying them to more flexible agents that might be learning on-the-fly. In particular, we may want to model the decision making process of a human decision-maker, whose actions we may see executed over time - this is clearly of great use in a range of fields including economics and medicine (Hunink et al., 2014; Zavadskas & Turskis, 2011). In the context of machine learning, they would be considered to be undergoing a process of online learning (Hoi et al., 2018); training while interacting with the environment, where it is to be expected that beliefs and strategies are constantly adapted based on accumulating experience (Elio & Pelletier, 1997). Additionally, such belief updates are often imperfect; for example, people are known to a priori over-weight the potential impact of rare events when they first start a task, before proceeding to neglect them when they do not occur in their experience (Hertwig & Erev, 2009). If our goal is to analyse human decision-making, it is thus important to be able to model non-stationary policies that are potentially imperfect in two senses: both marginally at each time-step, and in the way that they become adapted to new information.
+
+In this work, instead of relying on the assumption that an agent follows a stationary optimal policy, we consider their actions to be consistent within an online leaning framework where they learn from an incoming sequence of examples (Hoi et al., 2018) and update their policies accordingly, as in Figure 1. Thus, instead of assuming that actions are globally optimal, we assume that at any exact time the agent thought that the action was better than the alternatives. As a natural consequence, we frame the problem of decision modelling as that of inverse online learning; that is, we aim to uncover
+
+![](images/25e4484fbc1013936292d02ced632f187093863375d8cac28645f14174eb07ea.jpg)  
+Figure 1: Capturing online learning behaviour. We assume the agent models the response surfaces in their mind, and that their belief changes each time step in response to the actions they take and the consequent observations they make. The response surface they model need not converge on the true surface since the update method of the agent may be biased and not appropriately use the information available to them.
+
+![](images/d49d1119ef929eb5eb4367750f30b61464ea0919224474e34668413420d25f53.jpg)
+
+![](images/81bd173a4394a8fb06572bfbb1a88d964883d482a0dbc760d9e8fc39a9af2927.jpg)
+
+![](images/3cf56dc723ac9faee05183dd586dd3d488f478cb289e8d9c67ab84ab4d59d050.jpg)
+
+the process through which an agent is learning online. Note that this does not involve coming up with a new way to do online learning itself, rather we recover how it appears a given agent may have learnt. We do this by modelling the (non-stationary) policy of the agent using a deep state-space model (Krishnan et al., 2017), allowing us to track the memory of the agent while allowing policies to evolve in a non-parametric and non-prescribed fashion. We construct an interpretable mapping from the accrued memory to policies by framing the decision-making process within a potential-outcome setting (Rubin, 2005); letting actions correspond to interventions (or treatments), and assuming that agents make decisions by choosing an action which they perceive to have the highest (treatment) effect.
+
+Contributions. In this work we make a number of contributions: First, we formalise the inverse online learning problem and a connection between decision modelling, online learning and potential outcomes, highlighting how they complement each other to result in an interpretable understanding of policies (Section 2); Second, we introduce a practical method to estimate the nonstationary policy and update method of an agent given demonstrations (Section 4); And third, we demonstrate how we can uncover useful practical insights into the decision making process of agents with a case study on the acceptance of liver donation offers (Section 5). Code is made available at https://github.com/XanderJC/inverse-online, along with the group codebase at https://github.com/vanderschaarlab/mlforhealthlabpub.
+
+# 2 PROBLEM FORMALISATION
+
+Preliminaries. Assume we observe a data trajectory $^1$  of the form  $\mathcal{D} = \{(X_t, A_t, Y_t)\}_{t=1}^T$ . Here,  $X_t \in \mathcal{X} \subset \mathbb{R}^d$ , denotes a context vector of possible confounders;  $A_t \in \mathcal{A} = \{0,1\}$ , a binary action or intervention; and  $Y_t \in \mathcal{V}$ , a binary or continuous outcome of interest. The subscript  $t$  indicates a time-ordering of observed triplets; any time-step  $t$  is generated by (i) arrival of context  $x$ , (ii) an intervention  $a$  being performed by an agent according to some policy  $\pi_t(x)$  and (iii) a corresponding outcome  $y$  being observed.
+
+Goal. We are interested in recovering the agent's non-stationary policy  $\pi_t(x) = P(A = 1|X = x, H_t = h_t)$ , which depends on the observed context  $x$  and can change over time due to the observed history  $H_t = \{(X_k, A_k, Y_k)\}_{k=1}^{t-1}$ . We make the key assumption that the agent acts with the intention to choose an intervention  $a$  leading to the best (largest) potential outcome  $Y(a)$ , but that the perception of the optimal intervention may change over time throughout the agent's learning process about their environment. This problem formalisation naturally places us at the in
+
+![](images/d0eba3c1b8dccd57dd2699de8a7f62a4d9db6c6bcad2c366c1a998723b3538ae.jpg)  
+Figure 2: Simplified inverse process. Our task is to infer the evolving perceived effect over time (the blue line) given only realisations of the agent's behaviour.
+
+tersection of treatment effect inference within the Neyman-Rubin potential outcomes (PO) framework
+
+(Rubin, 2005) with the classical (inverse) reinforcement learning (RL) paradigm (Abbeel & Ng, 2004). More specifically though the RL-subproblem of logged contextual bandits (Swaminathan & Joachims, 2015; Atan et al., 2018), where the setting is offline and the contexts arrive independently.
