Add files using upload-large-folder tool

parse/train/0cmMMy8J5q/0cmMMy8J5q.md

@@ -0,0 +1,357 @@
+# ZERO-COST PROXIES FOR LIGHTWEIGHT NAS
+
+Mohamed S. Abdelfattah1, Abhinav Mehrotra1, Łukasz Dudziak1, Nicholas D. Lane1,2   
+1 Samsung AI Center, Cambridge · 2 University of Cambridge   
+mohamed1.a@samsung.com
+
+# ABSTRACT
+
+Neural Architecture Search (NAS) is quickly becoming the standard methodology to design neural network models. However, NAS is typically compute-intensive because multiple models need to be evaluated before choosing the best one. To reduce the computational power and time needed, a proxy task is often used for evaluating each model instead of full training. In this paper, we evaluate conventional reduced-training proxies and quantify how well they preserve ranking between neural network models during search when compared with the rankings produced by final trained accuracy. We propose a series of zero-cost proxies, based on recent pruning literature, that use just a single minibatch of training data to compute a model’s score. Our zero-cost proxies use 3 orders of magnitude less computation but can match and even outperform conventional proxies. For example, Spearman’s rank correlation coefficient between final validation accuracy and our best zero-cost proxy on NAS-Bench-201 is 0.82, compared to 0.61 for EcoNAS (a recently proposed reduced-training proxy). Finally, we use these zerocost proxies to enhance existing NAS search algorithms such as random search, reinforcement learning, evolutionary search and predictor-based search. For all search methodologies and across three different NAS datasets, we are able to significantly improve sample efficiency, and thereby decrease computation, by using our zero-cost proxies. For example on NAS-Bench-101, we achieved the same accuracy $4 \times$ quicker than the best previous result. Our code is made public at: https://github.com/mohsaied/zero-cost-nas.
+
+# 1 INTRODUCTION
+
+Instead of manually designing neural networks, neural architecture search (NAS) algorithms are used to automatically discover the best ones (Tan & Le, $2 0 1 9 \mathrm { a }$ ; Liu et al., 2019; Bender et al., 2018). Early work by Zoph & Le (2017) proposed using a reinforcement learning (RL) controller that constructs candidate architectures, these are evaluated and then feedback is provided to the controller based on the performance of the candidate. One major problem with this basic NAS methodology is that each evaluation is very costly – typically on the order of hours or days to train a single neural network fully. We focus on this evaluation phase – we propose using proxies that require a single minibatch of data and a single forward/backward propagation pass to score a neural network. This is inspired by recent pruning-at-initialization work by Lee et al. (2019), Wang et al. (2020) and Tanaka et al. (2020) wherein a per-parameter saliency metric is computed before training to inform parameter pruning. Can we use such saliency metrics to score an entire neural network? Furthermore, can we use these “single minibatch” metrics to rank and compare multiple neural networks for use within NAS? If so, how do we best integrate these metrics within existing NAS algorithms such as RL or evolutionary search? These are the questions that we hope to (empirically) tackle in this work with the goal of making NAS less compute-hungry. Our contributions are:
+
+• Zero-cost proxies We adapt pruning-at-initialization metrics for use with NAS. This requires these metrics to operate at the granularity of an entire network rather than individual parameters – we devise and validate approaches that aggregate parameter-level metrics in a manner suitable for ranking candidates during NAS search.
+
+• Comparison to conventional proxies We perform a detailed comparison between zerocost and conventional NAS proxies that use a form of reduced-computation training. First, we quantify the rank consistency of conventional proxies on large-scale datasets: $1 5 \mathrm { k }$ models vs. 50 models used in (Zhou et al., 2020). Second, we show that zero-cost proxies can match or exceed the rank consistency of conventional proxies.
+
+• Ablations on NAS benchmarks We perform ablations of our zero-cost proxies on five different NAS benchmarks (NAS-Bench-101/201/NLP/ASR and PyTorchCV) to both test the zero-cost metrics under different settings, and expose properties of successful metrics.
+
+• Integration with NAS Finally, we propose two ways to use zero-cost metrics effectively within NAS algorithms: random search, reinforcement learning, aging evolution and predictor-based search. For all algorithms and three NAS datasets we show significant speedups, up to $4 \times$ for NAS-Bench-101 compared to current state-of-the-art.
+
+# 2 RELATED WORK
+
+NAS Efficiency To decrease NAS search time, various techniques were used in the literature. Pham et al. (2018) and Cai et al. (2018) use weight sharing between candidate models to decrease the training time during evaluation. Liu et al. (2019) and others use smaller datasets (CIFAR-10) as a proxy to the full task (ImageNet1k). In EcoNAS, Zhou et al. (2020) extensively investigated reduced-training proxies wherein input size, model size, number of training samples and number of epochs were reduced in the NAS evaluation phase. We compare to EcoNAS in this work to elucidate how well our zero-cost proxies perform compared to familiar and widely-used conventional proxies.
+
+Pruning The goal is to reduce the number of parameters in a neural network, one way to do this is by identifying a saliency (importance) metric for each parameter, and the less-important parameters are removed. For example, Han et al. (2015), Frankle & Carbin (2019) and others use parameter magnitudes as the criterion while LeCun et al. (1990), Hassibi & Stork (1993) and Molchanov et al. (2017) use gradients. However, the aforementioned works require training before computing the saliency criterion. A new class of pruning-at-initialization algorithms, that require no training, were introduced by Lee et al. (2019) and extended by Wang et al. (2020) and Tanaka et al. (2020). A single forward/backward propagation pass is used to compute a saliency criterion which is successfully used to heavily prune neural networks before training. We extend these pruning-at-initialization criteria towards scoring entire neural networks and we investigate their use with NAS algorithms.
+
+Intersection between pruning and NAS Concepts from pruning have been used within NAS multiple times. For example, Mei et al. (2020) use channel pruning in their AtomNAS work to arrive at customized multi-kernel-size convolutions (mixconvs as introduced by Tan & Le (2019b)). In their Blockswap work, Turner et al. (2020) use Fisher information at initialization to score different lightweight primitives that are substituted into a neural network to decrease computation. This is the earliest work we could find that attempts to perform a type of NAS by scoring neural networks without training using a pruning criterion, More recently, Mellor et al. (2020) introduced a new metric for scoring neural networks at initialization based on the correlation of Jacobians with different inputs. They perform “NAS without training” by performing random search with their zero-cost metric (jacob cov) to rank neural networks instead of using accuracy. We include jacob cov in our analysis and we introduce five more zero-cost metrics in this work.
+
+# 3 PROXIES FOR NEURAL NETWORK ACCURACY
+
+# 3.1 CONVENTIONAL NAS PROXIES (ECONAS)
+
+In conventional sample-based NAS, a proxy training regime is often used to predict a model’s accuracy instead of full training. Zhou et al. (2020) investigate conventional proxies in depth by computing the Spearman rank correlation coefficient (Spearman $\rho$ ) of a proxy task to final test accuracy. The proxy used is a reduced-computation training, wherein one of the following four variables is reduced: (1) number of epochs, (2) number of training samples, (3) input resolution (4) model size (controlled through the number of channels after the first convolution). Even though such proxies were used in many prior works, EcoNAS is the first systematic study of conventional proxy tasks that we found. One main finding by Zhou et al. (2020) is that using approximately $\textstyle { \frac { 1 } { 4 } }$ of the model size and input resolution, all training samples, and $\textstyle { \frac { 1 } { 1 0 } }$ the number of epochs was a reasonable proxy which yielded the best results for their experiment (Zhou et al., 2020).
+
+# 3.2 ZERO-COST NAS PROXIES
+
+We present alternative proxies for network accuracy that can be used to speed up NAS. A simple proxy that we use is grad norm in which we sum the Euclidean norm of the gradients after a single minibatch of training data. Other metrics listed below were previously introduced in the context of parameter pruning at the granularity of a single parameter – a saliency is computed to rank parameters and remove the least important ones. We adapt these metrics to score and rank entire neural network models for NAS.
+
+# 3.2.1 SNIP, GRASP AND SYNAPTIC FLOW
+
+In their snip work, Lee et al. (2019) proposed performing parameter pruning based on a saliency metric computed at initialization using a single minibatch of data. This saliency criteria approximates the change in loss when a specific parameter is removed. Wang et al. (2020) attempted to improve on the snip metric by approximating the change in gradient norm (instead of loss) when a parameter is pruned in their grasp objective. Finally, Tanaka et al. (2020) generalized these so-called synaptic saliency scores and proposed a modified version (synflow) which avoids layer collapse when performing parameter pruning. Instead of using a minibatch of training data and cross-entropy loss (as in snip or grasp), with synflow we compute a loss which is simply the product of all parameters in the network; therefore, no data is needed to compute this loss or the synflow metric itself. These are the three metrics:
+
+$$
+\mathrm { s n i p } : S _ { P } ( \theta ) = \left| \frac { \partial \mathcal { L } } { \partial \theta } \odot \theta \right| , \quad \mathrm { g r a s s p } : S _ { P } ( \theta ) = - ( H \frac { \partial \mathcal { L } } { \partial \theta } ) \odot \theta , \quad \mathrm { ~ s y n f 1 o w : ~ } S _ { P } ( \theta ) = \frac { \partial \mathcal { L } } { \partial \theta } \odot \theta .
+$$
+
+where $\mathcal { L }$ is the loss function of a neural network with parameters $\theta , H$ is the Hessian1, $S _ { p }$ is the per-parameter saliency and $\odot$ is the Hadamard product. We extend these saliency metrics to score an entire neural network by summing over all parameters $N$ in the model: $\begin{array} { r } { S _ { n } = \dot { \sum } _ { i } ^ { N } S _ { p } ( \theta ) _ { i } } \end{array}$ .
+
+# 3.2.2 FISHER
+
+Theis et al. (2018) perform channel pruning by removing activation channels (and their corresponding parameters) that are estimated to have the least effect on the loss. They build on the work of Molchanov et al. (2017) and Figurnov et al. (2016). More recently, Turner et al. (2020) aggregated this fisher metric for all channels in a convolution primitive to quantify the importance of that primitive when it is replaced by a more efficient alternative. We further aggregate the fisher metric for all layers in a neural network to score an entire network as shown in the following equations:
+
+$$
+{ \mathrm { \bf { f i s h e r : } } } \ S _ { z } ( z ) = \left( { \frac { \partial { \mathcal { L } } } { \partial z } } z \right) ^ { 2 } , \quad \ S _ { n } = \sum _ { i = 1 } ^ { M } S _ { z } ( z _ { i } )
+$$
+
+where $S _ { z }$ is the saliency per activation $z$ , and $M$ is the length of the vectorized feature map.
+
+# 3.2.3 JACOBIAN COVARIANCE
+
+This metric was purpose-designed to score neural networks in the context of NAS – we refer the reader to the original paper for detailed reasoning and derivation of the metric which we call jacob cov (Mellor et al., 2020). In brief, this metric captures the correlation of activations within a network when subject to different inputs within a minibatch of data – the lower the correlation, the better the network is expected to perform as it can differentiate between different inputs well.
+
+# 4 EMPIRICAL EVALUATION OF PROXY TASKS
+
+Generally, most of the proxies presented in the previous section try to capture how trainable a neural network is by inspecting the gradients at the beginning of training. In this work, we refrain from attempting to explain precisely why each metric works (or does not work) and instead focus on the empirical evaluation of those metrics in different scenarios. We use the Spearman rank correlation coefficient (Spearman $\rho$ ) to quantify how well a proxy ranks models compared to the ground-truth ranking produced by final test accuracy (Daniel, 1990).
+
+![](images/7a77fc951f37c2e4eacba79dd827fcbfb9921946b9a054f990371082151ad0b4.jpg)  
+Figure 1: Evaluation of different econas proxies on NAS-Bench-201 CIFAR-10. FLOPS and runtime are normalized to the FLOPS/runtime of a single baseline (full training) epoch.
+
+# 4.1 NAS-BENCH-201
+
+NAS-Bench-201 is a purpose-built benchmark for prototyping NAS algorithms (Dong & Yang, 2020). It contains 15,625 CNN models from a cell-based search space and corresponding training statistics. We first use NAS-Bench-201 to evaluate conventional proxies from EcoNAS, then we evaluate our zero-cost proxies and compare the two approaches.
+
+# 4.1.1 ECONAS PROXY ON NAS-BENCH-201
+
+Even though Zhou et al. (2020) thoroughly investigated reduced-training proxies, they only evaluated a small model zoo consisting of 50 models. To study EcoNAS more extensively we evaluate it on all 15,625 models in NAS-Bench-201 search space (training details in A.1). The full configuration training of NAS-Bench-201 on CIFAR-10 uses input resolution $\scriptstyle 1 = 3 2$ , number of channels in the stem convolution $\scriptstyle { \mathrm { c = } } 1 6$ and number of epochs $\scriptstyle { \mathrm { e } = 2 0 0 }$ – we summarize this as: $r _ { 3 2 } c _ { 1 6 } e _ { 2 0 0 }$ .
+
+According to the EcoNAS study, the most effective configuration divides both the input resolution and stem channels by ${ \sim } 4$ and the number of epochs by 10, that is, $r _ { 8 } c _ { 4 } e _ { 2 0 }$ for NAS-Bench-201 models. Keeping that in mind we investigate $r _ { 8 } c _ { 4 }$ in Fig. 1 (labeled econas); however, this proxy training seems to suffer from overfitting as correlation to final accuracy started to drop after 20 epochs. Additionally, the Spearman $\rho$ was a modest 0.61 when evaluated on all 15,625 models in NAS-Bench-201 – a far cry from the 0.87 achieved on the 50 models in the EcoNAS paper (Zhou et al., 2020). We additionally explore $r _ { 8 } c _ { 8 }$ , $r _ { 1 6 } c _ { 4 }$ and $r _ { 1 6 } c _ { 8 }$ and find a very good proxy with $r _ { 1 6 } c _ { 8 } e _ { 1 5 }$ , labeled in Fig. 1 as econas+. From the plots in Fig. 1, we would like to highlight that:
+
+1. A reduced-training proxy that works well on one search space may not work well on another as highlighted by the difference in Spearman $\rho$ between econas and econas+. This occurs even though both tasks in this case were CIFAR-10 image classification. 2. Even though EcoNAS-style proxies reduce computation load by a large factor (as seen in the middle plot in Fig. 1, this does not translate fully into actual runtime improvement when run on a nominal desktop $\mathrm { G P U } ^ { 2 }$ . We therefore plot actual GPU speedup in the third subplot in Fig. 1. For example, notice that the point labeled econas $( r _ { 8 } c _ { 4 } e _ { 2 0 } )$ has the same FLOPS as $\sim \frac { 1 } { 1 0 }$ of a full training epoch, but when measured on a GPU, takes time equivalent to 5 full training epochs – a $5 0 \times$ gap between theoretical and actual speedup.
+
+# 4.1.2 ZERO-COST PROXIES ON NAS-BENCH-201
+
+We now shift our focus towards our zero-cost NAS proxies which rely on gradient computations using a single minibatch of data at initialization. A clear advantage of zero-cost proxies is that they take very little time to compute – the forward/backward pass using a single minibatch of data. We ran the zero-cost proxies on all 15,625 models in NAS-Bench-201 for three image classification datasets and we summarize the results in Table 1.
+
+The synflow metric performed the best on all three datasets with a Spearman $\rho$ consistently above 0.73, jacob cov was second best but was also very well-correlated to final accuracy. Next came grad norm and snip with a Spearman $\rho$ close to 0.6. We add another metric that we simply label with vote that takes a majority vote between the three metrics synflow, jacob cov and snip when ranking two models. This performed better than any single metric with a Spearman $\rho$ consistently above 0.8. At the cost of just 3 minibatches instead of $\sim 1 0 0 0$ , this is already performing slightly better than econas+, and much better than econas as shown in Fig. 2a. In Fig. 2 we also plot the rank correlation of validation accuracy (without any reduced training) over the first 10 epochs of training for the three datasets available in NAS-Bench-201.
+
+![](images/7ba8056c8e8423389e4f34d51829906cbd2390da0a81b89b2ac805c97e58eb6c.jpg)  
+Figure 2: Correlation of validation accuracy to final test accuracy during the first 12 epochs of training for three datasets on the NAS-Bench-201 search space. Zero-cost and EcoNAS proxies are also labeled for comparison.
+
+Table 1: Spearman $\rho$ of zero-cost proxies on NAS-Bench-201.   
+
+<table><tr><td>Dataset</td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td><td>vote</td></tr><tr><td>CIFAR-10</td><td>0.58</td><td>0.58</td><td>0.48</td><td>0.36</td><td>0.74</td><td>0.73</td><td>0.82</td></tr><tr><td>CIFAR-100</td><td>0.64</td><td>0.63</td><td>0.54</td><td>0.39</td><td>0.76</td><td>0.71</td><td>0.83</td></tr><tr><td>ImageNet16-120</td><td>0.58</td><td>0.58</td><td>0.56</td><td>0.33</td><td>0.75</td><td>0.71</td><td>0.82</td></tr></table>
+
+Having set a comparison point with EcoNAS and reduced-training proxies, we have shown that zero-cost proxies can match and outperform these conventional methods in a large-scale empirical analysis. However, different NAS search spaces may behave differently, so in the remainder of this section, we test the zero-cost proxies on different search spaces.
+
+# 4.2 MODELS IN THE WILD (PYTORCHCV)
+
+To study zero-cost proxies in a different setting, we scored the models in the PyTorchCV database (Semery, 2020). PytorchCV contains common state-of-the-art neural networks such as ResNets (He ´ et al., 2016), DenseNets (Huang et al., 2017), MobileNets (Howard et al., 2017) and EfficientNets (Tan & Le, 2019a) – a representative assortment of top-performing models. We evaluated $\mathord { \sim } 5 0$ models for CIFAR-10, CIFAR-100 (Krizhevsky, 2009) and SVHN (Netzer et al., 2011), and $\sim 2 0 0$ models for ImageNet1k (Deng et al., 2009). Fig. 3 shows the resulting correlation for the zero-cost metrics. synflow, snip, fisher and grad norm all perform similarly well on all datasets, with the exception of SVHN where synflow outperforms other metrics by a large margin. However, grasp failed in this setting completely as shown by the low mean Spearman $\rho$ and high variance as shown in Fig. 3. Curiously, jacob cov also failed in this setting even though it performed well on NASBench-201. This suggests that this metric is better at scoring models from within a search space (similar topology and size), but becomes worse when scoring unrelated models.
+
+# 4.3 OTHER SEARCH SPACES
+
+We investigate our zero-cost metrics with other NAS benchmarks. Our goal is to empirically find a good metric to speed up NAS algorithms reliably on different tasks and datasets.
+
+• NAS-Bench-101: This is the first and largest NAS benchmark available with over $4 2 3 \mathrm { k }$ CNN models and training statistics on CIFAR-10 (Ying et al., 2019). • NAS-Bench-NLP: Klyuchnikov et al. (2020) investigate the architectures of 14k different recurrent cells in natural language processing (NLP) tasks such as next word prediction.
+
+![](images/138fb731b163de8b76d3bfe072e485a8f3a5bcdb12a4c66055945c882985b9fc.jpg)  
+Figure 3: Performance of zero-cost metrics on PyTorchCV models (averaged over 5 seeds).
+
+• NAS-Bench-ASR: This is our in-house dataset for convolution-based automatic speech recognition models evaluated on the TIMIT dataset (Garofolo et al., 1993). The search space includes linear, convolution, zeroize and skip-connections, forming 8242 models (Mehrotra et al., 2021).
+
+Compared to NAS-Bench-201, these datasets are either much larger (NAS-Bench-101) or based on a different task (NAS-Bench-NLP/ASR). From Table 2 we would like to highlight that the synflow metric (highlighted in bold) is the only consistent one across all analyzed benchmarks. Additionally, even for the synflow metric, rank correlation is quite a bit lower than that for NAS-Bench-201 ( ${ \sim } 0 . 3$ vs. ${ \sim } 0 . 8 )$ . Other than global rank correlation, we posit that ranking of top models from a search space is also critically important for NAS algorithms – this is because we ultimately care about finding those top models. In Section A.4 we perform an analysis of how top models are ranked by zero-cost proxies. Additionally, local rank correlation of top models could be important for NAS algorithms when two good models are compared using their proxy metric value. Tables 9 and 10 show that the only metric that maintains correct ranking among top models consistently across all NAS benchmarks is synflow. In Section 5 we deliberately evaluate 3 benchmarks that exhibit different levels of rank correlation: NAS-Bench-201/101/ASR to see if we can integrate synflow within NAS and achieve consistent gains for all three search spaces.
+
+Table 2: Spearman $\rho$ of zero-cost proxies on other NAS search spaces.   
+
+<table><tr><td></td><td> grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>NAS-Bench-101</td><td>0.20</td><td>0.16</td><td>0.45</td><td>0.26</td><td>0.37</td><td>0.38</td></tr><tr><td>NAS-Bench-NLP</td><td>-0.21</td><td>-0.19</td><td>0.16</td><td>1</td><td>0.34</td><td>0.56</td></tr><tr><td>NAS-Bench-ASR</td><td>0.07</td><td>0.01</td><td>1</td><td>0.02</td><td>0.40</td><td>-0.37</td></tr></table>
+
+# 5 ZERO-COST NAS
+
+Mellor et al. (2020) proposed using jacob cov to score a set of randomly-sampled models and to greedily choose the model with the highest score. This “NAS without training” methodology is very attractive thanks to its simplicity and low computational cost. In this section, we evaluate our metrics in this setting that we simply call “random search” (RAND). We extend this methodology slightly: instead of just training the top model, we keep training models (from best to worst as ranked by the zero-cost metric) until the desired accuracy is achieved. However, this approach can only produce results that are as good as the metric being used – and we have no guarantees (just empirical evidence) that these metrics will perform well on all datasets. Therefore, we also investigate how to integrate zero-cost metrics within existing NAS algorithms such as reinforcement learning (RL) (Zoph & Le, 2017), aging evolution (AE) search (Real et al., 2019) and predictor-based search (Dudziak et al., 2020). More specifically, we investigate enhancing these search algorithms through either (a) zero-cost warmup phase or (b) zero-cost move proposal.
+
+![](images/e65525014f0c52cb4d2bcb35e1d6d77c99e829704057ce0c628fdd095cccadbb.jpg)  
+Figure 4: Search speedup with the synflow zero-cost proxy on NAS-Bench-201 CIFAR-100.
+
+# 5.1 ZERO-COST WARMUP
+
+Generally speaking, by warmup we mean using the zero-cost proxies at the beginning of the search process to initialize the search algorithm without training any models or using accuracy. The main parameter in zero-cost warmup is the number of models for which we compute and use the zero-cost metric $( N )$ , and the potential gain comes from the fact that this number can be usually much larger than the number of models we can afford to train $( T \ll N )$ ).
+
+Aging Evolution We score $N$ random models with our proxy metric and choose the ones ranked highest as the initial population (pool) in the aging evolution (AE) algorithm (Real et al., 2019).
+
+Reinforcement Learning In the REINFORCE algorithm (Zoph & Le (2017)), we sample $N$ random models and use their zero-cost scores to reward the controller, thus biasing it towards selecting architectures which are likely to have higher values of the chosen metrics. During warmup, reward for the controller is calculated by linearly normalizing values returned by the proxy functions to the range $[ - 1 , 1 ]$ (with online adjustment of min and max).
+
+Binary Predictor We warm up a binary graph convolutional network (GCN) predictor from Dudziak et al. (2020) by training it to predict relative performance of two models by considering their zero-cost scores instead of accuracy. For $N$ warmup points, we use the relative rankings (according to the zero-cost metric) of all pairs of models $( 0 . 5 N ( N - 1 )$ pairs) when performing warmup training for the predictor. As in (Dudziak et al., 2020), models ranked by the predictor after each training round (including the warmup phase) and the top models are evaluated.
+
+# 5.2 ZERO-COST MOVE PROPOSAL
+
+Whereas warmup tries to leverage global correlation of the proxy metrics to the accuracy of models, move proposal focuses on a local neighborhood at each step. A common parameter for move proposal algorithms is denoted as $R$ and means sample ratio, i.e., how many models can be checked using zero-cost metrics each time we select a model to train.
+
+Aging Evolution The algorithm is enhanced by performing “guided” mutations. More specifically, each time a model is being mutated (in the baseline algorithm this is done randomly) we consider all possible mutations with edit distance 1 from the current model, score them using the zero-cost proxies and select the best one to add to the pool.
+
+![](images/4c41c4c13da8362ef0acf7473e758981477421401a26c1539f0aaef070ae261e.jpg)  
+Figure 5: Search speedup with the synflow zero-cost proxy on NAS-Bench-ASR TIMIT.
+
+Reinforcement Learning In the case of REINFORCE, move proposal is similar to warmup – instead of rewarding a controller $N$ time before the search begins, we interleave $R$ zero-cost rewards for each accuracy reward $( R \ll N$ ).
+
+# 5.3 RESULTS
+
+For all NAS experiments, we repeat experiments 32 times and we plot the median and shade between the lower/upper quartiles. Our baselines are already heavily tuned and achieve the same or better results than those reported in the original NAS-Bench-101/201 papers. When adding zero-cost warmup or move proposal with synflow, we leave all search hyper-parameters unchanged.
+
+NAS-Bench-201 The global/top- $10 \%$ rank correlations of synflow for this dataset are (0.76/0.42) so we expect this proxy to perform quite well. Indeed, as Figure 4 and Table 7 show, we improve search speed on all four types of searches using zero-cost warmup and move proposal. RAND and RL are both significantly improved, both in terms of sample efficiency and final achieved accuracy. But even more powerful algorithms like AE and BP exhibit $5 . 6 \times$ and $2 . 3 \times$ speedups respectively to arrive at $7 3 . 5 \%$ accuracy. Generally, the more zero-cost warmup, the better the results. This holds true for all algorithms except RL which degrades at $1 5 \mathrm { k }$ warmup points, suggesting that the controller is overfitting to the synflow metric instead of learning to optimize for accuracy.
+
+NAS-Bench-101 This dataset is an order of magnitude larger than NAS-Bench-201 and has lower global/top- $10 \%$ rank correlations of (0.37/0.14). In many ways, this provides a true test as to whether these lower correlations are still useful with zero-cost warmup and move proposal. Table 3 shows a summary of the results and Figure 7 (in Section A.6) shows the full plots. As the table shows, even with modest correlations, there is a major boost to all searching algorithms thus outperforming the best previously published result by a large margin and setting a new state-of-the-art result on this dataset. However, it is worth noting that the binary predictor exhibits no improvement (but also no degradation). Perhaps this is because it was already very sample-efficient and synflow warmup couldn’t help further due to its relatively poor correlation on this dataset.
+
+Table 3: Comparison to prior work on NAS-Bench-101 dataset.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Wen et al. (2019)</td><td rowspan="2">Wei et al. (2020)</td><td rowspan="2">Dudziak et al. (2020)</td><td colspan="3">Ours</td></tr><tr><td>RL+M(100)</td><td>AE+W (15k)</td><td>RAND+W (3k)</td></tr><tr><td># Trained Models</td><td>256</td><td>150</td><td>140</td><td>51</td><td>50</td><td>34</td></tr><tr><td>Test Accuracy [%]</td><td>94.17</td><td>94.14</td><td>94.22</td><td>94.22</td><td>94.22</td><td>94.22</td></tr></table>
+
+NAS-Bench-ASR We repeat our evaluation on NAS-Bench-ASR with global/top- $10 \%$ correlations (0.40/0.40). Even though this is a different task (speech recognition), synflow warmup and move proposal both yield large improvements in search speeds compared to all baselines in Figure 5 and Table 8. For example, to achieve a phoneme error rate (PER) of $2 1 . 3 \%$ , baseline RAND and RL required $> 1 0 0 0$ trained models, and AE required 138 trained models; however, this is reduced to 68, 173 and 87 trained models with 2000 models of zero-cost warmup.
+
+# 6 DISCUSSION
+
+In this section we investigate why zero-cost NAS is effective in improving the sample efficiency of NAS algorithms by looking more closely at how top models are selected by the synflow proxy.
+
+Warmup Table 4 shows the number of top- $5 \%$ most-accurate models ranked within the top 64 models by the synflow metric. If we compare random warmup versus zero-cost warmup with synflow, random warmup will only return $5 \%$ or $\sim 3$ models out of 64 that are within the top $5 \%$ of models whereas synflow warmup returns a higher number of top- $5 \%$ models as listed in Table 4. This is key to the improvements observed when adding zero-cost warmup to algorithms like random search or AE. For example, with AE, the numbers in Table 4 are indicative of the models that may end up in the initial AE pool. By initializing the AE pool with many good models, it becomes more likely that a random mutation will lead to an even better model, thus allowing the search to find a top model more quickly. Note that synflow is able to rank many good models in its top 64 models even when global/local correlation is low (as it is the case for NAS-Bench-ASR).
+
+Table 4: Number of top- $5 \%$ most-accurate models within the top 64 models returned by synflow.   
+
+<table><tr><td colspan="3">NAS-Bench-201</td><td rowspan="2">NAS-Bench-101</td><td rowspan="2">NAS-Bench-ASR</td></tr><tr><td>CIFAR-10</td><td>CIFAR-100</td><td>ImageNet16-120</td></tr><tr><td>44</td><td>54</td><td>56</td><td>12</td><td>16</td></tr></table>
+
+Move Proposal For a search algorithm like AE, search moves consist of random mutations (with edit distance 1 for our experiments) for a model from the AE pool. Zero-cost move proposal enhances this by trying out all possible mutations and selecting the best one according to synflow. To investigate how this improves search efficiency, we took 1000 random points and explored their local neighbourhood cluster of possible mutations. Table 5 shows the probability that the synflow proxy correctly identifies the top model. Indeed, synflow improves the chance of selecting the best mutation from $\sim 4 \%$ to $> 3 0 \%$ for NAS-Bench-201 and $12 \%$ for NAS-Bench-101. Even for NAS-Bench-ASR a random mutation has a $7 . 7 \%$ chance $( = 1 / 1 3 )$ to select the best mutation, but this increases to $10 \%$ with the synflow proxy thus speeding up convergence to top models.
+
+Table 5: For 1000 clusters of models with edit distance 1, we empirically measure the probability that the synflow proxy will select the most accurate model from each cluster.   
+
+<table><tr><td rowspan="2"></td><td colspan="3">NAS-Bench-201</td><td rowspan="2">NAS-Bench-101</td><td rowspan="2">NAS-Bench-ASR</td></tr><tr><td>CIFAR-10</td><td>CIFAR-100</td><td>ImageNet16-120</td></tr><tr><td>Top Model Match</td><td>32%</td><td>35%</td><td>33%</td><td>12%</td><td>10%</td></tr><tr><td>Average Cluster Size</td><td>25</td><td>25</td><td>25</td><td>26</td><td>13</td></tr></table>
+
+# 7 CONCLUSION
+
+In this paper, we introduced six zero-cost proxies, mainly based on recent pruning-at-initialization work, that are used to rank neural network models in NAS. First, we compared to conventional proxies (EcoNAS) that perform reduced-computation training and we found that zero-cost proxies such as synflow can outperform EcoNAS in maintaining rank consistency. Next, we verified our zerocost metrics on four additional datasets of varying sizes and tasks and found that indeed out of the six initially-considered zero-cost metrics, only synflow was robust across all datasets for both global and top- $10 \%$ rank correlation. Finally, we proposed two ways to integrate synflow within NAS algorithms: zero-cost warmup and zero-cost move proposal. Both methods demonstrated significant speedups across four search algorithms and three NAS benchmarks, setting new state-of-the-art results for both NAS-Bench-101 and NAS-Bench-201 datasets. Our strong and consistent empirical results suggest that the synflow metric, when combined with warmup and move proposal can be an effective and reliable methodology for speeding up different NAS algorithms. We hope that our work lays a foundation for further zero-cost techniques that expose favourable model properties with little computation thus making NAS more readily accessible without exorbitant computing resources. The most immediate open question for future investigation is why the synflow proxy works well – analytical insights will enable further research in zero-cost NAS proxies.
+
+# REFERENCES
+
+Gabriel Bender, Pieter-Jan Kindermans, Barret Zoph, Vijay Vasudevan, and Quoc V. Le. Understanding and simplifying one-shot architecture search. In International Conference on Machine Learning (ICML), 2018.
+
+Han Cai, Tianyao Chen, Weinan Zhang, Yong Yu, and Jun Wang. Efficient architecture search by network transformation. In AAAI Conference on Artificial Intelligence (AAAI), 2018.
+
+Wayne W. Daniel. Applied Nonparametric Statistics. Boston: PWS-Kent, 1990.
+
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. ImageNet: A Large-Scale Hierarchical Image Database. In CVPR, 2009.
+
+Xuanyi Dong and Yi Yang. NAS-Bench-201: Extending the Scope of Reproducible Neural Architecture Search. In International Conference on Learning Representations (ICLR), 2020.
+
+Łukasz Dudziak, Thomas Chau, Mohamed S. Abdelfattah, Royson Lee, Hyeji Kim, and Nicholas D. Lane. BRP-NAS: Prediction-based NAS using GCNs. In Neural Information Processing Systems (NeurIPS), 2020.
+
+Mikhail Figurnov, Aizhan Ibraimova, Dmitry P Vetrov, and Pushmeet Kohli. Perforatedcnns: Acceleration through elimination of redundant convolutions. In Neural Information Processing Systems (NeurIPS), 2016.
+
+Jonathan Frankle and Michael Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks. In International Conference on Learning Representations (ICLR), 2019.
+
+John S. Garofolo, Lori F. Lamel, William M. Fisher, Jonathan G. Fiscus, David S. Pallett, Nancy L. Dahlgren, and Victor Zue. Timit acoustic phonetic continuous speech corpus. Linguistic Data Consortium, 1993. URL https://catalog.ldc.upenn.edu/LDC93S1.
+
+Song Han, Jeff Pool, John Tran, and William Dally. Learning both weights and connections for efficient neural network. In Neural Information Processing Systems (NeurIPS), 2015.
+
+Babak Hassibi and David G. Stork. Second order derivatives for network pruning: Optimal brain surgeon. In Advances in Neural Information Processing Systems 5, 1993.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Computer Vision and Pattern Recognition (CVPR), June 2016.
+
+Andrew G. Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam. MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications. arXiv preprint arXiv:1704.04861, 2017.
+
+Gao Huang, Zhuang Liu, Laurens van der Maaten, and Kilian Q. Weinberger. Densely connected convolutional networks. In Computer Vision and Pattern Recognition (CVPR), July 2017.
+
+Nikita Klyuchnikov, Ilya Trofimov, Ekaterina Artemova, Mikhail Salnikov, Maxim Fedorov, and Evgeny Burnaev. Nas-bench-nlp: Neural architecture search benchmark for natural language processing. arXiv preprint arXiv:2006.07116, 2020.
+
+Alex Krizhevsky. Learning Multiple Layers of Features from Tiny Images, 2009. URL https: //www.cs.toronto.edu/˜kriz/cifar.html.
+
+Yann LeCun, John S. Denker, and Sara A. Solla. Optimal brain damage. In Neural Information Processing Systems (NeurIPS), 1990.
+
+Namhoon Lee, Thalaiyasingam Ajanthan, and Philip HS Torr. Snip: Single-shot network pruning based on connection sensitivity. In International Conference on Learning Representations (ICLR), 2019.
+
+Hanxiao Liu, Karen Simonyan, and Yiming Yang. DARTS: Differentiable architecture search. In International Conference on Learning Representations (ICLR), 2019.
+
+Abhinav Mehrotra, Alberto Gil Ramos, Sourav Bhattacharya, Łukasz Dudziak, Ravichander Vipperla, Thomas Chau, Mohamed S. Abdelfattah, Samin Ishtiaq, and Nicholas D. Lane. NASBench-ASR: Reproducible Neural Architecture Search for Speech Recognition. In International Conference on Learning Representations (ICLR), 2021.
+
+Jieru Mei, Yingwei Li, Xiaochen Lian, Xiaojie Jin, Linjie Yang, Alan Yuille, and Jianchao Yang. AtomNAS: Fine-grained end-to-end neural architecture search. In International Conference on Learning Representations (ICLR), 2020.
+
+Joseph Mellor, Jack Turner, Amos Storkey, and Elliot J. Crowley. Neural architecture search without training. arXiv preprint arXiv:2006.04647, 2020.
+
+Pavlo Molchanov, Stephen Tyree, Tero Karras, Timo Aila, and Jan Kautz. Pruning convolutional neural networks for resource efficient inference. In International Conference on Learning Representations (ICLR), 2017.
+
+Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y. Ng. Reading Digits in Natural Images with Unsupervised Feature Learning. In NeurIPS Workshop on Deep Learning and Unsupervised Feature Learning, 2011.
+
+Barak A. Pearlmutter. Fast Exact Multiplication by the Hessian. Neural Computation, 1993.
+
+Hieu Pham, Melody Y. Guan, Barret Zoph, Quoc V. Le, and Jeff Dean. Efficient neural architecture search via parameter sharing, 2018.
+
+Esteban Real, Alok Aggarwal, Yanping Huang, and Quoc V. Le. Regularized Evolution for Image Classifier Architecture Search. In AAAI Conference on Artificial Intelligence (AAAI), 2019.
+
+Oleg Semery. PyTorchCV Convolutional neural networks for computer vision, August 2020. URL ´ https://github.com/osmr/imgclsmob.
+
+Mingxing Tan and Quoc Le. EfficientNet: Rethinking model scaling for convolutional neural networks. In International Conference on Machine Learning (ICML), 2019a.
+
+Mingxing Tan and Quoc V. Le. Mixconv: Mixed depthwise convolutional kernels, 2019b.
+
+Hidenori Tanaka, Daniel Kunin, Daniel L. K. Yamins, and Surya Ganguli. Pruning neural networks without any data by iteratively conserving synaptic flow. arXiv preprint arXiv:2006.05467, 2020.
+
+Lucas Theis, Iryna Korshunova, Alykhan Tejani, and Ferenc Huszar. Faster gaze prediction with ´ dense networks and fisher pruning. arXiv:1801.05787, 2018.
+
+Jack Turner, Elliot J. Crowley, Michael O’Boyle, Amos Storkey, and Gavin Gray. Blockswap: Fisher-guided block substitution for network compression on a budget. In International Conference on Learning Representations (ICLR), 2020.
+
+Chaoqi Wang, Guodong Zhang, and Roger Grosse. Picking winning tickets before training by preserving gradient flow. In International Conference on Learning Representations (ICLR), 2020.
+
+Chen Wei, Chuang Niu, Yiping Tang, and Jimin Liang. NPENAS: Neural predictor guided evolution for neural architecture search. arXiv 2003.12857, 03 2020.
+
+Wei Wen, Hanxiao Liu, Hai Li, Yiran Chen, Gabriel Bender, and Pieter-Jan Kindermans. Neural predictor for neural architecture search. arXiv:1912.00848, 2019.
+
+Chris Ying, Aaron Klein, Eric Christiansen, Esteban Real, Kevin Murphy, and Frank Hutter. NASbench-101: Towards reproducible neural architecture search. In International Conference on Machine Learning (ICML), 2019.
+
+Dongzhan Zhou, Xinchi Zhou, Wenwei Zhang, Chen Change Loy, Shuai Yi, Xuesen Zhang, and Wanli Ouyang. Econas: Finding proxies for economical neural architecture search. In Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.
+
+Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. In International Conference on Learning Representations (ICLR), 2017.
+
+# A APPENDIX
+
+Because this paper is empirically-driven, there are many more results than what we presented in the main text of the paper. In the appendix we list many important results that support our main arguments and hypotheses in the main text of this paper.
+
+# A.1 EXPERIMENTAL DETAILS
+
+In Table 6 we list the hyper-parameters used in training the EcoNAS proxies to produce Figure 1. The only difference to the standard NAS-Bench-201 training pipeline (Dong & Yang, 2020) is our use of fewer epochs for the learning rate annealing schedule – we anneal the learning rate to zero over 40 epochs instead of 200. This is a common technique used in speeding up convergence for training proxies Zhou et al. (2020). We acknowledge that slightly better correlations could have been achieved for econas and econas $^ +$ proxies in Figure 1 if the learning rate was annealed to zero over fewer epochs (20 and 15 epochs respectively). However, we do not anticipate the results to change significantly.
+
+Table 6: EcoNAS training hyper-parameters for NAS-Bench-201.   
+
+<table><tr><td>optimizer Nesterov momentum weight decay</td><td>SGD √ 0.9 0.0005 (p=0.5)</td><td>initial LR final LR LR schedule epochs batch size</td><td>0.1 0 cosine 40 256</td></tr></table>
+
+One additional comment regarding Figure 1 in the main paper. While we run the training ourselves for all EcoNAS variants in the plot, we take the data for the line labeled baseline directly from the NAS-Bench-201 dataset. We are not sure why the line is not smooth like the lines for the EcoNAS variants that we trained but assume that this is an artifact of averaging over multiple seeds in the NAS-Bench-201 dataset. In any case, we do not anticipate that this would change any conclusions or observations that we draw from this plot.
+
+Finally, we would like to note some details about our NAS experiments in Section 5. NAS datasets provide multiple seeds of results for each model, so whenever we “train” a model, we query a random seed from the database to mimic a real NAS pipeline without caching. We refer the reader to (Dudziak et al., 2020), specifically Section S3.2 for more details on this.
+
+# A.2 GPU RUNTIME FOR ECONAS
+
+Figure 6 shows the speedup of different EcoNAS proxies compared to baseline training. Even though $r _ { 8 } c _ { 4 }$ has $6 4 \times$ less computation compared to $r _ { 3 2 } c _ { 1 6 }$ , it achieves a maximum of $4 \times$ real speedup even when the batch size is increased.
+
+![](images/4ff85890acab96fe33b81848c95232390efa880f144e2225ec6a0ababe18d8d2.jpg)  
+Figure 6: Higher batch sizes when training econas proxies have diminishing returns in terms of measured speedup. This measurement is done for 10 randomly-sampled NAS-Bench-201 models on the CIFAR-10 dataset.
+
+# A.3 TABULATED RESULTS
+
+This subsection contains tabulated results from Figures 4 and 5 to facilitate comparisons with future work. Tables 7 and 8 highlight important data points about the NAS searches we conducted with NAS-Bench-201 and NAS-Bench-ASR respectively. We highlight results in two ways: First, we show the accuracy of the best model found after 50 trained models. Second, we indicate the number of trained models needed for each search method to reach a specific accuracy $7 3 . 5 \%$ CIFAR-10 classification accuracy for NAS-Bench-201 and $2 1 . 3 \%$ TIMIT PER.) We colour the best results (red) and the second best (blue) results in each table.
+
+Table 7: Zero-cost NAS comparison with baseline algorithms on NAS-Bench-201 CIFAR-100. We show accuracy after 50 trained models and the number of models to reach $7 3 . 5 \%$ accuracy.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Baseline</td><td colspan="3">Warmup</td><td colspan="2">Move</td></tr><tr><td>1000 (BP=256)</td><td>3000 (BP=512)</td><td>15k</td><td>10</td><td>100</td></tr><tr><td>RAND</td><td>71.31/1000+</td><td>72.98 /1000+</td><td>73.18 /1000+</td><td>73.75 /8</td><td></td><td>1</td></tr><tr><td>RL</td><td>71.08 /1000+</td><td>72.76 /145</td><td>73.14/84</td><td>73.21/107</td><td>71.16/289</td><td>73.34 / 70</td></tr><tr><td>AE</td><td>71.53 /139</td><td>72.91 / 115</td><td>73.40 /71</td><td>73.63 / 25</td><td>71.3 /77</td><td></td></tr><tr><td>BP</td><td>72.74 /93</td><td>73.32 / 66</td><td>73.85 / 40</td><td>1</td><td></td><td>1</td></tr></table>
+
+Table 8: Zero-cost NAS comparison with baseline algorithms on NAS-Bench-ASR. We show PER after 50 trained models and the number of models to reach PER $= 2 1 . 3 \%$ .   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Baseline</td><td colspan="2">Warmup</td><td colspan="2">Move</td></tr><tr><td>500</td><td>2000</td><td>10</td><td>100</td></tr><tr><td>RAND</td><td>21.65 /1000+</td><td>21.38 /1000+</td><td>21.35 / 68</td><td></td><td></td></tr><tr><td>RL</td><td>21.66 /1000+</td><td>21.48 /1000+</td><td>21.45 / 173</td><td>21.62 / 169</td><td>21.43 / 161</td></tr><tr><td>AE</td><td>21.62 / 138</td><td>21.40 / 115</td><td>21.36 / 87</td><td></td><td>21.74 / 112</td></tr></table>
+
+# A.4 ANALYSIS OF THE TOP $10 \%$ OF MODELS
+
+In the main text we pointed to the fact that only synflow achieves consistent rank correlation for the top- $10 \%$ of models across different datasets. Here, in Table 9 we provide the full results. Additionally, we hypothesized that a successful metric will rank many of the most-accurate models in its top models. In Table 10 we enumerate the percentage of top- $10 \%$ most accurate models ranked as top- $10 \%$ by each proxy metric. Again, synflow is the only consistent metric for all datasets, and performs best on average.
+
+Table 9: Spearman $\rho$ of zero-cost proxies for the top $10 \%$ of points on all NAS search spaces.   
+
+<table><tr><td></td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>NB2-CIFAR-10</td><td>-0.38</td><td>-0.38</td><td>-0.37</td><td>-0.38</td><td>0.18</td><td>0.17</td></tr><tr><td>NB2-CIFAR-100</td><td>-0.09</td><td>-0.09</td><td>-0.11</td><td>-0.16</td><td>0.42</td><td>0.08</td></tr><tr><td>NB2-ImageNet16-120</td><td>0.13</td><td>0.13</td><td>0.10</td><td>0.02</td><td>0.55</td><td>0.05</td></tr><tr><td>NAS-Bench-101</td><td>0.05</td><td>-0.01</td><td>-0.01</td><td>0.07</td><td>0.14</td><td>0.08</td></tr><tr><td>NAS-Bench-NLP</td><td>-0.03</td><td>-0.02</td><td>0.04</td><td>1</td><td>0.10</td><td>0.04</td></tr><tr><td>NAS-Bench-ASR</td><td>0.25</td><td>0.13</td><td>1</td><td>-0.07</td><td>0.40</td><td>-0.03</td></tr></table>
+
+# A.5 ANALYSIS OF WARMUP AND MOVE PROPOSAL
+
+This section provides more results relevant to our discussion in Section 6. Table 11 shows the number of top- $5 \%$ models ranked in the top 64 models by each metric. This is an extension to Table 4 in the main text that only shows the results for synflow. As shown in the table, synflow is the most powerful metric that we tried.
+
+Table 12 shows the rank correlation coefficient of models within 1000 randomly-sampled local clusters of models. This result highlights that both grad norm and jacob cov work well in distinguishing between very similar models. However, synflow still consistently the best metric in this analysis. Furthermore, we measure the percentage of times that a metric correctly predicts the top model within a local cluster of models in Table 13 This is an extension to Table 5 in the main text. The results are averaged over 1000 randomly-sampled local clusters. Again, synflow has the highest probability of selecting the top model compared to other zero-cost metrics.
+
+Table 10: Percentage of top- $10 \%$ most-accurate models within the top- $10 \%$ of models ranked by each zero-cost metric.   
+
+<table><tr><td></td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>NB2-CIFAR-10</td><td>30%</td><td>31%</td><td>30%</td><td>5%</td><td>46%</td><td>25%</td></tr><tr><td>NB2-CIFAR-100</td><td>35%</td><td>36%</td><td>34%</td><td>4%</td><td>50%</td><td>24%</td></tr><tr><td>NB2-ImageNet16-120</td><td>31%</td><td>31%</td><td>32%</td><td>5%</td><td>44%</td><td>30%</td></tr><tr><td>NAS-Bench-101</td><td>2%</td><td>3%</td><td>26%</td><td>3%</td><td>23%</td><td>2%</td></tr><tr><td>NAS-Bench-NLP</td><td>10%</td><td>10%</td><td>4%</td><td>1</td><td>22%</td><td>38%</td></tr><tr><td>NAS-Bench-ASR</td><td>0%</td><td>0%</td><td>1</td><td>0%</td><td>15%</td><td>46%</td></tr></table>
+
+Table 11: Number of top- $5 \%$ most-accurate models within the top-64 models returned by each metric.   
+
+<table><tr><td></td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td>jacob_cov</td></tr><tr><td>NB2-CIFAR-10</td><td>0</td><td>0</td><td>0</td><td>0</td><td>44</td><td>15</td></tr><tr><td>NB2-CIFAR-100</td><td>4</td><td>4</td><td>4</td><td>0</td><td>54</td><td>16</td></tr><tr><td>NB2-ImageNet16-120</td><td>13</td><td>13</td><td>14</td><td>0</td><td>56</td><td>15</td></tr><tr><td>NAS-Bench-101</td><td>0</td><td>0</td><td>6</td><td>0</td><td>12</td><td>0</td></tr><tr><td>NAS-Bench-ASR</td><td>1</td><td>0</td><td>1</td><td>1</td><td>16</td><td>13</td></tr></table>
+
+Table 12: Rank correlation coefficient for the local neighbourhoods (edit distance $= 1$ ) of 1000 clusters in each search space.   
+
+<table><tr><td></td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>NB2-CIFAR-10</td><td>0.51</td><td>0.51</td><td>0.37</td><td>0.37</td><td>0.66</td><td>0.62</td></tr><tr><td>NB2-CIFAR-100</td><td>0.58</td><td>0.58</td><td>0.44</td><td>0.41</td><td>0.69</td><td>0.61</td></tr><tr><td>NB2-ImageNet16-120</td><td>0.56</td><td>0.57</td><td>0.5</td><td>0.4</td><td>0.67</td><td>0.61</td></tr><tr><td>NAS-Bench-101</td><td>0.23</td><td>0.21</td><td>0.44</td><td>0.27</td><td>0.36</td><td>0.37</td></tr><tr><td>NAS-Bench-ASR</td><td>0.59</td><td>0.4</td><td>1</td><td>0.56</td><td>0.38</td><td>0.28</td></tr></table>
+
+Table 13: For 1000 clusters of points with edit distance $= 1$ . We count the number of times wherein the top model returned by a zero-cost metric matches the top model according to validation accuracy. This represents the probability that zero-cost move proposal will perform the best possible mutation.   
+
+<table><tr><td></td><td> grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>NB2-CIFAR-10</td><td>14.8%</td><td>14.8%</td><td>12.7%</td><td>5.7%</td><td>32.2%</td><td>14.5%</td></tr><tr><td>NB2-CIFAR-100</td><td>19.1%</td><td>18.5%</td><td>14.2%</td><td>6.0%</td><td>35.4%</td><td>13.8%</td></tr><tr><td>NB2-ImageNet16-120</td><td>17.5%</td><td>18.5%</td><td>15.7%</td><td>5.5%</td><td>33.4%</td><td>16.7%</td></tr><tr><td>NAS-Bench-101</td><td>0.4%</td><td>0.9%</td><td>7.4%</td><td>0.5%</td><td>12.3%</td><td>0.5%</td></tr><tr><td>NAS-Bench-ASR</td><td>11.0%</td><td>9.8%</td><td>1</td><td>10.3%</td><td>10.3%</td><td>10.5%</td></tr></table>
+
+# A.6 NAS-BENCH-101 SEARCH PLOTS
+
+Figure 7 shows the NAS search curves for all considered algorithms on NAS-Bench-101 dataset.   
+Important points from this plot are summarized in Table 3 in the main text.
+
+![](images/6921e93248b0f0a2cf8e20a79aadc988ab5cb39b97d407f1e89b2d694a5c6471.jpg)  
+Figure 7: Search speedup with the synflow zero-cost proxy on NAS-Bench-101 CIFAR-10.
+
+# A.7 SENSITIVITY ANALYSIS
+
+We performed some sensitivity analysis to investigate how the zero-cost metrics perform on all points within NAS-Bench-201 with different initialization seed, initialization method and minibatch size. We comment on each table in its caption; however, to summarize, all metrics seem to be relatively unaffected when initialization and minibatch size are varied. The one exception can be seen in Table 15 where fisher benefits when biases are initialized with zeroes.
+
+Table 14: All metrics remain fairly constant when varying the initialization seed – the variations are only observed at the third significant digit. Dataload is random with 128 samples and initialization is done with default PyTorch initialization scheme.   
+
+<table><tr><td>seed</td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>1</td><td>0.578</td><td>0.581</td><td>0.487</td><td>0.361</td><td>0.737</td><td>0.735</td></tr><tr><td>2</td><td>0.580</td><td>0.583</td><td>0.488</td><td>0.354</td><td>0.740</td><td>0.728</td></tr><tr><td>3</td><td>0.582</td><td>0.584</td><td>0.486</td><td>0.358</td><td>0.738</td><td>0.726</td></tr><tr><td>4</td><td>0.581</td><td>0.584</td><td>0.491</td><td>0.356</td><td>0.738</td><td>0.73</td></tr><tr><td>5</td><td>0.581</td><td>0.583</td><td>0.486</td><td>0.356</td><td>0.738</td><td>0.727</td></tr><tr><td>Average</td><td>0.580</td><td>0.583</td><td>0.488</td><td>0.357</td><td>0.738</td><td>0.729</td></tr></table>
+
+Table 15: fisher becomes noticeably better when biases are initialized to zero; otherwise, metrics seem to perform independently of initialization method. Results averaged over 3 seeds.   
+
+<table><tr><td>Weights init</td><td>Bias init</td><td> grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>default</td><td>default</td><td>0.580</td><td>0.583</td><td>0.488</td><td>0.357</td><td>0.738</td><td>0.729</td></tr><tr><td>kaiming</td><td>default</td><td>0.548</td><td>0.558</td><td>0.364</td><td>0.332</td><td>0.731</td><td>0.723</td></tr><tr><td>xavier</td><td>default</td><td>0.543</td><td>0.568</td><td>0.424</td><td>0.345</td><td>0.736</td><td>0.729</td></tr><tr><td>default</td><td>zero</td><td>0.581</td><td>0.583</td><td>0.488</td><td>0.509</td><td>0.738</td><td>0.729</td></tr><tr><td>kaiming</td><td>zero</td><td>0.542</td><td>0.551</td><td>0.370</td><td>0.479</td><td>0.730</td><td>0.723</td></tr><tr><td>xavier</td><td>zero</td><td>0.540</td><td>0.566</td><td>0.412</td><td>0.495</td><td>0.735</td><td>0.730</td></tr></table>
+
+Table 16: Surprisingly, grasp becomes worse with more (random) data, while grad norm and snip degrade very slightly. Other metrics seem to perform independently of the number of samples in the minibatch. Initialization is done with default PyTorch initialization scheme.   
+
+<table><tr><td>Number of Samples</td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>32</td><td>0.595</td><td>0.596</td><td>0.511</td><td>0.362</td><td>0.737</td><td>0.732</td></tr><tr><td>64</td><td>0.589</td><td>0.59</td><td>0.509</td><td>0.361</td><td>0.737</td><td>0.735</td></tr><tr><td>128</td><td>0.578</td><td>0.581</td><td>0.487</td><td>0.361</td><td>0.737</td><td>0.735</td></tr><tr><td>256</td><td>0.564</td><td>0.569</td><td>0.447</td><td>0.361</td><td>0.737</td><td>0.731</td></tr><tr><td>512</td><td>0.547</td><td>0.552</td><td>0.381</td><td>0.361</td><td>0.737</td><td>0.724</td></tr></table>
+
+# A.8 RESULTS FOR ALL ZERO-COST METRICS
+
+Here we provide some NAS search results using all considered metrics for both RAND and AE searches on NAS-Bench-101/201 datasets. Our experiments point to synflow as the only effective zero-cost metric across different datasets; however, we provide the plots below for the reader to inspect how poorer metrics perform in NAS.
+
+![](images/0e75bfa7f50cd7cd06f49cf98450eea07a6395bd2f9913745380c4a4bccdd3c2.jpg)  
+(d) NAS-Bench-101 CIFAR-10   
+Figure 8: Evaluation of all zero-cost proxies on different datasets and search algorithms: random search (RAND) and aging evolution (AE). RAND benefits greatly from a strong metric (such as synflow) but may deteriorate with a weaker metric as shown in the plot. However, AE benefits when a strong metric is used and is resilient to weaker metrics as well – it is able to recover and achieve the top accuracy in most cases.

parse/train/0cmMMy8J5q/0cmMMy8J5q_content_list.json

@@ -0,0 +1,1900 @@
+[
+    {
+        "type": "text",
+        "text": "ZERO-COST PROXIES FOR LIGHTWEIGHT NAS",
+        "text_level": 1,
+        "bbox": [
+            178,
+            90,
+            735,
+            112
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Mohamed S. Abdelfattah1, Abhinav Mehrotra1, Łukasz Dudziak1, Nicholas D. Lane1,2   \n1 Samsung AI Center, Cambridge · 2 University of Cambridge   \nmohamed1.a@samsung.com ",
+        "bbox": [
+            184,
+            131,
+            785,
+            174
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
+            544,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural Architecture Search (NAS) is quickly becoming the standard methodology to design neural network models. However, NAS is typically compute-intensive because multiple models need to be evaluated before choosing the best one. To reduce the computational power and time needed, a proxy task is often used for evaluating each model instead of full training. In this paper, we evaluate conventional reduced-training proxies and quantify how well they preserve ranking between neural network models during search when compared with the rankings produced by final trained accuracy. We propose a series of zero-cost proxies, based on recent pruning literature, that use just a single minibatch of training data to compute a model’s score. Our zero-cost proxies use 3 orders of magnitude less computation but can match and even outperform conventional proxies. For example, Spearman’s rank correlation coefficient between final validation accuracy and our best zero-cost proxy on NAS-Bench-201 is 0.82, compared to 0.61 for EcoNAS (a recently proposed reduced-training proxy). Finally, we use these zerocost proxies to enhance existing NAS search algorithms such as random search, reinforcement learning, evolutionary search and predictor-based search. For all search methodologies and across three different NAS datasets, we are able to significantly improve sample efficiency, and thereby decrease computation, by using our zero-cost proxies. For example on NAS-Bench-101, we achieved the same accuracy $4 \\times$ quicker than the best previous result. Our code is made public at: https://github.com/mohsaied/zero-cost-nas. ",
+        "bbox": [
+            233,
+            238,
+            764,
+            529
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            550,
+            336,
+            565
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Instead of manually designing neural networks, neural architecture search (NAS) algorithms are used to automatically discover the best ones (Tan & Le, $2 0 1 9 \\mathrm { a }$ ; Liu et al., 2019; Bender et al., 2018). Early work by Zoph & Le (2017) proposed using a reinforcement learning (RL) controller that constructs candidate architectures, these are evaluated and then feedback is provided to the controller based on the performance of the candidate. One major problem with this basic NAS methodology is that each evaluation is very costly – typically on the order of hours or days to train a single neural network fully. We focus on this evaluation phase – we propose using proxies that require a single minibatch of data and a single forward/backward propagation pass to score a neural network. This is inspired by recent pruning-at-initialization work by Lee et al. (2019), Wang et al. (2020) and Tanaka et al. (2020) wherein a per-parameter saliency metric is computed before training to inform parameter pruning. Can we use such saliency metrics to score an entire neural network? Furthermore, can we use these “single minibatch” metrics to rank and compare multiple neural networks for use within NAS? If so, how do we best integrate these metrics within existing NAS algorithms such as RL or evolutionary search? These are the questions that we hope to (empirically) tackle in this work with the goal of making NAS less compute-hungry. Our contributions are: ",
+        "bbox": [
+            173,
+            578,
+            825,
+            785
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• Zero-cost proxies We adapt pruning-at-initialization metrics for use with NAS. This requires these metrics to operate at the granularity of an entire network rather than individual parameters – we devise and validate approaches that aggregate parameter-level metrics in a manner suitable for ranking candidates during NAS search. ",
+        "bbox": [
+            215,
+            794,
+            823,
+            849
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• Comparison to conventional proxies We perform a detailed comparison between zerocost and conventional NAS proxies that use a form of reduced-computation training. First, we quantify the rank consistency of conventional proxies on large-scale datasets: $1 5 \\mathrm { k }$ models vs. 50 models used in (Zhou et al., 2020). Second, we show that zero-cost proxies can match or exceed the rank consistency of conventional proxies. ",
+        "bbox": [
+            220,
+            854,
+            823,
+            922
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• Ablations on NAS benchmarks We perform ablations of our zero-cost proxies on five different NAS benchmarks (NAS-Bench-101/201/NLP/ASR and PyTorchCV) to both test the zero-cost metrics under different settings, and expose properties of successful metrics. ",
+        "bbox": [
+            218,
+            94,
+            823,
+            137
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Integration with NAS Finally, we propose two ways to use zero-cost metrics effectively within NAS algorithms: random search, reinforcement learning, aging evolution and predictor-based search. For all algorithms and three NAS datasets we show significant speedups, up to $4 \\times$ for NAS-Bench-101 compared to current state-of-the-art. ",
+        "bbox": [
+            218,
+            141,
+            823,
+            196
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            213,
+            344,
+            229
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "NAS Efficiency To decrease NAS search time, various techniques were used in the literature. Pham et al. (2018) and Cai et al. (2018) use weight sharing between candidate models to decrease the training time during evaluation. Liu et al. (2019) and others use smaller datasets (CIFAR-10) as a proxy to the full task (ImageNet1k). In EcoNAS, Zhou et al. (2020) extensively investigated reduced-training proxies wherein input size, model size, number of training samples and number of epochs were reduced in the NAS evaluation phase. We compare to EcoNAS in this work to elucidate how well our zero-cost proxies perform compared to familiar and widely-used conventional proxies. ",
+        "bbox": [
+            174,
+            241,
+            825,
+            339
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Pruning The goal is to reduce the number of parameters in a neural network, one way to do this is by identifying a saliency (importance) metric for each parameter, and the less-important parameters are removed. For example, Han et al. (2015), Frankle & Carbin (2019) and others use parameter magnitudes as the criterion while LeCun et al. (1990), Hassibi & Stork (1993) and Molchanov et al. (2017) use gradients. However, the aforementioned works require training before computing the saliency criterion. A new class of pruning-at-initialization algorithms, that require no training, were introduced by Lee et al. (2019) and extended by Wang et al. (2020) and Tanaka et al. (2020). A single forward/backward propagation pass is used to compute a saliency criterion which is successfully used to heavily prune neural networks before training. We extend these pruning-at-initialization criteria towards scoring entire neural networks and we investigate their use with NAS algorithms. ",
+        "bbox": [
+            174,
+            349,
+            825,
+            489
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Intersection between pruning and NAS Concepts from pruning have been used within NAS multiple times. For example, Mei et al. (2020) use channel pruning in their AtomNAS work to arrive at customized multi-kernel-size convolutions (mixconvs as introduced by Tan & Le (2019b)). In their Blockswap work, Turner et al. (2020) use Fisher information at initialization to score different lightweight primitives that are substituted into a neural network to decrease computation. This is the earliest work we could find that attempts to perform a type of NAS by scoring neural networks without training using a pruning criterion, More recently, Mellor et al. (2020) introduced a new metric for scoring neural networks at initialization based on the correlation of Jacobians with different inputs. They perform “NAS without training” by performing random search with their zero-cost metric (jacob cov) to rank neural networks instead of using accuracy. We include jacob cov in our analysis and we introduce five more zero-cost metrics in this work. ",
+        "bbox": [
+            174,
+            501,
+            825,
+            652
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 PROXIES FOR NEURAL NETWORK ACCURACY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            670,
+            588,
+            685
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3.1 CONVENTIONAL NAS PROXIES (ECONAS) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            696,
+            514,
+            712
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In conventional sample-based NAS, a proxy training regime is often used to predict a model’s accuracy instead of full training. Zhou et al. (2020) investigate conventional proxies in depth by computing the Spearman rank correlation coefficient (Spearman $\\rho$ ) of a proxy task to final test accuracy. The proxy used is a reduced-computation training, wherein one of the following four variables is reduced: (1) number of epochs, (2) number of training samples, (3) input resolution (4) model size (controlled through the number of channels after the first convolution). Even though such proxies were used in many prior works, EcoNAS is the first systematic study of conventional proxy tasks that we found. One main finding by Zhou et al. (2020) is that using approximately $\\textstyle { \\frac { 1 } { 4 } }$ of the model size and input resolution, all training samples, and $\\textstyle { \\frac { 1 } { 1 0 } }$ the number of epochs was a reasonable proxy which yielded the best results for their experiment (Zhou et al., 2020). ",
+        "bbox": [
+            174,
+            718,
+            825,
+            861
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3.2 ZERO-COST NAS PROXIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            873,
+            401,
+            887
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We present alternative proxies for network accuracy that can be used to speed up NAS. A simple proxy that we use is grad norm in which we sum the Euclidean norm of the gradients after a single minibatch of training data. Other metrics listed below were previously introduced in the context of parameter pruning at the granularity of a single parameter – a saliency is computed to rank parameters and remove the least important ones. We adapt these metrics to score and rank entire neural network models for NAS. ",
+        "bbox": [
+            176,
+            896,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            94,
+            825,
+            150
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2.1 SNIP, GRASP AND SYNAPTIC FLOW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            165,
+            473,
+            180
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In their snip work, Lee et al. (2019) proposed performing parameter pruning based on a saliency metric computed at initialization using a single minibatch of data. This saliency criteria approximates the change in loss when a specific parameter is removed. Wang et al. (2020) attempted to improve on the snip metric by approximating the change in gradient norm (instead of loss) when a parameter is pruned in their grasp objective. Finally, Tanaka et al. (2020) generalized these so-called synaptic saliency scores and proposed a modified version (synflow) which avoids layer collapse when performing parameter pruning. Instead of using a minibatch of training data and cross-entropy loss (as in snip or grasp), with synflow we compute a loss which is simply the product of all parameters in the network; therefore, no data is needed to compute this loss or the synflow metric itself. These are the three metrics: ",
+        "bbox": [
+            173,
+            186,
+            825,
+            327
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f147f6f756e5cac0333fac1d2e174af5b50e5cc15b7f3c0831086dd281c96a6b.jpg",
+        "text": "$$\n\\mathrm { s n i p } : S _ { P } ( \\theta ) = \\left| \\frac { \\partial \\mathcal { L } } { \\partial \\theta } \\odot \\theta \\right| , \\quad \\mathrm { g r a s s p } : S _ { P } ( \\theta ) = - ( H \\frac { \\partial \\mathcal { L } } { \\partial \\theta } ) \\odot \\theta , \\quad \\mathrm { ~ s y n f 1 o w : ~ } S _ { P } ( \\theta ) = \\frac { \\partial \\mathcal { L } } { \\partial \\theta } \\odot \\theta .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            351,
+            818,
+            386
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L }$ is the loss function of a neural network with parameters $\\theta , H$ is the Hessian1, $S _ { p }$ is the per-parameter saliency and $\\odot$ is the Hadamard product. We extend these saliency metrics to score an entire neural network by summing over all parameters $N$ in the model: $\\begin{array} { r } { S _ { n } = \\dot { \\sum } _ { i } ^ { N } S _ { p } ( \\theta ) _ { i } } \\end{array}$ . ",
+        "bbox": [
+            174,
+            400,
+            825,
+            445
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2.2 FISHER ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            459,
+            281,
+            473
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theis et al. (2018) perform channel pruning by removing activation channels (and their corresponding parameters) that are estimated to have the least effect on the loss. They build on the work of Molchanov et al. (2017) and Figurnov et al. (2016). More recently, Turner et al. (2020) aggregated this fisher metric for all channels in a convolution primitive to quantify the importance of that primitive when it is replaced by a more efficient alternative. We further aggregate the fisher metric for all layers in a neural network to score an entire network as shown in the following equations: ",
+        "bbox": [
+            173,
+            481,
+            825,
+            565
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d37286ff8ff2888b37a977f6f3de90eb566f49d16d7c13a8b171116e7738623f.jpg",
+        "text": "$$\n{ \\mathrm { \\bf { f i s h e r : } } } \\ S _ { z } ( z ) = \\left( { \\frac { \\partial { \\mathcal { L } } } { \\partial z } } z \\right) ^ { 2 } , \\quad \\ S _ { n } = \\sum _ { i = 1 } ^ { M } S _ { z } ( z _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            583,
+            668,
+            626
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $S _ { z }$ is the saliency per activation $z$ , and $M$ is the length of the vectorized feature map. ",
+        "bbox": [
+            173,
+            637,
+            772,
+            652
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2.3 JACOBIAN COVARIANCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            666,
+            397,
+            681
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This metric was purpose-designed to score neural networks in the context of NAS – we refer the reader to the original paper for detailed reasoning and derivation of the metric which we call jacob cov (Mellor et al., 2020). In brief, this metric captures the correlation of activations within a network when subject to different inputs within a minibatch of data – the lower the correlation, the better the network is expected to perform as it can differentiate between different inputs well. ",
+        "bbox": [
+            174,
+            688,
+            825,
+            758
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 EMPIRICAL EVALUATION OF PROXY TASKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            779,
+            566,
+            795
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Generally, most of the proxies presented in the previous section try to capture how trainable a neural network is by inspecting the gradients at the beginning of training. In this work, we refrain from attempting to explain precisely why each metric works (or does not work) and instead focus on the empirical evaluation of those metrics in different scenarios. We use the Spearman rank correlation coefficient (Spearman $\\rho$ ) to quantify how well a proxy ranks models compared to the ground-truth ranking produced by final test accuracy (Daniel, 1990). ",
+        "bbox": [
+            174,
+            808,
+            825,
+            892
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/7a77fc951f37c2e4eacba79dd827fcbfb9921946b9a054f990371082151ad0b4.jpg",
+        "image_caption": [
+            "Figure 1: Evaluation of different econas proxies on NAS-Bench-201 CIFAR-10. FLOPS and runtime are normalized to the FLOPS/runtime of a single baseline (full training) epoch. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            90,
+            816,
+            248
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 NAS-BENCH-201 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            296,
+            338,
+            309
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "NAS-Bench-201 is a purpose-built benchmark for prototyping NAS algorithms (Dong & Yang, 2020). It contains 15,625 CNN models from a cell-based search space and corresponding training statistics. We first use NAS-Bench-201 to evaluate conventional proxies from EcoNAS, then we evaluate our zero-cost proxies and compare the two approaches. ",
+        "bbox": [
+            174,
+            318,
+            825,
+            373
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1.1 ECONAS PROXY ON NAS-BENCH-201 ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            383,
+            500,
+            398
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Even though Zhou et al. (2020) thoroughly investigated reduced-training proxies, they only evaluated a small model zoo consisting of 50 models. To study EcoNAS more extensively we evaluate it on all 15,625 models in NAS-Bench-201 search space (training details in A.1). The full configuration training of NAS-Bench-201 on CIFAR-10 uses input resolution $\\scriptstyle 1 = 3 2$ , number of channels in the stem convolution $\\scriptstyle { \\mathrm { c = } } 1 6$ and number of epochs $\\scriptstyle { \\mathrm { e } = 2 0 0 }$ – we summarize this as: $r _ { 3 2 } c _ { 1 6 } e _ { 2 0 0 }$ . ",
+        "bbox": [
+            174,
+            405,
+            825,
+            474
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "According to the EcoNAS study, the most effective configuration divides both the input resolution and stem channels by ${ \\sim } 4$ and the number of epochs by 10, that is, $r _ { 8 } c _ { 4 } e _ { 2 0 }$ for NAS-Bench-201 models. Keeping that in mind we investigate $r _ { 8 } c _ { 4 }$ in Fig. 1 (labeled econas); however, this proxy training seems to suffer from overfitting as correlation to final accuracy started to drop after 20 epochs. Additionally, the Spearman $\\rho$ was a modest 0.61 when evaluated on all 15,625 models in NAS-Bench-201 – a far cry from the 0.87 achieved on the 50 models in the EcoNAS paper (Zhou et al., 2020). We additionally explore $r _ { 8 } c _ { 8 }$ , $r _ { 1 6 } c _ { 4 }$ and $r _ { 1 6 } c _ { 8 }$ and find a very good proxy with $r _ { 1 6 } c _ { 8 } e _ { 1 5 }$ , labeled in Fig. 1 as econas+. From the plots in Fig. 1, we would like to highlight that: ",
+        "bbox": [
+            174,
+            477,
+            825,
+            588
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. A reduced-training proxy that works well on one search space may not work well on another as highlighted by the difference in Spearman $\\rho$ between econas and econas+. This occurs even though both tasks in this case were CIFAR-10 image classification. 2. Even though EcoNAS-style proxies reduce computation load by a large factor (as seen in the middle plot in Fig. 1, this does not translate fully into actual runtime improvement when run on a nominal desktop $\\mathrm { G P U } ^ { 2 }$ . We therefore plot actual GPU speedup in the third subplot in Fig. 1. For example, notice that the point labeled econas $( r _ { 8 } c _ { 4 } e _ { 2 0 } )$ has the same FLOPS as $\\sim \\frac { 1 } { 1 0 }$ of a full training epoch, but when measured on a GPU, takes time equivalent to 5 full training epochs – a $5 0 \\times$ gap between theoretical and actual speedup. ",
+        "bbox": [
+            212,
+            597,
+            825,
+            727
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1.2 ZERO-COST PROXIES ON NAS-BENCH-201 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            738,
+            529,
+            752
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We now shift our focus towards our zero-cost NAS proxies which rely on gradient computations using a single minibatch of data at initialization. A clear advantage of zero-cost proxies is that they take very little time to compute – the forward/backward pass using a single minibatch of data. We ran the zero-cost proxies on all 15,625 models in NAS-Bench-201 for three image classification datasets and we summarize the results in Table 1. ",
+        "bbox": [
+            174,
+            758,
+            825,
+            828
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The synflow metric performed the best on all three datasets with a Spearman $\\rho$ consistently above 0.73, jacob cov was second best but was also very well-correlated to final accuracy. Next came grad norm and snip with a Spearman $\\rho$ close to 0.6. We add another metric that we simply label with vote that takes a majority vote between the three metrics synflow, jacob cov and snip when ranking two models. This performed better than any single metric with a Spearman $\\rho$ consistently above 0.8. At the cost of just 3 minibatches instead of $\\sim 1 0 0 0$ , this is already performing slightly better than econas+, and much better than econas as shown in Fig. 2a. In Fig. 2 we also plot the rank correlation of validation accuracy (without any reduced training) over the first 10 epochs of training for the three datasets available in NAS-Bench-201. ",
+        "bbox": [
+            174,
+            830,
+            823,
+            886
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/7ba8056c8e8423389e4f34d51829906cbd2390da0a81b89b2ac805c97e58eb6c.jpg",
+        "image_caption": [
+            "Figure 2: Correlation of validation accuracy to final test accuracy during the first 12 epochs of training for three datasets on the NAS-Bench-201 search space. Zero-cost and EcoNAS proxies are also labeled for comparison. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            89,
+            825,
+            280
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/1424ca5984ae6f6743070cb5ecd54ef43ae30e9c5d47f2253513d628950039ad.jpg",
+        "table_caption": [
+            "Table 1: Spearman $\\rho$ of zero-cost proxies on NAS-Bench-201. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td><td>vote</td></tr><tr><td>CIFAR-10</td><td>0.58</td><td>0.58</td><td>0.48</td><td>0.36</td><td>0.74</td><td>0.73</td><td>0.82</td></tr><tr><td>CIFAR-100</td><td>0.64</td><td>0.63</td><td>0.54</td><td>0.39</td><td>0.76</td><td>0.71</td><td>0.83</td></tr><tr><td>ImageNet16-120</td><td>0.58</td><td>0.58</td><td>0.56</td><td>0.33</td><td>0.75</td><td>0.71</td><td>0.82</td></tr></table>",
+        "bbox": [
+            205,
+            367,
+            790,
+            439
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            446,
+            825,
+            516
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Having set a comparison point with EcoNAS and reduced-training proxies, we have shown that zero-cost proxies can match and outperform these conventional methods in a large-scale empirical analysis. However, different NAS search spaces may behave differently, so in the remainder of this section, we test the zero-cost proxies on different search spaces. ",
+        "bbox": [
+            174,
+            518,
+            825,
+            575
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 MODELS IN THE WILD (PYTORCHCV) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            590,
+            482,
+            604
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To study zero-cost proxies in a different setting, we scored the models in the PyTorchCV database (Semery, 2020). PytorchCV contains common state-of-the-art neural networks such as ResNets (He ´ et al., 2016), DenseNets (Huang et al., 2017), MobileNets (Howard et al., 2017) and EfficientNets (Tan & Le, 2019a) – a representative assortment of top-performing models. We evaluated $\\mathord { \\sim } 5 0$ models for CIFAR-10, CIFAR-100 (Krizhevsky, 2009) and SVHN (Netzer et al., 2011), and $\\sim 2 0 0$ models for ImageNet1k (Deng et al., 2009). Fig. 3 shows the resulting correlation for the zero-cost metrics. synflow, snip, fisher and grad norm all perform similarly well on all datasets, with the exception of SVHN where synflow outperforms other metrics by a large margin. However, grasp failed in this setting completely as shown by the low mean Spearman $\\rho$ and high variance as shown in Fig. 3. Curiously, jacob cov also failed in this setting even though it performed well on NASBench-201. This suggests that this metric is better at scoring models from within a search space (similar topology and size), but becomes worse when scoring unrelated models. ",
+        "bbox": [
+            174,
+            613,
+            825,
+            780
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 OTHER SEARCH SPACES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            797,
+            383,
+            811
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We investigate our zero-cost metrics with other NAS benchmarks. Our goal is to empirically find a good metric to speed up NAS algorithms reliably on different tasks and datasets. ",
+        "bbox": [
+            174,
+            820,
+            823,
+            848
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "• NAS-Bench-101: This is the first and largest NAS benchmark available with over $4 2 3 \\mathrm { k }$ CNN models and training statistics on CIFAR-10 (Ying et al., 2019). • NAS-Bench-NLP: Klyuchnikov et al. (2020) investigate the architectures of 14k different recurrent cells in natural language processing (NLP) tasks such as next word prediction. ",
+        "bbox": [
+            217,
+            859,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/138fb731b163de8b76d3bfe072e485a8f3a5bcdb12a4c66055945c882985b9fc.jpg",
+        "image_caption": [
+            "Figure 3: Performance of zero-cost metrics on PyTorchCV models (averaged over 5 seeds). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            217,
+            90,
+            776,
+            280
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• NAS-Bench-ASR: This is our in-house dataset for convolution-based automatic speech recognition models evaluated on the TIMIT dataset (Garofolo et al., 1993). The search space includes linear, convolution, zeroize and skip-connections, forming 8242 models (Mehrotra et al., 2021). ",
+        "bbox": [
+            217,
+            314,
+            825,
+            369
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Compared to NAS-Bench-201, these datasets are either much larger (NAS-Bench-101) or based on a different task (NAS-Bench-NLP/ASR). From Table 2 we would like to highlight that the synflow metric (highlighted in bold) is the only consistent one across all analyzed benchmarks. Additionally, even for the synflow metric, rank correlation is quite a bit lower than that for NAS-Bench-201 ( ${ \\sim } 0 . 3$ vs. ${ \\sim } 0 . 8 )$ . Other than global rank correlation, we posit that ranking of top models from a search space is also critically important for NAS algorithms – this is because we ultimately care about finding those top models. In Section A.4 we perform an analysis of how top models are ranked by zero-cost proxies. Additionally, local rank correlation of top models could be important for NAS algorithms when two good models are compared using their proxy metric value. Tables 9 and 10 show that the only metric that maintains correct ranking among top models consistently across all NAS benchmarks is synflow. In Section 5 we deliberately evaluate 3 benchmarks that exhibit different levels of rank correlation: NAS-Bench-201/101/ASR to see if we can integrate synflow within NAS and achieve consistent gains for all three search spaces. ",
+        "bbox": [
+            173,
+            385,
+            825,
+            565
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/ae2475278b44d41307c379fcb41c9a7248e32f244f744e721ed78fad0f913264.jpg",
+        "table_caption": [
+            "Table 2: Spearman $\\rho$ of zero-cost proxies on other NAS search spaces. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td> grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>NAS-Bench-101</td><td>0.20</td><td>0.16</td><td>0.45</td><td>0.26</td><td>0.37</td><td>0.38</td></tr><tr><td>NAS-Bench-NLP</td><td>-0.21</td><td>-0.19</td><td>0.16</td><td>1</td><td>0.34</td><td>0.56</td></tr><tr><td>NAS-Bench-ASR</td><td>0.07</td><td>0.01</td><td>1</td><td>0.02</td><td>0.40</td><td>-0.37</td></tr></table>",
+        "bbox": [
+            223,
+            607,
+            772,
+            679
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 ZERO-COST NAS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            724,
+            352,
+            739
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Mellor et al. (2020) proposed using jacob cov to score a set of randomly-sampled models and to greedily choose the model with the highest score. This “NAS without training” methodology is very attractive thanks to its simplicity and low computational cost. In this section, we evaluate our metrics in this setting that we simply call “random search” (RAND). We extend this methodology slightly: instead of just training the top model, we keep training models (from best to worst as ranked by the zero-cost metric) until the desired accuracy is achieved. However, this approach can only produce results that are as good as the metric being used – and we have no guarantees (just empirical evidence) that these metrics will perform well on all datasets. Therefore, we also investigate how to integrate zero-cost metrics within existing NAS algorithms such as reinforcement learning (RL) (Zoph & Le, 2017), aging evolution (AE) search (Real et al., 2019) and predictor-based search (Dudziak et al., 2020). More specifically, we investigate enhancing these search algorithms through either (a) zero-cost warmup phase or (b) zero-cost move proposal. ",
+        "bbox": [
+            173,
+            757,
+            825,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/e65525014f0c52cb4d2bcb35e1d6d77c99e829704057ce0c628fdd095cccadbb.jpg",
+        "image_caption": [
+            "Figure 4: Search speedup with the synflow zero-cost proxy on NAS-Bench-201 CIFAR-100. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            93,
+            758,
+            440
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 ZERO-COST WARMUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            472,
+            367,
+            486
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Generally speaking, by warmup we mean using the zero-cost proxies at the beginning of the search process to initialize the search algorithm without training any models or using accuracy. The main parameter in zero-cost warmup is the number of models for which we compute and use the zero-cost metric $( N )$ , and the potential gain comes from the fact that this number can be usually much larger than the number of models we can afford to train $( T \\ll N )$ ). ",
+        "bbox": [
+            174,
+            494,
+            825,
+            564
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Aging Evolution We score $N$ random models with our proxy metric and choose the ones ranked highest as the initial population (pool) in the aging evolution (AE) algorithm (Real et al., 2019). ",
+        "bbox": [
+            173,
+            578,
+            823,
+            607
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement Learning In the REINFORCE algorithm (Zoph & Le (2017)), we sample $N$ random models and use their zero-cost scores to reward the controller, thus biasing it towards selecting architectures which are likely to have higher values of the chosen metrics. During warmup, reward for the controller is calculated by linearly normalizing values returned by the proxy functions to the range $[ - 1 , 1 ]$ (with online adjustment of min and max). ",
+        "bbox": [
+            174,
+            619,
+            825,
+            690
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Binary Predictor We warm up a binary graph convolutional network (GCN) predictor from Dudziak et al. (2020) by training it to predict relative performance of two models by considering their zero-cost scores instead of accuracy. For $N$ warmup points, we use the relative rankings (according to the zero-cost metric) of all pairs of models $( 0 . 5 N ( N - 1 )$ pairs) when performing warmup training for the predictor. As in (Dudziak et al., 2020), models ranked by the predictor after each training round (including the warmup phase) and the top models are evaluated. ",
+        "bbox": [
+            174,
+            704,
+            825,
+            787
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 ZERO-COST MOVE PROPOSAL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            803,
+            423,
+            818