+
+Perception Versus Reality. As is standard within the PO framework, we assume that any observation has two potential outcomes  $Y(a)$ , associated with each intervention (treatment-level), which are generated from some true distribution inherent to the environment, i.e.  $Y(0), Y(1) \sim \mathbb{P}^e(\cdot | X = x)$ , yet only the potential outcome associated with the chosen treatment is revealed to the agent. If the agent had full knowledge of the environment, its optimal policy would be to choose an action as  $\arg \max_a \mu_a^e(x)$  where  $\mu_a^e(x) = \mathbb{E}_{\mathbb{P}^e}[Y(a) | X = x]$ , or conversely to choose to perform the intervention only when its expected conditional average treatment effect (CATE)  $\tau^e(x) = \mu_1^e(x) - \mu_0^e(x)$  is positive. In general, we assume that such perfect knowledge of the environment is not available to the agent in many scenarios of practical interest. Instead, we assume that at any time  $t$ , the agent acts according to some perceived model  $\tilde{\mathbb{P}}$  of its environment, which induces the perceived expected potential outcomes  $\tilde{\mu}_a(x; t)$  and associated treatment effect  $\tilde{\tau}(x; t)$ . Here,  $\tilde{\mu}_a(x; t) = \mathbb{E}_{\tilde{\mathbb{P}}} [Y(a) | X = x, H_t = h_t]$ , is a function not only of the observed context but also the history, such that the agent can update its beliefs according to observed outcomes. Note that we allow the agent's model of the environment to be misspecified; in this case  $\tilde{\mu}_a(x; t)$  would not converge to  $\mu_a^e(x)$  even for very long trajectories. We consider this an important feature, as we assume that human decision-makers generally need to make simplifications (Griffiths et al., 2015). Formally, we make the following assumption about the agent's decision making process:
+
+Assumption 1 Mutual Observability: The observable space  $\mathcal{X}$  contains only and all the information available to the agent at the point of assigning an action.
+
+Assumption 2 Perceived Optimality: An agent assigns the action they think will maximise the outcome. For a deterministic policy  $\pi_t$ , the agent believes  $\pi_t(x) = 1 \iff \tilde{\mu}_1(x; t) > \tilde{\mu}_0(x; t)$ . Given a stochastic agent, the policy  $\pi_t(x) = P(A_t = 1|x)$  is a monotonic increasing function of the agent's belief over the treatment effect  $\tilde{\tau}(x; t)$ .
+
+Assumption 3 Continual Adaptation: The agent may continually adjust their strategy based on experience. The index in the dataset  $\mathcal{D}$  represents a temporal ordering which  $\pi$  is non-stationary with respect to, i.e.  $\pi_i(x)$  may not equal  $\pi_j(x), i \neq j \geq 1$  where  $\pi_i$  represents the policy at time step  $i$ .
+
+Assumptions 1 and 2 are generally unverifiable in practice, yet, they are necessary: there would be little opportunity to learn anything meaningful about a non-stationary policy without them. Mutual observability is crucial, as assuming that different information streams are available to the observer and the agent would make it impossible to accurately describe the decision-making process. Assumption 2, albeit untestable, appears very reasonable: in the medical setting, for example, a violation would imply malicious intent – i.e. that a clinician purposefully chooses the intervention with a sub-optimal expected outcome. Nonetheless, this does mean that we do not consider exploratory agents that are routinely willing to sacrifice outcomes for knowledge, a constraint we do not consider too restrictive within the medical applications we have in mind. Assumption 3, on the other hand, is not at all restrictive; on the contrary, it explicitly specifies a more flexible framework than is usual in imitation or inverse reinforcement learning as the set of policies it describes contains stationary policies.
+
+# 3 RELATED WORK
+
+Our problem setting is related to, but distinct from, a number of problems considered within related literature. Effectively, we assume that we observe *logged data* generated by a learning agent that acts according to its evolving belief over effects of its actions, and aim to solve the inverse problem by making inferences over said belief given the data. Below, we discuss our relationship to work studying both the forward and the inverse problem of decision modelling, with main points summarised in Table 1.
+
+The Forward Problem: Inferring Prescriptive Models for Behaviour. The agent, whose (nonstationary) policy we aim to understand, actively learns online through repeated interaction with an environment and thus effectively solves an online learning problem (Hoi et al., 2018). The most prominent example of such an agent would be a contextual bandit (Agarwal et al., 2014), which learns to assign sequentially arriving contexts to (treatment) arms. Designing agents that update
+
+policies online is thus the inverse of the problem we consider here. Another seemingly related problem within the RL context is learning optimal policies from logged bandit data (Swaminathan & Joachims, 2015); this is different from our setting as it is prescriptive rather than descriptive.
+
+If the goal was to infer the true  $\tau^e (x)$  from the (logged) observational data  $\mathcal{D}$  (instead of the effect perceived by the agent), this would constitute a standard static CATE estimation problem, for which many solutions have been proposed in the recent ML literature (Alaa & van der Schaar, 2018; Shalit et al., 2017; Künzel et al., 2019). Note that within this literature, the treatment-assignment policy (the so-called propensity score  $\pi (x) = \mathbb{P}(A = 1|X = x)$ ) is usually assumed to be stationary and considered a nuisance parameter that is not of primary interest. Estimating CATE can also be a pretext task to the problem of developing optimal treatment rules (OTR) (Zhang et al., 2012a;b; Zhao et al., 2012), as  $g(x) = \mathbb{1}(\tau (x) > 0)$  is such an optimal rule (Zhang et al., 2012b).
+
+The Backward Problem: Obtaining a Descriptive Summary of an Agent. When no outcomes are observed (unlike in the bandit problem) the task of learning a policy becomes imitation learning (IL), itself a large field that aims to match the policy of a demonstrator (Bain & Sammut, 1995; Ho & Ermon, 2016). Note that, compared to the bandit problem, there is a subtle shift from obtaining a policy that is optimal with respect to some true notion of outcome to one that minimises some divergence from the demonstrator. While IL is often used in the forward problem to find an optimal policy – thereby implicitly assuming that the demonstrator is themselves acting optimally –, it can also, if done interpretably, be used in the backward problem to very effectively reason about the goals and preferences of the agent (Hüyük et al., 2021a). One way to achieve this is through inverse reinforcement learning (IRL), which aims to recover the reward function that is seemingly maximised by the agent (Ziebart et al., 2008; Fu et al., 2018; Chan & van der Schaar, 2021). This need not be the true reward function, and can be interpreted as a potentially more compact way to describe a policy, which is also more easily portable given shifts in environment dynamics.
+
+Our problem formalisation is conceptually closely related to the counterfactual inverse reinforcement learning (CIRL) work of Bica et al. (2021). There the authors similarly aim to explain decisions based on the PO framework, specifically by augmenting a max-margin IRL (Abbeel & Ng, 2004) approach to parameterise the learnt reward as a weighted sum of counterfactual outcomes. However, CIRL involves a pre-processing counterfactual estimation step that focuses on estimating the true treatment effects, and implicitly assumes that these are identical to the ones perceived by the agent (i.e. it assumes that the agent has perfect knowledge of the environment dynamics generating the potential outcomes), marking a significant departure from our (much weaker) assumptions. Without the focus on treatment effects, our method could be seen as a generalised non-parametric extension of the inverse contextual bandits of Huyuk et al. (2021b). In general, the backward problem is hard to evaluate empirically since information about the true beliefs of agents is not normally available in the data, thus relying on simulation to validate (Chan et al., 2021).