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Whereas warmup tries to leverage global correlation of the proxy metrics to the accuracy of models, move proposal focuses on a local neighborhood at each step. A common parameter for move proposal algorithms is denoted as $R$ and means sample ratio, i.e., how many models can be checked using zero-cost metrics each time we select a model to train. ",
+        "bbox": [
+            174,
+            825,
+            823,
+            881
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Aging Evolution The algorithm is enhanced by performing “guided” mutations. More specifically, each time a model is being mutated (in the baseline algorithm this is done randomly) we consider all possible mutations with edit distance 1 from the current model, score them using the zero-cost proxies and select the best one to add to the pool. ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/4c41c4c13da8362ef0acf7473e758981477421401a26c1539f0aaef070ae261e.jpg",
+        "image_caption": [
+            "Figure 5: Search speedup with the synflow zero-cost proxy on NAS-Bench-ASR TIMIT. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            92,
+            813,
+            234
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            265,
+            823,
+            294
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement Learning In the case of REINFORCE, move proposal is similar to warmup – instead of rewarding a controller $N$ time before the search begins, we interleave $R$ zero-cost rewards for each accuracy reward $( R \\ll N$ ). ",
+        "bbox": [
+            174,
+            308,
+            825,
+            349
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            364,
+            279,
+            378
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For all NAS experiments, we repeat experiments 32 times and we plot the median and shade between the lower/upper quartiles. Our baselines are already heavily tuned and achieve the same or better results than those reported in the original NAS-Bench-101/201 papers. When adding zero-cost warmup or move proposal with synflow, we leave all search hyper-parameters unchanged. ",
+        "bbox": [
+            174,
+            387,
+            823,
+            443
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "NAS-Bench-201 The global/top- $10 \\%$ rank correlations of synflow for this dataset are (0.76/0.42) so we expect this proxy to perform quite well. Indeed, as Figure 4 and Table 7 show, we improve search speed on all four types of searches using zero-cost warmup and move proposal. RAND and RL are both significantly improved, both in terms of sample efficiency and final achieved accuracy. But even more powerful algorithms like AE and BP exhibit $5 . 6 \\times$ and $2 . 3 \\times$ speedups respectively to arrive at $7 3 . 5 \\%$ accuracy. Generally, the more zero-cost warmup, the better the results. This holds true for all algorithms except RL which degrades at $1 5 \\mathrm { k }$ warmup points, suggesting that the controller is overfitting to the synflow metric instead of learning to optimize for accuracy. ",
+        "bbox": [
+            174,
+            457,
+            825,
+            568
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "NAS-Bench-101 This dataset is an order of magnitude larger than NAS-Bench-201 and has lower global/top- $10 \\%$ rank correlations of (0.37/0.14). In many ways, this provides a true test as to whether these lower correlations are still useful with zero-cost warmup and move proposal. Table 3 shows a summary of the results and Figure 7 (in Section A.6) shows the full plots. As the table shows, even with modest correlations, there is a major boost to all searching algorithms thus outperforming the best previously published result by a large margin and setting a new state-of-the-art result on this dataset. However, it is worth noting that the binary predictor exhibits no improvement (but also no degradation). Perhaps this is because it was already very sample-efficient and synflow warmup couldn’t help further due to its relatively poor correlation on this dataset. ",
+        "bbox": [
+            174,
+            580,
+            825,
+            707
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/511178df4c3a3d6cac96a3f2998e1de4fea2f613a53956f743653532268cd3cf.jpg",
+        "table_caption": [
+            "Table 3: Comparison to prior work on NAS-Bench-101 dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Wen et al. (2019)</td><td rowspan=\"2\">Wei et al. (2020)</td><td rowspan=\"2\">Dudziak et al. (2020)</td><td colspan=\"3\">Ours</td></tr><tr><td>RL+M(100)</td><td>AE+W (15k)</td><td>RAND+W (3k)</td></tr><tr><td># Trained Models</td><td>256</td><td>150</td><td>140</td><td>51</td><td>50</td><td>34</td></tr><tr><td>Test Accuracy [%]</td><td>94.17</td><td>94.14</td><td>94.22</td><td>94.22</td><td>94.22</td><td>94.22</td></tr></table>",
+        "bbox": [
+            187,
+            741,
+            810,
+            814
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "NAS-Bench-ASR We repeat our evaluation on NAS-Bench-ASR with global/top- $10 \\%$ correlations (0.40/0.40). Even though this is a different task (speech recognition), synflow warmup and move proposal both yield large improvements in search speeds compared to all baselines in Figure 5 and Table 8. For example, to achieve a phoneme error rate (PER) of $2 1 . 3 \\%$ , baseline RAND and RL required $> 1 0 0 0$ trained models, and AE required 138 trained models; however, this is reduced to 68, 173 and 87 trained models with 2000 models of zero-cost warmup. ",
+        "bbox": [
+            173,
+            839,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            93,
+            310,
+            109
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section we investigate why zero-cost NAS is effective in improving the sample efficiency of NAS algorithms by looking more closely at how top models are selected by the synflow proxy. ",
+        "bbox": [
+            176,
+            119,
+            825,
+            147
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Warmup Table 4 shows the number of top- $5 \\%$ most-accurate models ranked within the top 64 models by the synflow metric. If we compare random warmup versus zero-cost warmup with synflow, random warmup will only return $5 \\%$ or $\\sim 3$ models out of 64 that are within the top $5 \\%$ of models whereas synflow warmup returns a higher number of top- $5 \\%$ models as listed in Table 4. This is key to the improvements observed when adding zero-cost warmup to algorithms like random search or AE. For example, with AE, the numbers in Table 4 are indicative of the models that may end up in the initial AE pool. By initializing the AE pool with many good models, it becomes more likely that a random mutation will lead to an even better model, thus allowing the search to find a top model more quickly. Note that synflow is able to rank many good models in its top 64 models even when global/local correlation is low (as it is the case for NAS-Bench-ASR). ",
+        "bbox": [
+            174,
+            157,
+            825,
+            296
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/2dd66129f8a8370483c001629ff3c420a7ba262d396eb6c5633a1537ac3225e6.jpg",
+        "table_caption": [
+            "Table 4: Number of top- $5 \\%$ most-accurate models within the top 64 models returned by synflow. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">NAS-Bench-201</td><td rowspan=\"2\">NAS-Bench-101</td><td rowspan=\"2\">NAS-Bench-ASR</td></tr><tr><td>CIFAR-10</td><td>CIFAR-100</td><td>ImageNet16-120</td></tr><tr><td>44</td><td>54</td><td>56</td><td>12</td><td>16</td></tr></table>",
+        "bbox": [
+            228,
+            327,
+            761,
+            387
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Move Proposal For a search algorithm like AE, search moves consist of random mutations (with edit distance 1 for our experiments) for a model from the AE pool. Zero-cost move proposal enhances this by trying out all possible mutations and selecting the best one according to synflow. To investigate how this improves search efficiency, we took 1000 random points and explored their local neighbourhood cluster of possible mutations. Table 5 shows the probability that the synflow proxy correctly identifies the top model. Indeed, synflow improves the chance of selecting the best mutation from $\\sim 4 \\%$ to $> 3 0 \\%$ for NAS-Bench-201 and $12 \\%$ for NAS-Bench-101. Even for NAS-Bench-ASR a random mutation has a $7 . 7 \\%$ chance $( = 1 / 1 3 )$ to select the best mutation, but this increases to $10 \\%$ with the synflow proxy thus speeding up convergence to top models. ",
+        "bbox": [
+            173,
+            406,
+            825,
+            534
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/92e301d83e0d57345bb0d0e62e8d3589913b1cbdb4be29f102c40654e3405a62.jpg",
+        "table_caption": [
+            "Table 5: For 1000 clusters of models with edit distance 1, we empirically measure the probability that the synflow proxy will select the most accurate model from each cluster. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"3\">NAS-Bench-201</td><td rowspan=\"2\">NAS-Bench-101</td><td rowspan=\"2\">NAS-Bench-ASR</td></tr><tr><td>CIFAR-10</td><td>CIFAR-100</td><td>ImageNet16-120</td></tr><tr><td>Top Model Match</td><td>32%</td><td>35%</td><td>33%</td><td>12%</td><td>10%</td></tr><tr><td>Average Cluster Size</td><td>25</td><td>25</td><td>25</td><td>26</td><td>13</td></tr></table>",
+        "bbox": [
+            173,
+            577,
+            826,
+            650
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            675,
+            318,
+            691
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduced six zero-cost proxies, mainly based on recent pruning-at-initialization work, that are used to rank neural network models in NAS. First, we compared to conventional proxies (EcoNAS) that perform reduced-computation training and we found that zero-cost proxies such as synflow can outperform EcoNAS in maintaining rank consistency. Next, we verified our zerocost metrics on four additional datasets of varying sizes and tasks and found that indeed out of the six initially-considered zero-cost metrics, only synflow was robust across all datasets for both global and top- $10 \\%$ rank correlation. Finally, we proposed two ways to integrate synflow within NAS algorithms: zero-cost warmup and zero-cost move proposal. Both methods demonstrated significant speedups across four search algorithms and three NAS benchmarks, setting new state-of-the-art results for both NAS-Bench-101 and NAS-Bench-201 datasets. Our strong and consistent empirical results suggest that the synflow metric, when combined with warmup and move proposal can be an effective and reliable methodology for speeding up different NAS algorithms. We hope that our work lays a foundation for further zero-cost techniques that expose favourable model properties with little computation thus making NAS more readily accessible without exorbitant computing resources. The most immediate open question for future investigation is why the synflow proxy works well – analytical insights will enable further research in zero-cost NAS proxies. ",
+        "bbox": [
+            173,
+            702,
+            825,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            93,
+            285,
+            108
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gabriel Bender, Pieter-Jan Kindermans, Barret Zoph, Vijay Vasudevan, and Quoc V. Le. Understanding and simplifying one-shot architecture search. In International Conference on Machine Learning (ICML), 2018. ",
+        "bbox": [
+            173,
+            117,
+            823,
+            160
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Han Cai, Tianyao Chen, Weinan Zhang, Yong Yu, and Jun Wang. Efficient architecture search by network transformation. In AAAI Conference on Artificial Intelligence (AAAI), 2018. ",
+        "bbox": [
+            173,
+            169,
+            823,
+            199
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Wayne W. Daniel. Applied Nonparametric Statistics. Boston: PWS-Kent, 1990. ",
+        "bbox": [
+            173,
+            207,
+            699,
+            223
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. ImageNet: A Large-Scale Hierarchical Image Database. In CVPR, 2009. ",
+        "bbox": [
+            173,
+            232,
+            816,
+            261
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xuanyi Dong and Yi Yang. NAS-Bench-201: Extending the Scope of Reproducible Neural Architecture Search. In International Conference on Learning Representations (ICLR), 2020. ",
+        "bbox": [
+            173,
+            271,
+            821,
+            300
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Łukasz Dudziak, Thomas Chau, Mohamed S. Abdelfattah, Royson Lee, Hyeji Kim, and Nicholas D. Lane. BRP-NAS: Prediction-based NAS using GCNs. In Neural Information Processing Systems (NeurIPS), 2020. ",
+        "bbox": [
+            173,
+            309,
+            825,
+            353
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mikhail Figurnov, Aizhan Ibraimova, Dmitry P Vetrov, and Pushmeet Kohli. Perforatedcnns: Acceleration through elimination of redundant convolutions. In Neural Information Processing Systems (NeurIPS), 2016. ",
+        "bbox": [
+            173,
+            362,
+            825,
+            405
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jonathan Frankle and Michael Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks. In International Conference on Learning Representations (ICLR), 2019. ",
+        "bbox": [
+            169,
+            415,
+            823,
+            444
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "John S. Garofolo, Lori F. Lamel, William M. Fisher, Jonathan G. Fiscus, David S. Pallett, Nancy L. Dahlgren, and Victor Zue. Timit acoustic phonetic continuous speech corpus. Linguistic Data Consortium, 1993. URL https://catalog.ldc.upenn.edu/LDC93S1. ",
+        "bbox": [
+            179,
+            453,
+            823,
+            497
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Song Han, Jeff Pool, John Tran, and William Dally. Learning both weights and connections for efficient neural network. In Neural Information Processing Systems (NeurIPS), 2015. ",
+        "bbox": [
+            173,
+            506,
+            823,
+            535
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Babak Hassibi and David G. Stork. Second order derivatives for network pruning: Optimal brain surgeon. In Advances in Neural Information Processing Systems 5, 1993. ",
+        "bbox": [
+            174,
+            544,
+            821,
+            574
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Computer Vision and Pattern Recognition (CVPR), June 2016. ",
+        "bbox": [
+            173,
+            583,
+            820,
+            612
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Andrew G. Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam. MobileNets: Efficient Convolutional Neural Networks for Mobile Vision Applications. arXiv preprint arXiv:1704.04861, 2017. ",
+        "bbox": [
+            174,
+            621,
+            825,
+            665
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Gao Huang, Zhuang Liu, Laurens van der Maaten, and Kilian Q. Weinberger. Densely connected convolutional networks. In Computer Vision and Pattern Recognition (CVPR), July 2017. ",
+        "bbox": [
+            174,
+            674,
+            825,
+            704
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Nikita Klyuchnikov, Ilya Trofimov, Ekaterina Artemova, Mikhail Salnikov, Maxim Fedorov, and Evgeny Burnaev. Nas-bench-nlp: Neural architecture search benchmark for natural language processing. arXiv preprint arXiv:2006.07116, 2020. ",
+        "bbox": [
+            176,
+            713,
+            825,
+            756
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Krizhevsky. Learning Multiple Layers of Features from Tiny Images, 2009. URL https: //www.cs.toronto.edu/˜kriz/cifar.html. ",
+        "bbox": [
+            171,
+            765,
+            820,
+            795
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yann LeCun, John S. Denker, and Sara A. Solla. Optimal brain damage. In Neural Information Processing Systems (NeurIPS), 1990. ",
+        "bbox": [
+            171,
+            804,
+            823,
+            833
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Namhoon Lee, Thalaiyasingam Ajanthan, and Philip HS Torr. Snip: Single-shot network pruning based on connection sensitivity. In International Conference on Learning Representations (ICLR), 2019. ",
+        "bbox": [
+            173,
+            842,
+            823,
+            885
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hanxiao Liu, Karen Simonyan, and Yiming Yang. DARTS: Differentiable architecture search. In International Conference on Learning Representations (ICLR), 2019. ",
+        "bbox": [
+            173,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Abhinav Mehrotra, Alberto Gil Ramos, Sourav Bhattacharya, Łukasz Dudziak, Ravichander Vipperla, Thomas Chau, Mohamed S. Abdelfattah, Samin Ishtiaq, and Nicholas D. Lane. NASBench-ASR: Reproducible Neural Architecture Search for Speech Recognition. In International Conference on Learning Representations (ICLR), 2021. ",
+        "bbox": [
+            174,
+            94,
+            825,
+            151
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jieru Mei, Yingwei Li, Xiaochen Lian, Xiaojie Jin, Linjie Yang, Alan Yuille, and Jianchao Yang. AtomNAS: Fine-grained end-to-end neural architecture search. In International Conference on Learning Representations (ICLR), 2020. ",
+        "bbox": [
+            176,
+            160,
+            821,
+            202
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Joseph Mellor, Jack Turner, Amos Storkey, and Elliot J. Crowley. Neural architecture search without training. arXiv preprint arXiv:2006.04647, 2020. ",
+        "bbox": [
+            171,
+            210,
+            823,
+            241
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Pavlo Molchanov, Stephen Tyree, Tero Karras, Timo Aila, and Jan Kautz. Pruning convolutional neural networks for resource efficient inference. In International Conference on Learning Representations (ICLR), 2017. ",
+        "bbox": [
+            173,
+            248,
+            823,
+            291
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y. Ng. Reading Digits in Natural Images with Unsupervised Feature Learning. In NeurIPS Workshop on Deep Learning and Unsupervised Feature Learning, 2011. ",
+        "bbox": [
+            174,
+            300,
+            825,
+            343
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Barak A. Pearlmutter. Fast Exact Multiplication by the Hessian. Neural Computation, 1993. ",
+        "bbox": [
+            169,
+            352,
+            779,
+            367
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Hieu Pham, Melody Y. Guan, Barret Zoph, Quoc V. Le, and Jeff Dean. Efficient neural architecture search via parameter sharing, 2018. ",
+        "bbox": [
+            173,
+            376,
+            823,
+            405
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Esteban Real, Alok Aggarwal, Yanping Huang, and Quoc V. Le. Regularized Evolution for Image Classifier Architecture Search. In AAAI Conference on Artificial Intelligence (AAAI), 2019. ",
+        "bbox": [
+            173,
+            412,
+            823,
+            443
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Oleg Semery. PyTorchCV Convolutional neural networks for computer vision, August 2020. URL ´ https://github.com/osmr/imgclsmob. ",
+        "bbox": [
+            173,
+            452,
+            823,
+            481
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Mingxing Tan and Quoc Le. EfficientNet: Rethinking model scaling for convolutional neural networks. In International Conference on Machine Learning (ICML), 2019a. ",
+        "bbox": [
+            173,
+            489,
+            823,
+            518
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Mingxing Tan and Quoc V. Le. Mixconv: Mixed depthwise convolutional kernels, 2019b. ",
+        "bbox": [
+            169,
+            526,
+            763,
+            542
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Hidenori Tanaka, Daniel Kunin, Daniel L. K. Yamins, and Surya Ganguli. Pruning neural networks without any data by iteratively conserving synaptic flow. arXiv preprint arXiv:2006.05467, 2020. ",
+        "bbox": [
+            173,
+            550,
+            825,
+            580
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Lucas Theis, Iryna Korshunova, Alykhan Tejani, and Ferenc Huszar. Faster gaze prediction with ´ dense networks and fisher pruning. arXiv:1801.05787, 2018. ",
+        "bbox": [
+            169,
+            588,
+            823,
+            617
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jack Turner, Elliot J. Crowley, Michael O’Boyle, Amos Storkey, and Gavin Gray. Blockswap: Fisher-guided block substitution for network compression on a budget. In International Conference on Learning Representations (ICLR), 2020. ",
+        "bbox": [
+            176,
+            626,
+            821,
+            670
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chaoqi Wang, Guodong Zhang, and Roger Grosse. Picking winning tickets before training by preserving gradient flow. In International Conference on Learning Representations (ICLR), 2020. ",
+        "bbox": [
+            173,
+            678,
+            823,
+            708
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chen Wei, Chuang Niu, Yiping Tang, and Jimin Liang. NPENAS: Neural predictor guided evolution for neural architecture search. arXiv 2003.12857, 03 2020. ",
+        "bbox": [
+            171,
+            715,
+            823,
+            744
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Wei Wen, Hanxiao Liu, Hai Li, Yiran Chen, Gabriel Bender, and Pieter-Jan Kindermans. Neural predictor for neural architecture search. arXiv:1912.00848, 2019. ",
+        "bbox": [
+            169,
+            752,
+            823,
+            782
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chris Ying, Aaron Klein, Eric Christiansen, Esteban Real, Kevin Murphy, and Frank Hutter. NASbench-101: Towards reproducible neural architecture search. In International Conference on Machine Learning (ICML), 2019. ",
+        "bbox": [
+            176,
+            791,
+            825,
+            833
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Dongzhan Zhou, Xinchi Zhou, Wenwei Zhang, Chen Change Loy, Shuai Yi, Xuesen Zhang, and Wanli Ouyang. Econas: Finding proxies for economical neural architecture search. In Conference on Computer Vision and Pattern Recognition (CVPR), June 2020. ",
+        "bbox": [
+            173,
+            842,
+            823,
+            886
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Barret Zoph and Quoc V. Le. Neural architecture search with reinforcement learning. In International Conference on Learning Representations (ICLR), 2017. ",
+        "bbox": [
+            173,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            93,
+            299,
+            108
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Because this paper is empirically-driven, there are many more results than what we presented in the main text of the paper. In the appendix we list many important results that support our main arguments and hypotheses in the main text of this paper. ",
+        "bbox": [
+            174,
+            119,
+            825,
+            161
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            172,
+            395,
+            188
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Table 6 we list the hyper-parameters used in training the EcoNAS proxies to produce Figure 1. The only difference to the standard NAS-Bench-201 training pipeline (Dong & Yang, 2020) is our use of fewer epochs for the learning rate annealing schedule – we anneal the learning rate to zero over 40 epochs instead of 200. This is a common technique used in speeding up convergence for training proxies Zhou et al. (2020). We acknowledge that slightly better correlations could have been achieved for econas and econas $^ +$ proxies in Figure 1 if the learning rate was annealed to zero over fewer epochs (20 and 15 epochs respectively). However, we do not anticipate the results to change significantly. ",
+        "bbox": [
+            173,
+            194,
+            825,
+            306
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/65dca82edc3d320a47299656e8a5f2e77a84e8538a9290cc729517481fb3aa01.jpg",
+        "table_caption": [
+            "Table 6: EcoNAS training hyper-parameters for NAS-Bench-201. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>optimizer Nesterov momentum weight decay</td><td>SGD √ 0.9 0.0005 (p=0.5)</td><td>initial LR final LR LR schedule epochs batch size</td><td>0.1 0 cosine 40 256</td></tr></table>",
+        "bbox": [
+            321,
+            337,
+            673,
+            429
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "One additional comment regarding Figure 1 in the main paper. While we run the training ourselves for all EcoNAS variants in the plot, we take the data for the line labeled baseline directly from the NAS-Bench-201 dataset. We are not sure why the line is not smooth like the lines for the EcoNAS variants that we trained but assume that this is an artifact of averaging over multiple seeds in the NAS-Bench-201 dataset. In any case, we do not anticipate that this would change any conclusions or observations that we draw from this plot. ",
+        "bbox": [
+            173,
+            441,
+            826,
+            525
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Finally, we would like to note some details about our NAS experiments in Section 5. NAS datasets provide multiple seeds of results for each model, so whenever we “train” a model, we query a random seed from the database to mimic a real NAS pipeline without caching. We refer the reader to (Dudziak et al., 2020), specifically Section S3.2 for more details on this. ",
+        "bbox": [
+            174,
+            527,
+            825,
+            584
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 GPU RUNTIME FOR ECONAS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            595,
+            424,
+            609
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 shows the speedup of different EcoNAS proxies compared to baseline training. Even though $r _ { 8 } c _ { 4 }$ has $6 4 \\times$ less computation compared to $r _ { 3 2 } c _ { 1 6 }$ , it achieves a maximum of $4 \\times$ real speedup even when the batch size is increased. ",
+        "bbox": [
+            174,
+            617,
+            825,
+            659
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/4ff85890acab96fe33b81848c95232390efa880f144e2225ec6a0ababe18d8d2.jpg",
+        "image_caption": [
+            "Figure 6: Higher batch sizes when training econas proxies have diminishing returns in terms of measured speedup. This measurement is done for 10 randomly-sampled NAS-Bench-201 models on the CIFAR-10 dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            316,
+            685,
+            656,
+            859
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 TABULATED RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            94,
+            370,
+            109
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This subsection contains tabulated results from Figures 4 and 5 to facilitate comparisons with future work. Tables 7 and 8 highlight important data points about the NAS searches we conducted with NAS-Bench-201 and NAS-Bench-ASR respectively. We highlight results in two ways: First, we show the accuracy of the best model found after 50 trained models. Second, we indicate the number of trained models needed for each search method to reach a specific accuracy $7 3 . 5 \\%$ CIFAR-10 classification accuracy for NAS-Bench-201 and $2 1 . 3 \\%$ TIMIT PER.) We colour the best results (red) and the second best (blue) results in each table. ",
+        "bbox": [
+            173,
+            116,
+            825,
+            214
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/82c2352de3079451407538156c333239fa0510a890641cc24e1406995c531462.jpg",
+        "table_caption": [
+            "Table 7: Zero-cost NAS comparison with baseline algorithms on NAS-Bench-201 CIFAR-100. We show accuracy after 50 trained models and the number of models to reach $7 3 . 5 \\%$ accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Baseline</td><td colspan=\"3\">Warmup</td><td colspan=\"2\">Move</td></tr><tr><td>1000 (BP=256)</td><td>3000 (BP=512)</td><td>15k</td><td>10</td><td>100</td></tr><tr><td>RAND</td><td>71.31/1000+</td><td>72.98 /1000+</td><td>73.18 /1000+</td><td>73.75 /8</td><td></td><td>1</td></tr><tr><td>RL</td><td>71.08 /1000+</td><td>72.76 /145</td><td>73.14/84</td><td>73.21/107</td><td>71.16/289</td><td>73.34 / 70</td></tr><tr><td>AE</td><td>71.53 /139</td><td>72.91 / 115</td><td>73.40 /71</td><td>73.63 / 25</td><td>71.3 /77</td><td></td></tr><tr><td>BP</td><td>72.74 /93</td><td>73.32 / 66</td><td>73.85 / 40</td><td>1</td><td></td><td>1</td></tr></table>",
+        "bbox": [
+            178,
+            260,
+            818,
+            351
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/2af0e83a0f01b30e0c4459388a2d9a5540cc3ddf49d2706f4722523a1bf57755.jpg",
+        "table_caption": [
+            "Table 8: Zero-cost NAS comparison with baseline algorithms on NAS-Bench-ASR. We show PER after 50 trained models and the number of models to reach PER $= 2 1 . 3 \\%$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Baseline</td><td colspan=\"2\">Warmup</td><td colspan=\"2\">Move</td></tr><tr><td>500</td><td>2000</td><td>10</td><td>100</td></tr><tr><td>RAND</td><td>21.65 /1000+</td><td>21.38 /1000+</td><td>21.35 / 68</td><td></td><td></td></tr><tr><td>RL</td><td>21.66 /1000+</td><td>21.48 /1000+</td><td>21.45 / 173</td><td>21.62 / 169</td><td>21.43 / 161</td></tr><tr><td>AE</td><td>21.62 / 138</td><td>21.40 / 115</td><td>21.36 / 87</td><td></td><td>21.74 / 112</td></tr></table>",
+        "bbox": [
+            235,
+            409,
+            761,
+            488
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 ANALYSIS OF THE TOP $10 \\%$ OF MODELS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            512,
+            501,
+            526
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the main text we pointed to the fact that only synflow achieves consistent rank correlation for the top- $10 \\%$ of models across different datasets. Here, in Table 9 we provide the full results. Additionally, we hypothesized that a successful metric will rank many of the most-accurate models in its top models. In Table 10 we enumerate the percentage of top- $10 \\%$ most accurate models ranked as top- $10 \\%$ by each proxy metric. Again, synflow is the only consistent metric for all datasets, and performs best on average. ",
+        "bbox": [
+            174,
+            534,
+            825,
+            618
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/5f0759be79bd3548513bbf7f89552d73741a57d96f3144294333a454a8d7572d.jpg",
+        "table_caption": [
+            "Table 9: Spearman $\\rho$ of zero-cost proxies for the top $10 \\%$ of points on all NAS search spaces. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>NB2-CIFAR-10</td><td>-0.38</td><td>-0.38</td><td>-0.37</td><td>-0.38</td><td>0.18</td><td>0.17</td></tr><tr><td>NB2-CIFAR-100</td><td>-0.09</td><td>-0.09</td><td>-0.11</td><td>-0.16</td><td>0.42</td><td>0.08</td></tr><tr><td>NB2-ImageNet16-120</td><td>0.13</td><td>0.13</td><td>0.10</td><td>0.02</td><td>0.55</td><td>0.05</td></tr><tr><td>NAS-Bench-101</td><td>0.05</td><td>-0.01</td><td>-0.01</td><td>0.07</td><td>0.14</td><td>0.08</td></tr><tr><td>NAS-Bench-NLP</td><td>-0.03</td><td>-0.02</td><td>0.04</td><td>1</td><td>0.10</td><td>0.04</td></tr><tr><td>NAS-Bench-ASR</td><td>0.25</td><td>0.13</td><td>1</td><td>-0.07</td><td>0.40</td><td>-0.03</td></tr></table>",
+        "bbox": [
+            209,
+            650,
+            787,
+            763
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.5 ANALYSIS OF WARMUP AND MOVE PROPOSAL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            787,
+            542,
+            801
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This section provides more results relevant to our discussion in Section 6. Table 11 shows the number of top- $5 \\%$ models ranked in the top 64 models by each metric. This is an extension to Table 4 in the main text that only shows the results for synflow. As shown in the table, synflow is the most powerful metric that we tried. ",
+        "bbox": [
+            174,
+            809,
+            825,
+            864
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 12 shows the rank correlation coefficient of models within 1000 randomly-sampled local clusters of models. This result highlights that both grad norm and jacob cov work well in distinguishing between very similar models. However, synflow still consistently the best metric in this analysis. Furthermore, we measure the percentage of times that a metric correctly predicts the top model within a local cluster of models in Table 13 This is an extension to Table 5 in the main text. The results are averaged over 1000 randomly-sampled local clusters. Again, synflow has the highest probability of selecting the top model compared to other zero-cost metrics. ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/a09889f1fc65864b49ba66f1b060479af7bee4e2b874498a234040c885ff47ef.jpg",
+        "table_caption": [
+            "Table 10: Percentage of top- $10 \\%$ most-accurate models within the top- $10 \\%$ of models ranked by each zero-cost metric. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>NB2-CIFAR-10</td><td>30%</td><td>31%</td><td>30%</td><td>5%</td><td>46%</td><td>25%</td></tr><tr><td>NB2-CIFAR-100</td><td>35%</td><td>36%</td><td>34%</td><td>4%</td><td>50%</td><td>24%</td></tr><tr><td>NB2-ImageNet16-120</td><td>31%</td><td>31%</td><td>32%</td><td>5%</td><td>44%</td><td>30%</td></tr><tr><td>NAS-Bench-101</td><td>2%</td><td>3%</td><td>26%</td><td>3%</td><td>23%</td><td>2%</td></tr><tr><td>NAS-Bench-NLP</td><td>10%</td><td>10%</td><td>4%</td><td>1</td><td>22%</td><td>38%</td></tr><tr><td>NAS-Bench-ASR</td><td>0%</td><td>0%</td><td>1</td><td>0%</td><td>15%</td><td>46%</td></tr></table>",
+        "bbox": [
+            210,
+            125,
+            782,
+            238
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            246,
+            825,
+            287
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/0511542460e2646da05eb5708c695fc8382798a032aef1cbe8f1d997f97a2421.jpg",
+        "table_caption": [
+            "Table 11: Number of top- $5 \\%$ most-accurate models within the top-64 models returned by each metric. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td>jacob_cov</td></tr><tr><td>NB2-CIFAR-10</td><td>0</td><td>0</td><td>0</td><td>0</td><td>44</td><td>15</td></tr><tr><td>NB2-CIFAR-100</td><td>4</td><td>4</td><td>4</td><td>0</td><td>54</td><td>16</td></tr><tr><td>NB2-ImageNet16-120</td><td>13</td><td>13</td><td>14</td><td>0</td><td>56</td><td>15</td></tr><tr><td>NAS-Bench-101</td><td>0</td><td>0</td><td>6</td><td>0</td><td>12</td><td>0</td></tr><tr><td>NAS-Bench-ASR</td><td>1</td><td>0</td><td>1</td><td>1</td><td>16</td><td>13</td></tr></table>",
+        "bbox": [
+            210,
+            334,
+            782,
+            433
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/9d40ddde224ea9a9437231f85ff3f60091220f9091b007dd92a148779950eea8.jpg",
+        "table_caption": [
+            "Table 12: Rank correlation coefficient for the local neighbourhoods (edit distance $= 1$ ) of 1000 clusters in each search space. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>NB2-CIFAR-10</td><td>0.51</td><td>0.51</td><td>0.37</td><td>0.37</td><td>0.66</td><td>0.62</td></tr><tr><td>NB2-CIFAR-100</td><td>0.58</td><td>0.58</td><td>0.44</td><td>0.41</td><td>0.69</td><td>0.61</td></tr><tr><td>NB2-ImageNet16-120</td><td>0.56</td><td>0.57</td><td>0.5</td><td>0.4</td><td>0.67</td><td>0.61</td></tr><tr><td>NAS-Bench-101</td><td>0.23</td><td>0.21</td><td>0.44</td><td>0.27</td><td>0.36</td><td>0.37</td></tr><tr><td>NAS-Bench-ASR</td><td>0.59</td><td>0.4</td><td>1</td><td>0.56</td><td>0.38</td><td>0.28</td></tr></table>",
+        "bbox": [
+            210,
+            491,
+            782,
+            590
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/27330f26d11a0e417f39b4fd9b863c651aaa3daa0ed04d75633e82fe32f17363.jpg",
+        "table_caption": [
+            "Table 13: For 1000 clusters of points with edit distance $= 1$ . We count the number of times wherein the top model returned by a zero-cost metric matches the top model according to validation accuracy. This represents the probability that zero-cost move proposal will perform the best possible mutation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td> grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>NB2-CIFAR-10</td><td>14.8%</td><td>14.8%</td><td>12.7%</td><td>5.7%</td><td>32.2%</td><td>14.5%</td></tr><tr><td>NB2-CIFAR-100</td><td>19.1%</td><td>18.5%</td><td>14.2%</td><td>6.0%</td><td>35.4%</td><td>13.8%</td></tr><tr><td>NB2-ImageNet16-120</td><td>17.5%</td><td>18.5%</td><td>15.7%</td><td>5.5%</td><td>33.4%</td><td>16.7%</td></tr><tr><td>NAS-Bench-101</td><td>0.4%</td><td>0.9%</td><td>7.4%</td><td>0.5%</td><td>12.3%</td><td>0.5%</td></tr><tr><td>NAS-Bench-ASR</td><td>11.0%</td><td>9.8%</td><td>1</td><td>10.3%</td><td>10.3%</td><td>10.5%</td></tr></table>",
+        "bbox": [
+            197,
+            661,
+            795,
+            760
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.6 NAS-BENCH-101 SEARCH PLOTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            94,
+            455,
+            108
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 shows the NAS search curves for all considered algorithms on NAS-Bench-101 dataset.   \nImportant points from this plot are summarized in Table 3 in the main text. ",
+        "bbox": [
+            169,
+            116,
+            823,
+            145
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/6921e93248b0f0a2cf8e20a79aadc988ab5cb39b97d407f1e89b2d694a5c6471.jpg",
+        "image_caption": [
+            "Figure 7: Search speedup with the synflow zero-cost proxy on NAS-Bench-101 CIFAR-10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            160,
+            816,
+            568
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.7 SENSITIVITY ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            94,
+            385,
+            109
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We performed some sensitivity analysis to investigate how the zero-cost metrics perform on all points within NAS-Bench-201 with different initialization seed, initialization method and minibatch size. We comment on each table in its caption; however, to summarize, all metrics seem to be relatively unaffected when initialization and minibatch size are varied. The one exception can be seen in Table 15 where fisher benefits when biases are initialized with zeroes. ",
+        "bbox": [
+            173,
+            116,
+            825,
+            185
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/1c379e46b79d38edb17a2824c15eac2f9d7d400cca25f5a3a7b3e862e1b7d7cf.jpg",
+        "table_caption": [
+            "Table 14: All metrics remain fairly constant when varying the initialization seed – the variations are only observed at the third significant digit. Dataload is random with 128 samples and initialization is done with default PyTorch initialization scheme. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>seed</td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>1</td><td>0.578</td><td>0.581</td><td>0.487</td><td>0.361</td><td>0.737</td><td>0.735</td></tr><tr><td>2</td><td>0.580</td><td>0.583</td><td>0.488</td><td>0.354</td><td>0.740</td><td>0.728</td></tr><tr><td>3</td><td>0.582</td><td>0.584</td><td>0.486</td><td>0.358</td><td>0.738</td><td>0.726</td></tr><tr><td>4</td><td>0.581</td><td>0.584</td><td>0.491</td><td>0.356</td><td>0.738</td><td>0.73</td></tr><tr><td>5</td><td>0.581</td><td>0.583</td><td>0.486</td><td>0.356</td><td>0.738</td><td>0.727</td></tr><tr><td>Average</td><td>0.580</td><td>0.583</td><td>0.488</td><td>0.357</td><td>0.738</td><td>0.729</td></tr></table>",
+        "bbox": [
+            253,
+            243,
+            745,
+            363
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/816861563723e6613cc86a0bbd1e4e7c41ed212653a4ba228ed2fe461ea42624.jpg",
+        "table_caption": [
+            "Table 15: fisher becomes noticeably better when biases are initialized to zero; otherwise, metrics seem to perform independently of initialization method. Results averaged over 3 seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Weights init</td><td>Bias init</td><td> grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>default</td><td>default</td><td>0.580</td><td>0.583</td><td>0.488</td><td>0.357</td><td>0.738</td><td>0.729</td></tr><tr><td>kaiming</td><td>default</td><td>0.548</td><td>0.558</td><td>0.364</td><td>0.332</td><td>0.731</td><td>0.723</td></tr><tr><td>xavier</td><td>default</td><td>0.543</td><td>0.568</td><td>0.424</td><td>0.345</td><td>0.736</td><td>0.729</td></tr><tr><td>default</td><td>zero</td><td>0.581</td><td>0.583</td><td>0.488</td><td>0.509</td><td>0.738</td><td>0.729</td></tr><tr><td>kaiming</td><td>zero</td><td>0.542</td><td>0.551</td><td>0.370</td><td>0.479</td><td>0.730</td><td>0.723</td></tr><tr><td>xavier</td><td>zero</td><td>0.540</td><td>0.566</td><td>0.412</td><td>0.495</td><td>0.735</td><td>0.730</td></tr></table>",
+        "bbox": [
+            204,
+            420,
+            794,
+            534
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/2818ba1aeae8d5b87747bbefd538b0af540d0dc203f242d6a6f33beb9e4e4039.jpg",
+        "table_caption": [
+            "Table 16: Surprisingly, grasp becomes worse with more (random) data, while grad norm and snip degrade very slightly. Other metrics seem to perform independently of the number of samples in the minibatch. Initialization is done with default PyTorch initialization scheme. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Number of Samples</td><td>grad_norm</td><td>snip</td><td>grasp</td><td>fisher</td><td>synflow</td><td> jacob_cov</td></tr><tr><td>32</td><td>0.595</td><td>0.596</td><td>0.511</td><td>0.362</td><td>0.737</td><td>0.732</td></tr><tr><td>64</td><td>0.589</td><td>0.59</td><td>0.509</td><td>0.361</td><td>0.737</td><td>0.735</td></tr><tr><td>128</td><td>0.578</td><td>0.581</td><td>0.487</td><td>0.361</td><td>0.737</td><td>0.735</td></tr><tr><td>256</td><td>0.564</td><td>0.569</td><td>0.447</td><td>0.361</td><td>0.737</td><td>0.731</td></tr><tr><td>512</td><td>0.547</td><td>0.552</td><td>0.381</td><td>0.361</td><td>0.737</td><td>0.724</td></tr></table>",
+        "bbox": [
+            215,
+            604,
+            781,
+            704
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.8 RESULTS FOR ALL ZERO-COST METRICS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            729,
+            504,
+            744
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Here we provide some NAS search results using all considered metrics for both RAND and AE searches on NAS-Bench-101/201 datasets. Our experiments point to synflow as the only effective zero-cost metric across different datasets; however, we provide the plots below for the reader to inspect how poorer metrics perform in NAS. ",
+        "bbox": [
+            173,
+            752,
+            825,
+            809
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/0e75bfa7f50cd7cd06f49cf98450eea07a6395bd2f9913745380c4a4bccdd3c2.jpg",
+        "image_caption": [
+            "(d) NAS-Bench-101 CIFAR-10 ",
+            "Figure 8: Evaluation of all zero-cost proxies on different datasets and search algorithms: random search (RAND) and aging evolution (AE). RAND benefits greatly from a strong metric (such as synflow) but may deteriorate with a weaker metric as shown in the plot. However, AE benefits when a strong metric is used and is resilient to weaker metrics as well – it is able to recover and achieve the top accuracy in most cases. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            212,
+            54,
+            784,
+            845
+        ],
+        "page_idx": 16
+    }
+]

parse/train/A5VV3UyIQz/A5VV3UyIQz.md

@@ -0,0 +1,391 @@
+# EXPLAINABLE DEEP ONE-CLASS CLASSIFICATION
+
+Philipp Liznerski1∗ Lukas Ruff2∗ Robert A. Vandermeulen2∗   
+Billy Joe Franks1 Marius Kloft1 Klaus-Robert Muller ¨ 2 3 4 5   
+$^ { 1 } \mathrm { M L }$ group, Technical University of Kaiserslautern, Germany   
+$^ { 2 } \mathbf { M } \mathbf { L }$ group, Technical University of Berlin, Germany   
+3Google Research, Brain Team, Berlin, Germany   
+4Department of Artificial Intelligence, Korea University, Seoul, Republic of Korea   
+5Max Planck Institute for Informatics, Saarbrucken, Germany ¨   
+{liznerski, franks, kloft}@cs.uni-kl.de   
+{lukas.ruff, vandermeulen, klaus-robert.mueller}@tu-berlin.de
+
+# ABSTRACT
+
+Deep one-class classification variants for anomaly detection learn a mapping that concentrates nominal samples in feature space causing anomalies to be mapped away. Because this transformation is highly non-linear, finding interpretations poses a significant challenge. In this paper we present an explainable deep one-class classification method, Fully Convolutional Data Description (FCDD), where the mapped samples are themselves also an explanation heatmap. FCDD yields competitive detection performance and provides reasonable explanations on common anomaly detection benchmarks with CIFAR-10 and ImageNet. On MVTec-AD, a recent manufacturing dataset offering ground-truth anomaly maps, FCDD sets a new state of the art in the unsupervised setting. Our method can incorporate ground-truth anomaly explanations during training and using even a few of these $( \sim 5 )$ improves performance significantly. Finally, using FCDD’s explanations, we demonstrate the vulnerability of deep one-class classification models to spurious image features such as image watermarks.1
+
+# 1 INTRODUCTION
+
+Anomaly detection (AD) is the task of identifying anomalies in a corpus of data (Edgeworth, 1887; Barnett and Lewis, 1994; Chandola et al., 2009; Ruff et al., 2021). Powerful new anomaly detectors based on deep learning have made AD more effective and scalable to large, complex datasets such as high-resolution images (Ruff et al., 2018; Bergmann et al., 2019). While there exists much recent work on deep AD, there is limited work on making such techniques explainable. Explanations are needed in industrial applications to meet safety and security requirements (Berkenkamp et al., 2017; Katz et al., 2017; Samek et al., 2020), avoid unfair social biases (Gupta et al., 2018), and support human experts in decision making (Jarrahi, 2018; Montavon et al., 2018; Samek et al., 2020). One typically makes anomaly detection explainable by annotating pixels with an anomaly score and, in some applications, such as finding tumors in cancer detection (Quellec et al., 2016), these annotations are the primary goal of the detector.
+
+One approach to deep AD, known as Deep Support Vector Data Description (DSVDD) (Ruff et al., 2018), is based on finding a neural network that transforms data such that nominal data is concentrated to a predetermined center and anomalous data lies elsewhere. In this paper we present Fully Convolutional Data Description (FCDD), a modification of DSVDD so that the transformed samples are themselves an image corresponding to a downsampled anomaly heatmap. The pixels in this heatmap that are far from the center correspond to anomalous regions in the input image. FCDD does this by only using convolutional and pooling layers, thereby limiting the receptive field of each output pixel. Our method is based on the one-class classification paradigm (Moya et al., 1993; Tax, 2001; Tax and Duin, 2004; Ruff et al., 2018), which is able to naturally incorporate known anomalies Ruff et al. (2021), but is also effective when simply using synthetic anomalies.
+
+We show that FCDD’s anomaly detection performance is close to the state of the art on the standard AD benchmarks with CIFAR-10 and ImageNet while providing transparent explanations. On MVTecAD, an AD dataset containing ground-truth anomaly maps, we demonstrate the accuracy of FCDD’s explanations (see Figure 1), where FCDD sets a new state of the art. In further experiments we find that deep one-class classification models (e.g. DSVDD) are prone to the “Clever Hans” effect (Lapuschkin et al., 2019) where a detector fixates on spurious features such as image watermarks. In general, we find that the generated anomaly heatmaps are less noisy and provide more structure than the baselines, including gradient-based methods (Simonyan et al., 2013; Sundararajan et al., 2017) and autoencoders (Sakurada and Yairi, 2014; Bergmann et al., 2019).
+
+![](images/7e591794677ee59acac5f101829093892ce279479865b0a9f1aada8b399ebe4c.jpg)  
+Figure 1: FCDD explanation heatmaps for MVTec-AD (Bergmann et al., 2019). Rows from top to bottom show: (1) nominal samples (2) anomalous samples (3) FCDD anomaly heatmaps (4) ground-truth anomaly maps.
+
+# 2 RELATED WORK
+
+Here we outline related works on deep AD focusing on explanation approaches. Classically deep AD used autoencoders (Hawkins et al., 2002; Sakurada and Yairi, 2014; Zhou and Paffenroth, 2017; Zhao et al., 2017). Trained on a nominal dataset autoencoders are assumed to reconstruct anomalous samples poorly. Thus, the reconstruction error can be used as an anomaly score and the pixel-wise difference as an explanation (Bergmann et al., 2019), thereby naturally providing an anomaly heatmap. Recent works have incorporated attention into reconstruction models that can be used as explanations (Venkataramanan et al., 2019; Liu et al., 2020). In the domain of videos, Sabokrou et al. (2018) used a pre-trained fully convolutional architecture in combination with a sparse autoencoder to extract 2D features and provide bounding boxes for anomaly localization. One drawback of reconstruction methods is that they offer no natural way to incorporate known anomalies during training.
+
+More recently, one-class classification methods for deep AD have been proposed. These methods attempt to separate nominal samples from anomalies in an unsupervised manner by concentrating nominal data in feature space while mapping anomalies to distant locations (Ruff et al., 2018; Chalapathy et al., 2018; Goyal et al., 2020). In the domain of NLP, DSVDD has been successfully applied to text, which yields a form of interpretation using attention mechanisms (Ruff et al., 2019). For images, Kauffmann et al. (2020) have used a deep Taylor decomposition (Montavon et al., 2017) to derive relevance scores.
+
+Some of the best performing deep AD methods are based on self-supervision. These methods transform nominal samples, train a network to predict which transformation was used on the input, and provide an anomaly score via the confidence of the prediction (Golan and El-Yaniv, 2018; Hendrycks et al., 2019b). Hendrycks et al. (2019a) have extended this to incorporate known anomalies as well. No explanation approaches have been considered for these methods so far.
+
+Finally, there exists a great variety of explanation methods in general, for example model-agnostic methods (e.g. LIME (Ribeiro et al., 2016)) or gradient-based techniques (Simonyan et al., 2013; Sundararajan et al., 2017). Relating to our work, we note that fully convolutional architectures have been used for supervised segmentation tasks where target segmentation maps are required during training (Long et al., 2015; Noh et al., 2015).
+
+# 3 EXPLAINING DEEP ONE-CLASS CLASSIFICATION
+
+We review one-class classification and fully convolutional architectures before presenting our method.
+
+Deep One-Class Classification Deep one-class classification (Ruff et al., 2018; 2020b) performs anomaly detection by learning a neural network to map nominal samples near a center c in output space, causing anomalies to be mapped away. For our method we use a Hypersphere Classifier (HSC) (Ruff et al., 2020a), a recently proposed modification of Deep SAD (Ruff et al., 2020b), a semi-supervised version of DSVDD (Ruff et al., 2018). Let $X _ { 1 } , \ldots , X _ { n }$ denote a collection of samples and $y _ { 1 } , \ldots , y _ { n }$ be labels where $y _ { i } = 1$ denotes an anomaly and $y _ { i } = 0$ denotes a nominal sample. Then the HSC objective is
+
+$$
+\operatorname* { m i n } _ { \mathcal { W } , \mathbf { c } } \quad \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( 1 - y _ { i } ) h ( \phi ( X _ { i } ; \mathcal { W } ) - \mathbf { c } ) - y _ { i } \log \left( 1 - \exp \left( - h ( \phi ( X _ { i } ; \mathcal { W } ) - \mathbf { c } ) \right) \right) ,
+$$
+
+where $\mathbf { c } \in \mathbb { R } ^ { d }$ is the center, and $\phi : \mathbb { R } ^ { c \times h \times w }  \mathbb { R } ^ { d }$ a neural network with weights $\mathcal { W }$ . Here $h$ is the pseudo-Huber loss (Huber et al., 1964), $h ( \mathbf { a } ) = \sqrt { \left\| \mathbf { a } \right\| _ { 2 } ^ { 2 } + 1 } - 1$ , which is a robust loss that interpolates from quadratic to linear penalization. The HSC loss encourages $\phi$ to map nominal samples near c and anomalous samples away from the center c. In our implementation, the center $\mathbf { c }$ corresponds to the bias term in the last layer of our networks, i.e. is included in the network $\phi$ , which is why we omit c in the FCDD objective below.
+
+Fully Convolutional Architecture Our method uses a fully convolutional network (FCN) (Long et al., 2015; Noh et al., 2015) that maps an image to a matrix of features, i.e. $\phi : \overline { { \mathbb { R } ^ { c \times h \times w } } }  \mathbb { R } ^ { 1 \times u \times v }$ by using alternating convolutional and pooling layers only, and does not contain any fully connected layers. In this context, pooling can be seen as a special kind of convolution with fixed parameters.
+
+![](images/9a58387c81d6a4afca4104a3f580bfae592f922e24d1aad7a5cf277a86b4bf79.jpg)  
+Figure 2: Visualization of a $3 \times 3$ convolution followed by a $3 \times 3$ transposed convolution with a Gaussian kernel, both using a stride of 2.
+
+A core property of a convolutional layer is that each pixel of its output only depends on a small region of its input, known as the output pixel’s receptive field. Since the output of a convolution is produced by moving a filter over the input image, each output pixel has the same relative position as its associated receptive field in the input. For instance, the lower-left corner of the output representation has a corresponding receptive field in the lower-left corner of the input image, etc. (see Figure 2 left side). The outcome of several stacked convolutions also has receptive fields of limited size and consistent relative position, though their size grows with the amount of layers. Because of this an FCN preserves spatial information.
+
+Fully Convolutional Data Description Here we introduce our novel explainable AD method Fully Convolutional Data Description (FCDD). By taking advantage of FCNs along with the HSC above, we propose a deep one-class method where the output features preserve spatial information and also serve as a downsampled anomaly heatmap. For situations where one would like to have a full-resolution heatmap, we include a methodology for upsampling the low-resolution heatmap based on properties of receptive fields.
+
+![](images/9ee4b72090ad4bcd644c597c5a19bc57155b05c7822d9e874e7c76f8e2abac0a.jpg)  
+Figure 3: Visualization of the overall procedure to produce full-resolution anomaly heatmaps with FCDD. $X$ denotes the input, $\phi$ the network, $A$ the produced anomaly heatmap and $A ^ { \prime }$ the upsampled version of $A$ using a transposed Gaussian convolution.
+
+FCDD is trained using samples that are labeled as nominal or anomalous. As before, let $X _ { 1 } , \ldots , X _ { n }$ denote a collection of samples with labels $y _ { 1 } , \ldots , y _ { n }$ where $y _ { i } = 1$ denotes an anomaly and $y _ { i } = 0$ denotes a nominal sample. Anomalous samples can simply be a collection of random images which are not from the nominal collection, e.g. one of the many large collections of images which are freely available like 80 Million Tiny Images (Torralba et al., 2008) or ImageNet (Deng et al., 2009). The use of such an auxiliary corpus has been recommended in recent works on deep AD, where it is termed Outlier Exposure (OE) (Hendrycks et al., 2019a;b). When one has access to “true” examples of the anomalous dataset, i.e. something that is likely to be representative of what will be seen at test time, we find that even using a few examples as the corpus of labeled anomalies performs exceptionally well. Furthermore, in the absence of any sort of known anomalies, one can generate synthetic anomalies, which we find is also very effective.
+
+With an $\operatorname { F C N } \phi : \mathbb { R } ^ { c \times h \times w } \to \mathbb { R } ^ { u \times v }$ the FCDD objective utilizes a pseudo-Huber loss on the FCN output matrix $A ( X ) = \left( { \sqrt { \phi ( X ; \mathcal { W } ) ^ { 2 } + 1 } } - 1 \right)$ , where all operations are applied element-wise. The FCDD objective is then defined as (cf., (1)):
+
+$$
+\operatorname* { m i n } _ { \mathcal { W } } \quad \frac { 1 } { n } \sum _ { i = 1 } ^ { n } ( 1 - y _ { i } ) \frac { 1 } { u \cdot v } \left\| A ( X _ { i } ) \right\| _ { 1 } - y _ { i } \log \left( 1 - \exp \left( - \frac { 1 } { u \cdot v } \left\| A ( X _ { i } ) \right\| _ { 1 } \right) \right) .
+$$
+
+Here $\| A ( X ) \| _ { 1 }$ is the sum of all entries in $A ( X )$ , which are all positive. FCDD is the utilization of an FCN in conjunction with the novel adaptation of the HSC loss we propose in (2). The objective maximizes $\| A ( X ) \| _ { 1 }$ for anomalies and minimizes it for nominal samples, thus we use $\| A ( X ) \| _ { 1 }$ as the anomaly score. Entries of $A ( X )$ that contribute to $\| A ( X ) \| _ { 1 }$ correspond to regions of the input image that add to the anomaly score. The shape of these regions depends on the receptive field of the FCN. We include a sensitivity analysis on the size of the receptive field in Appendix A, where we find that performance is not strongly affected by the receptive field size. Note that $A ( X )$ has spatial dimensions $u \times v$ and is smaller than the original image dimensions $h \times w$ . One could use $A ( X )$ directly as a low-resolution heatmap of the image, however it is often desirable to have full-resolution heatmaps. Because we generally lack ground-truth anomaly maps in an AD setting during training, it is not possible to train an FCN in a supervised way to upsample the low-resolution heatmap $A ( X )$ (e.g. as in (Noh et al., 2015)). For this reason we introduce an upsampling scheme based on the properties of receptive fields.
+
+Heatmap Upsampling Since we generally do not have access to ground-truth pixel annotations in anomaly detection during training, we cannot learn how to upsample using a deconvolutional type of structure. We derive a principled way to upsample our lower resolution anomaly heatmap instead. For every output pixel in $A ( \bar { X ( ) }$ there is a unique input pixel which lies at the center of its receptive field. It has been observed before that the effect of the receptive field for an output pixel decays in a Gaussian manner as one moves away from the center of the receptive field (Luo et al.,
+
+# Algorithm 1 Receptive Field Upsampling
+
+Input: $A \in \mathbb { R } ^ { u \times v }$ (low-res anomaly heatmap)   
+Output: $A ^ { \prime } \in \mathbb { R } ^ { h \times w }$ (full-res anomaly heatmap)   
+Define: $\begin{array} { r } { [ G _ { 2 } ( \mu , \sigma ) ] _ { x , y } \triangleq \frac { 1 } { 2 \pi \sigma ^ { 2 } } \exp \left( - \frac { ( x - \mu _ { 1 } ) ^ { 2 } + ( y - \mu _ { 2 } ) ^ { 2 } } { 2 \sigma ^ { 2 } } \right) } \end{array}$ $A ^ { \prime }  0$ for all output pixels $a$ in $A$ do $f \gets$ receptive field of $a$ $c \gets$ center of field $f$ $A ^ { \prime }  A ^ { \prime } + a \cdot G _ { 2 } ( \overset { . } { c } , \sigma )$ end for return $A ^ { \prime }$
+
+2016). We use this fact to upsample $A ( X )$ by using a strided transposed convolution with a fixed Gaussian kernel (see Figure 2 right side). We describe this operation and procedure in Algorithm 1 which simply corresponds to a strided transposed convolution. The kernel size is set to the receptive field range of FCDD and the stride to the cumulative stride of FCDD. The variance of the distribution can be picked empirically (see Appendix B for details). Figure 3 shows a complete overview of our FCDD method and the process of generating full-resolution anomaly heatmaps.
+
+# 4 EXPERIMENTS
+
+In this section, we experimentally evaluate the performance of FCDD both quantitatively and qualitatively. For a quantitative evaluation, we use the Area Under the ROC Curve (AUC) (Spackman, 1989) which is the commonly used measure in AD. For a qualitative evaluation, we compare the heatmaps produced by FCDD to existing deep AD explanation methods. As baselines, we consider gradient-based methods (Simonyan et al., 2013) applied to hypersphere classifier (HSC) models (Ruff et al., 2020a) with unrestricted network architectures (i.e. networks that also have fully connected layers) and autoencoders (Bergmann et al., 2019) where we directly use the pixel-wise reconstruction error as an explanation heatmap. We slightly blur the heatmaps of the baselines with the same Gaussian kernel we use for FCDD, which we found results in less noisy, more interpretable heatmaps. We include heatmaps without blurring in Appendix G. We adjust the contrast of the heatmaps per method to highlight interesting features; see Appendix C for details. For our experiments we don’t consider model-agnostic explanations, such as LIME (Ribeiro et al., 2016) or anchors (Ribeiro et al., 2018), because they are not tailored to the AD task and performed poorly.
+
+# 4.1 STANDARD ANOMALY DETECTION BENCHMARKS
+
+We first evaluate FCDD on the Fashion-MNIST, CIFAR-10, and ImageNet datasets. The common AD benchmark is to utilize these classification datasets in a one-vs-rest setup where the “one” class is used as the nominal class and the rest of the classes are used as anomalies at test time. For training, we only use nominal samples as well as random samples from some auxiliary Outlier Exposure (OE) (Hendrycks et al., 2019a) dataset, which is separate from the ground-truth anomaly classes following Hendrycks et al. (2019a;b). We report the mean AUC over all classes for each dataset.
+
+Fashion-MNIST We consider each of the ten Fashion-MNIST (Xiao et al., 2017) classes in a one-vs-rest setup. We train Fashion-MNIST using EMNIST (Cohen et al., 2017) or grayscaled CIFAR-100 (Krizhevsky et al., 2009) as OE. We found that the latter slightly outperforms the former ${ \sim } 3$ AUC percent points). On Fashion-MNIST, we use a network that consists of three convolutional layers with batch normalization, separated by two downsampling pooling layers.
+
+CIFAR-10 We consider each of the ten CIFAR-10 (Krizhevsky et al., 2009) classes in a one-vs-rest setup. As OE we use CIFAR-100, which does not share any classes with CIFAR-10. We use a model similar to LeNet-5 (LeCun et al., 1998), but decrease the kernel size to three, add batch normalization, and replace the fully connected layers and last max-pool layer with two further convolutions.
+
+ImageNet We consider 30 classes from ImageNet1k (Deng et al., 2009) for the one-vs-rest setup following Hendrycks et al. (2019a). For OE we use ImageNet22k with ImageNet1k classes removed (Hendrycks et al., 2019a). We use an adaptation of VGG11 (Simonyan and Zisserman, 2015) with batch normalization, suitable for inputs resized to $2 2 4 \times 2 2 4$ (see Appendix D for model details).
+
+State-of-the-art Methods We report results from state-of-the-art deep anomaly detection methods. Methods that do not incorporate known anomalies are the autoencoder (AE), DSVDD (Ruff et al., 2018), Geometric Transformation based AD (GEO) (Golan and El-Yaniv, 2018), and a variant of GEO by Hendrycks et al. (2019b) $\mathrm { ( G E O + ) }$ ). Methods that use OE are a Focal loss classifier (Hendrycks et al., 2019b), also $\mathrm { G E O + }$ , Deep SAD (Ruff et al., 2020b), and HSC (Ruff et al., 2020a).
+
+Table 1: Mean AUC (over all classes and 5 seeds per class) for Fashion-MNIST, CIFAR-10, and ImageNet. Results from existing literature are marked with an asterisk (Bergman and Hoshen, 2020; Golan and El-Yaniv, 2018; Hendrycks et al., 2019b; Ruff et al., 2020a).   
+
+<table><tr><td></td><td colspan="4">without OE</td><td colspan="5">with OE</td></tr><tr><td>Dataset</td><td>AE</td><td>DSVDD*</td><td>GEO*</td><td>Geo+*</td><td>Focal*</td><td>Geo+*</td><td>Deep SAD*</td><td>HSC*</td><td>FCDD</td></tr><tr><td>Fashion-MNIST</td><td>0.82</td><td>0.93</td><td>0.94</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td><td>0.89</td></tr><tr><td>CIFAR-10</td><td>0.59*</td><td>0.65</td><td>0.86</td><td>0.90</td><td>0.87</td><td>0.96</td><td>0.95</td><td>0.96</td><td>0.95</td></tr><tr><td>ImageNet</td><td>0.56</td><td>×</td><td>×</td><td>×</td><td>0.56</td><td>0.86</td><td>0.97</td><td>0.97</td><td>0.94</td></tr></table>
+
+Quantitative Results The mean AUC detection performance on the three AD benchmarks are reported in Table 1. We can see that FCDD, despite using a restricted FCN architecture to improve explainability, achieves a performance that is close to state-of-the-art methods and outperforms autoencoders, which yield a detection performance close to random on more complex datasets. We provide detailed results for all individual classes in Appendix F.
+
+![](images/8fdc9dc9ae1be14c2f55ebfd22c5df5fbc20efe5bfd5ba4cccd8f40103734d6b.jpg)  
+Figure 4: Anomaly heatmaps for anomalous test samples of a Fashion-MNIST model trained on nominal class “trousers” (nominal samples are shown in (a)). In (b) CIFAR-100 was used for OE and in (c) EMNIST. Columns are ordered by increasing anomaly score from left to right, i.e. what FCDD finds the most nominal looking anomaly on the left to the most anomalous looking anomaly on the right.
+
+Qualitative Results Figures 4 and 5 show the heatmaps for Fashion-MNIST and ImageNet respectively. For a Fashion-MNIST model trained on the nominal class “trousers,” the heatmaps show that FCDD correctly highlights horizontal elements as being anomalous, which makes sense since trousers are vertically aligned. For an ImageNet model trained on the nominal class “acorns,” we observe that colors seem to be fairly relevant features with green and brown areas tending to be seen as more nominal, and other colors being deemed anomalous, for example the red barn or the white snow. Nonetheless, the method also seems capable of using more semantic features, for example it recognizes the green caterpillar as being anomalous and it distinguishes the acorn to be nominal despite being against a red background.
+
+Figure 6 shows heatmaps for CIFAR-10 models with varying amount of OE, all trained on the nominal class “airplane.” We can see that, as the number of OE samples increases, FCDD tends to concentrate the explanations more on the primary object in the image, i.e. the bird, ship, and truck. We provide further heatmaps for additional classes from all datasets in Appendix G.
+
+![](images/8f49793370255db3ed64cf44d650763a62506609e270c927b6fbb4b96ad60301.jpg)  
+Figure 5: Anomaly heatmaps of an ImageNet model trained on nominal class “acorns.” Here (a) are nominal samples and (b) are anomalous samples. Columns are ordered by increasing anomaly score from left to right, i.e. what FCDD finds the most nominal looking on the left to the most anomalous looking on the right for (a) nominal samples and (b) anomalies.
+
+![](images/ba472e202b2d17fa8bcddd98fa4f39c791b7576bb1838877ccfced36a7b34434.jpg)  
+Figure 6: Anomaly heatmaps for three anomalous test samples on a CIFAR-10 model trained on nominal class “airplane.” The second, third, and fourth blocks show the heatmaps of FCDD, gradientbased heatmaps of HSC, and AE heatmaps respectively. For Ours and Grad, we grow the number of OE samples from 2, 8, 128, 2048 to full OE. AE is not able to incorporate OE.
+
+Baseline Explanations We found the gradient-based heatmaps to mostly produce centered blobs which lack spatial context (see Figure 6) and thus are not useful for explaining. The AE heatmaps, being directly tied to the reconstruction error anomaly score, look reasonable. We again note, however, that it is not straightforward how to include auxiliary OE samples or labeled anomalies into an AE approach, which leaves them with a poorer detection performance (see Table 1). Overall we find that the proposed FCDD anomaly heatmaps yield a good and consistent visual interpretation.
+
+# 4.2 EXPLAINING DEFECTS IN MANUFACTURING
+
+Here we compare the performance of FCDD on the MVTec-AD dataset of defects in manufacturing (Bergmann et al., 2019). This datasets offers annotated ground-truth anomaly segmentation maps for testing, thus allowing a quantitative evaluation of model explanations. MVTec-AD contains 15 object classes of high-resolution RGB images with up to $1 0 2 4 \times 1 0 2 4$ pixels, where anomalous test samples are further categorized in up to 8 defect types, depending on the class. We follow Bergmann et al. (2019) and compute an AUC from the heatmap pixel scores, using the given (binary) anomaly segmentation maps as ground-truth pixel labels. We then report the mean over all samples of this “explanation” AUC for a quantitative evaluation. For FCDD, we use a network that is based on a VGG11 network pre-trained on ImageNet, where we freeze the first ten layers, followed by additional fully convolutional layers that we train.
+
+Synthetic Anomalies OE with a natural image dataset like ImageNet is not informative for MVTec-AD since anomalies here are subtle defects of the nominal class, rather than being out of class (see Figure 1). For this reason, we generate synthetic anomalies using a sort of “confetti noise,” a simple noise model that inserts colored blobs into images and reflects the local nature of anomalies. See Figure 7 for an example.
+
+![](images/da34c9801ea08c72e19de02de1dfa1d394df111b41da2ebf0cfce6563117aaa6.jpg)  
+Figure 7: Confetti noise.
+
+Semi-Supervised FCDD A major advantage of FCDD in comparison to reconstruction-based methods is that it can be readily used in a semi-supervised AD setting (Ruff et al., 2020b). To see the effect of having even only a few labeled anomalies and their corresponding ground-truth anomaly maps available for training, we pick for each MVTec-AD class just one true anomalous sample per defect type at random and add it to the training set. This results in only 3–8 anomalous training samples. To also take advantage of the ground-truth heatmaps, we train a model on a pixel level. Let $X _ { 1 } , \ldots , X _ { n }$ again denote a batch of inputs with corresponding ground-truth heatmaps $Y _ { 1 } , \dots , Y _ { n }$ , each having $m = h \cdot w$ number of pixels. Let $A ( X )$ also again denote the corresponding output anomaly heatmap of $X$ . Then, we can formulate a pixel-wise objective by the following:
+
+$$
+\operatorname* { m i n } _ { W } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( \frac { 1 } { m } \sum _ { j = 1 } ^ { m } ( 1 - ( Y _ { i } ) _ { j } ) A ^ { \prime } ( X _ { i } ) _ { j } \right) - \log \left( 1 - \exp \left( - \frac { 1 } { m } \sum _ { j = 1 } ^ { m } ( Y _ { i } ) _ { j } A ^ { \prime } ( X _ { i } ) _ { j } \right) \right) .
+$$
+
+Results Figure 1 in the introduction shows heatmaps of FCDD trained on MVTec-AD. The results of the quantitative explanation are shown in Table 2. We can see that FCDD outperforms its competitors in the unsupervised setting and sets a new state of the art of 0.92 pixel-wise mean AUC. In the semi-supervised setting —using only one anomalous sample with corresponding anomaly map per defect class— the explanation performance improves further to 0.96 pixel-wise mean AUC. FCDD also has the most consistent performance across classes.
+
+Table 2: Pixel-wise mean AUC scores for all classes of the MVTec-AD dataset (Bergmann et al., 2019). For competitors we include the baselines presented in the original MVTec-AD paper and previously published works from peer-reviewed venues that include the MVTec-AD benchmark. The competitors are Self-Similarity and L2 Autoencoder (Bergmann et al., 2019), AnoGAN (Schlegl et al., 2017; Bergmann et al., 2019), CNN Feature Dictionaries (Napoletano et al., 2018; Bergmann et al., 2019), Visually Explained Variational Autoencoder (Liu et al., 2020), Superpixel Masking and Inpainting (Li et al., 2020), Gradient Descent Reconstruction with VAEs (Dehaene et al., 2020), and Encoding Structure-Texture Relation with P-Net for AD (Zhou et al., 2020).   
+
+<table><tr><td colspan="10">unsupervised</td><td rowspan="2">semi-supervised FCDD</td></tr><tr><td></td><td>AE-SS*</td><td>AE-L2*</td><td>AnoGAN*</td><td>CNNFD*</td><td>VEVAE*</td><td>SMAI*</td><td>GDR*</td><td>P-NET*</td><td>FCDD</td></tr><tr><td>Bottle</td><td>0.93</td><td>0.86</td><td>0.86</td><td>0.78</td><td>0.87</td><td>0.86</td><td>0.92</td><td>0.99</td><td>0.97</td><td>0.96</td></tr><tr><td>Cable</td><td>0.82</td><td>0.86</td><td>0.78</td><td>0.79</td><td>0.90</td><td>0.92</td><td>0.91</td><td>0.70</td><td>0.90</td><td>0.93</td></tr><tr><td>Capsule</td><td>0.94</td><td>0.88</td><td>0.84</td><td>0.84</td><td>0.74</td><td>0.93</td><td>0.92</td><td>0.84</td><td>0.93</td><td>0.95</td></tr><tr><td>Carpet</td><td>0.87</td><td>0.59</td><td>0.54</td><td>0.72</td><td>0.78</td><td>0.88</td><td>0.74</td><td>0.57</td><td>0.96</td><td>0.99</td></tr><tr><td>Grid</td><td>0.94</td><td>0.90</td><td>0.58</td><td>0.59</td><td>0.73</td><td>0.97</td><td>0.96</td><td>0.98</td><td>0.91</td><td>0.95</td></tr><tr><td>Hazelnut</td><td>0.97</td><td>0.95</td><td>0.87</td><td>0.72</td><td>0.98</td><td>0.97</td><td>0.98</td><td>0.97</td><td>0.95</td><td>0.97</td></tr><tr><td>Leather</td><td>0.78</td><td>0.75</td><td>0.64</td><td>0.87</td><td>0.95</td><td>0.86</td><td>0.93</td><td>0.89</td><td>0.98</td><td>0.99</td></tr><tr><td>Metal Nut</td><td>0.89</td><td>0.86</td><td>0.76</td><td>0.82</td><td>0.94</td><td>0.92</td><td>0.91</td><td>0.79</td><td>0.94</td><td>0.98</td></tr><tr><td>Pill</td><td>0.91</td><td>0.85</td><td>0.87</td><td>0.68</td><td>0.83</td><td>0.92</td><td>0.93</td><td>0.91</td><td>0.81</td><td>0.97</td></tr><tr><td>Screw</td><td>0.96</td><td>0.96</td><td>0.80</td><td>0.87</td><td>0.97</td><td>0.96</td><td>0.95</td><td>1.00</td><td>0.86</td><td>0.93</td></tr><tr><td>Tile</td><td>0.59</td><td>0.51</td><td>0.50</td><td>0.93</td><td>0.80</td><td>0.62</td><td>0.65</td><td>0.97</td><td>0.91</td><td>0.98</td></tr><tr><td>Toothbrush</td><td>0.92</td><td>0.93</td><td>0.90</td><td>0.77</td><td>0.94</td><td>0.96</td><td>0.99</td><td>0.99</td><td>0.94</td><td>0.95</td></tr><tr><td>Transistor</td><td>0.90</td><td>0.86</td><td>0.80</td><td>0.66</td><td>0.93</td><td>0.85</td><td>0.92</td><td>0.82</td><td>0.88</td><td>0.90</td></tr><tr><td>Wood</td><td>0.73</td><td>0.73</td><td>0.62</td><td>0.91</td><td>0.77</td><td>0.80</td><td>0.84</td><td>0.98</td><td>0.88</td><td>0.94</td></tr><tr><td>Zipper</td><td>0.88</td><td>0.77</td><td>0.78</td><td>0.76</td><td>0.78</td><td>0.90</td><td>0.87</td><td>0.90</td><td>0.92</td><td>0.98</td></tr><tr><td>Mean</td><td>0.86</td><td>0.82</td><td>0.74</td><td>0.78</td><td>0.86</td><td>0.89</td><td>0.89</td><td>0.89</td><td>0.92</td><td>0.96</td></tr><tr><td>0</td><td>0.10</td><td>0.13</td><td>0.13</td><td>0.10</td><td>0.09</td><td>0.09</td><td>0.09</td><td>0.12</td><td>0.04</td><td>0.02</td></tr></table>
+
+# 4.3 THE CLEVER HANS EFFECT
+
+Lapuschkin et al. (2016; 2019) revealed that roughly one fifth of all horse images in PASCAL VOC (Everingham et al., 2010) contain a watermark in the lower left corner. They showed that a classifier recognizes this as the relevant class pattern and fails if the watermark is removed. They call this the “Clever Hans” effect in memory of the horse Hans, who could correctly answer math problems by reading its master2. We adapt this experiment to one-class classification by swapping our standard setup and train FCDD so that the “horse” class is anomalous and use ImageNet as nominal samples. We choose this setup so that one would expect FCDD to highlight horses in its heatmaps and so that any other highlighting makes FCDD reveal a Clever Hans effect.
+
+![](images/4ef3e33e0ae5f069848d77020dcea1e9a57d353d9a09f798c3060e84d7f96915.jpg)  
+Figure 8: Heatmaps for horses on PASCAL VOC. Here (a) shows anomalous samples ordered from most nominal to most anomalous from left to right, and (b) shows examples that indicate that the model is a “Clever Hans,” i.e. has learned a characterization based on spurious features (watermarks).
+
+Figure 8 (b) shows that a one-class model is indeed also vulnerable to learning a characterization based on spurious features: the watermarks in the lower left corner which have high scores whereas other regions have low scores. We also observe that the model yields high scores for bars, grids, and fences in Figure 8 (a). This is due to many images in the dataset containing horses jumping over bars or being in fenced areas. In both cases, the horse features themselves do not attain the highest scores because the model has no way of knowing that the spurious features, while providing good discriminative power at training time, would not be desirable upon deployment/test time. In contrast to traditional black-box models, however, transparent detectors like FCDD enable a practitioner to recognize and remedy (e.g. by cleaning or extending the training data) such behavior or other undesirable phenomena (e.g. to avoid unfair social bias).
+
+# 5 CONCLUSION
+
+In conclusion we find that FCDD, in comparison to previous methods, performs well and is adaptable to both semantic detection tasks (Section 4.1) and more subtle defect detection tasks (Section 4.2). Finally, directly tying an explanation to the anomaly score should make FCDD less vulnerable to attacks (Anders et al., 2020) in contrast to a posteriori explanation methods. We leave an analysis of this phenomenon for future work.
+
+# ACKNOWLEDGEMENTS
+
+MK, PL, and BJF acknowledge support by the German Research Foundation (DFG) award KL 2698/2- 1 and by the German Federal Ministry of Science and Education (BMBF) awards 01IS18051A, 031B0770E, and 01MK20014U. LR acknowledges support by the German Federal Ministry of Education and Research (BMBF) in the project ALICE III (01IS18049B). RV acknowledges support by the Berlin Institute for the Foundations of Learning and Data (BIFOLD) sponsored by the German Federal Ministry of Education and Research (BMBF). KRM was supported in part by the Institute of Information & Communications Technology Planning & Evaluation (IITP) grants funded by the Korea Government (No. 2017-0-00451 and 2019-0-00079) and was partly supported by the German Federal Ministry of Education and Research (BMBF) for the Berlin Center for Machine Learning (01IS18037A-I) and under the Grants 01IS14013A-E, 01GQ1115, 01GQ0850, 01IS18025A, and 031L0207A-D; the German Research Foundation (DFG) under Grant Math+, EXC 2046/1, Project ID 390685689. Finally, we thank all reviewers for their constructive feedback, which helped to improve this work.
+
+REFERENCES   
+C. J. Anders, P. Pasliev, A.-K. Dombrowski, K.-R. Muller, and P. Kessel. Fairwashing explanations with ¨ off-manifold detergent. In ICML, pages 6757–6766, 2020.   
+V. Barnett and T. Lewis. Outliers in Statistical Data. Wiley, 3rd edition, 1994.   
+L. Bergman and Y. Hoshen. Classification-based anomaly detection for general data. In ICLR, 2020.   
+P. Bergmann, M. Fauser, D. Sattlegger, and C. Steger. MVTec AD–A comprehensive real-world dataset for unsupervised anomaly detection. In CVPR, pages 9592–9600, 2019.   
+F. Berkenkamp, M. Turchetta, A. Schoellig, and A. Krause. Safe model-based reinforcement learning with stability guarantees. In NeurIPS, pages 908–918, 2017.   
+L. Bottou. Large-scale machine learning with stochastic gradient descent. In Proceedings of COMPSTAT’2010, pages 177–186. Springer, 2010.   
+R. Chalapathy, A. K. Menon, and S. Chawla. Anomaly detection using one-class neural networks. arXiv preprint arXiv:1802.06360, 2018.   
+V. Chandola, A. Banerjee, and V. Kumar. Anomaly detection: A survey. ACM Computing Surveys, 41(3):1–58, 2009.   
+G. Cohen, S. Afshar, J. Tapson, and A. Van Schaik. EMNIST: Extending MNIST to handwritten letters. In IJCNN, pages 2921–2926, 2017.   
+D. Dehaene, O. Frigo, S. Combrexelle, and P. Eline. Iterative energy-based projection on a normal data manifold for anomaly localization. In ICLR, 2020.   
+J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. ImageNet: A large-scale hierarchical image database. In CVPR, pages 248–255, 2009.   
+F. Y. Edgeworth. On discordant observations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 23(5):364–375, 1887.   
+M. Everingham, L. Van Gool, C. K. Williams, J. Winn, and A. Zisserman. The pascal visual object classes (VOC) challenge. International Journal of Computer Vision, 88(2):303–338, 2010.   
+I. Golan and R. El-Yaniv. Deep anomaly detection using geometric transformations. In NeurIPS, pages 9758–9769, 2018.   
+S. Goyal, A. Raghunathan, M. Jain, H. V. Simhadri, and P. Jain. DROCC: Deep robust one-class classification. In ICML, pages 11335–11345, 2020.   
+A. Gupta, J. Johnson, L. Fei-Fei, S. Savarese, and A. Alahi. Social GAN: Socially acceptable trajectories with generative adversarial networks. In CVPR, pages 2255–2264, 2018.   
+S. Hawkins, H. He, G. Williams, and R. Baxter. Outlier Detection Using Replicator Neural Networks. In DaWaK, volume 2454, pages 170–180, 2002.   
+D. Hendrycks, M. Mazeika, and T. G. Dietterich. Deep anomaly detection with outlier exposure. In ICLR, 2019a.   
+D. Hendrycks, M. Mazeika, S. Kadavath, and D. Song. Using self-supervised learning can improve model robustness and uncertainty. In NeurIPS, pages 15637–15648, 2019b.   
+P. J. Huber et al. Robust estimation of a location parameter. The Annals of Mathematical Statistics, 35(1): 73–101, 1964.   
+M. H. Jarrahi. Artificial intelligence and the future of work: Human-AI symbiosis in organizational decision making. Business Horizons, 61(4):577–586, 2018.   
+G. Katz, C. Barrett, D. L. Dill, K. Julian, and M. J. Kochenderfer. Reluplex: An efficient SMT solver for verifying deep neural networks. In International Conference on Computer Aided Verification, pages 97–117. Springer, 2017.   
+J. Kauffmann, K.-R. Muller, and G. Montavon. Towards Explaining Anomalies: A Deep Taylor Decomposition ¨ of One-Class Models. Pattern Recognition, 101:107198, 2020.   
+D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. In ICLR, 2015.   
+A. Krizhevsky, G. Hinton, et al. Learning multiple layers of features from tiny images. Technical report, 2009.   
+S. Lapuschkin, A. Binder, G. Montavon, K.-R. Muller, and W. Samek. Analyzing classifiers: Fisher vectors and deep neural networks. In CVPR, pages 2912–2920, 2016.   
+S. Lapuschkin, S. Waldchen, A. Binder, G. Montavon, W. Samek, and K.-R. M ¨ uller. Unmasking clever hans ¨ predictors and assessing what machines really learn. Nature Communications, 10:1096, 2019.   
+Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   
+Z. Li, N. Li, K. Jiang, Z. Ma, X. Wei, X. Hong, and Y. Gong. Superpixel masking and inpainting for selfsupervised anomaly detection. In BMVC, 2020.   
+W. Liu, R. Li, M. Zheng, S. Karanam, Z. Wu, B. Bhanu, R. J. Radke, and O. Camps. Towards visually explaining variational autoencoders. In CVPR, pages 8642–8651, 2020.   
+J. Long, E. Shelhamer, and T. Darrell. Fully convolutional networks for semantic segmentation. In CVPR, pages 3431–3440, 2015.   
+W. Luo, Y. Li, R. Urtasun, and R. Zemel. Understanding the effective receptive field in deep convolutional neural networks. In NeurIPS, pages 4898–4906, 2016.   
+G. Montavon, S. Lapuschkin, A. Binder, W. Samek, and K.-R. Muller. Explaining nonlinear classification ¨ decisions with deep Taylor decomposition. Pattern Recognition, 65:211–222, 2017.   
+G. Montavon, W. Samek, and K.-R. Muller. Methods for interpreting and understanding deep neural networks. ¨ Digital Signal Processing, 73:1–15, 2018.   
+M. M. Moya, M. W. Koch, and L. D. Hostetler. One-class classifier networks for target recognition applications. In World Congress on Neural Networks, pages 797–801, 1993.   
+P. Napoletano, F. Piccoli, and R. Schettini. Anomaly detection in nanofibrous materials by CNN-based selfsimilarity. Sensors, 18(1):209, 2018.   
+H. Noh, S. Hong, and B. Han. Learning deconvolution network for semantic segmentation. In ICCV, pages 1520–1528, 2015.   
+G. Quellec, M. Lamard, M. Cozic, G. Coatrieux, and G. Cazuguel. Multiple-instance learning for anomaly detection in digital mammography. IEEE Transactions on Medical Imaging, 35(7):1604–1614, 2016.   