+
+Table 1: Problems considered in related work, noting the relationship between both input and output.  
+
+<table><tr><td>Problem</td><td>Input</td><td>Target quantity</td></tr><tr><td>Online Learning of Optimal Polices (Agarwal et al., 2014)</td><td>(Xt, At, Yt)</td><td>πtopt(x)</td></tr><tr><td>Learning Optimal Policy from Logged Bandits (Swaminathan &amp; Joachims, 2015)</td><td>(Xi, Ai, Yt)</td><td>πopt(x)</td></tr><tr><td>Estimating Heterogeneous Treatment Effects (Alaa &amp; van der Schaar, 2018)</td><td>(Xi, Ai, Yt)</td><td>τe(x)</td></tr><tr><td>Learning Imitator Policies via Imitation Learning (Bain &amp; Sammut, 1995)</td><td>(Xi, Ai)</td><td>πobs(x)</td></tr><tr><td>Learning Reward Functions via IRL (Abbeel &amp; Ng, 2004)</td><td>(Xi, Ai)</td><td>Robs(x)</td></tr><tr><td>Learning Reward Functions via CIRL (Bica et al., 2021)</td><td>(Xi, Ai, Yt)</td><td>Robs(x)</td></tr><tr><td>Modelling Dynamic Policies via Inverse Online Learning</td><td>(Xt, At, Yt)</td><td>πtobs(x)</td></tr></table>
+
+# 4 INFERRING THE ONLINE LEARNING PROCESS
+
+Preliminaries: The Forward Problem. In the forward context, the problem we consider amounts to an agent attempting to, at each time-step, take the action they believe will maximise the outcome, without being aware of the true effect of their intervention. We assume that the agent believes the potential outcomes are a linear combination of the contextual features such that  $\tilde{\mu}_a(X_t; t) = \langle X_t, \omega_a^t \rangle$ , with  $\omega_a^t \in \Omega$  a set of weights for the action  $a$  at time  $t$ , and  $\langle \cdot, \cdot \rangle$  denoting the inner product. Normally we may consider a linear model an oversimplification, but
+
+![](images/4688337ff0ef3b51e02f8f3a86f276b0ab19b5c10b39875de76fba0e019cf792.jpg)  
+Figure 3: Model. Solid black lines indicate single time step dependencies. Given covariates  $X_{i}$  and an observed history then treatment effects are estimated in the mind of the agent. An action  $A_{i}$  is selected based on these effect and an outcome  $Y_{i}$  is observed. The dashed grey lines then indicate information flow over time.
+
+when modelling the thought process of an agent it appears much more reasonable - for example, does a doctor consider higher order terms when weighing up potential outcomes of a patient? It seems unlikely, especially as clinical guidelines are often given as linear cutoffs (Burgers et al., 2003). The considered forward problem thus proceeds as follows for every given time step  $t$ :
+
+1. A context  $X_{t} \sim P_{\mathcal{X}}$  arrives independently over time. We make no assumptions on the form of  $P_{\mathcal{X}}$ , in particular noting that there is no need for any level of stationarity.  
+2. Given  $X_{t}$  and the current set of weights  $\{\omega_1^t, \omega_0^t\}$ , the agent predicts their perceived potential outcomes  $\tilde{\mu}_1(X_t; t)$  and  $\tilde{\mu}_0(X_t; t)$ .  
+3. Action  $A_{t}$  is assigned by some probabilistic optimal treatment rule  $\pi : \mathcal{Y}^2 \mapsto [0,1]$  such that  $\mathbb{P}(A_t = 1 | X_t) = \pi (\tilde{\mu}_1(X_t; t), \tilde{\mu}_0(X_t; t))$ , before the associated true outcome is observed.  
+4. Based on this new information the agent (potentially stochastically) updates their belief over the response surface for the taken action according to some update function  $f: \Omega^2 \times \mathcal{X} \times \mathcal{A} \times \mathcal{Y} \mapsto \triangle(\Omega^2)$  such that  $\omega_1^{t+1}, \omega_0^{t+1} \sim f(\omega_1^t, \omega_0^t, X_t, A_t, Y_t)$ .
+
+To maximise outcomes it is crucial to employ an update function  $f$  that maximally captures the available information at each step of the process. This then allows for an appropriate determination of  $\pi$  that aims to maximise the total (potentially future discounted) outcomes.
+
+# 4.1 A MODEL OF INVERSE ONLINE LEARNING
+
+Here, we focus on the inverse to the problem outlined above, which, to the best of our knowledge, has received little to no attention in related work. That is, after observing an agent's actions, can we recover how they arrived at their decisions, and how this changed over time? In the following, we use vector notation  $\vec{x}_{t:t'} = \{x_i\}_{i=t}^{t'}$  with  $\vec{x} = \vec{x}_{1:T}$  for all realised variables, with histories  $\vec{h}_{t:t'} = \{(x_i, a_i, y_i)\}_{i=t}^{t'}$ . Note that the previously described forward model assumes a factorisation of the full generative distribution of the observed data given by:
+
+$$
+p (\vec {x}, \vec {a}, \vec {y}) = \prod_ {t = 1} ^ {T} p \left(y _ {t} \mid a _ {t}, x _ {t}\right) \times p \left(a _ {t} \mid \vec {x} _ {1: t - 1}, \vec {a} _ {1: t - 1}, \vec {y} _ {1: t - 1}\right) \times p \left(x _ {t}\right). \tag {1}
+$$
+
+We assume that  $p(x_{t})$  and  $p(y_{t}|a_{t},x_{t})$  are governed by some true environment dynamics, independent from the perception of the agent and thus not part of our modelling problem. This seems trivially true, as the perception of the agent will neither affect the contexts that arrive nor the true outcome conditional on the action being taken.