+M. T. Ribeiro, S. Singh, and C. Guestrin. “Why should i trust you?” Explaining the predictions of any classifier. In KDD, pages 1135–1144, 2016.   
+M. T. Ribeiro, S. Singh, and C. Guestrin. Anchors: High-precision model-agnostic explanations. In AAAI, volume 18, pages 1527–1535, 2018.   
+L. Ruff, R. A. Vandermeulen, N. Gornitz, L. Deecke, S. A. Siddiqui, A. Binder, E. M ¨ uller, and M. Kloft. Deep ¨ one-class classification. In ICML, volume 80, pages 4390–4399, 2018.   
+L. Ruff, Y. Zemlyanskiy, R. Vandermeulen, T. Schnake, and M. Kloft. Self-attentive, multi-context one-class classification for unsupervised anomaly detection on text. In ACL, pages 4061–4071, 2019.   
+L. Ruff, R. A. Vandermeulen, B. J. Franks, K.-R. Muller, and M. Kloft. Rethinking assumptions in deep anomaly ¨ detection. arXiv preprint arXiv:2006.00339, 2020a.   
+L. Ruff, R. A. Vandermeulen, N. Gornitz, A. Binder, E. M ¨ uller, K.-R. M ¨ uller, and M. Kloft. Deep semi-supervised ¨ anomaly detection. In ICLR, 2020b.   
+L. Ruff, J. R. Kauffmann, R. A. Vandermeulen, G. Montavon, W. Samek, M. Kloft, T. G. Dietterich, and K.-R. Muller. A unifying review of deep and shallow anomaly detection. ¨ Proceedings of the IEEE, 2021. doi: 10.1109/JPROC.2021.3052449.   
+M. Sabokrou, M. Fayyaz, M. Fathy, Z. Moayed, and R. Klette. Deep-anomaly: Fully convolutional neural network for fast anomaly detection in crowded scenes. Computer Vision and Image Understanding, 172: 88–97, 2018.
+
+M. Sakurada and T. Yairi. Anomaly detection using autoencoders with nonlinear dimensionality reduction. In Proceedings of the MLSDA 2014 2nd Workshop on Machine Learning for Sensory Data Analysis, pages 4–11, 2014.
+
+W. Samek, G. Montavon, S. Lapuschkin, C. J. Anders, and K.-R. Muller. Toward interpretable machine learning: ¨ Transparent deep neural networks and beyond. arXiv preprint arXiv:2003.07631, 2020.
+
+T. Schlegl, P. Seebock, S. M. Waldstein, U. Schmidt-Erfurth, and G. Langs. Unsupervised anomaly detection ¨ with generative adversarial networks to guide marker discovery. In International conference on information processing in medical imaging, pages 146–157. Springer, 2017.
+
+K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. In ICLR, 2015.
+
+K. Simonyan, A. Vedaldi, and A. Zisserman. Deep inside convolutional networks: Visualising image classification models and saliency maps. arXiv preprint arXiv:1312.6034, 2013.
+
+K. A. Spackman. Signal detection theory: Valuable tools for evaluating inductive learning. In Proceedings of the Sixth International Workshop on Machine Learning, pages 160–163, 1989.
+
+M. Sundararajan, A. Taly, and Q. Yan. Axiomatic attribution for deep networks. In ICML, pages 3319–3328, 2017.
+
+I. Sutskever, J. Martens, G. Dahl, and G. Hinton. On the importance of initialization and momentum in deep learning. In ICML, pages 1139–1147, 2013.
+
+D. M. J. Tax. One-class classification. PhD thesis, Delft University of Technology, 2001.
+
+D. M. J. Tax and R. P. W. Duin. Support Vector Data Description. Machine Learning, 54(1):45–66, 2004.
+
+A. Torralba, R. Fergus, and W. T. Freeman. 80 million tiny images: A large data set for nonparametric object and scene recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 30(11):1958–1970, 2008.
+
+S. Venkataramanan, K.-C. Peng, R. V. Singh, and A. Mahalanobis. Attention guided anomaly detection and localization in images. arXiv preprint arXiv:1911.08616, 2019.
+
+H. Xiao, K. Rasul, and R. Vollgraf. Fashion-MNIST: A novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017.
+
+Y. Zhao, B. Deng, C. Shen, Y. Liu, H. Lu, and X.-S. Hua. Spatio-temporal autoencoder for video anomaly detection. In Proceedings of the 25th ACM International Conference on Multimedia, pages 1933–1941, 2017.
+
+C. Zhou and R. C. Paffenroth. Anomaly detection with robust deep autoencoders. In KDD, pages 665–674, 2017.
+
+K. Zhou, Y. Xiao, J. Yang, J. Cheng, W. Liu, W. Luo, Z. Gu, J. Liu, and S. Gao. Encoding structure-texture relation with p-net for anomaly detection in retinal images. In ECCV, 2020.
+
+# A RECEPTIVE FIELD SENSITIVITY ANALYSIS
+
+The receptive field has an impact on both detection performance and explanation quality. Here we provide some heatmaps and AUC scores for networks with different receptive field sizes. We observe that the detection performance is only minimally affected, but larger receptive fields cause the explanation heatmap to become less concentrated and more “blobby.” For MVTec-AD we see that this can also negatively affect pixel-wise AUC scores, see Table 4.
+
+CIFAR-10 For CIFAR-10 we create eight different network architectures to study the impact of the receptive field size. Each architecture has four convolutional layers and two max-pool layers. To change the receptive field we vary the kernel size of the first convolutional layer between 3 and 17. When this kernel size is 3 then the receptive field contains approximately one quarter of the image; for a kernel size of 17 the receptive field is the entire image. Table 3 shows the detection performance of the networks. Figure 9 contains example heatmaps.
+
+Table 3: Mean AUC (over all classes and 5 seeds per class) for CIFAR-10 and neural networks with varying receptive field size.   
+
+<table><tr><td>Receptive field size</td><td>18</td><td>20</td><td>22</td><td>24</td><td>26</td><td>28</td><td>30</td><td>32</td></tr><tr><td>AUC</td><td>0.9328</td><td>0.9349</td><td>0.9344</td><td>0.9320</td><td>0.9303</td><td>0.9283</td><td>0.9257</td><td>0.9235</td></tr></table>
+
+![](images/fdca83b9c1174f7067d4d09a9c2b3f4473b1f199c57cf13099f4a4f663f3a3cc.jpg)  
+Figure 9: Anomaly heatmaps for three anomalous test samples on CIFAR-10 models trained on nominal class “airplane.” We grow the receptive field size from 18 (left) to 32 (right).
+
+![](images/6795dc9db134e184a4f4ce16fab58907b97025f90dca52c950e17bf40d469e25.jpg)  
+Figure 10: Anomaly heatmaps for seven anomalous test samples of MVTec-AD. We grow the receptive field size from 53 (left) to 243 (right).
+
+Table 4: Pixel-wise mean AUC (over all classes and 5 seeds per class) for MVTec-AD and neural networks with varying receptive field size.   
+
+<table><tr><td>Receptive field size</td><td>53</td><td>91</td><td>129</td><td>167</td><td>205</td><td>243</td></tr><tr><td>AUC</td><td>0.88</td><td>0.85</td><td>0.79</td><td>0.76</td><td>0.75</td><td>0.75</td></tr></table>
+
+MVTec-AD We create six different network architectures for MVTec-AD. They have six convolutional layers and three max-pool layers. We vary the kernel size for all of the convolutional layers between 3 and 13, which corresponds to a receptive field containing $1 / 1 6$ of the image to the full image respectively. Table 4 shows the explanation performance of the networks in terms of pixel-wise mean AUC. Figure 10 contains some example heatmaps. We observe that a smaller receptive field yields better explanation performance.
+
+# B IMPACT OF THE GAUSSIAN VARIANCE
+
+Using the proposed heatmap upsampling in Section 3 FCDD provides full-resolution anomaly heatmaps. However, this upsampling involves the choice of $\sigma$ for the Gaussian kernel. In this section, we demonstrate the effect of this hyperparameter on the explanation performance of FCDD on MVTec-AD. Table 5 shows the pixel-wise mean AUC, Figure 11 corresponding heatmaps.
+
+![](images/a13fe6bb7645f491c9b8b829154cf4eb4c3ae22f9ce13df5ab73928e24a659b9.jpg)  
+Figure 11: Anomaly heatmaps for seven anomalous test samples of MVTec-AD. We grow $\sigma$ from 4 (left) to 16 (right).
+
+Table 5: Pixel-wise mean AUC (over all classes and 5 seeds per class) for MVTec-AD and different $\sigma$ .   
+
+<table><tr><td>0</td><td>4</td><td>6</td><td>8</td><td>10</td><td>12</td><td>14</td><td>16</td></tr><tr><td>AUC</td><td>0.8567</td><td>0.8836</td><td>0.9030</td><td>0.9124</td><td>0.9164</td><td>0.9217</td><td>0.9208</td></tr></table>
+
+# C ANOMALY HEATMAP VISUALIZATION
+
+For anomaly heatmap visualization, the FCDD anomaly scores $A ^ { \prime } ( X )$ need to be rescaled to values in $[ 0 , 1 ]$ . Instead of applying standard min-max scaling that would divide all heatmap entries by max $A ^ { \prime } ( X )$ , we use anomaly score quantiles to adjust the contrast in the heatmaps. For a collection of inputs ${ \mathcal { X } } = \{ X _ { 1 } , \ldots , X _ { n } \}$ with corresponding full-resolution anomaly heatmaps $\mathcal { A } = \{ A ^ { \prime } ( X _ { 1 } ) , \ldots , A ^ { \prime } ( X _ { n } ) \}$ , the normalized heatmap $I ( X )$ for some $A ^ { \prime } ( X )$ is computed as
+
+$$
+I ( X ) _ { j } = \operatorname* { m i n } \left\{ \frac { A ^ { \prime } ( X ) _ { j } - \operatorname* { m i n } ( A ) } { q _ { \eta } ( \{ A ^ { \prime } - \operatorname* { m i n } ( A ) \mid A ^ { \prime } \in \mathcal { A } \} ) } , 1 \right\} ,
+$$
+
+where $j$ denotes the $j$ -th pixel and $q _ { \eta }$ the $\eta$ -th percentile over all pixels and examples in $\mathcal { A }$ . The subtraction and min operation are applied on a pixel level, i.e. the minimum is extracted over all pixels and all samples of $\mathcal { A }$ and subtraction is then applied elementwise. Using the $\eta$ -th percentile might leave some of the values above 1, which is why we finally clamp the pixels at 1.
+
+The specific choice of $\eta$ and set of samples $\mathcal { X }$ differs per figure. We select them to highlight different properties of the heatmaps. In general, the lower $\eta$ the more red (anomalous) regions we have in the heatmaps because more values are left above one (before clamping to 1) and vice versa. The choice of $\mathcal { X }$ ranges from just one sample $X$ , such that $A ^ { \prime } ( X )$ is normalized only w.r.t. to its own scores (highlighting the most anomalous regions within the image), to the complete dataset (highlighting which regions look anomalous compared to the whole dataset). For the latter visualization we rebalance the dataset so that $\mathcal { X }$ contains an equal amount of nominal and anomalous images to maintain consistent scaling. The choice of $\eta$ and $\mathcal { X }$ is consistent per figure. In the following we list the choices made for the individual figures.
+
+MVTec-AD Figures 1, 10, and 11 use $\eta = 0 . 9 7$ and set $\mathcal { X }$ to $X$ for each heatmap $I ( X )$ to show relative anomalies. So each image is normalized with respect to itself only.
+
+Fashion-MNIST Figure 4 uses $\eta = 0 . 8 5$ and sets $\mathcal { X }$ to the complete balanced test set.
+
+CIFAR-10 Figures 6 and 9 use $\eta = 0 . 8 5$ and set $\mathcal { X }$ to $X$ for each heatmap $I ( X )$ to show relative anomalies. So each image is normalized with respect to itself only.
+
+ImageNet Figure 5 uses $\eta = 0 . 9 7$ and sets $\mathcal { X }$ to the complete balanced test set.
+
+Pascal VOC Figure 8 uses $\eta = 0 . 9 9$ and sets $\mathcal { X }$ to the complete balanced test set.
+
+Heatmap Upsampling For the Gaussian kernel heatmap upsampling described in Algorithm 1, we set $\sigma$ to 1.2 for CIFAR-10 and Fashion-MNIST, to 8 for ImageNet and Pascal VOC, and to 12 for MVTec-AD.
+
+# D DETAILS ON THE NETWORK ARCHITECTURES
+
+Here we provide the complete FCDD network architectures we used on the different datasets.
+
+# Fashion-MNIST
+
+<table><tr><td>Layer (type) Output Shape Param #</td></tr><tr><td>Conv2d-1 [-1, 128, 28, 28] 3,328</td></tr><tr><td>BatchNorm2d-2 [-1， 128, 28， 28] 256</td></tr><tr><td>LeakyReLU-3 [-1， 128， 28， 28] 0</td></tr><tr><td>MaxPool2d-4 [-1， 128, 14， 14] 0</td></tr><tr><td>Conv2d-5 [-1， 128， 14， 14] 409,728</td></tr><tr><td>MaxPool2d-6 [-1，128, 7， 71 0</td></tr><tr><td>Conv2d-7 [-1，1， 7， 71 129</td></tr></table>
+
+Total params: 413,441 Trainable params: 413,441 Non-trainable params: 0 Receptive field (pixels): 16 x 16
+
+CIFAR-10   
+
+<table><tr><td>Layer (type) Output Shape Param #</td></tr><tr><td>Conv2d-1 32， 32] 3,584</td></tr><tr><td>[-1， 128， BatchNorm2d-2 [-1， 128， 32， 32] 256</td></tr><tr><td>LeakyReLU-3 [-1， 128, 32， 32] 0</td></tr><tr><td>MaxPool2d-4 [-1， 128， 16, 16] 0</td></tr><tr><td>Conv2d-5 [-1， 256， 16, 16] 295,168</td></tr><tr><td>BatchNorm2d-6 [-1， 256, 16, 16] 512</td></tr><tr><td>LeakyReLU-7 [-1， 256， 16, 16] 0</td></tr><tr><td>Conv2d-8 [-1， 256， 16, 16] 590,080</td></tr><tr><td>BatchNorm2d-9 [-1， 256, 16, 16] 512</td></tr><tr><td>LeakyReLU-10 [-1， 256， 16, 16] 0</td></tr><tr><td>MaxPool2d-11 [-1， 256, 8， 8] 0</td></tr><tr><td>Conv2d-12 [-1，128， 8， 8] 295,040</td></tr><tr><td>Conv2d-13 [-1，1， 8， 8] 129</td></tr></table>
+
+Total params: 1,185,281 Trainable params: 1,185,281 Non-trainable params: 0 Receptive field (pixels): 22 x 22
+
+# ImageNet, MVTec-AD, and Pascal VOC
+
+<table><tr><td rowspan=1 colspan=7>Layer(type)                 OutputShape         Param #</td></tr><tr><td rowspan=1 colspan=7>Conv2d-1         [-1，64，224，224]            1,792</td></tr><tr><td rowspan=1 colspan=7>BatchNorm2d-2         [-1，64，224，224]              128</td></tr><tr><td rowspan=1 colspan=7>ReLU-3         [-1，64，224，224]                 0</td></tr><tr><td rowspan=1 colspan=1>MaxPool2d-4</td><td rowspan=1 colspan=6>[-1，64，112，112]                 0</td></tr><tr><td rowspan=1 colspan=1>Conv2d-5</td><td rowspan=1 colspan=5>[-1，128,112，112]</td><td rowspan=1 colspan=1>73,856</td></tr><tr><td rowspan=1 colspan=1>BatchNorm2d-6</td><td rowspan=1 colspan=4>[-1，128，112，</td><td rowspan=1 colspan=1>112]</td><td rowspan=1 colspan=1>256</td></tr><tr><td rowspan=1 colspan=1>ReLU-7</td><td rowspan=1 colspan=4>[-1，128，112，</td><td rowspan=1 colspan=1>112]</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>MaxPool2d-8</td><td rowspan=1 colspan=4>[-1，128，56,</td><td rowspan=1 colspan=1>56]</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Conv2d-9</td><td rowspan=1 colspan=4>[-1，256，56,</td><td rowspan=1 colspan=1>56]</td><td rowspan=1 colspan=1>295,168</td></tr><tr><td rowspan=1 colspan=1>BatchNorm2d-10</td><td rowspan=1 colspan=4>[-1，256，56,</td><td rowspan=1 colspan=1>56]</td><td rowspan=1 colspan=1>512</td></tr><tr><td rowspan=1 colspan=1>ReLU-11</td><td rowspan=1 colspan=4>[-1，256,56，</td><td rowspan=1 colspan=1>56]</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1>Conv2d-12</td><td rowspan=1 colspan=4>[-1，256,56,</td><td rowspan=1 colspan=1>56]</td><td rowspan=1 colspan=1>590,080</td></tr><tr><td rowspan=1 colspan=1>BatchNorm2d-13</td><td rowspan=1 colspan=4>[-1，256，56,</td><td rowspan=1 colspan=1>56]</td><td rowspan=1 colspan=1>512</td></tr><tr><td rowspan=2 colspan=1>ReLU-14MaxPool2d-15</td><td rowspan=1 colspan=4>[-1，256,56,</td><td rowspan=1 colspan=1>56]</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>[-1，256，</td><td rowspan=1 colspan=1>28,</td><td rowspan=1 colspan=1>28]</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=7 colspan=1>Conv2d-16BatchNorm2d-17ReLU-18Conv2d-19BatchNorm2d-20ReLU-21Conv2d-22</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>[-1，512,</td><td rowspan=1 colspan=1>28，</td><td rowspan=1 colspan=1>28]</td><td rowspan=1 colspan=1>1,180,160</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>[-1，512，</td><td rowspan=1 colspan=1>28,</td><td rowspan=1 colspan=1>28]</td><td rowspan=1 colspan=1>1,024</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>[-1，512，</td><td rowspan=1 colspan=1>28，</td><td rowspan=1 colspan=1>28]</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[-1，</td><td rowspan=1 colspan=1>512，</td><td rowspan=1 colspan=2>28,28]</td><td rowspan=1 colspan=1>2,359,808</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[-1，</td><td rowspan=1 colspan=1>512，</td><td rowspan=1 colspan=2>28,28]</td><td rowspan=1 colspan=1>1,024</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[-1，</td><td rowspan=1 colspan=1>512,</td><td rowspan=1 colspan=2>28,28]</td><td rowspan=1 colspan=1>0</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>[-1，</td><td rowspan=1 colspan=2>1，28，28]</td><td rowspan=1 colspan=1>513</td></tr></table>
+
+Total params: 4,504,833 Trainable params: 4,504,833 Non-trainable params: 0 Receptive field (pixels): 62 x 62
+
+# E TRAINING AND OPTIMIZATION
+
+Here we provide the training and optimization details for the individual experiments from Section 4.
+
+We apply common pre-processing (e.g. data normalization) and data augmentation steps in our data loading pipeline. To sample auxiliary anomalies in an online manner during training, each nominal sample of a batch has a $50 \%$ chance of being replaced by a randomly picked auxiliary anomaly. This leads to balanced training batches for sufficiently large batch sizes. One epoch in our implementation still refers to the original nominal data training set size, so that approximately $50 \%$ of the nominal samples have been seen per training epoch. Below, we list further details for the specific datasets.
+
+Fashion-MNIST We train for 400 epochs using a batch size of 128 samples. We optimize the network parameters using SGD (Bottou, 2010) with Nesterov momentum $\mathbf { \chi } _ { \mathcal { \mu } } = 0 . 9$ ) (Sutskever et al., 2013), weight decay of $1 0 ^ { - 6 }$ and an initial learning rate of 0.01, which decreases the previous learning rate per epoch by a factor of 0.98. The pre-processing pipeline is: (1) Random crop to size 28 with beforehand zero-padding of 2 pixels on all sides (2) random horizontal flipping with a chance of $50 \%$ (3) data normalization.
+
+CIFAR-10 We train for 600 epochs using a batch size of 200 samples. We optimize the network using Adam (Kingma and Ba, 2015) $( \beta = ( 0 . 9 , 0 . 9 9 9 ) )$ with weight decay $\mathrm { \dot { 1 } 0 ^ { - 6 } }$ and an initial learning rate of 0.001 which is decreased by a factor of 10 at epoch 400 and 500. The pre-processing pipeline is: (1) Random color jitter with all parameters3 set to 0.01 (2) random crop to size 32 with beforehand zero-padding of 4 pixels on all sides (3) random horizontal flipping with a chance of $50 \%$ (4) additive Gaussian noise with $\sigma = 0 . 0 0 1$ (5) data normalization.
+
+ImageNet We use the same setup as in CIFAR-10, but resize all images to size $2 5 6 \times 2 5 6$ before forwarding them through the pipeline and change the random crop to size 224 with no padding. Test samples are center cropped to a size of 224 before being normalized.
+
+Pascal VOC We use the same setup as in CIFAR-10, but resize all images to size $2 2 4 \times 2 2 4$ before forwarding them through the pipeline and remove the Random Crop step.
+
+MVTec-AD For MVTec-AD we redefine an epoch to be ten times an iteration of the full dataset because this improves the computational performance of the data pipeline. We train for 200 epochs using SGD with Nesterov momentum $\langle \mu = 0 . 9 \rangle$ ), weight decay $1 0 ^ { - \bar { 4 } }$ , and an initial learning rate of 0.001, which decreases per epoch by a factor of 0.985. The pre-processing pipeline is: (1) Resize to $2 4 0 \times 2 4 0$ pixels (2) random crop to size 224 with no padding (3) random color jitter with either all parameters set to 0.04 or 0.0005, randomly chosen (4) $50 \%$ chance to apply additive Gaussian noise (5) data normalization.
+
+# F QUANTITATIVE DETECTION RESULTS FOR INDIVIDUAL CLASSES
+
+Table 6 shows the class-wise results on Fashion-MNIST for AE, Deep Support Vector Data Description (DSVDD) (Ruff et al., 2018; Bergman and Hoshen, 2020) and Geometric Transformation based AD (GEO) (Golan and El-Yaniv, 2018).
+
+Table 6: AUC scores for all classes of Fashion-MNIST (Xiao et al., 2017).   
+
+<table><tr><td></td><td colspan="3">without OE</td><td>with OE</td></tr><tr><td></td><td>AE</td><td>DSVDD*</td><td>Geo*</td><td>FCDD</td></tr><tr><td>T-Shirt/Top</td><td>0.85</td><td>0.98</td><td>0.99</td><td>0.82</td></tr><tr><td>Trouser</td><td>0.91</td><td>0.90</td><td>0.98</td><td>0.98</td></tr><tr><td>Pullover</td><td>0.78</td><td>0.91</td><td>0.91</td><td>0.84</td></tr><tr><td>Dress</td><td>0.88</td><td>0.94</td><td>0.90</td><td>0.92</td></tr><tr><td>Coat</td><td>0.88</td><td>0.89</td><td>0.92</td><td>0.87</td></tr><tr><td>Sandal</td><td>0.45</td><td>0.92</td><td>0.93</td><td>0.90</td></tr><tr><td>Shirt</td><td>0.70</td><td>0.83</td><td>0.83</td><td>0.75</td></tr><tr><td>Sneaker</td><td>0.96</td><td>0.99</td><td>0.99</td><td>0.99</td></tr><tr><td>Bag</td><td>0.87</td><td>0.92</td><td>0.91</td><td>0.86</td></tr><tr><td>Ankle Boot</td><td>0.96</td><td>0.99</td><td>0.99</td><td>0.94</td></tr><tr><td>Mean</td><td>0.82</td><td>0.93</td><td>0.94</td><td>0.89</td></tr></table>
+
+In Table 7 the class-wise results for CIFAR-10 are reported. Competitors without OE are AE (Ruff et al., 2018), DSVDD (Ruff et al., 2018), GEO (Golan and El-Yaniv, 2018) and an adaptation of GEO $\mathrm { ( G E O + ) }$ (Hendrycks et al., 2019b). Competitors with OE are the focal loss classifier (Hendrycks et al., 2019b), again $\mathrm { G E O + }$ (Hendrycks et al., 2019b), Deep Semi-supervised Anomaly Detection (Deep SAD) (Ruff et al., 2020b;a) and the hypersphere Classifier (Ruff et al., 2020a).
+
+In Table 8 the class-wise results for Imagenet are shown, where competitors are the AE, the focal loss classifier (Hendrycks et al., 2019b), Geo $^ +$ (Hendrycks et al., 2019b), Deep SAD (Ruff et al., 2020b) and HSC (Ruff et al., 2020a). Results from the literature are marked with an asterisk.
+
+Table 7: AUC scores for all classes of CIFAR-10 (Krizhevsky et al., 2009).   
+
+<table><tr><td></td><td colspan="4">without OE</td><td colspan="5">with OE</td></tr><tr><td></td><td>AE*</td><td>DSVDD*</td><td>GEO*</td><td>Geo+*</td><td>Focal*</td><td>Geo+*</td><td>Deep SAD*</td><td>HSC*</td><td>FCDD</td></tr><tr><td>Airplane</td><td>0.59</td><td>0.62</td><td>0.75</td><td>0.78</td><td>0.88</td><td>0.90</td><td>0.94</td><td>0.97</td><td>0.95</td></tr><tr><td>Automobile</td><td>0.57</td><td>0.66</td><td>0.96</td><td>0.97</td><td>0.94</td><td>0.99</td><td>0.98</td><td>0.99</td><td>0.96</td></tr><tr><td>Bird</td><td>0.49</td><td>0.51</td><td>0.78</td><td>0.87</td><td>0.79</td><td>0.94</td><td>0.90</td><td>0.93</td><td>0.91</td></tr><tr><td>Cat</td><td>0.58</td><td>0.59</td><td>0.72</td><td>0.81</td><td>0.80</td><td>0.88</td><td>0.87</td><td>0.90</td><td>0.90</td></tr><tr><td>Deer</td><td>0.54</td><td>0.61</td><td>0.88</td><td>0.93</td><td>0.82</td><td>0.97</td><td>0.95</td><td>0.97</td><td>0.94</td></tr><tr><td>Dog</td><td>0.62</td><td>0.66</td><td>0.88</td><td>0.90</td><td>0.86</td><td>0.94</td><td>0.93</td><td>0.94</td><td>0.93</td></tr><tr><td>Frog</td><td>0.51</td><td>0.68</td><td>0.83</td><td>0.91</td><td>0.93</td><td>0.97</td><td>0.97</td><td>0.98</td><td>0.97</td></tr><tr><td>Horse</td><td>0.59</td><td>0.67</td><td>0.96</td><td>0.97</td><td>0.88</td><td>0.99</td><td>0.97</td><td>0.98</td><td>0.96</td></tr><tr><td>Ship</td><td>0.77</td><td>0.76</td><td>0.93</td><td>0.95</td><td>0.93</td><td>0.99</td><td>0.97</td><td>0.98</td><td>0.97</td></tr><tr><td>Truck</td><td>0.67</td><td>0.73</td><td>0.91</td><td>0.93</td><td>0.92</td><td>0.99</td><td>0.96</td><td>0.97</td><td>0.96</td></tr><tr><td>Mean</td><td>0.59</td><td>0.65</td><td>0.86</td><td>0.90</td><td>0.87</td><td>0.96</td><td>0.95</td><td>0.96</td><td>0.95</td></tr></table>
+
+Table 8: AUC scores for 30 classes of ImageNet (Deng et al., 2009).   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">without OE AE</td><td colspan="5">with OE</td></tr><tr><td>Focal*</td><td>Geo+*</td><td>Deep SAD*</td><td>HSC*</td><td>FCDD</td></tr><tr><td>Acorn</td><td>0.45</td><td>×</td><td>×</td><td>0.99</td><td>0.99</td><td>0.97</td></tr><tr><td>Airliner</td><td>0.80</td><td>×</td><td>×</td><td>0.97</td><td>1.00</td><td>0.98</td></tr><tr><td>Ambulance</td><td>0.25</td><td>×</td><td>×</td><td>0.99</td><td>1.00</td><td>0.99</td></tr><tr><td>American alligator</td><td>0.61</td><td>×</td><td>×</td><td>0.93</td><td>0.98</td><td>0.97</td></tr><tr><td>Banjo</td><td>0.45</td><td>×</td><td>×</td><td>0.97</td><td>0.98</td><td>0.91</td></tr><tr><td>Barn</td><td>0.59</td><td>×</td><td>×</td><td>0.99</td><td>1.00</td><td>0.97</td></tr><tr><td>Bikini</td><td>0.46</td><td>×</td><td>×</td><td>0.97</td><td>0.99</td><td>0.94</td></tr><tr><td>Digital clock</td><td>0.63</td><td>×</td><td>×</td><td>0.99</td><td>0.97</td><td>0.92</td></tr><tr><td>Dragonfly</td><td>0.62</td><td>×</td><td>×</td><td>0.99</td><td>0.98</td><td>0.98</td></tr><tr><td>Dumbbell</td><td>0.42</td><td>×</td><td>×</td><td>0.93</td><td>0.92</td><td>0.88</td></tr><tr><td>Forklift</td><td>0.28</td><td>×</td><td>×</td><td>0.91</td><td>0.99</td><td>0.94</td></tr><tr><td>Goblet</td><td>0.63</td><td>×</td><td>×</td><td>0.92</td><td>0.94</td><td>0.90</td></tr><tr><td>Grand piano</td><td>0.45</td><td>×</td><td>×</td><td>1.00</td><td>0.97</td><td>0.95</td></tr><tr><td>Hotdog</td><td>0.48</td><td>×</td><td>×</td><td>0.96</td><td>0.99</td><td>0.97</td></tr><tr><td>Hourglass</td><td>0.58</td><td>×</td><td>×</td><td>0.96</td><td>0.97</td><td>0.92</td></tr><tr><td>Manhole cover</td><td>0.70</td><td>×</td><td>×</td><td>0.99</td><td>1.00</td><td>1.00</td></tr><tr><td>Mosque</td><td>0.72</td><td>×</td><td>×</td><td>0.99</td><td>0.99</td><td>0.97</td></tr><tr><td>Nail</td><td>0.57</td><td>×</td><td>×</td><td>0.93</td><td>0.94</td><td>0.92</td></tr><tr><td>Parking meter</td><td>0.45</td><td>×</td><td>×</td><td>0.99</td><td>0.93</td><td>0.87</td></tr><tr><td>Pillow</td><td>0.40</td><td>×</td><td>×</td><td>0.99</td><td>0.94</td><td>0.94</td></tr><tr><td>Revolver</td><td>0.60</td><td>×</td><td>×</td><td>0.98</td><td>0.98</td><td>0.93</td></tr><tr><td>Rotary dial telephone</td><td>0.58</td><td>×</td><td>×</td><td>0.90</td><td>0.98</td><td>0.91</td></tr><tr><td>Schooner</td><td>0.65</td><td>×</td><td>×</td><td>0.99</td><td>0.99</td><td>0.96</td></tr><tr><td>Snowmobile</td><td>0.54</td><td>×</td><td>×</td><td>0.98</td><td>0.99</td><td>0.97</td></tr><tr><td>Soccer ball</td><td>0.46</td><td>×</td><td>×</td><td>0.97</td><td>0.93</td><td>0.86</td></tr><tr><td>Stingray</td><td>0.84</td><td>×</td><td>×</td><td>0.99</td><td>0.99</td><td>0.97</td></tr><tr><td>Strawberry</td><td>0.44</td><td>×</td><td>×</td><td>0.98</td><td>0.99</td><td>0.97</td></tr><tr><td>Tank</td><td>0.57</td><td>×</td><td>×</td><td>0.97</td><td>0.99</td><td>0.96</td></tr><tr><td>Toaster</td><td>0.59</td><td>×</td><td>×</td><td>0.98</td><td>0.92</td><td>0.79</td></tr><tr><td>Volcano</td><td>0.90</td><td>×</td><td>×</td><td>0.90</td><td>1.00</td><td>0.97</td></tr><tr><td>Mean</td><td>0.56</td><td>0.56</td><td>0.86</td><td>0.97</td><td>0.97</td><td>0.94</td></tr></table>
+
+# G FURTHER QUALITATIVE ANOMALY HEATMAP RESULTS
+
+In this section we report some further anomaly heatmaps, unblurred baseline heatmaps, as well as class-wise heatmaps for all datasets.
+
+Unblurred Anomaly Heatmap Baselines Here we show unblurred baseline heatmaps for the figures in Section 4.1. Figures 12, 13, and 14 show the unblurred heatmaps for Fashion-MNIST, ImageNet, and CIFAR-10 respectively.
+
+![](images/bf566bbf15807b34e65eff014a272ca78a6e5d933dfaed726d62c23048b288a0.jpg)  
+Figure 12: Anomaly heatmaps for anomalous test samples of a Fashion-MNIST model trained on nominal class “trousers.” In (a) CIFAR-100 was used for OE and in (b) EMNIST.
+
+![](images/713d6ce56af5fd035e9117c7eecb9cc85e72e5503d064f1a8bede1f1f2047577.jpg)  
+Figure 13: Anomaly heatmaps of an ImageNet model trained on nominal class “acorns.” (a) are nominal and (b) anomalous samples.
+
+![](images/69e2db4c8878829d358fef26b17bf0d24957b48c626d9d19f7eb4e1fdd0d362f.jpg)  
+Figure 14: Anomaly heatmaps for three anomalous test samples (Input left) on a CIFAR-10 model trained on nominal class “airplane.” The second, third, and fourth blocks show the heatmaps of FCDD (Ours), gradient-based heatmaps of HSC, and AE heatmaps respectively. For Ours and Grad, we grow the number of OE samples from 2, 8, 128, 2048 to full OE. AE is not able to incorporate OE.
+
+Class-wise Anomaly Heatmaps Due to space restrictions we have only shown heatmaps for some of the classes in the main paper. Here we also report a collection of heatmaps for all classes.
+
+We show heatmaps with adjusted contrast curves by setting $\mathcal { X }$ to the balanced set of all samples for all datasets in this section. Further, we set $\eta = 0 . 8 5$ for Fashion-MNIST and CIFAR-10, $\eta = 0 . 9 9$ for MVTec-AD, and $\eta = 0 . 9 7$ for ImageNet. Note that, to keep the heatmaps for different classes comparable, we use a unified normalization for all heatmaps in one figure. However, since for each class a separate anomaly detector is trained, this yields suboptimal visualizations for some of the classes (for example, the “toothbrush” images for MVTec-AD in Figure 18 where the heatmaps just show a huge red blob). Tweaking the normalization for such classes reveals that the heatmaps actually tend to mark the correct anomalous regions, which in the case of “toothbrushes” can be seen in the explanation performance evaluation in Table 2.
+
+The rows in all heatmaps show the following: (1) Input samples (2) FCDD heatmaps (3) gradient heatmaps with HSC (4) autoencoder reconstruction heatmaps. Heatmaps for MVTec-AD add a fifth row containing the ground-truth anomaly map.
+
+Heatmaps for Fashion-MNIST using auxiliary anomalies from CIFAR-100 are in Figure 15, using EMNIST for OE instead are in Figure 16. CIFAR-10 heatmaps are in Figure 17, and heatmaps for all classes of MVTec-AD are in Figure 18. Finally, we present ImageNet heatmaps in Figures 19 and 20.
+
+![](images/854aa9570e335413ba17ef874c3a0cb9974abf27b25e955ce03903436f802fee.jpg)  
+Figure 15: Anomaly heatmaps for anomalous test samples in Fashion-MNIST using CIFAR-100 OE. Columns are ordered by increasing anomaly score from left to right. The subcaptions refer to the nominal class that each model is trained on, for which some examples are also displayed as a separate column on the left.
+
+![](images/f54d932778369151a3b9d1920f3740580596a2e2695240eac99aabb47f46d6de.jpg)  
+Figure 16: Anomaly heatmaps for anomalous test samples in Fashion-MNIST using EMNIST OE. Columns are ordered by increasing anomaly score from left to right. The subcaptions refer to the nominal class that each model is trained on, for which some examples are also displayed as a separate column on the left.
+
+![](images/2e42bbcc25bf012d9d62333d47002ead8f77d5ae7c05d7bc37f7ba94e199af2c.jpg)  
+Figure 17: Anomaly heatmaps for anomalous test samples in CIFAR-10. Columns are ordered by increasing anomaly score from left to right. The subcaptions refer to the nominal class that each model is trained on, for which some examples are also displayed as a separate column on the left.
+
+![](images/113883b930f8b320ea27b6b2b719b8f98177f875aff107c8266adcd2c7f79d83.jpg)  
+Figure 18: Anomaly heatmaps for anomalous test samples in MVTec-AD. Columns are ordered by increasing anomaly score from left to right. The subcaptions refer to the nominal class that each model is trained on.
+
+![](images/17a3e5bdba9ad747b544a39a0ef48421ac4e4156dcaa4559e0f8297b20cbb47b.jpg)  
+Figure 19: Anomaly heatmaps for anomalous test samples in ImageNet, where classes 1-21 are shown. Columns are ordered by increasing anomaly score from left to right. The subcaptions refer to the nominal class that each model is trained on, for which some examples are also displayed as a separate column on the left.
+
+![](images/d8c33e1273c400c98359f72275cd39060342ef42cba535ecc7e2e2c162d54a4e.jpg)  
+Figure 20: Anomaly heatmaps for anomalous test samples in ImageNet, where classes 22-30 are shown. Columns are ordered by increasing anomaly score from left to right. The subcaptions refer to the nominal class that each model is trained on, for which some examples are also displayed as a separate column on the left.

parse/train/ByJDAIe0b/ByJDAIe0b.md

@@ -0,0 +1,487 @@
+# INTEGRATING EPISODIC MEMORY INTO A REINFORCEMENT LEARNING AGENT USING RESERVOIR SAMPLING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Episodic memory is a psychology term which refers to the ability to recall specific events from the past. We suggest one advantage of this particular type of memory is the ability to easily assign credit to a specific state when remembered information is found to be useful. Inspired by this idea, and the increasing popularity of external memory mechanisms to handle long-term dependencies in deep learning systems, we propose a novel algorithm which uses a reservoir sampling procedure to maintain an external memory consisting of a fixed number of past states. The algorithm allows a deep reinforcement learning agent to learn online to preferentially remember those states which are found to be useful to recall later on. Critically this method allows for efficient online computation of gradient estimates with respect to the write process of the external memory. Thus unlike most prior mechanisms for external memory it is feasible to use in an online reinforcement learning setting.
+
+Much of reinforcement learning (RL) theory is based on the assumption that the environment has the Markov property, meaning that future states are independent of past states given the present state. This implies the agent has all the information it needs to make an optimal decision at each time and therefore has no need to remember the past. This is however not realistic in general, realistic problems often require significant information from the past to make an informed decision in the present, and there is often no obvious way to incorporate the relevant information into an expanded present state. It is thus desirable to establish techniques for learning a representation of the relevant details of the past (e.g. a memory, or learned state) to facilitate decision making in the present.
+
+A popular approach to integrate information from the past into present decision making is to use some variant of a recurrent neural network, possibly coupled to some form of external memory, trained with backpropagation through time. This can work well for many tasks, but generally requires backpropagating many steps into the past which is not practical in an online RL setting. In purely recurrent architectures one way to make online training practical is to simply truncate gradients after a fixed number of steps. In architectures which include some form of external memory however it is not clear that this is a viable option as the intent of the external memory is generally to capture long term dependencies which would be difficult for a recurrent architecture alone to handle, especially when trained with truncated gradients. Truncating gradients to the external memory would likely greatly hinder this capability.
+
+In this work we explore a method for adding external memory to a reinforcement learning architecture which can be efficiently trained online. We liken our method to the idea of episodic memory from psychology. In this approach the information stored in memory is constrained to consist of a finite set of past states experienced by the agent. In this work, by states we mean observations explicitly provided by the environment. In general, states could be more abstract, such as the internal state of an RNN or predictions generated by something like the Horde architecture of Sutton et al. (2011). By storing states explicitly we enforce that the information recorded also provides the context in which it was recorded. We can therefore assign credit to the recorded state without explicitly backpropagating through time between when the information proves useful and when it was recorded. If a recorded state is found to be useful we train the agent to preferentially remember similar states in the future.
+
+In our approach the set of states in memory at a given time is drawn from a distribution over all $n$ -subsets (subsets of size $n$ ) of visited states, parameterized by a weight value assigned to each state by a trained model. To allow us to draw from such a distribution without maintaining all visited states in memory we introduce a reservoir sampling technique. Reservoir sampling refers to a class of algorithms for sampling from a distribution over $n$ -subsets of items from a larger set streamed one item at a time. The goal is to ensure, through specific add and drop probabilities, that the $n$ items in the reservoir at each time-step correspond to a sample from the desired distribution over $n$ -subsets of all observed items. Two important examples which sample from different distributions are found in Chao (1982) and Efraimidis & Spirakis (2006). In this work we will define our own distribution and sampling procedure to suit our needs.
+
+# 1 RELATED WORK
+
+Deep learning systems which make use of an external memory have received a lot of interest lately. Two prototypical examples are found in Graves et al. (2014) and the follow-up Graves et al. (2016). These systems use an LSTM controller attached to read and write heads of a fully differentiable external memory and train the combined system to perform algorithmic tasks. Contrary to our approach, training is done entirely by backpropagation through time. See also Zaremba & Sutskever (2015), Joulin & Mikolov (2015), Sukhbaatar et al. (2015), Gulcehre et al. (2017) and Kaiser et al. (2017) for more examples of deep learning systems with integrated external memory.
+
+More directly related to the present work is the application of deep RL to non-markov tasks, in particular Oh et al. (2016). They experiment with architectures using a combination of key-value memory and a recurrent neural networks. The memory saves keys and values corresponding to the last $N$ observations for some integer $N$ , thus it is inherently limited in temporal extent but does not require any mechanism for information triage. They test on problems in the Minecraft domain which could provide compelling testbeds for a potential follow-up to the present work. See also Bakker et al. (2003), Wierstra et al. (2010), Zhang et al. (2016) and Hausknecht & Stone (2015) for more examples of applying deep RL to non-markov tasks.
+
+# 2 ARCHITECTURE
+
+Our model is based around an advantage actor critic architecture (Mnih et al., 2016) consisting of separate value and policy networks. In addition we include an external memory $\mathcal { M }$ consisting of a set of $n$ past visited states $( S _ { t _ { 0 } } , . . , S _ { t _ { n - 1 } } )$ with associated importance weights $( w _ { t _ { 0 } } , . . . , w _ { t _ { n - 1 } } )$ . The query network $q ( S _ { t } )$ outputs a vector of size equal to the state size with tanh activation. At each time step a single item $S _ { t _ { i } }$ is drawn from the memory to condition the policy according to:
+
+$$
+Q ( S _ { t _ { i } } | { \cal M } _ { t } ) = \exp \left( \langle q ( S _ { t } ) | S _ { t _ { i } } \rangle / \tau \right) \Biggl / \sum _ { j = 0 } ^ { n - 1 } \exp \left( \langle q ( S _ { t } ) | S _ { t _ { j } } \rangle / \tau \right)
+$$
+
+where $\tau$ is a positive learnable temperature parameter. The state, $m _ { t }$ , selected from memory is given as input to the policy network along with the current state, both of which condition the resulting policy. Finally the write network takes the current state as input and outputs a single value with sigmoid activation. This value is used to determine how likely the present state is to be written to and subsequently retained in the memory according to the distribution in equation 7 which will be throughly explained in section 3. An illustration of this architecture is shown in figure 1.
+
+# 3 ALGORITHM
+
+For the most part our model is trained using standard stochastic gradient descent on common RL loss functions. The value network is trained by gradient descent on the squared one step temporal different error $\delta _ { t } ^ { 2 }$ where $\delta _ { t } = r _ { t + 1 } + V ( S _ { t + 1 } ) - V ( S _ { t } )$ , and the gradient is passed only through $V ( S _ { t } )$ . The policy is trained using the advantage loss $- \bar { \delta } _ { t } \log ( \pi ( a _ { t } \mathbf { \bar { | } } S _ { t } , m _ { t } ) )$ with gradients passed only through $\pi ( a _ { t } | S _ { t } , m _ { t } )$ . The query network is trained similarly on the loss $- \delta _ { t } \log ( Q ( m _ { t } | S _ { t } ) )$ with gradients passed only through $Q ( { m } _ { t } | S _ { t } )$ . We train online, performing one update per timestep with no experience replay. The main innovation of this work is in the training method for the write network which is described in sections 3.1 and 3.2. There are two main desiderata we wish to satisfy with the write network. First we want to use the weights $w ( S _ { t } )$ generated by the network in a reservoir sampling algorithm such that the probability of a particular state $S _ { \hat { t } }$ being present in memory at any given future time $t > \hat { t }$ is proportional to the associated weight $w ( S _ { \hat { t } } )$ . Second we want to obtain estimates of the gradient of the return with respect to the weight of each item in memory such that we can perform approximate gradient descent on the generated weights.
+
+![](images/48a3bd31a1808f05a63b4278bda820c6d3633ba9aba13cf190db9cd0aa30849a.jpg)  
+Figure 1: Episodic memory architecture, each grey circle represents a neural network module. Input state (S) is given separately to the query (q), write (w), value (V) and policy $( \pi )$ networks at each time step. The query network outputs a vector of size equal to the input state size which is used (via equation 1) to choose a past state from the memory $( m _ { 1 } \mathrm { ~ } , m _ { 2 }$ or $m _ { 3 }$ in the above diagram) to condition the policy. The write network assigns a weight to each new state determining how likely it is to stay in memory. The policy network assigns probabilities to each action conditioned on current state and recalled state. The value network estimates expected return (value) from the current state.
+
+# 3.1 GRADIENT ESTIMATE
+
+For brevity, in this section we will use the notation $E _ { t } [ x ]$ to denote $E [ x | S _ { 0 } , . . . , S _ { t } ]$ i.e. the expectation conditioned on the entire history of state visitation up to time $t$ . Similarly $P _ { t } ( x )$ will represent probability conditioned on the entire history of state visitation. All expectations and probabilities are assumed to be with respect to the current policy, query and write network. Let $\mathcal { A }$ represent the set of available actions and $A _ { t }$ the action selected at time $t$ .
+
+# 3.1.1 ONE-STATE MEMORY CASE
+
+To introduce the idea we first present our gradient estimation procedure for the case when our memory can store just one state, and thus there is no need to query. Here $m _ { t }$ represents the state in memory at time $t$ and thus, in the one-state memory case, the state read from memory by the agent at time $t$ . Assume the stored memory is drawn from a distribution parameterized as follows by a set of weights $\{ w _ { i } | i \in \{ 0 , . . . , t - 1 \} \}$ associated with each state $S _ { i }$ when the state is first visited:
+
+$$
+P _ { t } ( m _ { t } = S _ { i } ) = { w _ { i } \Bigg / \sum _ { j = 0 } ^ { t - 1 } w _ { j } }
+$$
+
+We can then write the expected return $R _ { t } = r _ { t + 1 } + \gamma r _ { t + 2 } + \ldots$ as follows:
+
+$$
+E _ { t } [ R _ { t } ] = \sum _ { k = 0 } ^ { t - 1 } P _ { t } ( m _ { t } = S _ { k } ) \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { k } ) E _ { t } [ R _ { t } | A _ { t } = a ]
+$$
+
+In order to perform gradient descent on the weights $w _ { i }$ we wish to estimate the gradient of this expectation value with respect to each weight. In particular we will derive an estimate of this gradient using an actor critic method which is unbiased if the critic’s evaluation is correct. Additionally our estimate will be non-zero only for the $w _ { i }$ associated with the index $i$ such that $m _ { t } ~ = ~ S _ { i }$ . This means if our weights $w _ { i }$ are generated by a neural network, we will only have to propagate gradients through the single stored state. This is crucial to allow our algorithm to run online, as otherwise we would need to store every visited state to compute the gradient estimate.
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { \partial } { \partial w _ { i } } E _ { t } [ R _ { t } ] = \sum _ { k = 0 } ^ { t - 1 } \Bigg ( \frac { \partial P _ { t } ( m _ { t } = S _ { k } ) } { \partial w _ { i } } \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { k } ) E _ { t } [ R _ { t } | A _ { t } = a ] } \\ & { \quad \quad \quad \quad \quad + P _ { t } ( m _ { t } = S _ { k } ) \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { k } ) \frac { \partial E _ { t } [ R _ { t } | A _ { t } = a ] } { \partial w _ { i } } \Bigg ) } \end{array}
+$$
+
+We can rewrite this as:
+
+$$
+\begin{array} { r l r } {  { \frac { \partial } { \partial w _ { i } } E _ { t } [ R _ { t } ] = \frac { 1 } { w _ { i } } P _ { t } ( m _ { t } = S _ { i } ) ( \sum _ { a \in A } \pi ( a | S _ { t } , S _ { i } ) E _ { t } [ R _ { t } | A _ { t } = a ] - E _ { t } [ R _ { t } ] ) } } \\ & { } & { + \gamma E _ { t } [ \frac { \partial } { \partial w _ { i } } E _ { t + 1 } [ R _ { t + 1 } ] ] } \end{array}
+$$
+
+See appendix A for a detailed derivation of this expression. We will use a policy gradient approach, similar to REINFORCE (Williams, 1992), to estimate the this gradient using an estimator $G _ { i , t }$ such that $\begin{array} { r } { E _ { t } [ \sum _ { \hat { t } \ge t } \gamma ^ { \hat { t } - t } G _ { i , \hat { t } } ] \approx { \frac { \partial E _ { t } [ R _ { t } ] } { \partial w _ { i } } } } \end{array}$ ∂Et[Rt]∂w , thus the second term is estimated recursively on subsequent time-steps. In the present work we will focus on the undiscounted episodic case with the start-state value objective, for which it suffices to follow the first term in the above gradient expression for each visited state. This is also true in the continuing case with an average-reward objective. See Sutton et al. (2000) for further discussion of this distinction. Consider the gradient estimator:
+
+$$
+G _ { i , t } = { \left\{ \begin{array} { l l } { \delta _ { t } / w _ { i } ~ } & { { \mathrm { i f } } \ m _ { t } = S _ { i } } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. }
+$$
+
+which has expectation:
+
+$$
+\begin{array} { c l } { \displaystyle E _ { t } [ G _ { i , t } ] = \frac { 1 } { w _ { i } } P _ { t } ( m _ { t } = S _ { i } ) \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { i } ) E _ { t } [ r _ { t + 1 } + \gamma V ( S _ { t + 1 } ) - V ( S _ { t } ) | A _ { t } = a ] } \\ { \displaystyle \approx \frac { 1 } { w _ { i } } P _ { t } ( m _ { t } = S _ { i } ) \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { i } ) ( E _ { t } [ r _ { t + 1 } + \gamma E _ { t + 1 } [ R _ { t + 1 } ] | A _ { t } = a ] - E _ { t } [ R _ { t } ] ) } \\ { \displaystyle = \frac { 1 } { w _ { i } } P _ { t } ( m _ { t } = S _ { i } ) \left( \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { i } ) E _ { t } [ R _ { t } | A _ { t } = a ] - E _ { t } [ R _ { t } ] \right) } \end{array}
+$$
+
+Where the approximation is limited by the accuracy of our value function. In conventional policy gradient subtracting the state value (e.g. using $\delta _ { t } \doteq r _ { t + 1 } + \gamma V ( S _ { t + 1 } ) - V ( S _ { t } )$ instead of $r _ { t + 1 } +$ $\gamma V ( S _ { t + 1 } ) )$ is a means of variance reduction. Here it is critical to avoid computing gradients with respect to the denominator of equation 2, which allows our algorithm to run online while computing the gradient with respect to only the weight stored in memory.
+
+Given these estimated gradients with respect to $w _ { i }$ we apply the chain rule to compute $\begin{array} { r } { \frac { \partial R _ { t } } { \partial \theta _ { w } } = } \end{array}$ $\begin{array} { r } { \frac { \partial R _ { t } } { \partial w _ { i } } \frac { \partial w _ { i } } { \partial \theta _ { w } } \approx \sum _ { \hat { t } \geq t } G _ { i , \hat { t } } \frac { \partial w ( S _ { i } ) } { \partial \theta _ { w } } } \end{array}$ ,tˆ∂w(Si)∂θ , for each parameter θw of the write network. This gradient estimate is used in a gradient descent procedure to emphasize retention of states which improve the return.
+
+Gradient estimates are generated based on the stored values of $w _ { i }$ in memory but applied to the parameters of the network at the present time. With online updating, this introduces a potential multiple timescale issue which we conjecture will vanish in the limit of small learning rate, but leave further investigation to future work.
+
+There are a number of ways to extend the distribution defined in equation 2 to the case where multiple elements of a set must be selected (see for example Efraimidis & Spirakis (2006)). We will focus on a generalization which is less explored but which we will see in the following section results in gradient estimates which are an elegant generalization of the single-state memory case.
+
+# 3.1.2 MULTIPLE-STATE MEMORY CASE
+
+In this section and those that follow we will routinely use the notation $\textstyle { { \binom { Z } { n } } }$ where $Z$ is a set and $n$ an integer to indicate the set of all $n$ -subsets of $Z$ . Note that ${ \binom { Z } { 0 } } = \{ \emptyset \}$ and we adopt the convention Q $x = 1$ thus $\Sigma \ : \ : \ : \prod \ : x = 1$ which will be important in a few places in what follows. $x \in \varnothing$ $\hat { Z } \in \left( \begin{array} { l } { Z } \\ { 0 } \end{array} \right) x \in \hat { Z }$
+
+We will introduce some notation to facilitate reasoning about sets of states. Let $T _ { t } = \{ t ^ { \prime } : 0 \leq t ^ { \prime } \leq$ $t - 1 \}$ be the set of all time indices from 0 to $t - 1$ . Let ${ \hat { T } } \in \left( { \begin{array} { c } { T _ { t } } \\ { n } \end{array} } \right)$ be a set of $n$ indices chosen from $T _ { t }$ where $n$ is the memory size. Let $S _ { \hat { T } }$ be the set of states $\{ S _ { \hat { t } } : \hat { t } \in \hat { T } \}$ . Let $\mathcal { M } _ { t }$ be the set of states in memory at time $t$ . Let $Q ( S _ { \hat { t } } | S _ { \hat { T } } ^ { ' } )$ be the probability of querying $S _ { \hat { t } }$ given $\mathcal { M } _ { t } = S _ { \hat { T } }$ . The probability for a particular set of states being contained in memory is defined to be the following:
+
+$$
+P _ { t } ( \mathcal { M } _ { t } = S _ { \hat { T } } ) = \prod _ { i \in \hat { T } } w _ { i } \Bigg / \sum _ { \tilde { T } \in \binom { T _ { t } } { n } i \in \tilde { T } } \prod _ { i \in \tilde { T } } w _ { i }
+$$
+
+A straightforward extension of the derivation of the equation 5 shows that this choice results in a gradient estimate which is an elegant extension of the one-state memory case. The derivation is given in appendix B, the result for $\backslash \frac { \partial } { \partial w _ { i } } E _ { t } [ R _ { t } ]$ is:
+
+$$
+\frac { \partial } { \partial w _ { i } } E _ { t } [ R _ { t } ] = \sum _ { \hat { T } \in ( \binom { { T _ { t } } } { n } \ni i ) } \frac { 1 } { w _ { i } } P _ { t } ( \mathcal { M } _ { t } = S _ { \hat { T } } ) \left( \sum _ { j \in \hat { T } } Q ( S _ { j } | S _ { \hat { T } } ) \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { j } ) E _ { t } [ R _ { t } | A _ { t } = a ] \right) ,
+$$
+
+As in the single-state memory case we recursively handle the second term. To estimate the first term we could choose the following estimator $G _ { i , t }$ :
+
+$$
+G _ { i , t } = \left\{ { \begin{array} { l l } { \delta _ { t } / w _ { i } } & { { \mathrm { ~ i f ~ } } S _ { i } \in { \mathcal { M } } _ { t } } \\ { 0 } & { { \mathrm { ~ o t h e r w i s e } } } \end{array} } \right.
+$$
+
+This estimator is unbiased under the assumption the critic is perfect, however it scales poorly in terms of both variance and computation time as the memory size increases. This is because it requires updating every state in memory regardless of whether it was queried, spreading credit assignment and requiring compute time proportional to the product of the number of states in memory with the number of parameters in the write network. Instead we will further approximate the second term and perform an update only for the queried item. We rewrite the first term of equation 8 as follows:
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { 1 } { w _ { i } } P _ { t } ( \mathcal { M } _ { t } \ni S _ { i } ) \Bigg ( P _ { t } ( m _ { t } = S _ { i } | \mathcal { M } _ { t } \ni S _ { i } ) \left( \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { j } ) E _ { t } [ R _ { t } | A _ { t } = a ] - E _ { t } [ R _ { t } ] \right) } \\ & { \displaystyle + P _ { t } ( m _ { t } \neq S _ { i } | \mathcal { M } _ { t } \ni S _ { i } ) \left( \sum _ { a \in \mathcal { A } } P _ { t } ( A _ { t } = a | \mathcal { M } _ { t } \ni S _ { i } , m _ { t } \neq S _ { i } ) E _ { t } [ R _ { t } | A _ { t } = a ] - E _ { t } [ R _ { t } ] \right) \Bigg ) } \\ & { \displaystyle \approx \frac { 1 } { w _ { i } } P _ { t } ( m _ { t } = S _ { i } ) \left( \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { j } ) E _ { t } [ R _ { t } | A _ { t } = a ] - E _ { t } [ R _ { t } ] \right) } \end{array}
+$$
+
+This approximation is accurate to the extent that our query network is able to accurately select useful states. To see this, note that if querying a state when it’s in memory helps to generate a better expected return a well trained query network should do it with high probability and hence $P _ { t } ( m _ { t } \neq \boldsymbol { S } _ { i } | \mathcal { M } _ { t } \ni \boldsymbol { S } _ { i } )$ will be low. On the other hand if querying a state in memory is unhelpful $\displaystyle \left( \sum _ { a \in { \cal A } } P _ { t } ( A _ { t } = a | { \cal M } _ { t } \ni S _ { i } , m _ { t } \neq S _ { i } ) E _ { t } [ R _ { t } | A _ { t } = a ] - E _ { t } [ R _ { t } ] \right)$ will generally be small. With this approximation the gradient estimate becomes identical to the one-state memory case:
+
+$$
+G _ { i , t } = { \left\{ \begin{array} { l l } { \delta _ { t } / w _ { i } ~ } & { { \mathrm { i f } } \ m _ { t } = S _ { i } } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. }
+$$
+
+While this justification is not rigorous, this approximation should significantly improve computational and sample efficiency, and is used in our experiments in section 4.
+
+# 3.2 RESERVOIR SAMPLING PROCEDURE
+
+<table><tr><td>1: Ω ← Zeros(n+1)</td><td>1: function UPDATE(w,t)</td></tr><tr><td>2:←Zeros(n)</td><td>2: ε↑ε 3:</td></tr><tr><td>3:W ←Zeros(n)</td><td>T↑t</td></tr><tr><td>4:T ← Zeros(n)</td><td>for0≤i≤n-1do</td></tr><tr><td>5:for time0≤t≤n-1do</td><td>←[i]+ω·Ω2[]</td></tr><tr><td>6: Receive Wt</td><td>if i=n-1 then</td></tr><tr><td>W[t]←Wt 7:</td><td>&quot;← Ω[i+1]</td></tr><tr><td>8: T[←t</td><td>else</td></tr><tr><td>9: end for</td><td>Ω&quot;←Ω[i+1]+ω·Ω[𝑖+1]</td></tr><tr><td>Apply equivalent random permutation to W 10:</td><td>end if &quot;[]</td></tr><tr><td>and T</td><td>P←1-+1 11:</td></tr><tr><td>11: Ω[n] ←1</td><td>12: Swap ω with W[i] and T with T[i]</td></tr><tr><td>12: Ω[n-1] ←1</td><td>with probability P</td></tr><tr><td>forn-1≥i≥0do 13:</td><td>13: Ω[i]←Ω&#x27;</td></tr><tr><td>14: 2[i]=W[i]·Ω2[i+1]</td><td>14: end for</td></tr><tr><td>15: end for</td><td>15: forn-2≥i≥0do</td></tr><tr><td>16: forn-2≥i≥0do</td><td>16: Ω[]=[i+1]+W[i]·Ω[i+1]</td></tr><tr><td>17: {[i]=2[i+1]+W[i]·Ω2[i+1]</td><td>17: end for</td></tr><tr><td>18: end for</td><td>18:end function</td></tr><tr><td>19: for all time t ≥ n do</td><td></td></tr><tr><td>20: Receive Wt</td><td></td></tr><tr><td>21: UPDATE(Wt,t)</td><td></td></tr><tr><td>22: end for</td><td></td></tr><tr><td></td><td></td></tr></table>
+
+In the previous section we derived a gradient estimator for our desired memory distribution. in this section we introduce a method for sampling from this distribution online. Specifically we will formulate a reservoir sampling algorithm for drawing a subset $\hat { T }$ of size $n$ from a set of indices $T = \{ 0 , . . . , t - 1 \}$ according to a distribution parameterized by a weight $w _ { i }$ for each index $i \in T$ . Following equation 7 the probability for a given subset $\hat { T }$ is defined as follows:
+
+$$
+\tilde { P } ( \hat { T } ; T , n ) = \prod _ { i \in \hat { T } } w _ { i } \Bigg / \sum _ { \tilde { T } \in \binom { T } { n } } \prod _ { i \in \tilde { T } } w _ { i }
+$$
+
+This distribution can be sampled from by selecting members sequentially for $i \in \{ 0 , . . , n - 1 \}$ with the following conditional probabilities:
+
+$$
+\hat { P } ( \hat { T } [ i ] | \hat { T } [ 0 : i - 1 ] ; T , n ) = w _ { \hat { T } [ i ] } \sum _ { \hat { T } \in \binom { T \setminus \hat { T } [ 0 : i ] } { n - i - 1 } } \prod _ { j \in \bar { T } } w _ { j } \left/ \left( ( n - i ) \sum _ { \hat { T } \in \binom { T \setminus \hat { T } [ 0 : i - 1 ] } { n - i } , j \in \bar { T } } \prod _ { j \in \bar { T } } w _ { j } \right) \right.
+$$
+
+We abuse notation slightly and use $\hat { T }$ to refer to both an ordered vector and the set of its elements.
+
+Lemma 1. Selecting elements sequentially according to equation $^ { 1 2 }$ will result in a vector $\hat { T }$ whose elements correspond to a sample drawn from equation $1 l$ .
+
+Proof. See appendix C.
+
+We use lemma 1 to derive a reservoir sampling procedure which works online to update the reservoir $\hat { T }$ at each time-step when a new index is added to $T$ along with an associated weight. The result is algorithm 1. At each time-step UPDATE moves through $\hat { T }$ starting from index 0 and chooses whether to swap the item and weight at each index with the ones currently contained in a buffer ( $\mathit { \Pi } _ { \tau }$ and $\omega$ , initially set to contain the newly added item and associated weight). The probability of swapping is chosen such that it corrects the conditional probability of the item at each index (conditioned on the items before it) to compensate for the item in the buffer being added to the set of possible items for that index. After doing this sequentially at each index the overall probability of $\hat { T }$ will be correct with the newly added item. Computing the necessary swap probabilities is nontrivial in itself, however we show that it is possible to do this in $O ( n )$ time per time-step (where here $\mathbf { n }$ is the memory size) by iteratively updating two vectors $\Omega$ and $\tilde { \Omega }$ .
+
+Theorem 1. In algorithm $^ { l }$ let $t$ refer to the parameter of the call to UPDATE, $\hat { T } _ { t } [ i ]$ refer to the value of ${ \hat { T } } [ i ]$ when that call is made, and $T _ { t } = \{ t ^ { \prime } : 0 \leq t ^ { \prime } \leq t - 1 \}$ refer to the set of all time indices from 0 to $t - 1 . \forall t \geq n , 0 \leq i \leq n - 1 .$ :
+
+$$
+P ( \hat { T } _ { t } [ i ] = t _ { i } | \hat { T } _ { t } [ 0 : i - 1 ] = [ t _ { 0 } , . . . , t _ { i - 1 } ] ) = \hat { P } ( t _ { i } | [ t _ { 0 } , . . . , t _ { i - 1 } ] ; T _ { t } , n )
+$$
+
+where $[ t _ { 0 } , . . . , t _ { i } ]$ is any arbitrary vector of unique elements of $T _ { t }$ .
+
+Proof. See appendix D.
+
+Corollary 1.1. At the call to UPDATE with parameter $t$ $, \forall t \geq n , \hat { T } \in \binom { T _ { t } } { n }$
+
+$$
+P ( \{ \hat { T } _ { t } [ 0 ] , . . . , \hat { T } _ { t } [ n - 1 ] \} = \hat { T } ) = \tilde { P } ( \hat { T } ; T _ { t } , n )
+$$
+
+Proof. The proof follows from theorem 1 and lemma 1.
+
+Corollary 1.1 tells us that for any given time-step algorithm 1 produces reservoirs which are a valid sample from the distribution of equation 11. Note that algorithm 1 runs in $O ( n )$ time per time-step where $n$ is the size of the memory. We use this algorithm along with the weights generated by our write network to manage updating the memory on each new state visitation.
+
+The careful reader will notice that with the use of reservoir sampling equation 7 no longer holds explicitly. This is because certain parts of the history may strongly correlate with certain states being in memory at a particular time in the past, which under reservoir sampling will effect the distribution of the present memory. We do not account for this in this work and simply assume for the purpose of estimating gradients that the history of state visitation arises independently of the history of memory content. Further analysis of the implications of this assumption, and whether it can be weakened is left to future work.
+
+# 4 EXPERIMENTS AND RESULTS
+
+We test our algorithm on a toy problem we call “the secret informant problem”. The problem is intended to highlight the kinds of sharp, long-term dependencies that are often difficult for recurrent models. The problem is such that in order to behave optimally an agent must remember specific, initially unknown past states. An instance of the problem is shown in figure 2 and a detailed explanation of the problem structure is available in the associated caption. In each training episode a new random instance of the problem is created (with the chain length, number of actions and number of decisions held fixed for a particular training run). This consists of randomly choosing the rewarding action sequence, the location of the informative state for each decision, and the implied action and decision state for each of the uninformative states.
+
+All experiments with our episodic memory architecture are run for 3 repetitions with error bars indicating standard error in the mean over these 3 runs. The architecture and hyper-parameters used in each experiment are identical. We use the architecture from section 2 with 1 hidden layer for the value, query and write networks and 2 hidden layers for policy. The value network and query network outputs each use tanh activation, the policy uses softmax, and the write network uses sigmoid. Each hidden layer has 10 units. We train using ordinary stochastic gradient descent with learning rate 0.005 and gradient estimates generated as specified in section 3.
+
+![](images/4ef6f81c4f7c878cb0c154df357169e596c468733d74987efb3a3d3c1a3f772c.jpg)
+
+Figure 2: (a) shows an instance of the secret informant problem with 3 actions $\alpha \ =$ $\{ u p , f o r w a r d , d o w n \} \rangle$ and 2 decision states. The start state uniquely contains all zeros. In the final 2 states of an episode (which we call decision states), the agent must select the right sequence of actions to receive a $+ 1$ reward, any other action sequence gives reward 0. At all other states the forward action leads forward along the chain while other actions keep the agent in the same state. The correct action (i.e. the one that leads towards the reward) at each decision state is indicated by a one hot encoding on bits 1-3 of a certain informative state where the pattern in bits 6 and 7 matches those of the decision state itself. Informative states are distinguished from uninformative states by bits 4 and 5. To succeed the agent must learn to remember informative states with the pattern 10 in bits 4 and 5 and subsequently query them at the associated decision state. (b) shows a particular state of the problem. The first 3 bits are action indicators, a one-hot encoding of the action the state is suggesting should be taken at the associated decision state. The next two bits are informative and uninformative indicators. If these bits are 01 the state is uninformative, meaning the decision state and associated action it suggests are uniformly random and give no indications of a correct action. If these bits are 10 then the state is informative and the associated action it suggests is on the path toward the reward at the decision state it indicates. The next two bits are decision state identifiers, in an informative state they indicate it provides information about the decision state with matching identifier, in a decision state they serve as an identifier for that decision state. Thus, the correct thing for an agent to do in each decision state is to take the action suggested by the informative state with matching values of bits 6 and 7. The next bit is a decision state indicator and will be 1 if and only if the state is a decision state. The final bit is a correct path indicator and indicates for a decision state whether all the decisions made so far have been correct. This is necessary for our current system because without it the final decision states all look the same and it is not possible to learn via one step updates which decision is correct at the first decision state, in future work we would like to investigate eliminating the need for information like this by using multi-step updates or eligibility traces. The particular state shown above is informative, it indicates that the correct action for the second decision state will be the up action. Versions of this problem can be created with variable length (which we use to refer to the number of informative states plus the number of uninformative states), number of actions, and number of decision states by modifying the above description in the obvious way.
+
+![](images/08d8c5fee64bb7f6c224065d59853dfffadcec5b8aa690baa07e4c6e850c27d4.jpg)  
+Figure 3: Experiment with environment length 10, 1 decision and a 1 state memory. In the 1 state memory case the query module is unnecessary. (a) shows average write weight assigned to informative states (•) and uninformative states $( \_ )$ . (b) shows average return with episodic memory learner (•) and recurrent baseline ().
+
+![](images/4e794754e2844346e11cde5a4e320deffee9dfb303b65b26c790a1224ce76088.jpg)  
+Figure 4: Experiment with environment length 10, 1 decision and a 3 state memory. In this case the query module is necessarily. (a) shows average write weight assigned to informative states (•) and uninformative states $( \_ )$ . (b) shows average return with episodic memory learner (•) and recurrent baseline $( \mathbb { L } )$ . (c) shows the value of several relevant query vector elements in the decision state: the uninformative indicator (N), the informative indicator $( \mathbb { L } )$ and the first decision state identifier (•) (unnecessary here since there is only one decision state, but included for uniformity).
+
+![](images/af0585a33da28591c5dc3854016a597094622d2fa3753d95c87f7bc976488255.jpg)  
+Figure 5: Experiment with environment length 10, 2 decisions and a 3 state memory. (a) shows average write weight assigned to informative states (•) and uninformative states $( \_ )$ . (b) shows average return with episodic memory learner (•) and recurrent baseline (). (c) shows the value of several relevant query vector elements in the first decision state: the uninformative indicator (N), the informative indicator $( \sqcup )$ , the first decision state identifier (•), and the second decision state identifier ${ \bf ( \omega ) }$ . (d) shows the same thing but for the query generated in the second decision state.
+
+We also ran a recurrent baseline with full backpropagation through time which we found required more fine-tuning to train with online updates. To make comparison as meaningful as possible the recurrent baseline used essentially the same architecture but with the entire memory module replaced by a basic GRU (Cho et al., 2014) network of 10 units. Stochastic gradient descent alone was found to give very poor results with the recurrent learner so RMSProp was used instead. Additionally to obtain reasonable results with the recurrent learner, and avoid catastrophic looping behavior, it was necessary to add a discount factor $\gamma = 0 . 9$ , applied only for learning purposes and not used in computing the plotted return) as well as entropy regularization on the policy with a relatively low weight of 0.0005. Perhaps surprisingly neither of these were necessary with the episodic memory based system as it tended to proceed quickly through the chain without significant looping even without discounting or entropy regularization. We tuned the learning rate and layer width (including the number of recurrent units) for each of the 2 environments on which a recurrent baseline was trained according to highest average performance over the last 100 episodes of a single training run of 25000 episodes for the 1 decision environment, and 50000 episodes for the 2 decision environment. In each case learning rate was selected from $\{ 0 . 0 5 \cdot 2 ^ { - x } : x \in \{ 0 , . . . , 9 \} \}$ and layer width was selected from $\{ 5 , 1 0 , 1 5 , 2 0 \}$ . For the 1 decision environment we ran the recurrent baseline for 3 repeats, for the 2 decision environment due to higher variance we ran it for 10 repeats. This baseline is not intended to be representative of the performance of all possible architectures based on RNN variants trained with backpropagation through time, but merely to provide context for the main experimental results of this work.
+
+Results and descriptions of the experiments are shown in figures 3, 4 and 5, in each plot the $\mathbf { X }$ -axis shows number of training episodes. One additional experiment with twice the environment length is shown in appendix E. Notice that in each case the episodic memory learner was is able to learn a good query policy, drive the write weight of the uninformative states to near 0 while keeping the value for informative states much larger, and obtain close to perfect average return. Comparing figures 3 and 4 it appears that the addition of redundant memory size may accelerate initial learning though it has little effect on the overall convergence time. Comparing figures 4 and 5 the number of episodes to converge appears to roughly double from approximately 25, 000 to 50, 000 with the addition of the extra decision state but the training remains quite stable.
+
+# 5 CONCLUSION
+
+We present a novel algorithm for integrating a form of external memory with trainable reading and writing into a RL agent. The method depends on the observation that if we restrict the information stored in memory to be a set of past visited states, the information recorded also provides the context in which it was recorded. This means it is possible to assign credit to useful information without needing to backpropagate through time to when it was recorded. To achieve this we devise a reservoir sampling technique which uses a sampling procedure we introduce to generate a distribution over memory configurations for which we can derive gradient estimates. The whole algorithm is ${ \mathrm { O } } ( { \mathrm { n } } )$ in both the number of trainable parameters and the size of the memory. In particular neither memory required nor computation time increase with history length, making it feasible to run in an online RL setting. We show that the resulting algorithm is able to achieve good performance on a toy problem we introduce designed to have sharp long-term dependencies which can be problematic for recurrent models.
+
+# ACKNOWLEDGEMENTS
+
+We acknowledge the support of the Natural Sciences and Engineering Council of Canada (NSERC).
+
+# REFERENCES
+
+Bram Bakker, Viktor Zhumatiy, Gabriel Gruener, and Jurgen Schmidhuber. A robot that ¨ reinforcement-learns to identify and memorize important previous observations. In International Conference on Intelligent Robots and Systems, 2003.   
+Min-Te Chao. A general purpose unequal probability sampling plan. Biometrika, 69(3):653–656, 1982.   
+Kyunghyun Cho, Bart Van Merrienboer, Dzmitry Bahdanau, and Yoshua Bengio. On the properties ¨ of neural machine translation: Encoder-decoder approaches. In Workshop on Syntax, Semantics and Structure in Statistical Translation, 2014.   
+Pavlos S Efraimidis and Paul G Spirakis. Weighted random sampling with a reservoir. Information Processing Letters, 97(5):181–185, 2006.   
+Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. arXiv preprint arXiv:1410.5401, 2014.   
+Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka GrabskaBarwinska, Sergio G ´ omez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, ´ et al. Hybrid computing using a neural network with dynamic external memory. Nature, 538 (7626):471–476, 2016.   
+Caglar Gulcehre, Sarath Chandar, and Yoshua Bengio. Memory augmented neural networks with wormhole connections. arXiv preprint arXiv:1701.08718, 2017.   
+Matthew Hausknecht and Peter Stone. Deep recurrent q-learning for partially observable mdps. In AAAI Fall Symposium on Sequential Decision Making for Intelligent Agents, 2015.   
+Armand Joulin and Tomas Mikolov. Inferring algorithmic patterns with stack-augmented recurrent nets. In Neural Information Processing Systems, 2015.   
+Łukasz Kaiser, Ofir Nachum, Aurko Roy, and Samy Bengio. Learning to remember rare events. In International Conference on Representation Learning, 2017.
+
+Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International Conference on Machine Learning, 2016.
+
+Junhyuk Oh, Valliappa Chockalingam, Satinder P. Singh, and Honglak Lee. Control of memory, active perception, and action in minecraft. In International Conference on Machine Learning, 2016.
+
+Sainbayar Sukhbaatar, Arthur Szlam, Jason Weston, and Rob Fergus. End-to-end memory networks. In Neural Information Processing Systems, 2015.
+
+Richard S Sutton, David A McAllester, Satinder P Singh, and Yishay Mansour. Policy gradient methods for reinforcement learning with function approximation. In Neural Information Processing Systems, 2000.
+
+Richard S Sutton, Joseph Modayil, Michael Delp, Thomas Degris, Patrick M Pilarski, Adam White, and Doina Precup. Horde: A scalable real-time architecture for learning knowledge from unsupervised sensorimotor interaction. In International Conference on Autonomous Agents and Multiagent Systems, 2011.
+
+Daan Wierstra, Alexander Forster, Jan Peters, and J ¨ urgen Schmidhuber. Recurrent policy gradients. ¨ Logic Journal of IGPL, 18(5):620–634, 2010.
+
+Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992.
+
+Wojciech Zaremba and Ilya Sutskever. Reinforcement learning neural turing machines. arXiv preprint arXiv:1505.00521, 419, 2015.
+
+Marvin Zhang, Sergey Levine, Zoe McCarthy, Chelsea Finn, and Pieter Abbeel. Policy learning with continuous memory states for partially observed robotic control. In International Conference on Robotics and Automation, 2016.
+
+A DERIVATION OF EXPRESSION FOR GRADIENT FOR ONE-STATE MEMORY
+
+$$
+P _ { t } ( m _ { t } = S _ { i } ) = { w _ { i } \Bigg / \sum _ { j = 0 } ^ { t - 1 } w _ { j } }
+$$
+
+$$
+E _ { t } [ R _ { t } ] = \sum _ { k = 0 } ^ { t - 1 } P _ { t } ( m _ { t } = S _ { k } ) \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { k } ) E _ { t } [ R _ { t } | A _ { t } = a ]
+$$
+
+$$
+\begin{array} { l } { \displaystyle \frac { \partial } { \partial w _ { i } } E _ { t } [ R _ { t } ] = \sum _ { k = 0 } ^ { t - 1 } \Biggl ( \frac { \partial P _ { t } ( m _ { t } = S _ { k } ) } { \partial w _ { i } } \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { k } ) E _ { t } [ R _ { t } | A _ { t } = a ] } \\ { \displaystyle + P _ { t } ( m _ { t } = S _ { k } ) \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { k } ) \frac { \partial E _ { t } [ R _ { t } | A _ { t } = a ] } { \partial w _ { i } } \Biggr ) } \end{array}
+$$
+
+Working out the first term:
+
+$$
+\begin{array} { r l } &  \begin{array} { r l } &  \frac { 1 } { \sqrt { 2 } } \frac { \partial \xi ( \xi ( \xi ) = 0 , \xi ( \xi ) ) } { \partial \xi ( \xi ) } \underset { \xi \in \mathcal { R } _ { 0 } } { \overset { . . . . } { \sum } } \frac  \partial \xi ( \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi ( \xi ) , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \xi , \xi ( \xi ) , \end{array} \end{array}
+$$
+
+Working out the second term:
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { k = 0 } ^ { t - 1 } P _ { t } ( m _ { t } = S _ { k } ) \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { k } ) \frac { \partial E _ { t } [ R _ { t } | A _ { t } = a ] } { \partial w _ { i } } } \\ & { = \displaystyle \sum _ { k = 0 } ^ { t - 1 } P _ { t } ( m _ { t } = S _ { k } ) \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { k } ) \left( \frac { \partial E _ { t } [ r _ { t + 1 } | A _ { t } = a ] } { \partial w _ { i } } + \gamma \frac { \partial E _ { t } [ R _ { t + 1 } | A _ { t } = a ] } { \partial w _ { i } } \right) } \\ & { = \gamma E _ { t } \left[ \frac { \partial } { \partial w _ { i } } E _ { t + 1 } [ R _ { t + 1 } ] \right] } \end{array}
+$$
+
+Where we are able to drop ∂Et[rt+1|At=a] because the immediate reward is independent of the state in memory once conditioned on the action. Thus we finally arrive at:
+
+$$
+\begin{array} { r } { \cfrac { \partial } { \partial w _ { i } } E _ { t } [ R _ { t } ] = \cfrac { 1 } { w _ { i } } P _ { t } ( m _ { t } = S _ { i } ) \left( \displaystyle \sum _ { a \in A } \pi ( a | S _ { t } , S _ { i } ) E _ { t } [ R _ { t } | A _ { t } = a ] - E _ { t } [ R _ { t } ] \right) } \\ { + \left. \gamma E _ { t } \left[ \cfrac { \partial } { \partial w _ { i } } E _ { t + 1 } [ R _ { t + 1 } ] \right] \right. } \end{array}
+$$
+
+B DERIVATION OF EXPRESSION FOR GRADIENT FOR MULTIPLE-STATE MEMORY
+
+$$
+\begin{array} { c } { { P _ { t } ( \mathcal { M } _ { t } = S _ { \hat { T } } ) = { \displaystyle { \prod _ { i \in \hat { T } } w _ { i } } } \Bigg / { \displaystyle { \sum _ { \tilde { T } \in \binom { T _ { t } } { n } } \prod _ { i \in \tilde { T } } w _ { i } } } } } \\  { E _ { t } [ R _ { t } ] = \displaystyle { \sum _ { \hat { T } \in \binom { T _ { t } } { n } } P _ { t } ( \mathcal { M } _ { t } = S _ { \hat { T } } ) \displaystyle { \sum _ { j \in \hat { T } } Q ( S _ { j } | S _ { \hat { T } } ) \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { j } ) E _ { t } [ R _ { t } | A _ { t } = a ] } } } \end{array}
+$$
+
+$$
+\begin{array} { r } { \frac { \partial } { \partial w _ { i } } E _ { t } [ R _ { t } ] = \displaystyle \sum _ { \hat { T } \in \binom { T _ { t } } { n } } \left( \frac { \partial P _ { t } ( \mathcal { M } _ { t } = S _ { \hat { T } } ) } { \partial w _ { i } } \sum _ { j \in \hat { T } } Q ( S _ { j } | \mathcal { M } _ { t } = S _ { \hat { T } } ) \sum _ { a \in A } \pi ( a | S _ { t } , S _ { j } ) E _ { t } [ R _ { t } | A _ { t } = a ] \right. } \\ { \displaystyle \left. + P _ { t } ( \mathcal { M } _ { t } = S _ { \hat { T } } ) \sum _ { j \in \hat { T } } Q ( S _ { j } | S _ { \hat { T } } ) \sum _ { a \in A } \pi ( a | S _ { t } , S _ { j } ) \frac { \partial E _ { t } [ R _ { t } | A _ { t } = a ] } { \partial w _ { i } } \right) } \end{array}
+$$
+
+Working out the first term:
+
+$$
+\begin{array} { r l } & { \begin{array} { r l } & { \gamma _ { 1 } : 2 1 } \\ & { - \frac { 2 } { \sqrt { 2 } \pi \lambda } } \\ & { \gamma _ { 2 } : 2 1 } \\ & { - \frac { 2 } { \sqrt { 2 } \pi \lambda } ( \lambda - \lambda _ { 1 } ^ { 2 } ) ( \lambda - \lambda _ { 2 } ^ { 2 } ) \lambda _ { 1 } ( \lambda _ { 1 } ^ { 2 } - \lambda _ { 2 } ^ { 2 } ) } \end{array} \Biggr \} \frac { \sum _ { i = 1 } ^ { \infty } \lambda _ { i } ^ { 2 } } { 2 \sqrt { 2 } \pi \lambda } \lambda _ { i } ( \lambda _ { 1 } ^ { 2 } - \lambda _ { 2 } ^ { 2 } ) \lambda _ { 2 } ( \lambda _ { 2 } ^ { 2 } - \lambda _ { 1 } ^ { 2 } ) } \\ & { \qquad \quad \times \frac { \lambda } { 2 } \sum _ { i = 1 } ^ { \infty } \lambda _ { i } ^ { 2 } ( \lambda _ { 1 } ^ { 2 } - \lambda _ { 2 } ^ { 2 } ) } \\ & { \qquad \quad + \sum _ { i \neq 2 } ^ { \infty } \lambda _ { i } ^ { 2 } \lambda _ { 1 } ^ { 2 } \lambda _ { 1 } ^ { 2 } \lambda _ { 2 } ^ { 2 } \lambda _ { 1 } ^ { 3 } \lambda _ { 2 } ^ { 3 } \lambda _ { 1 } ^ { 4 } \lambda _ { 2 } ^ { 4 } \lambda _ { 1 } ^ { 4 } \lambda _ { 2 } ^ { 4 } \lambda _ { 2 } ^ { 4 } } \\ & { \qquad \quad + \sum _ { i \neq 2 } ^ { \infty } \lambda _ { i } ^ { 2 } \lambda _ { 1 } ^ { 2 } \lambda _ { 2 } ^ { 4 } \lambda _ { 1 } ^ { 4 } \lambda _ { 1 } ^ { 4 } \lambda _ { 2 } ^ { 4 } \lambda _ { 2 } ^ { 4 } \lambda _ { 2 } ^ { 4 } \lambda _ { 1 } ^ { 4 } \lambda _ { 1 } ^ { 4 } \lambda _ { 2 } ^ { 4 } \lambda _ { 2 } ^ { 4 } } \\ &  \qquad \quad \times \frac { \lambda } { 2 } \sum _ { i \neq 3 } ^ { \infty } \lambda _ { i } ^ { 2 } ( \lambda _ { 1 } ^ { 2 } - \lambda _ { 2 } ^ { 2 } ) \lambda _ { i } ^ { 2 } \lambda _ { 2 } ^ { 4 } \lambda _ { 3 } ^ { 2 } \lambda _ { 2 } ^ { 4 } \lambda _ { 3 } ^ { 2 } \lambda _ { 2 } ^ { 4 } \lambda _ { 3 } ^ { 2 } \lambda _ { 2 } \end{array}
+$$
+
+Working out the second term:
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { \bar { T } \in \{ \binom { T } { n } } } P _ { t } ( M _ { t } = S _ { \hat { T } } ) \displaystyle \sum _ { j \in \bar { T } } Q ( S _ { j } | S _ { \hat { T } } ) \displaystyle \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { j } ) \displaystyle \frac { \partial E _ { t } [ R _ { t } | A _ { t } = a ] } { \partial w _ { i } }  \\ & { \displaystyle = \sum _ { \hat { T } \in \binom { T } { n } } P _ { t } ( M _ { t } = S _ { \hat { T } } ) \displaystyle \sum _ { j \in \bar { T } } Q ( S _ { j } | S _ { \hat { T } } ) \displaystyle \sum _ { a \in \mathcal { A } } \pi ( a | S _ { t } , S _ { j } ) \left( \displaystyle \frac { \partial E _ { t } [ r _ { t + 1 } | A _ { t } = a ] } { \partial w _ { i } } \right. } \\ & { \quad \quad \quad \quad \quad \quad \left. + \gamma \displaystyle \frac { \partial E _ { t } [ R _ { t + 1 } | A _ { t } = a ] } { \partial w _ { i } } \right) } \\ & { \displaystyle = \gamma E _ { t } \left[ \displaystyle \frac { \partial } { \partial w _ { i } } E _ { t + 1 } [ R _ { t + 1 } ] \right] } \end{array}
+$$
+
+So all together we get:
+
+$$
+\begin{array} { r } { \frac { \partial } { \partial w _ { i } } E _ { t } [ R _ { t } ] = \displaystyle \sum _ { \hat { T } \in ( \binom { T _ { t } } { n } \ni i ) } \frac { 1 } { w _ { i } } P _ { t } ( \boldsymbol { M } _ { t } = \boldsymbol { S } _ { \hat { T } } ) \Bigg ( \displaystyle \sum _ { j \in \hat { T } } Q ( \boldsymbol { S } _ { j } | \boldsymbol { S } _ { \hat { T } } ) \sum _ { a \in \mathcal { A } } \boldsymbol { \pi } ( a | \boldsymbol { S } _ { t } , \boldsymbol { S } _ { j } ) E _ { t } [ R _ { t } | \boldsymbol { A } _ { t } = a ] } \\ { - E _ { t } [ R _ { t } ] \Bigg ) + \gamma E _ { t } \left[ \frac { \partial } { \partial w _ { i } } E _ { t + 1 } [ R _ { t + 1 } ] \right] } \end{array}
+$$
+
+# C PROOF OF LEMMA 1
+
+Lemma 1. Selecting elements sequentially according to equation 12 will result in a vector $\hat { T }$ whose elements correspond to a sample drawn from equation $1 l$ .