+
+Our model for the evolving beliefs of an agent with respect to their interaction with the environment revolves around a specialised deep state-space model. Crucially, this takes the form of a latent random variable  $M_{t} \in \mathbb{R}^{d}$ , which captures the memory of the agent at time  $t$ , and can evolve in a flexible way based on the observed history. We consider the memory to be an efficient summary of the history, which can be interpreted as a sufficient statistic for the beliefs of the agent at a given time. Naturally, this memory is unobserved and must be learnt in an unsupervised
+
+manner. The introduction of this latent variable leads to an extended probabilistic model with a more structured factorisation of the conditional distribution of the actions, and a model given by:
+
+$$
+p _ {\theta} (\vec {a} | \vec {x}, \vec {y}) = \prod_ {t = 1} ^ {T} \underbrace {\pi_ {\theta} \left(a _ {t} \mid \tilde {\tau} _ {t}\right)} _ {\text {T r e a t m e n t R u l e}} \times \underbrace {\mathcal {O} _ {\theta} \left(\tilde {\tau} _ {t} \mid x _ {t} , m _ {t}\right)} _ {\text {O u t c o m e E s t i m a t i o n}} \times \underbrace {\mathcal {S} _ {\theta} \left(m _ {t} \mid \vec {x} _ {1 : t - 1} , \vec {a} _ {1 : t - 1} , \vec {y} _ {1 : t - 1}\right)} _ {\text {M e m o r y A g g r e g a t i o n}}. \tag {2}
+$$
+
+This model consists of three components: a memory summarisation network  $S_{\theta}:\mathcal{H}\mapsto \triangle (\mathcal{M})$ , which maps the history into a (distribution over) memory; a (perceived) potential outcome predictor network  $\mathcal{O}_{\theta}:\mathcal{X}\times \mathcal{M}\mapsto \mathbb{R}^{2}$ , that takes the memory and a context and predicts the perceived potential outcomes; and a treatment rule  $\pi_{\theta}:\mathcal{Y}^2\mapsto [0,1]$ , that given the potential outcomes (summarised by the perceived treatment effect) outputs a distribution over actions. Throughout,  $\theta$  denotes the full set of parameters of the generative model and notation is shared across components to reflect that they can all be optimised jointly. Below, we discuss each component in turn.
+
+Memory Aggregation. We define the base structure of  $S_{\theta}$  to be relatively simple and recursive, and assume the memory at a given time step to be distributed as:
+
+$$
+M _ {t} \sim \mathcal {N} \left(\boldsymbol {\mu} _ {t}, \boldsymbol {\Sigma} _ {t}\right) \quad \text {w i t h} \quad \boldsymbol {\mu} _ {t}, \boldsymbol {\Sigma} _ {t} = \mathrm {M L P} \left(\left[ M _ {t - 1}, X _ {t - 1}, A _ {t - 1}, Y _ {t - 1} \right]\right), \tag {3}
+$$
+
+where  $\pmb{\mu}_t$  and  $\pmb{\Sigma}_t$  are two output heads of the memory network  $S$  given the history  $h_{t-1}$ . It is important to note that the memory network does not get the  $t$ -th step information (including the context) when predicting the memory at time  $t$ . This means the network cannot provide any specific predictive power for a given context and is forced to model the general response surfaces.
+
+In many models of behaviour the update function  $f$  of the agent is modelled as perfect Bayesian inference (Kaelbling et al., 1998; Colombo & Hartmann, 2017). While this may seem natural on the surface given the logical consistency of Bayesian inference, it seems unlikely in practice that agents (and especially humans) will be capable of it. Not least because in all but the simplest cases it will become intractable for computational agents and humans themselves are well known to make simplifying approximations when dealing with past histories (Genewein et al., 2015). By employing a flexible neural architecture, we remove this relatively restrictive assumption on the memory update process. Our model then learns exactly how the agent appears to use previous information to update their predictions - allowing us to model common cognitive biases such as overly weighting recent or largely negative events (Hertwig & Erev, 2009).
+
+Outcome Estimation. Having obtained a memory state  $m_t$ , the second stage involves predicting the perceived potential outcomes for a context  $x_t$  with the network  $\mathcal{O}_{\theta}(\tilde{\tau}_t|x_t,m_t)$ . Given the forward model where the potential outcomes are considered a linear combination of the features this then proceeds again in three steps:
+
+$$
+\omega_ {1} ^ {t}, \omega_ {0} ^ {t} = \operatorname {M L P} \left(m _ {t}\right), \tag {4}
+$$
+
+$$
+\tilde {\mu} _ {1} \left(x _ {t}; t\right) = \left\langle x _ {t}, \omega_ {1} ^ {t} \right\rangle , \quad \tilde {\mu} _ {0} \left(x _ {t}; t\right) = \left\langle x _ {t}, \omega_ {0} ^ {t} \right\rangle \tag {5}
+$$
+
+$$
+\tilde {\tau} _ {t} = \tilde {\mu} _ {1} \left(x _ {t}; t\right) - \tilde {\mu} _ {0} \left(x _ {t}; t\right). \tag {6}
+$$
+
+First the memory  $m_{t}$  is decoded into a set of weights for each action using some standard feedforward multi-layer perceptron (MLP), before the potential outcomes are predicted by taking the inner product with the context. This is preferable to, for example, concatenating and passing both  $x_{t}$  and  $m_{t}$  through a network since, while it may be potentially more expressive, it loses the connection to the forward model as well as the interpretability that is given by linearity. As it is, while our model of the potential outcomes is linear at each time step, the memory allows this to flexibly change throughout the course of the history. The perceived treatment effect is then calculated as the difference between potential outcomes. However, given the linear nature of the predictions, there is no non-degenerate solution for the  $\omega$ s. Thus, to ensure identifiability, we set  $\omega_0^t = 0$  as a baseline.
+
+Treatment Rule. Under Assumption 2, the agent should be expected to take an action that maximises the outcome. There are different possible options for the exact treatment rule, the most obvious being  $\pi (a_t|\tilde{\tau}_t) = \mathbb{1}(\tilde{\tau}_t > 0)$ ; assuming that as long as the treatment effect is positive then the intervention will be taken. However, in order to maintain differentiability, as well as to allow for more modelling flexibility to capture the stochasticity of agents, we parameterise the policy as a soft version of the indicator function:
+
+$$
+\pi \left(a _ {t} \mid \tilde {\tau} _ {t}\right) = \frac {1}{1 + \exp \left(- \alpha \left(\tilde {\tau} _ {t} - \beta\right)\right)}. \tag {7}
+$$
+
+Assuming positive  $\alpha$ , this is clearly monotonically increasing, satisfying Assumption 2. These parameters are learnt by the model and thus allow us to model a flexible threshold with  $\beta$  as well as aleatoric uncertainty in the actions of the agent with  $\alpha$ .
+
+# 4.2 LEARNING WITH AN INFERENCE NETWORK
+
+As with all flavours of deep state space models, exact posterior inference is analytically intractable, making optimisation of the marginal likelihood equally so. As such, we aim to maximise an Evidence Lower BOund (ELBO) on this likelihood of the data following standardised variational principles (Blei et al., 2017; Hoffman et al., 2013). This is achieved by positing a parameterised approximate posterior distribution  $q_{\phi}(\vec{m}|\vec{h})$ , and with a simple application of Jensen's inequality arrive at:
+
+$$
+\log p _ {\theta} (\vec {h}) \geq \mathbb {E} _ {q _ {\phi}} \left[ \log p _ {\theta} (\vec {m}, \vec {h}) - \log q _ {\phi} (\vec {m} | \vec {h}) \right]. \tag {8}
+$$
+
+Maximising this bound is not only useful in terms of the marginal likelihood but also equivalently minimises the Kullback-Leibler divergence between the approximate and true posterior  $D_{KL}[q_{\phi}(\vec{m}|\vec{h})||p(\vec{m}|\vec{h})]$ , giving us more confidence in our inferred posterior.
+
+Factorising the Posterior for Efficient Inference. In order to accelerate learning and make inference quick, we factorise the approximate posterior in the same way as the true posterior:
+
+$$
+q _ {\phi} (\vec {m} | \vec {h}) = q _ {\phi} (m _ {1} | \vec {h}) \prod_ {t = 2} ^ {T} q _ {\phi} \left(m _ {t} \mid m _ {t - 1}, \vec {h} _ {t - 1: T}\right) \tag {9}
+$$
+
+Thus to estimate the approximate posterior, we use a backwards LSTM to create a summary  $b_{t} = \mathrm{LSTM}(h_{t:T})$ . This is then concatenated with the previous memory at time step  $t - 1$  before being passed through a fully connected layer to estimate parameters. Given that the memory can be seen as a sufficient statistic of the past within the model,
+
+it is well known that given this factorisation, conditioning on the memory leads to a Rao-Blackwellization of the more general network that would be conditioned on all variables (Krishnan et al., 2017; Alaa & van der Schaar, 2019). This accelerates learning not only by parameter sharing between  $\phi$  and  $\theta$  but also by reducing the variance of the gradient estimates during training. With the specified inference network  $q_{\phi}$ , this leads to an optimisation objective:
+
+![](images/276c20427fe2dfb01ece5da8d81564f4755a50d84125d997bf90799483e9f715.jpg)  
+Figure 4: Structure of the approximate posterior inference network.
+
+$$
+\begin{array}{l} \mathcal {F} (\phi , \theta) = \mathbb {E} _ {q _ {\phi} (\vec {m} | \vec {h})} [ p _ {\theta} (\vec {a} | \vec {h}) ] - D _ {K L} (q _ {\phi} (m _ {1} | \vec {x}) | | p _ {\theta} (m _ {1})) \tag {10} \\ - \sum_ {t = 2} ^ {T} \mathbb {E} _ {q _ {\phi} (m _ {t - 1} | \vec {h})} \left[ D _ {K L} \left(q _ {\phi} (m _ {t} | m _ {t - 1}, \vec {h} _ {t - 1: T}) | | p _ {\theta} (m _ {t} | m _ {t - 1})\right) \right] \\ \end{array}
+$$
+
+This can optimised in a straightforward manner using stochastic variational inference and reparameterised Monte Carlo samples from the approximate posterior distribution. At each step a sample is drawn from the posterior through which gradients can flow and the ELBO is evaluated. This can be backpropagated through to get gradients with respect to  $\phi$  and  $\theta$  which can then be updated as appropriate using an optimiser of the practitioner's choice.
+
+# 5 CASE STUDY: LIVER TRANSPLANTATION ACCEPTANCE DECISIONS
+
+In this section we explore the explainability benefits of our method for evaluating real decision making. Given space constraints, we relegate validation of the method on synthetic examples to the appendix, and focus here on a real medical example of accepting liver donation offers for transplantation. This is an important area where much has been done to improve decision support for clinicians (Volk et al., 2015; Berrevoets et al., 2020), but which often focuses on simply suggesting theoretically optimal actions. Here, by trying to understand better the decision making process, we hope to be able to provide more bespoke support in the future.
+
+![](images/f422fb83659ae24c5570dfeda5345be6b9294023f4a2a51bce869bbe613c008c.jpg)  
+Figure 5: Changing Focuses. Predicted weights of different covariates at two-year time intervals.
+
+![](images/992dc79b7463e4be846b4c1e5ae5dec3e7fcfc3a6fcf6bc36557d543776cf4bd.jpg)
+
+![](images/0782e741040ca26b718dd26e3434ef06b0bd23661650493ad134bd5d09cdced3.jpg)
+
+Organ transplantation is an extreme medical procedure, often considered the last resort for critically ill patients (Merion et al., 2005). Despite the significant risks associated with major surgery, transplantation can offer the chance of a relatively normal life (Starzl et al., 1982) and as a result demand for donor organs far outstrips supply (Wynn & Alexander, 2011). Yet, perhaps surprisingly, a large proportion of organ offers are turned down by clinicians (in consultation with the receiving patients) for various reasons including donor quality or perceived poor compatibility with the recipient (Volk et al., 2015). This decision faced by clinicians can be (under some simplification) described by the process: 1) A donor organ becomes available and is offered to a patient; 2) The clinician considers the state of the organ, the state of the patient, as well as other information such as the previous reject history of the organ (this is the context  $X$ ); 3) The clinician makes the binary decision to accept/reject the organ (the intervention  $A$ ); 4) The patient receives/does not receive the organ and a measure of their outcome is recorded - we measure their survival time as the main outcome of interest (the outcome  $Y$ ). Full details of the dataset, preprocessing steps, and experimental setup can be found in the appendix. Note that in this example we are not trying to capture the policy of any individual doctor, but rather the general policy of a treatment centre over time and how what they find important changes - this is due to the form of the data available, but the method could equally be applied to individual clinicians if that information was made available. Additionally, while survival time will not be recorded immediately, the most powerful signal in this case is usually a death that follows very soon after the decision, which will be effectively captured in the data.