+
+Proof. Selecting elements sequentially according to equation 12 gives the following probability for a given vector $\hat { T }$ .
+
+$$
+\begin{array} { r l } { P ( \hat { T } ) = P ( \hat { T } [ 0 ] ) P ( \hat { T } [ 1 ] | T [ 0 ] ) . . . P ( \hat { T } [ n - 1 ] | \hat { T } [ 0 : n - 2 ] ) } & { } \\ { \displaystyle } & { = \frac { w _ { \hat { T } [ 0 ] } } { \tilde { T } [ 0 ] } \sum _ { \tilde { T } \in \binom { T \hat { T } [ 0 ] } { n - 1 } , j \in \tilde { T } } \prod _ { \ell = \tilde { T } } w _ { \hat { T } [ 1 ] } \sum _ { \tilde { T } \in \binom { T \hat { T } [ 0 , 1 ] } { n - 2 } , j \in \tilde { T } } w _ { j } } \\ { \displaystyle } & { \qquad \sum _ { \tilde { T } \in \binom { T } { n } , j \in \tilde { T } } \prod _ { \ell = \tilde { T } } w _ { j } \cdot \frac { \tilde { T } \in \binom { T \hat { T } [ 0 , 1 ] } { n - 2 } \tilde { \jmath } \in \tilde { T } } { \tilde { T } \in \binom { T \hat { T } \hat { \imath } - \tilde { T } [ 0 ] } { n - 1 } \jmath \in \tilde { T } } \prod _ { \ell = \tilde { T } } w _ { j } \cdots \frac { w _ { \hat { T } [ n - 1 ] } } { \tilde { T } \in \binom { T \hat { T } \hat { \imath } - \tilde { \imath } - \tilde { \imath } - \tilde { \imath } - \tilde { \jmath } } { 1 } \jmath \in \tilde { T } } \prod _ { \ell = \tilde { T } } w _ { j } } \\ { \displaystyle } & { = \prod _ { \ell \in \tilde { T } } w _ { i } / ( n ! \sum _ { \tilde { T } \in \binom { T } { n } } \prod _ { \ell \in \tilde { T } } w _ { i } ) } \end{array}
+$$
+
+To complete the proof note that this is one of $n$ ! vectors with the same set of elements in different order, each of which will have equal probability, hence to obtain the probability for the corresponding set we simply have to multiply by $n !$ which gives us equation 11. □
+
+# D PROOF OF THEOREM 1
+
+We divide the proof into a series of lemmas. First we will define some notation to facilitate referencing algorithm 1 in the proofs below. Let $T _ { t }$ be the set $\{ 0 , . . . , t - 1 \}$ and $\hat { T } _ { t } [ 0 : i - 1 ]$ be the value of ${ \hat { T } } [ 0 : i - 1 ]$ at the call to UPDATE with parameter $t$ . Note that $\hat { T } _ { t } [ 0 : - 1 ] = \varnothing$ . Let $\Omega _ { t } [ i ]$ and $\tilde { \Omega } _ { t } [ i ]$ be the values of $\Omega [ i ]$ and $\tilde { \Omega } [ i ]$ respectively, at the call to UPDATE with parameter $t$ . Let $P _ { t , i }$ be the value of $\mathrm { \bf P }$ when it is set in loop index i of the loop starting at line 4 within the call to UPDATE with parameter t. Let $\Omega _ { t , i } ^ { \prime \prime }$ and $\Omega _ { t , i } ^ { \prime }$ be the values of $\Omega ^ { \prime \prime }$ and $\Omega ^ { \prime }$ respectively after they are set within index $i$ of the loop starting at line 4 within the UPDATE call with parameter $t$ . Let $\omega _ { t , i }$ and $\tau _ { t , i }$ be the values of $\omega$ and $\tau$ respectively at the beginning of loop index i of the loop starting at line 4 within the call to UPDATE with parameter t. Note that $\omega _ { t , i } = { w _ { \tau _ { t , i } } }$ .
+
+Lemma 2.
+
+$$
+\Big ( T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i - 1 ] \Big ) = \Big ( T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] \Big ) \cup \{ \tau _ { t , i } \}
+$$
+
+Proof. By design $T _ { t + 1 } = T _ { t } \cup \{ \tau _ { t , 0 } \}$ . Towards a proof by induction assume that after choosing whether or not to swap $\hat { T } _ { t } [ i ]$ we have:
+
+$$
+\Big ( T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i - 1 ] \Big ) = \Big ( T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] \Big ) \cup \{ \tau _ { t , i } \}
+$$
+
+Then on the next iteration either we swap ${ \hat { T } } [ i ]$ for $\tau _ { t , i }$ or we don’t. If we do swap then $\tau _ { t , i + 1 } = \hat { T } _ { t } [ i ]$ and $\hat { T } _ { t + 1 } [ i ] = \tau _ { t , i }$ thus
+
+$$
+\begin{array} { r l } { \Big ( T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i ] \Big ) = \Big ( T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i - 1 ] \Big ) \setminus \{ T _ { t + 1 } [ i ] \} } & { } \\ { = \Big ( T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] \Big ) \cup \{ \pi _ { t , i } \} \setminus \{ T _ { t + 1 } [ i ] \} } & { } \\ { = \Big ( T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] \Big ) \cup \{ \pi _ { t , i } \} \setminus \{ \pi _ { t , i } \} } & { } \\ { = \Big ( T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] \Big ) } & { } \\ { = \Big ( T _ { t } \setminus \hat { T } _ { t } [ 0 : i ] \Big ) \cup \{ T _ { t } [ i ] \} } & { } \\ { = \Big ( T _ { t } \setminus \hat { T } _ { t } [ 0 : i ] \Big ) \cup \{ T _ { t , i + 1 } \} } & { } \end{array}
+$$
+
+On the other hand if we do not swap then $\tau _ { t , i + 1 } = \tau _ { t , i }$ and $\hat { T } _ { t + 1 } [ i ] = \hat { T } _ { t } [ i ]$ thus
+
+$$
+\begin{array} { r l } & { \Big ( T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i ] \Big ) = \Big ( T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i - 1 ] \Big ) \setminus \{ T _ { t + 1 } [ i ] \} } \\ & { \qquad = \Big ( T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] \Big ) \cup \{ \tau _ { t , i } \} \setminus \{ T _ { t + 1 } [ i ] \} } \\ & { \qquad = \Big ( T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] \Big ) \cup \{ \tau _ { t + 1 , i } \} \setminus \{ T _ { t } [ i ] \} } \\ & { \qquad = \Big ( T _ { t } \setminus \hat { T } _ { t } [ 0 : i ] \Big ) \cup \{ \tau _ { t , i + 1 } \} } \end{array}
+$$
+
+Which suffices to complete the inductive proof.
+
+Lemma 3.
+
+$$
+\sum _ { \tilde { T } \in ( \binom { T \cup \{ \hat { t } \} } { m } \tilde { t } \in \tilde { T } ) } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } = \sum _ { \tilde { T } \in ( \binom { T } { m } \tilde { t } \in \tilde { T } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } + w _ { \hat { t } } \cdot \sum _ { \tilde { T } \in ( \binom { T } { m - 1 } \tilde { t } \in \tilde { T }  } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } }
+$$
+
+Proof. The proof follows from noting that the first sum on the right side includes every term of the left sum where $\tilde { T }$ does not contain $\hat { t }$ , while the second term on the right side is equivalent to summing over those $\tilde { T }$ in the left sum that do contain $\hat { t }$ . □
+
+Lemma 4. For $t \geq n$
+
+$$
+\Omega _ { t } [ i ] = \sum _ { \tilde { T } \in \binom { T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] } { n - i } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } , \forall i \in \{ 0 , . . . , n \}
+$$
+
+$$
+\tilde { \Omega } _ { t } [ i ] = \sum _ { \tilde { T } \in \binom { T _ { t } \setminus \tilde { T } _ { t } [ 0 : i - 1 ] } { n - i - 1 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } , \forall i \in \{ 0 , . . . , n - 1 \}
+$$
+
+Proof. Towards a proof by induction, assume the lemma holds at time $t$ , a quick trace through algorithm 1 will show that the update leading to $\Omega _ { t + 1 } [ i ]$ for $i \in \{ 0 , . . . , n - 1 \}$ is always:
+
+$$
+\Omega _ { t + 1 } [ i ] = \Omega _ { t } [ i ] + \omega _ { t , i } \cdot \tilde { \Omega } _ { t } [ i ]
+$$
+
+From here we can apply lemma 3, along with lemma 2 as follows to get the desired result:
+
+$$
+\begin{array} { r l } & { \Omega _ { t + 1 } [ i ] = \Omega _ { t } [ i ] + \omega _ { t , i } \cdot \tilde { \Omega } _ { t } [ i ] } \\ & { \quad \quad = \quad \quad \quad \sum _ { \tilde { T } \in \binom { T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] } { n - i } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } + \omega _ { t , i } \cdot \sum _ { \tilde { T } \in \binom { T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] } { n - i - 1 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } } \\ & { \quad \quad \quad = \quad \quad \quad \quad \quad \sum _ { \tilde { T } \in \binom { T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i - 1 ] } { n - i } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } } \\ & { \quad \quad \quad \quad \tilde { T } \in \binom { T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i - 1 ] } { n - i } \tilde { t } \in \tilde { T } } \end{array}
+$$
+
+Now note that $\tilde { \Omega } _ { t } [ n ] = 1$ for all $t$ which is also the correct value thus we have completed the induction step for the first half of the lemma.
+
+On the other hand the update leading to $\tilde { \Omega } _ { t + 1 } [ i ]$ is as follows:
+
+$$
+\tilde { \Omega } _ { t + 1 } [ i ] = \Omega _ { t + 1 } [ i + 1 ] + w _ { \hat { T } _ { t + 1 } [ i ] } \cdot \tilde { \Omega } _ { t + 1 } [ i + 1 ]
+$$
+
+As a second level of induction assume the lemma holds for $\tilde { \Omega } _ { t + 1 } [ i + 1 ]$ then applying lemma 3 we get:
+
+$$
+\begin{array} { r l } & { \tilde { \Omega } _ { t + 1 } [ i ] = \Omega _ { t + 1 } [ i + 1 ] + w _ { \hat { T } _ { t + 1 } [ i ] } \cdot \tilde { \Omega } _ { t + 1 } [ i + 1 ] } \\ & { \quad \quad = \quad \quad \displaystyle \sum _ { \tilde { T } \in \binom { T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i ] } { n - i - 1 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } + w _ { \hat { T } _ { t + 1 } [ i ] } \cdot \sum _ { \tilde { T } \in \binom { T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i ] } { n - i - 2 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } } \\ & { \quad \quad \quad = \quad \quad \displaystyle \sum _ { \tilde { T } \in \binom { T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i - 1 ] } { n - i - 1 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } } \\ & { \quad \quad \quad \quad \tilde { T } \in \binom { T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i - 1 ] } { n - i - 1 } \tilde { t } \in \tilde { T } } \end{array}
+$$
+
+As the base case for this second level of induction, note that $\tilde { \Omega } _ { t + 1 } [ n - 1 ] = 1$ for all $t$ which is also the correct value, thus assuming $\Omega _ { t + 1 }$ is correct we will also get the correct values for $\tilde { \Omega } _ { t + 1 }$ . This completes the induction step for the lemma, it remains to prove the base case.
+
+Note that $\Omega [ n ]$ and $\tilde { \Omega } [ n - 1 ]$ each start at 1. The beginning of the algorithm before line 19 is then intended to initialize all $\Omega$ and $\tilde { \Omega }$ values to have the correct value at time $t = n$ , to see that this is the case, first note $\forall i \in \{ 0 , . . . , n \}$ :
+
+$$
+\begin{array} { l } { { \displaystyle \Omega _ { n } [ i ] = \prod _ { j = i } ^ { n - 1 } w _ { \hat { T } _ { n } [ j ] } } } \\ { { \displaystyle = \sum _ { \tilde { T } \in \binom { T _ { n } \setminus \hat { T } _ { n } [ 0 : i - 1 ] } { n - i } \tilde { t } \in \tilde { T } } w _ { \tilde { t } } } } \end{array}
+$$
+
+And also, by induction on the trivial $i = n - 1$ case, $\forall i \in \{ 0 , . . . , n - 2 \} ;$ :
+
+$$
+\begin{array} { r l } & { \tilde { \Omega } _ { n } [ i ] = \Omega _ { n } [ i + 1 ] + w _ { \hat { T } _ { n } [ i ] } \cdot \tilde { \Omega } _ { n } [ i + 1 ] } \\ & { \quad \quad = \displaystyle \sum _ { \tilde { T } \in \binom { T _ { n } \setminus \hat { T } _ { n } [ 0 : i ] } { n - i - 1 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } + w _ { \hat { T } _ { n } [ i ] } \cdot \sum _ { \tilde { T } \in \binom { T _ { n } \setminus \hat { T } _ { n } [ 0 : i ] } { n - i - 2 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } } \\ & { \quad \quad = \displaystyle \sum _ { \tilde { T } \in \binom { T _ { n } \setminus \hat { T } _ { n } [ 0 : i - 1 ] } { n - i - 1 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } } \\ & { \qquad \tilde { T } \in \binom { T _ { n } \setminus \hat { T } _ { n } [ 0 : i - 1 ] } { n - i - 1 } \tilde { t } \in \tilde { T } } \end{array}
+$$
+
+Thus indeed the lemma holds at time $n$ which serves as a base case for the rest of the proof.
+
+Lemma 5. Pt,i = 1 − Pˆ(Tˆt[i]|Tˆt+1[0:i−1];Tt+1,n)
+
+Proof. Substituting the definition from equation 12, along with the value assigned to $P _ { t , i }$ and simplifying slightly what we wish to show is:
+
+$$
+\begin{array}{c} \frac { \Omega _ { t , i } ^ { \prime \prime } \Omega _ { t } \left[ i \right] } { \Omega _ { t , i } ^ { \prime } \Omega _ { t } \left[ i + 1 \right] } = \frac { \underset { n = 1 } { \overset { \sum } { } } \left( \underset { n = i - 1 } { \overset { \sum } { } } \right) \underset { n = 1 } { \overset { } { \prod } } w _ { j } } { \underset { n = 1 } { \overset { } { \sum } } \left( \underset { n = i - 1 } { \overset { } { \prod } } \right) \underset { j \in \tilde { T } } { \overset { } { \prod } } w _ { j } } \underset { \tilde { T } \in \left( \begin{array} { l } { T _ { t } \backslash \hat { T } _ { t } \left[ 0 : i - 1 \right] } \\ { n - i - 1 } \end{array} \right) } { \overset { } { \prod } } w _ { j }  \end{array} \overset { \prod } { } { \underset { n = i } { \overset { } { \sum } } } \left[ \underset { n = i - 1 } { \overset { } { \prod } } \right] \underset { j \in \tilde { T } } { \overset { } { \prod } } w _ { j }
+$$
+
+Substituting in the values of $\Omega _ { t } [ i ]$ and $\Omega _ { t + 1 } [ i ]$ from lemma 4 into the above formula, it will suffice to apply lemma 3 and lemma 4 to additionally show:
+
+$$
+\begin{array} { r l } & { \Omega _ { t , i } ^ { \prime } = \Omega _ { t } [ i ] + \omega _ { t , i } \cdot \tilde { \Omega } _ { t } [ i ] } \\ & { \quad = \quad \displaystyle \sum _ { \tilde { T } \in \binom { T _ { t } \backslash \hat { T } _ { t } [ 0 : i - 1 ] } { n - i } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } + \omega _ { t , i } \cdot \sum _ { \tilde { T } \in \binom { T _ { t } \backslash \hat { T } _ { t } [ 0 : i - 1 ] } { n - i - 1 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } } \\ & { \quad = \quad \displaystyle \sum _ { \tilde { T } \in \binom { T _ { t + 1 } \backslash \hat { T } _ { t + 1 } [ 0 : i - 1 ] } { n - i } } \prod _ { j \in \tilde { T } } w _ { j } } \\ & { \quad \quad \quad \tilde { T } \in \binom { T _ { t + 1 } \backslash \hat { T } _ { t + 1 } [ 0 : i - 1 ] } { n - i } j \in \tilde { T } } \end{array}
+$$
+
+and for $0 \leq i \leq n - 2$
+
+$$
+\begin{array} { r l } & { \Omega _ { t , i } ^ { \prime \prime } = \Omega _ { t } [ i + 1 ] + \omega _ { t , i } \cdot \tilde { \Omega } _ { t } [ i + 1 ] } \\ & { \quad = \quad \displaystyle \sum _ { \tilde { T } \in \binom { T _ { t } \setminus \tilde { T } _ { t } [ 0 : i ] } { n - i - 1 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } + \omega _ { t , i } \cdot \sum _ { \tilde { T } \in \binom { T _ { t } \setminus \tilde { T } _ { t } [ 0 : i ] } { n - i - 2 } } \prod _ { \tilde { t } \in \tilde { T } } w _ { \tilde { t } } } \\ & { \quad = \quad \displaystyle \sum _ { \tilde { T } \in \binom { T _ { t + 1 } \setminus ( \tilde { T } _ { t + 1 } [ 0 : i - 1 ] \cup \{ \tilde { T } _ { t } [ i ] \} ) } { n - i - 1 } } \prod _ { j \in \tilde { T } } w _ { j } } \\ & { \quad \quad \tilde { T } \in \binom { T _ { t + 1 } \setminus ( \tilde { T } _ { t + 1 } [ 0 : i - 1 ] \cup \{ \tilde { T } _ { t } [ i ] \} ) } { n - i - 1 } j \in \tilde { T } } \end{array}
+$$
+
+The first is a simple application of lemma 2. The second follows from a similar observation in addition to noting that $\hat { T } _ { t } [ i ]$ has been removed from each term. For $i = n - 1$ , $\Omega _ { t , i } ^ { \prime \prime } = 1$ which trivially obeys the same formula. □
+
+Lemma 6. $P _ { t , i } \geq 0$
+
+Proof.
+
+$$
+\begin{array}{c} P _ { t , i } = 1 - \frac { \tilde { T } \in \left( ^ { T _ { t + 1 } \setminus ( \hat { T } _ { t + 1 } \cup : i - 1 ] \cup \{ \hat { T } _ { t } [ i ] \} ) } \right) ^ { \top } } { \tilde { n } - i - 1 } \overset { \prod } { \underset { j \in \tilde { T } } { \prod } } w _ { j } \underset { \tilde { T } \in \left( ^ { T _ { t } \setminus \hat { T } _ { t } [ 0 : i - 1 ] } \right) } { \sum } \overset { \sum } { j \in \tilde { T } } \quad \underset { \tilde { m } _ { j } } { \sum } \quad \underset { \tilde { m } _ { j } } { \sum } \quad  \\ { \tilde { T } \in \left( ^ { T _ { t + 1 } \setminus \hat { T } _ { t + 1 } [ 0 : i - 1 ] } \right) j \in \tilde { T } \quad \ } & { \tilde { T } \in \left( ^ { T _ { t } \setminus \hat { T } _ { t } [ 0 : i ] } _ { n - i - 1 } \right) j \in \tilde { T } } \end{array}
+$$
+
+hence
+
+$$
+\begin{array} { r l } &  \quad \sum _ { j = 1 } ^ { n } \sum _ { i = 1 } ^ { n } \prod _ { j = 1 } ^ { n } \exp ( \frac { 1 }  \exp ( \frac  \displaystyle \sum _ { j = 1 } ^ { n } \frac  \sum _ { j = 1 } ^ { n } \frac  \sum _ { i = 1 } ^ { n } \frac  \sum _ { j = 1 } ^ { n } \frac  \sum _ { i = 1 } ^ { n } \frac  \sum _ { j = 1 } ^ { n } \frac  \sum _ { i = 1 } ^ { n } \frac  \sum _ { j = 1 } ^ { n } \frac  \sum _ { i = 1 } ^ { n } \frac  \sum _ { i = 1 } ^ { n } \frac  \sum _ { i = 1 } ^ { n } \frac  \sum _ { i = 1 } ^ { n } \frac  \sum { i } \sum _ { i = 1 } ^ { n } \frac  \sum _ { i = 1 } ^ { n } \frac  \sum { i } \frac { \sum _ { i = 1 } ^ { n } \frac { \sum { i } \sum _ { i = 1 } ^ { n } \frac { \sum { i } \frac { \sum _ { i = 1 } ^ { n } \frac { \sum { i } \sum _ { i = 1 } ^ { n } \frac { \sum { i } \frac { \sum _ { i = 1 } ^ { n } \sum \frac { \sum { i } \frac { \sum _ { i = 1 } { i } ^ \sum { n } \frac { \sum \sum _ { i = 1 } } \frac { \frac { \sum { i \sum _ { i = 1 } ^ { n } \sum \frac { i } { \sum \sum _ { i = 1 } ^ { i } \frac { n \sum \sum _ { i } \frac { i } { i \operatorname { n } \sum \frac { \sum _ { i = 1 } \mathbb { i } \mathbb \mathbb { E } } } { \sum \mathbb \mathbb { E } ( \frac { \frac { 1 1 } { \prod { E _ { i } } \mathbb \mathbb \mathbb { E ( \frac { 1 \sum \sum _ { i } \mathbb \mathbb { E } \mathbb ( \frac { 1 \sum \mathbb { E } \mathbb ( 1 \cdot \sum _ { E } \mathbb \mathbb { E } ) ( \frac { 1 \sum _ { E \mathbb } \mathbb { E [ \mathbb | \sum _ { E \mathbb } \mathbb { E | ( E \frac { 1 } \mathbb { E | \sum \mathbb _ { E } \mathbb ( E  [ \sum _ { i } \mathbb \mathbb { E | \mathbb \sum _ { E } \mathbb ( E  [ \frac { 1 \sum _ { i = 1 } ^ { n } \mathbb \mathbb { E }  \Bigg ) } } } } ) } }  } } } } } } } } } } } } } } } } } } } } \\ &   \end{array}
+$$$$
+\begin{array} { r l } & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ &  \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  \end{array}
+$$
+
+It is relatively straightforward to show that the lemma holds in this last form. To do so note that except for terms for which $\tilde { T }$ and ${ \tilde { T } } ^ { \prime }$ are identical on the left (which are not possible on the right), each term on the left of the inequality is also present on the right, however the number of repetitions of each term varies between the left and right. In the left sum if a term includes m values shared between $\tilde { T }$ and ${ \tilde { T } } ^ { \prime }$ , this term will appear $\binom { 2 \bar { ( } n - i - 1 - m ) } { n - i - 1 - m }$ times. This is because we can choose $n -$ $i - m$ non-duplicate values to be in $\tilde { T }$ and place the rest in ${ \tilde { T } } ^ { \prime }$ , each of these permutations will correspond to a term in the sum. On the other hand in the left sum if a term includes m values shared between $\tilde { T }$ and ${ \tilde { T } } ^ { \prime }$ , this term will appear 2(n−i−1−m)n−i−2−m 
 times. Similarly this is because in this case we can choose $n - i - 1 - m$ non-duplicate values to be in $\tilde { T }$ and place the rest in ${ \tilde { T } } ^ { \prime }$ , each of these permutations will correspond to a different term in the sum.
+
+Since $\binom { 2 N } { N } > \binom { 2 N } { N - 1 }$ , $\forall N$ every term which is present on the right side is present on the left with more repetitions and thus the left side must be greater than the right and the lemma holds. □
+
+This lemma shows that $P$ is indeed a valid probability, which means that swapping according to it in algorithm 1 is admissible.
+
+Theorem 1. $\forall t \geq n , 0 \leq i \leq n - 1$ :
+
+$$
+P ( \hat { T } _ { t } [ i ] = t _ { i } | \hat { T } _ { t } [ 0 : i - 1 ] = [ t _ { 0 } , . . . , t _ { i - 1 } ] ) = \hat { P } ( t _ { i } | [ t _ { 0 } , . . . , t _ { i - 1 } ] ; T _ { t } , n )
+$$
+
+where $[ t _ { 0 } , . . . , t _ { i } ]$ is an arbitrary vector of unique elements of $T _ { t }$ .
+
+Proof. Let $\hat { T } _ { i , t }$ be the value of $\hat { T }$ right before making the swap decision on line 12 in loop index i of the loop starting at line 4 within the call to UPDATE with parameter t. Towards a proof by induction assume that at time index $t$ directly prior to making the swap decision for ${ \hat { T } } [ i ]$ at line 12 of UPDATE we have for any arbitrary vector $[ t _ { 0 } , . . . , t _ { i - 1 } ]$ of unique elements of $T _ { t }$ :
+
+$$
+P ( \hat { T } _ { i , t } [ j ] = t _ { j } | \hat { T } _ { t } [ 0 : j - 1 ] = [ t _ { 0 } , . . . , t _ { j - 1 } ] ) = \hat { P } ( t _ { j } | [ t _ { 0 } , . . . , t _ { j - 1 } ] ; T _ { t } , n ) ,
+$$
+
+and for any arbitrary vector $[ t _ { 0 } ^ { \prime } , . . . , t _ { i - 1 } ^ { \prime } ]$ of unique elements of $T _ { t + 1 }$ :
+
+$$
+\begin{array} { r } { P ( \hat { T } _ { i , t } [ j ] = t _ { j } ^ { \prime } | \hat { T } _ { t + 1 } [ 0 : j - 1 ] = [ t _ { 0 } ^ { \prime } , . . . , t _ { j - 1 } ^ { \prime } ] ) = \hat { P } ( t _ { j } | [ t _ { 0 } ^ { \prime } , . . . , t _ { j - 1 } ^ { \prime } ] ; T _ { t + 1 } , n ) , \quad \quad } \\ { \forall j \ s . t . \ 0 \leq j \leq i - 1 } \end{array}
+$$
+
+That is all elements of $\hat { T }$ from 0 to $i - 1$ have the desired probability conditioned on proceeding elements of $\hat { T } _ { t + 1 }$ while all elements from $i$ to $n - 1$ still have the desired probability when conditioned on proceeding elements of $\hat { T } _ { t }$ . This is a natural inductive assumption given we have already made our swap decisions up to but not including $i$ and are just about to make our decision for $i$ .
+
+Consider two mutually possible selections of $\hat { T } _ { t + 1 } [ 0 : i - 1 ] = [ t _ { 0 } ^ { \prime } , . . . , t _ { i - 1 } ^ { \prime } ]$ and $\hat { T } _ { t } [ 0 : i - 1 ] =$ $[ t _ { 0 } , . . . , t _ { i - 1 } ]$ and note that together these uniquely determine the value $\tau _ { t , i }$ . Fixing these values, the only possible way to end up with $\hat { T } _ { t + 1 } [ i ] = \hat { t }$ for $\hat { t } \in T _ { t } \setminus \{ t _ { 0 } , . . . , t _ { j - 1 } \}$ is to have $\hat { T } _ { t } [ i ] = \hat { t }$ and then choose not to swap ${ \hat { T } } [ i ]$ at time $t$ . Thus, substituting in $1 - P _ { t , i }$ using the expression for $P _ { t , i }$ obtained in lemma 5 we get:
+
+$$
+\begin{array} { r l } & { P ( \hat { T } _ { t + 1 } [ i ] = \hat { t } | \hat { T } _ { t + 1 } [ 0 : i - 1 ] = [ t _ { 0 } ^ { \prime } , . . . , t _ { j - 1 } ^ { \prime } ] , \hat { T } _ { t } [ 0 : i - 1 ] = [ t _ { 0 } , . . . , t _ { j - 1 } ] ) } \\ & { = \hat { P } ( \hat { t } | [ t _ { 0 } , . . . , t _ { i - 1 } ] ; T _ { t } , n ) \frac { \hat { P } ( \hat { t } | [ t _ { 0 } ^ { \prime } , . . . , t _ { i - 1 } ^ { \prime } ] ; T _ { t + 1 } , n ) } { \hat { P } ( \hat { t } | [ t _ { 0 } , . . . , t _ { i - 1 } ] ; T _ { t } , n ) } } \\ & { = \hat { P } ( \hat { t } | [ t _ { 0 } ^ { \prime } , . . . , t _ { i - 1 } ^ { \prime } ] ; T _ { t + 1 } , n ) } \end{array}
+$$
+
+On the other hand to end up with $\hat { T } _ { t + 1 } [ i ] = \tau _ { t , i }$ we may start with any $\hat { t } \in T _ { t } \setminus \left( \{ t _ { 0 } , . . . , t _ { j - 1 } \} \right)$ and then choose to swap ${ \hat { T } } [ i ]$ at time $t$ , in this case we get:
+
+$$
+\begin{array} { r l } & { P ( \widehat { T } _ { t + 1 } | \{ i \} = \pi _ { t , i } | \widehat { T } _ { t + 1 } | 0 : i - 1 | = [ t _ { 0 } ^ { i } , \dots , t _ { j - 1 } ^ { j } ] , \widehat { T } _ { t } | 0 : i - 1 | = [ \mu _ { 0 } , \dots , t _ { j - 1 } ] ) } \\ & { = \underset { i \in \mathbb { Z } \times \{ i \} } { = } \underset { ( i \in \mathbb { Z } \times \{ i \} \{ [ 0 , \dots , t _ { j - 1 } ] : T _ { t } , n \} } { \sum } \left( 1 - \frac { \widehat { P } _ { i } ( \widehat { \mathbb { I } } | \{ \mathbb { I } } _ { 0 } ^ { t } , \dots , t _ { j - 1 } ^ { j } | \cdot \mathbb { T } _ { t + 1 } , n ) \right)} { \widehat { P } _ { i } ( \widehat { \mathbb { I } } | \{ \mu _ { 0 } , \dots , t _ { j - 1 } \} ; T _ { t } , n ) }   \\ & { = \underset { i \in \mathbb { Z } \times \{ i \} \{ \sum _ { j = 1 } ^ { n } \dots \dots \dots \dots \dots \nmid } { \widehat { P } } ( \widehat { i } | [ \mu _ { 0 } , \dots , t _ { j - 1 } ] ; T _ { t } , n ) - \widehat { P } ( \widehat { i } | [ \mu _ { 0 } ^ { j } , \dots , t _ { j - 1 } ^ { j } ] ; T _ { t + 1 } , n ) } \\ & { \overset { ( i \in \mathbb { Z } \times \{ i \} \setminus \{ i \} \{ 0 , \dots , j \} ~ 1 ) } { = } \widehat { P } ( \widehat { i } | [ \mu _ { 0 } ^ { j } , \dots , t _ { j - 1 } ^ { j } ] ; T _ { t + 1 } , n ) } \\ &  = 1 - \underset { i \in \mathbb { Z } \times \{ i \} \times \{ i \} = \dots , i \not { = 1 \} } { \overset { } { \sum } } \widehat { P } ( \widehat { \mathbb { I } } | \mu _ { 0 } ^ { t } , \dots , t _ { j - 1 } ^ { j } | ; T _ { t + 1 } , n ) } \\ &  = 1 - \underset { i \in \mathbb { Z } \times \{ i \} } { = } \underset { i \in \mathbb { Z } \times \{ i \} } { \overset { } { \sum } } \underset { i \in \mathbb { Z } \times \{ i \} } { \overset { } { \sum } } \widehat { P } ( \widehat { \mathbb { I } } | \mu _ { 0 } ^ { t }  \end{array}
+$$
+
+Thus for any fixed mutually possible selections of $\hat { T } _ { t + 1 } [ 0 : i - 1 ] = [ t _ { 0 } ^ { \prime } , . . . , t _ { j - 1 } ^ { \prime } ]$ and $\hat { T } _ { t } [ 0 : i - 1 ] =$ $[ t _ { 0 } , . . . , t _ { j - 1 } ]$ for both $\tau _ { t , i }$ and all other $\hat { t } \in T _ { t + 1 } \setminus \{ t _ { 0 } ^ { \prime } , . . . , t _ { j - 1 } ^ { \prime } \}$ we end up with correct conditional probabilities for $\hat { T } _ { t + 1 } [ i ]$ assuming they are correct for $\hat { T } _ { t } [ i ]$ . Now note that the resulting probabilities are ultimately independent of $[ t _ { 0 } , . . . , t _ { j - 1 } ]$ , hence we would get the same values by conditioning on $[ t _ { 0 } ^ { \prime } , . . . , t _ { j - 1 } ^ { \prime } ]$ alone. Thus if our inductive assumption holds we have for any arbitrary vector $[ t _ { 0 } ^ { \prime } , . . . , t _ { i - 1 } ^ { \prime } ]$ of unique elements of $T _ { t + 1 }$ :
+
+$$
+P ( \hat { T } _ { i + 1 , t } [ j ] = t _ { j } ^ { \prime } | \hat { T } _ { t + 1 } [ 0 : j - 1 ] = [ t _ { 0 } ^ { \prime } , . . . , t _ { j - 1 } ^ { \prime } ] ) = \hat { P } ( t _ { j } | [ t _ { 0 } ^ { \prime } , . . . , t _ { j - 1 } ^ { \prime } ] ; T _ { t + 1 } , n ) ,
+$$
+
+Which completes the inductive step. To prove the base case note that we initialize $\hat { T }$ such that at time $n$ it is filled with all available items in random order. It is easy to show that equation 12 implies that the probability of any ordering of a given set $\hat { T }$ is equal thus if the items in $T$ exactly fill $\hat { T }$ then ordering them at random will give the desired probabilities $\hat { P } ( \hat { T } _ { n } [ i ] | \hat { T } _ { t + 1 } [ 0 \ :$ $i - 1 ] ; T _ { t + 1 } , n )$ , $\forall i$ s.t. $0 \leq i \leq n - 1$ , which gives us the base case to complete the inductive proof. □
+
+# E LENGTH 20 ENVIRONMENT EXPERIMENT
+
+Figure 6 shows the results of an experiment on the secret informant problem with environment length 20. Comparing figures 5 and 6 the number of episodes to convergence increases from 50, 000 to around 80, 000 with a doubling of the environment length while training still remains stable.
+
+![](images/87b6d3610c54779c4555c7ee01bcd42324d24151fe9802f2c277901b4aa86769.jpg)  
+Figure 6: Experiment with environment length 20, 2 decisions and a 3 state memory. (a) shows average write weight assigned to informative states (•) and uninformative states $( \_ )$ . (b) shows average return with episodic memory learner (•), we did not train a recurrent baseline for this problem. (c) shows the value of several relevant query vector elements in the decision state. N corresponds to the uninformative indicator,  to the informative indicator, $\bullet$ to the first decision state identifier, $\vee$ to the second decision state identifier. (d) shows the same thing but for the query generated in the second decision state.

parse/train/H1fl8S9ee/H1fl8S9ee.md

@@ -0,0 +1,379 @@
+# LEARNING AND POLICY SEARCH IN STOCHASTIC DYNAMICAL SYSTEMS WITH BAYESIAN NEURAL NETWORKS
+
+Stefan Depeweg Siemens AG and Technical University of Munich stefan.depeweg@siemens.com
+
+José Miguel Hernández-Lobato University of Cambridge jmh233@cam.ac.uk
+
+Finale Doshi-Velez   
+Harvard University   
+finale@seas.harvard.edu   
+Steffen Udluft   
+Siemens AG   
+steffen.udluft@siemens.com
+
+# ABSTRACT
+
+We present an algorithm for policy search in stochastic dynamical systems using model-based reinforcement learning. The system dynamics are described with Bayesian neural networks (BNNs) that include stochastic input variables. These input variables allow us to capture complex statistical patterns in the transition dynamics (e.g. multi-modality and heteroskedasticity), which are usually missed by alternative modeling approaches. After learning the dynamics, our BNNs are then fed into an algorithm that performs random roll-outs and uses stochastic optimization for policy learning. We train our BNNs by minimizing $\alpha$ -divergences with $\alpha = 0 . 5$ , which usually produces better results than other techniques such as variational Bayes. We illustrate the performance of our method by solving a challenging problem where model-based approaches usually fail and by obtaining promising results in real-world scenarios including the control of a gas turbine and an industrial benchmark.