+
+Evolving Relative Importances. First, we now examine exactly how our method can explain a nonstationary policy over time. In Figure 5 we plot the relative weights for covariates in terms of the treatment effect for accepting the organ offer as they change at two-year periods. The Model for End-Stage Liver Disease (MELD) score (Kamath & Kim, 2007) was introduced in 2002 to give an overall rating of how sick a patient is. On introduction, it was the main consideration when making an acceptance decision, as can be seen clearly in Figure 5. However, over time this became seen as less important as clinicians started to not want to rely on a single score, with some clinicians even turning down offers for patients with high scores in hopes of a better match, with one study showing that 84
+
+![](images/835c82b8ad6668288c0326b7912ef14c1f557039632babdf33d110e4981a1956.jpg)  
+Figure 6: One-step policy shift densities after observing an outcome.
+
+$\%$  of the patients who died while waiting for an organ had MELD scores  $\geq 15$  and had previously declined at least one offer (Lai et al., 2012). This can also be seen reflected in Figure 5 as the importance of the MELD score in accepting the offer decreases and the policy stabilises.
+
+Reactionary Policies The most important aspect of our method is that we can model the reaction of the agent to their experience on the fly. For example we can see how observing various outcomes change the opinion of the agent moving forward. This is demonstrated in Figure 6 which plots density estimates of the policy shift (measured as the distance the decision boundary moves) after rejecting the offer and measuring a positive/negative outcome. Here it can be
+
+seen that when an offer is rejected then a negative outcome shifts the decision boundary downwards more than a positive outcome, making it less likely to reject an offer in the next time-step.
+
+Matching Clinical Decisions. While it should be emphasised that our method is (to the best of our knowledge) the only method with the aim or capability to model the evolving decision process of the agent, as mentioned in Section 3 there are a variety of methods that are tangentially related. First, we consider a few standard approaches to imitation learning with behavioural cloning (BC) (Bain & Sammut, 1995), where we consider both a linear (BC-Linear) and deep (BC-Deep) functionapproximator, representing both ends of the interpretability spectrum for BC. Continuing in imitation learning, we also compare alongside reward-regularised classification for apprenticeship learning (RCAL) (Piot et al., 2014), a method that regularises the reward implied by the learnt  $Q$ -values. We additionally consider an adaptation of counterfactual inverse reinforcement learning (CIRL) (Bica et al., 2021) to the bandit setting where we first estimate the true CATE of the interventions and thus the true potential outcomes. We use these as the reward and apply an optimal policy, which in this one-step case is trivially achieved by choosing the action that maximises the potential outcome.
+
+In Table 2 we compare the predictive power of our method and all benchmarks on the task of imitation learning - i.e. matching the actions of the real demonstrator in some held-out test set. This is the only task for which we can make meaningful comparison with the existing literature, given our divergent goals. Despite the fact that at each step our method only considers a linear decision boundary, we are still able to outperform the deep network architectures used in BC and RCAL in terms of AUC and APS. This can be put down to the flexible way in which the policy can change over time to allow for adaptation. Less surprisingly, we also outperform the stationary linear classifier as well as the version of CIRL, the poor performance of which is probably ex
+
+Table 2: Action prediction performance. Comparison of methods on transplant offer acceptances. Performance of the methods is evaluated on the quality of action matching against a held out test set of demonstrations. We report the area under the receiving operator characteristic curve (AUC) and the average precision score (APS).
+
+<table><tr><td>Method</td><td>AUC ↑</td><td>APS ↑</td></tr><tr><td>BC-Linear</td><td>0.798 ± 0.002</td><td>0.647 ± 0.001</td></tr><tr><td>BC-Deep</td><td>0.803 ± 0.002</td><td>0.655 ± 0.001</td></tr><tr><td>RCAL</td><td>0.797 ± 0.008</td><td>0.662 ± 0.003</td></tr><tr><td>CIRL</td><td>0.553 ± 0.008</td><td>0.510 ± 0.002</td></tr><tr><td>IOL (Ours)</td><td>0.824 ± 0.006</td><td>0.677 ± 0.011</td></tr></table>
+
+plained by the unrealistic nature of the causal assumptions in this setting. Further experimental results on action matching in on multiple additional datasets are provided in the appendix.
+
+# 6 DISCUSSION
+
+Limitations. By employing a deep architecture our method is able to learn a non-parametric and accurate form of both the non-stationary policy and update function, making far fewer assumptions on their true form than is normal in the literature. This does mean, however, that it works best when training data is abundant, and can otherwise be prone to over-fitting as is common with deep neural networks. Additionally, the assumption that agents are always taking the action they believe will maximise the outcome does mean our method is out-of-the-box unable to account for strategies that deliberately pick actions that could be sub-optimal but may yield useful information (i.e. exploratory strategies). An interesting future direction would include exploring this, potentially by augmenting the outcome to include a measure of expected information.
+
+Societal Impact and Ethics. Being able to better understand decision making processes could be highly useful in combating biases or correcting mistakes made by human decision-makers. Nonetheless, it should be noted that any method that aims to analyse and explain observed behaviour has the potential to be misused, for example to unfairly attribute incorrect intentions to someone. As such, it is important to emphasise that, as is the nature of inverse problems, we are only giving one plausible explanation for the behaviour of an agent and cannot suggest that this is exactly what has gone on in their mind.
+
+Conclusions. In this paper we have tackled the problem of inverse online learning – that is descriptively modelling the process through which an observed agent adapts their policy over time – by interpreting actions as targeted interventions, or treatments. We drew together policy learning and treatment effect estimation to present a practical algorithm for solving this problem in an interpretable and meaningful manner, before demonstrating how this could be using in the medical domain to analyse the decisions of clinicians when choosing whether to accept offers for transplantation.
+
+# ACKNOWLEDGEMENTS
+
+AJC would like to acknowledge and thank Microsoft Research for its support through its PhD Scholarship Program with the EPSRC. AC gratefully acknowledges funding from AstraZeneca. This work was additionally supported by the Office of Naval Research (ONR) and the NSF (Grant number: 1722516). We would like to thank all of the anonymous reviewers on OpenReview, alongside the many members of the van der Schaar lab, for their input, comments, and suggestions at various stages that have ultimately improved the manuscript.
+
+# REFERENCES
+
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+
+# APPENDIX
+
+Algorithm 1: Inverse Online Learning (IOL)  
+Result: Parameters  $\phi$  of variational distribution and  $\theta$  of generative model  
+Input:  $\mathcal{D},\mathcal{X},\mathcal{A},\mathcal{Y}$ , learning rate  $\eta$ ;  
+Initialise  $\phi, \theta$ ;  
+while not converged do  
+    Evaluate  $p_{\theta}(m_t|m_{t-1})$ ;  $\triangleright$  Calculate forward model next step priors  
+    Evaluate  $q_{\phi}(m_t|m_{t-1},\vec{h})$ ;  $\triangleright$  Inference network to get posterior sample  $\hat{\vec{m}} \sim q_{\phi}(\vec{m}|\vec{h})$ ;  $\triangleright$  Straight-through MC sample  $NLL_{loss} = \mathbb{E}_{q_{\phi}(\vec{m}|\vec{h})}[p_{\theta}(\vec{a}|\vec{m})]$ ;  
+     $KL_{loss} = D_KL(q_{\phi}(m_1|\vec{x})||p_{\theta}(m_1)) + \sum_{t=2}^{T}\mathbb{E}[D_KL(q_{\phi}(m_t|m_{t-1},\vec{h})||p_{\theta}(m_t|m_{t-1}))]$ ;  
+     $\mathcal{F}(\phi,\theta) = NLL_{loss} + KL_{loss}$ ;  
+     $(\phi',\theta') \gets (\phi,\theta) - \eta\nabla_{\phi,\theta}\mathcal{F}(\phi,\theta)$ ;  
+     $\phi,\theta \gets \phi',\theta'$   
+end  
+Return:  $\phi,\theta$
+
+# A EXPERIMENTAL DETAILS
+
+All experiments were performed on a 2016 MacBook Pro, using a 2.9 GHz Dual-Core Intel Core i5 with 8GB of LPDDR3 RAM and no GPU acceleration.
+
+Code was written in PyTorch Paszke et al. (2019) and no external code was used in the implementation of either the model nor the benchmarks. Hyperparameters were selected through grid search over a validation fold of the training data.
+
+All medical data used has undergone a de-identification process and consent was obtained by the relevant curators for their data to be publicly released.
+
+# A.1 UNOS DATA
+
+Our data focuses on data from the United Network for Organ Sharing (UNOS), the US non-profit that manages their national waiting list and maintains a database and record of transplantations that occur.
+
+We focus on liver-transplantations, where the donor is deceased. The data is made up of patient details from various centres around the United States. Of the multiple million records of individual transplantations available, once those with missing information on either the donor or recipient are removed we are left with around 500,000 patients spread across 31 centres. Available variables are listed in Table 3
+
+In our action matching experiments we split the data into Train/Validation/Test folds by separating the data by centre. In particular we use centre CTR23901 with 114,314 patients for training; CTR124 with 21,067 patients for validation; and finally CTR279 with 14,295 patients for testing.
+
+For the more exploratory experiments who's focus is on inference over some data and not extrapolated prediction, we simply train on centre CTR23901, and examine what is inferred about the treatment of those specific patients
+
+# B ADDITIONAL VALIDATION
+
+While we focused on an important case study in the main paper, we appreciate the need to provide additional evidence that our model can work outside of one specific case example. As such, we now report results on some experiments focusing first on a toy (but relevant) synthetic example, followed by performance results on multiple further real datasets.
+
+Table 3: UNOS Variables  
+
+<table><tr><td>AGE</td><td>Recipient&#x27;s age at transplant</td></tr><tr><td>AGE_DON</td><td>Donor&#x27;s age at transplant</td></tr><tr><td>ALBUMIN_TX</td><td>Recipient&#x27;s albumin concentration at transplant</td></tr><tr><td>ASCITES_TX</td><td>Recipient had ascites at the time of transplant</td></tr><tr><td>BMI_CALC</td><td>Recipient&#x27;s BMI</td></tr><tr><td>COD_CAD_DON</td><td>Cause of death of the donor</td></tr><tr><td>COLD_ISCH</td><td>Cold ischemia time - length of time from the donor being recovered until it is transplanted</td></tr><tr><td>CREAT_DON</td><td>Donor&#x27;s creatinine concentration</td></tr><tr><td>CREAT_TX</td><td>Recipient&#x27;s creatinine concentration at transplant</td></tr><tr><td>DIAL_TX</td><td>Whether recipient was on dialysis before transplant</td></tr><tr><td>ETHCAT</td><td>Recipient&#x27;s ethnicity</td></tr><tr><td>ETHCAT_DON</td><td>Donor&#x27;s ethnicity</td></tr><tr><td>EXC_HCC</td><td>Whether the recipient received a hepatocellular carcinoma exception point</td></tr><tr><td>FUNC_STAT_TRR</td><td>Recipient&#x27;s functional status at the time of transplant</td></tr><tr><td>HCV_SEROSTATUS</td><td>Whether the recipient has hepatitis C</td></tr><tr><td>INR_TX</td><td>Recipient&#x27;s INR value at the time transplant</td></tr><tr><td>LIFE_SUP_TRR</td><td>Whether the recipient was on life support at the time of transplant</td></tr><tr><td>MED_COND_TRR</td><td>Whether the recipient was in an icu, hospital bed, or came from home before transplant</td></tr><tr><td>MELD_PELDLAB_SCORE</td><td>Recipient MELD score</td></tr><tr><td>NON_HRT_DON</td><td>Whether the donor was a DCD (donation after cardiac death)</td></tr><tr><td>NUM_PREV_TX</td><td>Number of previous liver transplants the recipient has had</td></tr><tr><td>ON_VENT_TRR</td><td>Whether the recipient was on a mechanical ventilator before transplant</td></tr><tr><td>PORTAL_VEIN_TRR</td><td>Whether the recipient had a portal vein thrombosis at the time of transplant</td></tr><tr><td>PREV_ABSURG_TRR</td><td>Whether the recipient had prior abdominal surgery before the liver transplant</td></tr><tr><td>PREV_TX</td><td>Whether the recipient had any previous transplant</td></tr><tr><td>SHARE_TX</td><td>Whether the donor is local, regional, or national</td></tr><tr><td>TBILI_TX</td><td>Recipient&#x27;s bilirubin at the time of transplant</td></tr><tr><td>TXLIV</td><td>Whether the graft is whole or segmental</td></tr><tr><td>death_mech_don_group</td><td>Donor mechanism of death</td></tr><tr><td>deathcirc</td><td>Donor circumstance of death</td></tr><tr><td>diag1</td><td>Recipient&#x27;s primary cause of liver disease</td></tr><tr><td>meldstat</td><td>Whether the recipient is status 1a</td></tr><tr><td>statushcc</td><td>Whether the recipient has hcc</td></tr><tr><td>status1</td><td>Whether the recipient is status 1a</td></tr></table>
+
+# B.1 SYNTHETIC EXPERIMENT
+
+We consider a toy example of an agent that is learning online about the treatment effect of the interventions they are performing.
+
+We let  $\mathcal{X} = \mathbb{R}^5, \mathcal{A} = [0,1]$ , and  $\mathcal{V} = \mathbb{R}$ , with the true outcomes a linear combination of the features given the action (with weights stochastically samples). From a sampled prior set of weights in the agents mind, the agent then interacts, estimating the potential outcomes and choosing an action stochastically according to equation 7 with  $\alpha = 1, \beta = 0$ . After every context, action, outcome triple  $(x, a, y)$ , the agent adjusts its beliefs over the weights of the potential outcome functions using an online gradient descent method:
+
+$$
+\omega_ {a} = \omega_ {a} - \lambda ((\tilde {y} - y) \times x), \tag {11}
+$$
+
+with  $\lambda$  some learning rate. Clearly then the policy of the agent is nonstationary and changes over time in reaction to the observed outcomes.
+
+Results. We simulate 10000 trajectories of length 50. When applying IOL to this simulated data, we can achieve an accuracy of  $99\%$ , meaning  $99\%$  of the time we can accurately determine which of the potential outcomes the agent thins will be higher. This is compared to a stationary estimate of the CATE which can only reach  $52\%$  given the fact that the policy is detached from the truth of the treatment effect. Clearly this is an extreme example designed to fail the existing approaches but it does serve to show how stark the contrast is.
+
+# B.2 FURTHER PERFORMANCE ON REAL DATA
+
+Alongside the synthetic example above, we also provide further results on real medical data, demonstrating improved performance across a variety of domains. These datasets do not contain a recorded outcome, consequently so we do not include CIRL, and the next state is used as a substitute in IOL. Results are given in Table 2, and a description of the datasets follows:
+
+Intensive Care Unit. The ICU data covers the treatment of 23,106 of patients in the intensive care unit from Amsterdam UMC Elbers (2019). These patients are in a more critical state than those on the general wards while suffering similarly from a variety of conditions and consequently are monitored more frequently, with the database containing around 1 billion clinical observations at varying timesteps down to minute by minute recordings. The data is aggregated into one hour timesteps and models the treatment of the prescription of antibiotics. Included are 24 series features focusing on vital signs including heart rate, blood pressure, and various chemical blood concentration levels.
+
+Cystic Fibrosis. The  $CF$  data considers patients enrolled in the UK Cystic Fibrosis Registry Taylor-Robinson et al. (2018), coviering around 5,800 patients over the course of multiple years. Measurements were taken on average every 6 months, and the 79 features cover common comorbidities, test results, and infections that arise. The intervention corresponds to the prescription of cortico steroids.
+
+Hospital Wards. The Wards data is based on the care of 6,321 patients at the Ronald Reagan UCLA Medical Center in California who were treated on the general medicine floor between 2013-2016 Alaa et al. (2017). These patients were treated for a variety of conditions including pneumonia, sepsis, and fevers were in general stable and deterioration that required ICU care was rare. Measurements were taken roughly every 4 hours, with average stays lasting 9 days, and include common vital signs such as pulse and blood pressure alongside lab tests and results and 35 temporal features. The action space is taken as a choice of application of oxygen therapy.
+
+Table 4: Medical examples performance. Comparison of methods on three different medical datasets. Performance of the policy is evaluated on the quality of action matching against a held out test set of demonstrations. We report the area under the receiving operator characteristic curve (AUC) and average precision score (APS).  
+
+<table><tr><td></td><td colspan="2">ICU</td><td colspan="2">CF</td><td colspan="2">Wards</td></tr><tr><td>Metric</td><td>AUC</td><td>APS</td><td>AUC</td><td>APS</td><td>AUC</td><td>APS</td></tr><tr><td>BC-L</td><td>0.681 ± 0.001</td><td>0.655 ± 0.000</td><td>0.830 ± 0.010</td><td>0.797 ± 0.009</td><td>0.896 ± 0.006</td><td>0.853 ± 0.002</td></tr><tr><td>BC-D</td><td>0.668 ± 0.003</td><td>0.654 ± 0.001</td><td>0.952 ± 0.010</td><td>0.834 ± 0.003</td><td>0.845 ± 0.010</td><td>0.811 ± 0.013</td></tr><tr><td>RCAL</td><td>0.396 ± 0.018</td><td>0.563 ± 0.007</td><td>0.689 ± 0.022</td><td>0.745 ± 0.008</td><td>0.942 ± 0.003</td><td>0.920 ± 0.004</td></tr><tr><td>IOL</td><td>0.686 ± 0.092</td><td>0.631 ± 0.065</td><td>0.942 ± 0.031</td><td>0.943 ± 0.026</td><td>0.965 ± 0.031</td><td>0.941 ± 0.055</td></tr></table>
+
+# REFERENCES
+
+Alaa, A. M., Yoon, J., Hu, S., and Van der Schaar, M. (2017). Personalized risk scoring for critical care prognosis using mixtures of gaussian processes. IEEE Transactions on Biomedical Engineering, 65(1):207-218.  
+Elbers, P. W. G. (2019). AmsterdamUMCdb v1.0.2 ICU database.  
+Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., et al. (2019). Pytorch: An imperative style, high-performance deep learning library. In Advances in neural information processing systems, pages 8026-8037.  
+Taylor-Robinson, D., Archangelidi, O., Carr, S. B., Cosgriff, R., Gunn, E., Keogh, R. H., MacDougall, A., Newsome, S., Schluter, D. K., Stanojevic, S., et al. (2018). Data resource profile: the uk cystic fibrosis registry. International journal of epidemiology, 47(1):9-10e.

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