+
+# 1 INTRODUCTION
+
+In model-based reinforcement learning, an agent uses its experience to first learn a model of the environment and then uses that model to reason about what action to take next. We consider the case in which the agent observes the current state $\mathbf { s } _ { t }$ , takes some action a, and then observes the next state $\mathbf { s } _ { t + 1 }$ . The problem of learning the model corresponds then to learning a stochastic transition function $p ( \mathbf { s } _ { t + 1 } | \mathbf { s } _ { t } , \mathbf { a } )$ specifying the conditional distribution of $\mathbf { s } _ { t + 1 }$ given $\mathbf { s } _ { t }$ and a. Most classic control theory texts, e.g. Bertsekas (2002), will start with the most general model of dynamical systems:
+
+$$
+\mathbf { s } _ { t + 1 } = f ( \mathbf { s } _ { t } , \mathbf { a } , z , \mathcal { W } )
+$$
+
+where $f$ is some deterministic function parameterized by weights $\mathcal { W }$ that takes as input the current state $\mathbf { s } _ { t }$ , the control signal a, and some stochastic disturbance $z$ .
+
+However, to date, we have not been able to robustly learn dynamical system models to such a level of generality. Popular modes for transition functions include Gaussian processes (Rasmussen et al., 2003; Ko et al., 2007; Deisenroth & Rasmussen, 2011), fixed bases such as Laguerre functions (Wahlberg, 1991), and adaptive basis functions or neural networks (Draeger et al., 1995). All of these methods assume deterministic transition functions, perhaps with some addition of Gaussian observation noise. Thus, they are severely limited in the kinds of stochasticity—or transition noise—they can express. In many real-world scenarios stochasticity may often arise due to some unobserved environmental feature that can affect the dynamics in complex ways (such as unmeasured gusts of wind on a boat).
+
+In this work we use Bayesian neural networks (BNNs) in conjunction with a random input noise source $z$ to express stochastic dynamics. We take advantage of a very recent inference advance based on $\alpha$ -divergence minimization (Hernández-Lobato et al., 2016), with $\alpha = 0 . 5$ , to learn with high accuracy BNN transition functions that are both scalable and expressive in terms of stochastic patterns. Previous work achieved one but not both of these two characteristics.
+
+We focus our evaluation on the off-policy batch reinforcement learning scenario, in which we are given an initial batch of data from an already-running system and are asked to find a better (ideally near-optimal) policy. Such scenarios are common in real-world industry settings such as turbine control, where exploration is restricted to avoid possible damage to the system. We propose an algorithm that uses random roll-outs and stochastic optimization for learning an optimal policy from the predictions of BNNs. This method produces (to our knowledge) the first model-based solution of a 20-year-old benchmark problem: the Wet-Chicken (Tresp, 1994). We also obtain very promising results on a real-world application on controlling gas turbines and on an industrial benchmark.
+
+# 2 BACKGROUND
+
+# 2.1 MODEL-BASED REINFORCEMENT LEARNING
+
+We consider reinforcement learning problems in which an agent acts in a stochastic environment by sequentially choosing actions over a sequence of time steps, in order to minimize a cumulative cost. We assume that our environment has some true dynamics $T _ { \mathrm { t r u e } } ( \mathbf { s } _ { t + 1 } | \mathbf { s } , \mathbf { a } )$ , and we are given a cost function $c ( \mathbf { s } _ { t } )$ . In the model-based reinforcement learning setting, our goal is to learn an approximation $T _ { \mathrm { a p p r o x } } ( \mathbf { s } _ { t + 1 } | \mathbf { s } , \mathbf { a } )$ for the true dynamics based on collected samples $( \mathbf { s } _ { t } , \mathbf { a } , \mathbf { s } _ { t + 1 } )$ . The agent then tries to solve the control problem in which $T _ { \mathrm { a p p r o x } }$ is assumed to be the true dynamics.
+
+# 2.2 BAYESIAN NEURAL NETWORKS WITH STOCHASTIC INPUTS
+
+Given data $\mathcal { D } = \{ \mathbf { x } _ { n } , \mathbf { y } _ { n } \} _ { n = 1 } ^ { N }$ , formed by feature vectors $\mathbf { x } _ { n } \in \mathbb { R } ^ { D }$ and targets $\mathbf { y } _ { n } \in \mathbb { R } ^ { K }$ , we assume that ${ \bf y } _ { n } = f ( { \bf x } _ { n } , z _ { n } ; \mathcal { W } ) + \epsilon _ { n }$ , where $f ( \cdot , \cdot ; \mathcal W )$ is the output of a neural network with weights The network receives as input the feature vector ${ \bf x } _ { n }$ and the random disturbance $z _ { n } \sim \mathcal { N } ( 0 , \gamma )$ . The activation functions for the hidden layers are rectifiers: $\varphi ( x ) = \operatorname* { m a x } ( x , 0 )$ . The activation functions for the output layers are the identity function: $\varphi ( x ) = x$ . The network output is corrupted by the additive noise variable $\epsilon _ { n } \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { \tilde { \Sigma } } )$ with diagonal covariance matrix $\pmb { \Sigma }$ . The role of the noise disturbance $z _ { n }$ is to capture unobserved stochastic features that can affect the network’s output in complex ways. Without $z _ { n }$ , randomness is only given by the additive Gaussian observation noise $\epsilon _ { n }$ , which can only describe limited stochastic patterns. The network has $L$ layers, with $V _ { l }$ hidden units in layer $l$ , and $\mathcal { W } = \{ \mathbf { W } _ { l } \} _ { l = 1 } ^ { L }$ is the collection of $V _ { l } \times ( V _ { l - 1 } + 1 )$ weight matrices. The $+ 1$ is introduced here to account for the additional per-layer biases.
+
+One could argue why $\epsilon _ { n }$ is needed at all when we are already using the more flexible stochastic model based on $z _ { n }$ . The reason for this is that, in practice, we make predictions with the above model by averaging over a finite number of samples of $z _ { n }$ and $\mathcal { W }$ . By using $\epsilon _ { n }$ , we obtain a predictive distribution whose density is well defined and given by a mixture of Gaussians. If we eliminate $\epsilon _ { n }$ , the predictive density is degenerate and given by a mixture of delta functions.
+
+Let $\mathbf { Y }$ be an $N \times K$ matrix with the targets ${ \bf y } _ { n }$ and $\mathbf { X }$ be an $N \times D$ matrix of feature vectors ${ \bf x } _ { n }$ We denote by $\mathbf { z }$ the $N$ -dimensional vector with the values of the random disturbances $z _ { 1 } , \dots , z _ { N }$ that were used to generate the data. The likelihood function is
+
+$$
+p ( \mathbf { Y } \mid \mathcal { W } , \mathbf { z } , \mathbf { X } ) = \prod _ { n = 1 } ^ { N } p ( \mathbf { y } _ { n } \mid \mathcal { W } , \mathbf { z } , \mathbf { x } _ { n } ) = \prod _ { n = 1 } ^ { N } \prod _ { k = 1 } ^ { K } \mathcal { N } \big ( y _ { n , k } \mid f ( \mathbf { x } _ { n } , z _ { n } ; \mathcal { W } ) , \pmb { \Sigma } \big ) .
+$$
+
+The prior for each entry in $\mathbf { z }$ is $\mathcal { N } ( 0 , \gamma )$ . We also specify a Gaussian prior distribution for each entry in each of the weight matrices in $\mathcal { W }$ . That is,
+
+$$
+p ( \mathbf { z } ) = \prod _ { n = 1 } ^ { N } \mathcal { N } ( z _ { n } | 0 , \gamma ) , \qquad p ( \mathcal { W } ) = \prod _ { l = 1 } ^ { L } \prod _ { i = 1 } ^ { V _ { l } } \prod _ { j = 1 } ^ { V _ { l - 1 } + 1 } \mathcal { N } ( w _ { i j , l } | 0 , \lambda ) ,
+$$
+
+where $w _ { i j , l }$ is the entry in the $i$ -th row and $j$ -th column of $\mathbf { W } _ { l }$ and $\gamma$ and $\lambda$ are a prior variances. The posterior distribution for the weights $\mathcal { W }$ and the random disturbances $\mathbf { z }$ is given by Bayes’ rule:
+
+$$
+p ( \mathcal { W } , \mathbf { z } \mid \mathcal { D } ) = \frac { p ( \mathbf { Y } \mid \mathcal { W } , \mathbf { z } , \mathbf { X } ) p ( \mathcal { W } ) p ( \mathbf { z } ) } { p ( \mathbf { Y } \mid \mathbf { X } ) } .
+$$
+
+![](images/62d41847667e5f61a0c16af5a6179e44cadeb706fbfc28d39e46d6482e8d2e90.jpg)  
+Figure 1: Solution for the minimization of the $\alpha$ -divergence between the posterior $p$ (in blue) and the Gaussian approximation $q$ (in red and unnormalized). Figure source Minka (2005).
+
+Given a new input vector $\mathbf { x } _ { \star }$ , we can then make predictions for $\mathbf { y } _ { \star }$ using the predictive distribution
+
+$$
+p ( \mathbf { y } _ { \star } | \mathbf { x } _ { \star } , \mathcal { D } ) = \int \left[ \int \mathcal { N } ( y _ { \star } | f ( \mathbf { x } _ { \star } , z _ { \star } ; \mathcal { W } ) , \pm ) \mathcal { N } ( z _ { \star } | 0 , 1 ) d z _ { \star } \right] p ( \mathcal { W } , \mathbf { z } | \mathcal { D } ) d \mathcal { W } d \mathbf { z } .
+$$
+
+Unfortunately, the exact computation of (4) is intractable and we have to use approximations.
+
+# 2.3 $_ { \pmb { \alpha } }$ -DIVERGENCE MINIMIZATION
+
+We approximate the exact posterior distribution $p ( \mathcal { W } , \mathbf { z } \mid \mathcal { D } )$ with the factorized Gaussian distribution
+
+$$
+q ( \mathcal W , \mathbf z ) = \left[ \prod _ { l = 1 } ^ { L } \prod _ { i = 1 } ^ { V _ { l } } \prod _ { j = 1 } ^ { V _ { l - 1 } + 1 } \mathcal N ( w _ { i j , l } | m _ { i j , l } ^ { w } , v _ { i j , l } ^ { w } ) \right] \left[ \prod _ { n = 1 } ^ { N } \mathcal N ( z _ { n } | m _ { n } ^ { z } , v _ { n } ^ { z } ) \right] .
+$$
+
+The parameters mwij,l, $v _ { i j , l } ^ { w }$ and $m _ { n } ^ { z } , \ v _ { n } ^ { z }$ are determined by minimizing a divergence between $p ( \mathcal { W } , \mathbf { z } \vert \mathcal { D } )$ and the approximation $q$ . After fitting $q$ , we make predictions by replacing $p ( \mathcal { W } , \mathbf { z } \mid \mathcal { D } )$ with $q$ in (4) and approximating the integrals in (4) with empirical averages over samples of $\mathcal { W } \sim q$ .
+
+We aim to adjust the parameters of (5) by minimizing the $\alpha$ -divergence between $p ( \mathcal { W } , \mathbf { z } \vert \mathcal { D } )$ and $q ( \boldsymbol { \mathcal { W } } , { \mathbf z } )$ (Minka, 2005):
+
+$$
+\mathrm { D } _ { \alpha } [ p ( \mathcal { W } , \mathbf { z } | \mathcal { D } ) | | q ( \mathcal { W } , \mathbf { z } ) ] = \frac { 1 } { \alpha ( \alpha - 1 ) } \left( 1 - \int p ( \mathcal { W } , \mathbf { z } | \mathcal { D } ) ^ { \alpha } q ( \mathcal { W } , \mathbf { z } ) ^ { ( 1 - \alpha ) } \right) d \mathcal { W } d \mathbf { z } ,
+$$
+
+which includes a parameter $\alpha \in \mathbb { R }$ that controls the properties of the optimal $q$ . Figure 1 illustrates these properties for the one-dimensional case. When $\alpha \geq 1$ , $q$ tends to cover the whole posterior distribution $p$ . When $\alpha \leq 0$ , $q$ tends to fit a local mode in $p$ . The value $\alpha = 0 . 5$ is expected to achieve a balance between these two tendencies. Importantly, when $\alpha  0$ , the solution obtained is the same as with variational Bayes (VB) (Wainwright & Jordan, 2008).
+
+The direct minimization of (6) is infeasible in practice for arbitrary $\alpha$ . Instead, we follow HernándezLobato et al. (2016) and optimize an energy function whose minimizer corresponds to a local minimization of $\alpha$ -divergences, with one $\alpha$ -divergence for each of the $N$ likelihood factors in (1). Since $q$ is Gaussian and the priors $p ( \mathcal { W } )$ and $p ( \mathbf { z } )$ are also Gaussian, we represent $q$ as
+
+$$
+q ( \mathcal { W } , { \mathbf z } ) \propto \left[ \prod _ { n = 1 } ^ { N } f ( \mathcal { W } ) f _ { n } ( z _ { n } ) \right] p ( \mathcal { W } ) p ( { \mathbf z } ) ,
+$$
+
+where $f ( \mathcal W )$ is a Gaussian factor that approximates the geometric mean of the $N$ likelihood factors in (1) as a function of $\mathcal { W }$ . Each $f _ { n } ( z _ { n } )$ is also a Gaussian factor that approximates the $n$ -th likelihood factor in (1) as a function of $z _ { n }$ . We adjust $f ( \mathcal W )$ and the $f _ { n } ( z _ { n } )$ by minimizing local $\alpha$ -divergences. In particular, we minimize the energy function
+
+$$
+E _ { \alpha } ( q ) = - \log Z _ { q } - \frac { 1 } { \alpha } \sum _ { n = 1 } ^ { N } \log \mathbf { E } _ { \mathcal { W } , z _ { n } \sim q } \left[ \left( \frac { p ( \mathbf { y } _ { n } \mid \mathcal { W } , \mathbf { x } _ { n } , z _ { n } , \pmb { \Sigma } ) } { f ( \mathcal { W } ) f _ { n } ( z _ { n } ) } \right) ^ { \alpha } \right] ,
+$$
+
+(Hernández-Lobato et al., 2016), where $f ( \mathcal W )$ and $f _ { n } ( z _ { n } )$ are in exponential Gaussian form and parameterized in terms of the parameters of $q$ and the priors $p ( \mathcal { W } )$ and $p ( z _ { n } )$ , that is,
+
+$$
+\begin{array} { l } { { \displaystyle f ( \mathcal { W } ) = \exp \left\{ \sum _ { l = 1 } ^ { L } \sum _ { i = 1 } ^ { V _ { l } } \sum _ { j = 1 } ^ { V _ { l - 1 } + 1 } \frac { 1 } { N } \left( \frac { \lambda v _ { i , j , l } ^ { w } } { \lambda - v _ { i , j , l } ^ { w } } w _ { i , j , l } ^ { 2 } + \frac { m _ { i , j , l } ^ { w } } { v _ { i , j , l } ^ { w } } w _ { i , j , l } \right) \right\} \propto \left[ \frac { q ( \mathcal { W } ) } { p ( \mathcal { W } ) } \right] ^ { \frac { 1 } { N } } , } } \\ { { \displaystyle f _ { n } ( z _ { n } ) = \exp \left\{ \frac { \gamma v _ { n } ^ { z } } { \gamma - v _ { n } ^ { z } } z _ { n } ^ { 2 } + \frac { m _ { n } ^ { z } } { v _ { n } ^ { z } } z _ { n } \right\} \propto \frac { q ( z _ { n } ) } { p ( z _ { n } ) } } , }  \end{array}
+$$
+
+and $\log Z _ { q }$ is the logarithm of the normalization constant of the exponential Gaussian form of $q$ :
+
+$$
+\log Z _ { q } = \sum _ { l = 1 } ^ { L } \sum _ { i = 1 } ^ { V _ { l } } \sum _ { j = 1 } ^ { V _ { l - 1 } + 1 } \left[ \frac { 1 } { 2 } \log \left( 2 \pi v _ { i , j , l } ^ { w } \right) + \frac { \left( m _ { i , j , l } ^ { w } \right) ^ { 2 } } { v _ { i , j , l } ^ { w } } \right] + \sum _ { n = 1 } ^ { N } \left[ \frac { 1 } { 2 } \log \left( 2 \pi v _ { n } ^ { z } \right) + \frac { \left( m _ { n } ^ { z } \right) ^ { 2 } } { v _ { n } ^ { z } } \right] .
+$$
+
+The scalable optimization of (8) is done in practice by using stochastic gradient descent. For this, we subsample the sums for $n = 1 , \ldots , N$ in (8) and (11) using mini-batches and approximate the expectations over $q$ in (8) with an average over $K$ samples drawn from $q$ . We can then use the reparametrization trick (Kingma et al., 2015) to obtain gradients from the resulting stochastic approximator to (8). The hyper-parameters $\pmb { \Sigma }$ , $\lambda$ and $\gamma$ can also be tuned by minimizing (8). In practice we only tune $\pmb { \Sigma }$ and keep $\lambda = 1$ and $\gamma = d$ . The latter means that the prior scale of each $z _ { n }$ grows with the data dimensionality. This guarantees that, a priori, the effect of each $z _ { n }$ in the neural network’s output does not diminish when more and more features are available.
+
+Minimizing (8) when $\alpha  0$ is equivalent to running the method VB (Hernández-Lobato et al., 2016), which has recently been used to train Bayesian neural networks in reinforcement learning problems (Blundell et al., 2015; Houthooft et al., 2016; Gal et al., 2016). However, we propose to minimize (8) using $\alpha = 0 . 5$ , which often results in better test log-likelihood values.
+
+We have also observed $\alpha = 0 . 5$ to be more robust than VB when $q ( \mathbf { z } )$ is not fully optimized. In particular, $\alpha = 0 . 5$ can still capture complex stochastic patterns even when we do not learn $q ( \mathbf { z } )$ and instead keep it fixed to the prior $p ( \mathbf { z } )$ . By contrast, VB fails completely in this case (see Appendix A).
+
+# 3 POLICY SEARCH USING BNNS WITH STOCHASTIC INPUTS
+
+We now describe a gradient-based policy search algorithm that uses the BNNs with stochastic disturbances from the previous section. The motivation for our approach lies in its applicability to industrial systems: we wish to estimate a policy in parametric form, using only an available batch of state transitions obtained from an already-running system. We assume that the true dynamics present stochastic patterns that arise due to some unobserved process affecting the system in complex ways.
+
+Model-based policy search methods include two key parts (Deisenroth et al., 2013). The first part consists in learning a dynamics model from data in the form of state transitions $\left( \mathbf { s } _ { t } , \mathbf { a } _ { t } , \mathbf { s } _ { t + 1 } \right)$ , where $\mathbf { s } _ { t }$ denotes the current state, $\mathbf { a } _ { t }$ is the action applied and $\mathbf { s } _ { t + 1 }$ is the resulting state. The second part consists in learning the parameters $\mathcal { W } _ { \pi }$ of a deterministic policy function $\pi$ that returns the optimal action $\mathbf { a } _ { t } = \pi ( \mathbf { s } _ { t } ; \mathcal { W } _ { \pi } )$ as function of the current state $\mathbf { s } _ { t }$ . The policy function can be a neural network with deterministic weights given by $\mathcal { W } _ { \pi }$ .
+
+The first part in the aforementioned procedure is a standard regression task, which we solve by using the modeling approach from the previous section. We assume the dynamics to be stochastic with the following true transition model:
+
+$$
+\begin{array} { r } { \mathbf { s } _ { t } = f _ { \mathrm { t r u e } } \left( \mathbf { s } _ { t - 1 } , \mathbf { a } _ { t - 1 } , z _ { t } ; \mathcal { W } _ { \mathrm { t r u e } } \right) , \quad z _ { t } \sim \mathcal { N } ( 0 , \gamma ) . } \end{array}
+$$
+
+where the input disturbances $z _ { t } \sim \mathcal { N } ( 0 , \gamma )$ account for the stochasticity in the dynamics. When the Markov state $\mathbf { s } _ { t }$ is hidden and we are given only observations $\mathbf { o } _ { t }$ , we can use the time embedding theorem using a suitable window of length $n$ and approximate:
+
+$$
+\hat { \mathbf { s } } ( t ) = \left[ \mathbf { o } _ { t - n } , \cdot \cdot \cdot \mathbf { \sigma } , \mathbf { o } _ { t } \right] .
+$$
+
+The transition model in equation 12 specifies a probability distribution $p ( \mathbf { s } _ { t } | \mathbf { s } _ { t - 1 } , \mathbf { a } _ { t - 1 } )$ that we approximate using a BNN with stochastic inputs:
+
+$$
+p ( \mathbf { s } _ { t } | \mathbf { s } _ { t - 1 } , \mathbf { a } _ { t - 1 } ) \approx \int \mathcal { N } ( \mathbf { s } _ { t } | f ( \mathbf { s } _ { t - 1 } , \mathbf { a } _ { t - 1 } , z _ { t } ; \mathcal { W } ) , \pmb { \Sigma } ) q ( \mathcal { W } ) \mathcal { N } ( z _ { t } | 0 , \gamma ) d \mathcal { W } d z _ { t } ,
+$$
+
+Algorithm 1 Model-based policy search using Bayesian neural networks with stochastic inputs.   
+
+<table><tr><td>1: Input: D= {Sn,an,△n} for n ∈1..N 2: Fit q(W) and ∑ by optimizing (8). 3: function UNFOLD(So) 4: sample{W1,.,WK} from q(W) 5: C↑0 6: fork=1:Kdo 7: fort=O :Tdo 2+1~N(0,2） 8: 9: △t←f(st,π(st;Wπ),2t+1;Wk) e+1~N(0,∑) 10: 11:</td></tr></table>
+
+![](images/5585956c667bef16e2cc88bd3a8a1c016f2f1b236743a4501b1c456350bb7093.jpg)  
+Figure 2: Predictive distribution of $y _ { t }$ given by different methods in four different scenarios. Ground truth (red) is obtained by sampling from the real dynamics.
+
+where the feature vectors in our BNN are now $\mathbf { s } _ { t - 1 }$ and $\mathbf { a } _ { t - 1 }$ and the targets are given by $\mathbf { s } _ { t }$ . In this expression, the integration with respect to $\mathcal { W }$ accounts for stochasticity arising from lack of knowledge of the model parameters, while the integration with respect to $z _ { t }$ accounts for stochasticity arising from unobserved processes that cannot be modeled. In practice, these integrals are approximated by an average over samples of $z _ { t } \sim \mathcal { N } ( 0 , \gamma )$ and $\mathcal { W } \sim q$ .
+
+In the second part of our model-based policy search algorithm, we optimize the parameters $\mathcal { W } _ { \pi }$ of a policy that minimizes the sum of expected cost over a finite horizon $T$ with respect to our belief $q ( \mathcal { W } )$ . This expected cost is obtained by averaging over multiple virtual roll-outs. For each roll-out we sample $w _ { i } \sim q$ and then simulate state trajectories using the model $\mathbf { s } _ { t + 1 } = f ( \mathbf { s } _ { t } , \mathbf { a } _ { t } , z _ { t } ; \mathcal { W } _ { i } ) + \boldsymbol { \epsilon } _ { t + 1 }$ with policy $\mathbf { a } _ { t } = \pi ( \mathbf { s } _ { t } ; \mathcal { W } _ { \pi } )$ , input noise $z _ { t } \sim \mathcal { N } ( 0 , \gamma )$ and additive noise $\epsilon _ { t + 1 } \sim \mathcal { N } ( \mathbf { 0 } , \Sigma )$ . This procedure allows us to obtain estimates of the policy’s expected cost for any particular cost function. If model, policy and cost function are differentiable, we are then able to tune $\mathcal { W } _ { \pi }$ by stochastic gradient descent over the roll-out average.
+
+Given a cost function $c ( \mathbf { s } _ { t } )$ , the objective to be optimized by our policy search algorithm is
+
+$$
+\begin{array} { r } { J ( \mathcal { W } _ { \pi } ) = \mathbf { E } \left[ \sum _ { t = 1 } ^ { T } c ( \mathbf { s } _ { t } ) \right] . } \end{array}
+$$
+
+We approximate (15) by using (14), replacing $\mathbf { a } _ { t }$ with $\pi ( \mathbf { s } _ { t } ; \mathcal { W } _ { \pi } )$ and using sampling to approximate the expectations:
+
+$$
+\begin{array} { l } { { \displaystyle { \cal J } ( \mathcal { W } _ { \pi } ) = \int \left[ \displaystyle { \sum _ { t = 1 } ^ { T } c ( { \bf s } _ { t } ) } \right] \left[ \displaystyle { \prod _ { t = 1 } ^ { T } \int } \mathcal { N } ( { \bf s } _ { t } | f ( { \bf s } _ { t - 1 } , \pi ( { \bf s } _ { t - 1 } ; \mathcal { W } _ { \pi } ) , z _ { t } ; \mathcal { W } ) , { \bf \Sigma } ) q ( \mathcal { W } ) \mathcal { N } ( z _ { t } | 0 , \gamma ) d \mathcal { W } d z _ { t } \right] } } \\ { { \displaystyle p ( { \bf s } _ { 0 } ) d { \bf s } _ { 0 } \cdot \cdot \cdot \cdot d { \bf s } _ { T } } } \\  { \displaystyle ~ = \int \left[ \displaystyle { \sum _ { t = 1 } ^ { T } c ( { \bf s } _ { t } ^ { \mathcal { W } , \{ z _ { 1 } , \dots , z _ { t } \} , \{ { \bf s } _ { t } \} , \dots , { \bf s } _ { t } \} , \mathcal { W } _ { \pi } ) }  q ( \mathcal { W } ) d \mathcal { W } \left[ \displaystyle { \prod _ { t = 1 } ^ { T } \mathcal { N } ( \epsilon _ { t } | { \bf 0 } , { \bf \Sigma } ) \mathcal { N } ( z _ { t } | { \bf 0 } , \gamma ) d \epsilon _ { t } d z _ { t } } \right] p ( { \bf s } _ { 0 } ) d { \bf s } _ { 0 } } } \\  {\right] \displaystyle \approx \displaystyle \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \left[ \displaystyle { \sum _ { t = 1 } ^ { T } c ( { \bf s } _ { t } ^ { \mathcal { W } ^ { k } , \{ z _ { 1 } ^ { k } , \dots , z _ { t } ^ { k } \} , \{ { \bf s } _ { t } ^ { k } , \dots , { \bf s } _ { t } ^ { k } \} , \mathcal { W } _ { \pi } ) }  . } } \end{\right]array} \end{array}
+$$
+
+The first line in (16) is obtained by using the assumption that the dynamics are Markovian with respect to the current state and the current action and by replacing $p ( \mathbf { s } _ { t } | \mathbf { s } _ { t - 1 } , \mathbf { a } _ { t - 1 } )$ with the right-hand side of (14). In the second line, sW,{z1,...,zt},{1,...,t},Wπt is the state that is obtained at time t in a roll-out generated by using a policy with parameters $\mathcal { W } _ { \pi }$ , a transition function parameterized by $\mathcal { W }$ and input noise $z _ { 1 } , \dots , z _ { t }$ , with additive noise values $\epsilon _ { 1 } , \ldots , \epsilon _ { t }$ . In the last line we have approximated the integration with respect to $\mathcal { W } , z _ { 1 } , \ldots , z _ { T } , \epsilon _ { 1 } , \ldots , \epsilon _ { T }$ and $\mathbf { s } _ { 0 }$ by averaging over $K$ samples of these variables. To sample $\mathbf { s } _ { 0 }$ , we draw this variable uniformly from the available transitions $\left( \mathbf { s } _ { t } , \mathbf { a } _ { t } , \mathbf { s } _ { t + 1 } \right)$ .
+
+The expected cost (15) can then be optimized by stochastic gradient descent using the gradients of the Monte Carlo approximation given by the last line of (16). Algorithm 1 computes this Monte
+
+![](images/84707d6ab02c7350759998233bfcfcaf2a82dd4d7fcca965d2bb73ef7c1ee91e.jpg)  
+Figure 3: Visualization of three policies in state space. Waterfall is indicated by top black bar. Left: policy $\pi _ { V B }$ obtained with a BNN trained with VB. Avg. reward is $- 2 . 5 3$ . Middle: policy $\pi _ { \alpha = 0 . 5 }$ obtained with a BNN trained with $\alpha = 0 . 5$ . Avg. reward is $- 2 . 3 1$ . Right: policy $\pi _ { G P }$ obtained by using a Gaussian process model. Avg. reward is $- 2 . 9 4$ . Color and arrow indicate direction of paddling of policy when in state $\mathbf { s } _ { t }$ , arrow length indicates action magnitude. Best viewed in color.
+
+<table><tr><td>Dataset</td><td>MLP</td><td>VB</td><td>α=0.5</td><td>α=1.0</td><td>GP</td><td>PSO-P</td></tr><tr><td>Wetchicken</td><td>-2.71±0.09</td><td>-2.67±0.10</td><td>-2.37±0.01</td><td>-2.42±0.01</td><td>-3.05±0.06</td><td>-2.34</td></tr><tr><td>Turbine</td><td>-0.65±0.14</td><td>-0.45±0.02</td><td>-0.41±0.03</td><td>-0.55±0.08</td><td>-0.64±0.18</td><td>NA</td></tr><tr><td>Industrial</td><td>-183.5±4.1</td><td>-180.2±0.6</td><td>-174.2±1.1</td><td>-171.1±2.1</td><td>-285.2±20.5</td><td>-145.5</td></tr><tr><td>Avg. Rank</td><td>3.6±0.3</td><td>3.1±0.2</td><td>1.5±0.2</td><td>2.3±0.3</td><td>4.5±0.3</td><td></td></tr></table>
+
+Table 1: Policy performances over different benchmarks. Printed are average values over 5 runs with respective standard errors. Bottom row is the average rank over all $5 \times 3$ runs.
+
+Carlo approximation. The gradients can then be obtained using automatic differentiation tools such as Theano (Theano Development Team). Note that Algorithm 1 uses the BNNs to make predictions for the change in the state $\Delta _ { t } = { \bf s } _ { t + 1 } - { \bf s } _ { t }$ instead of for the next state $\mathbf { s } _ { t + 1 }$ since this approach often performs better in practice (Deisenroth & Rasmussen, 2011).
+
+# 4 EXPERIMENTS
+
+We now evaluate the performance of our algorithm for policy search in different benchmark problems. These problems are chosen based on two reasons. First, they contain complex stochastic dynamics and second, they represent real-world applications common in industrial settings. A theano implementation of algorithm 1 is available online1. See the appendix B for a short introduction to all methods we compare to and appendix C for the hyper-parameters used.
+
+# 4.1 WET-CHICKEN BENCHMARK
+
+The Wet-Chicken benchmark (Tresp, 1994) is a challenging problem for model-based policy search that presents both bi-modal and heteroskedastic transition dynamics. We use the two-dimensional version of the problem (Hans & Udluft, 2009) and extend it to the continuous case.
+
+In this problem, a canoeist is paddling on a two-dimensional river. The canoeist’s position at time $t$ is $( x _ { t } , y _ { t } )$ . The river has width $w = 5$ and length $l = 5$ with a waterfall at the end, that is, at $y _ { t } = l$ . The canoeist wants to move as close to the waterfall as possible because at time $t$ he gets reward $r _ { t } = - ( l - y _ { t } )$ . However, going beyond the waterfall boundary makes the canoeist fall down, having to start back again at the origin $( 0 , 0 )$ . At time $t$ the canoeist can choose an action $( a _ { t , x } , a _ { t , y } ) \in [ - 1 , 1 ] ^ { 2 }$ that represents the direction and magnitude of his paddling. The river dynamics have stochastic turbulences $s _ { t }$ and drift $v _ { t }$ that depend on the canoeist’s position on the $x$ axis. The larger $x _ { t }$ , the larger the drift and the smaller $x _ { t }$ , the larger the turbulences. The underlying dynamics are given by the following system of equations. The drift and the turbulence magnitude are given by $v _ { t } \stackrel { - } { = } 3 x _ { t } w ^ { - 1 }$ and $s _ { t } = 3 . 5 - v _ { t }$ , respectively. The new location $( x _ { t + 1 } , y _ { t + 1 } )$ is given by the current
+
+<table><tr><td>Dataset</td><td>MLP</td><td>VB</td><td>α=0.5</td><td>α=1.0</td><td>GP</td></tr><tr><td colspan="6">MSE</td></tr><tr><td>WetChicken</td><td>1.289±0.013</td><td>1.347±0.015</td><td>1.347±0.008</td><td>1.359±0.017</td><td>1.359±0.017</td></tr><tr><td>Turbine</td><td>0.16±0.001</td><td>0.21±0.003</td><td>0.192±0.002</td><td>0.237±0.004</td><td>0.492±0.026</td></tr><tr><td>Industrial</td><td>0.0186±0.0052</td><td>0.0182±0.0052</td><td>0.017±0.0046</td><td>0.0171±0.0047</td><td>0.0233±0.0049</td></tr><tr><td>Avg. Rank</td><td>2.0±0.34</td><td>3.1±0.24</td><td>2.4±0.23</td><td>2.9±0.36</td><td>4.6±0.23</td></tr><tr><td colspan="6">Log-Likelihood</td></tr><tr><td>WetChicken</td><td>-1.755±0.003</td><td>-1.140±0.033</td><td>-1.057±0.014</td><td>-1.070±0.011</td><td>-1.722±0.011</td></tr><tr><td>Turbine</td><td>-0.868±0.007</td><td>-0.775±0.004</td><td>-0.746±0.013</td><td>-0.774±0.015</td><td>-2.663±0.131</td></tr><tr><td>Industrial</td><td>0.767±0.047</td><td>1.132±0.064</td><td>1.328±0.108</td><td>1.326±0.098</td><td>0.724±0.04</td></tr><tr><td>Avg. Rank</td><td>4.3±0.12</td><td>2.6±0.16</td><td>1.3±0.15</td><td>2.1±0.18</td><td>4.7±0.12</td></tr></table>
+
+Table 2: Model test error and test log-likelihood for different benchmarks. Printed are average values over 5 runs with respective standard errors. Bottom row is the average rank over all $5 \times 3$ runs.
+
+location $( x _ { t } , y _ { t } )$ and current action $( a _ { t , x } , a _ { t , y } )$ using
+
+$$
+x _ { t + 1 } = \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f } \quad x _ { t } + a _ { t , x } < 0 } \\ { 0 } & { \mathrm { i f } \quad \hat { y } _ { t + 1 } > l } \\ { w } & { \mathrm { i f } \quad x _ { t } + a _ { t , x } > w } \\ { x _ { t } + a _ { t , x } } & { \mathrm { o t h e r w i s e } } \end{array} \right. , \qquad y _ { t + 1 } = \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f } \quad \hat { y } _ { t + 1 } < 0 } \\ { 0 } & { \mathrm { i f } \quad \hat { y } _ { t + 1 } > l } \\ { \hat { y } _ { t + 1 } } & { \mathrm { o t h e r w i s e } } \end{array} \right. ,
+$$
+
+where $\hat { y } _ { t + 1 } = y _ { t } + \left( a _ { t , y } - 1 \right) + v _ { t } + s _ { t } \tau _ { t }$ and $\tau _ { t } \sim \mathrm { U n i f } ( [ - 1 , 1 ] )$ is a random variable that represents the current turbulence. These dynamics result in rich transition distributions depending on the position as illustrated by the plots in Figure 2. As the canoeist moves closer to the waterfall, the distribution for the next state becomes increasingly bi-modal (see Figure 1c) because when he is close to the waterfall, the change in the current location can be large if the canoeist falls down the waterfall and starts again at $( 0 , 0 )$ . The distribution may also be truncated uniform for states close to the borders (see Figure 1d). Furthermore the system has heteroskedastic noise, the smaller the value of $x _ { t }$ the higher the noise variance (compare Figure 1a with 1b). Because of these properties, the Wet-Chicken problem is especially difficult for model-based reinforcement learning methods. To our knowledge it has only been solved using model-free approaches after a discretization of the state and action sets (Hans & Udluft, 2009). For model training we use a batch 2500 random state transitions.
+
+The predictive distributions of different models for $y _ { t + 1 }$ are shown in Figure 2 for specific choices of $( x _ { t } , y _ { t } )$ and $( a _ { x , t } , a _ { y , t } )$ . These plots show that BNNs with $\alpha = 0 . 5$ are very close to the ground-truth. While it is expected that Gaussian processes fail to model multi-modalities in Figure 1c, the FTIC approximation allows them to model the heteroskedasticity to an extent. VB captures the stochastic patterns on a global level, but often under or over-estimates the true probability density in specific regions. The test-loglikelihood and test MSE in $y$ -dimension are reported in Table 2 for all methods. (the transitions for $x$ are deterministic given $y$ ).
+
+After fitting the models, we train policies using Algorithm 1 with a horizon of size $T = 5$ . Table 1 shows the average reward obtained by each method. BNNs with $\alpha = 0 . 5$ perform best and produce policies that are very close to the optimal upper bound, as indicated by the performance of the particle swarm optimization policy (PSO-P). In this problem VB seems to lack robustness and has much larger empirical variance across experiment repetitions than $\alpha = 0 . 5$ or $\alpha = 1 . 0$ .
+
+Figure 3 shows three example policies, πVB , $\pi _ { \alpha = 0 . 5 }$ and $\pi _ { \mathrm { G P } }$ (Figure 3a,3b and 3c, respectively). The policies obtained by BNNs with random inputs (VB and $\alpha = 0 . 5$ ) show a richer selection of actions. The biggest differences are in the middle-right regions of the plots, where the drift towards the waterfall is large and the bi-modal transition for $y$ (missed by the GP) is more important.
+
+# 4.2 INDUSTRIAL APPLICATIONS
+
+We now present results on two industrial cases. First, we focus on data generated by a real gas turbine and second, we consider a recently introduced simulator called the "industrial benchmark", with code publicly available2 (Hein et al., 2016b). According to the authors: "The "industrial benchmark" aims at being realistic in the sense, that it includes a variety of aspects that we found to be vital in industrial applications."
+
+![](images/73a893d718f52a13f31cd7f47999e4a4d6a4fe0125923c611a0ccda9a0afe6c5.jpg)  
+Figure 4: Roll-outs of algorithm 1 for two starting states $\mathbf { s } _ { 0 }$ (top/bottom) using different types of BNNs (left to right) with $K = 7 5$ samples for $T = 7 5$ steps. Action sequence $A _ { 0 } , \cdots , A _ { T = 7 5 }$ given by dataset for each $\mathbf { s } _ { 0 }$ . From left to right: model trained using VB, $\alpha = 0 . 5$ and $\alpha = 1 . 0$ respectively. Red: trajectory observed in dataset, blue: sample average, light blue: individual samples.
+
+# 4.2.1 GAS TURBINE DATA
+
+For the experiment with gas turbine data we simulate a task with partial observability. To that end we use 40,000 observations of a 30 dimensional time-series of sensor recordings from a real gas turbine. We are also given a cost function that evaluates the performance of the current state of the turbine. The features in the time-series are grouped into three different sets: a set of environmental variables $E _ { t }$ (e.g. temperature and measurements from sensors in the turbine) that cannot be influenced by the agent, a set of variables relevant for the cost function $N _ { t }$ (e.g. the turbines current pollutant emission) and a set of steering variables $A _ { t }$ that can be manipulated to control the turbine.
+
+We first train a world model as a reflection of the real turbine dynamics. To that end we define the world model’s transitions for $N _ { t }$ to have the functional form $N _ { t } = f ( E _ { t - 5 } , . . , E _ { t } , A _ { t - 5 } , . . A _ { t } ) .$ . The world model assumes constant transitions for the environmental variables: $E _ { t + 1 } = E _ { t }$ . To make fair comparisons, our world model is given by a non-Bayesian neural network with deterministic weights and with additive Gaussian output noise.
+
+We then use the world model to generate an artificial batch of data for training the different methods. The inputs in this batch are still the same as in the original turbine data, but the outputs are now sampled from the world model. After generating the artificial data, we only keep a small subset of the original inputs to the world model. The aim of this experiment is to learn policies that are robust to noise in the dynamics. This noise would originate from latent factors that cannot be controlled, such as the missing features that were originally used to generate the outputs by the world model but which are no longer available. After training the models for the dynamics, we use algorithm 1 for policy optimization. The resulting policies are then finally evaluated in the world model.
+
+Tables 2 and 1 show the respective model and policy performances for each method. The experiment was repeated 5 times and we report average results. We observe that $\alpha = 0 . 5$ performs best in this scenario, having the highest test log-likelihood and best policy performance.
+
+# 4.2.2 INDUSTRIAL BENCHMARK
+
+In this benchmark the hidden Markov state space $\mathbf { s } _ { t }$ consists of 27 variables, whereas the observable state $\mathbf { o } _ { t }$ is only 5 dimensional. This observable state consists of 3 adjustable steering variables $A _ { t }$ : the velocity $v ( t )$ , the gain $g ( t )$ and the shift $s ( t )$ . We also observe the fatigue $f ( t )$ and consumption $c ( t )$ that together form the reward signal $R ( t ) = - ( 3 f ( t ) + c ( t ) )$ . Also visible is the setpoint $S$ , a constant hyper-parameter of the benchmark that indicates the complexity of the dynamics.
+
+For each setpoint $S \in \{ 1 0 , 2 0 , \cdots , 1 0 0 \}$ we generate 7 trajectories of length 1000 using random exploration. This batch with $7 0 , 0 0 0$ state transitions forms the training set. We use 30, 000 state transitions, consisting of 3 trajectories for each setpoint, as test set.
+
+For data preprocessing, in addition to the standard normalization process, we apply a log transformation to the reward variable. Because the reward is bounded in the interval $[ 0 , R _ { m a x } ]$ , we use a logit transformation to map this interval into the real line. We define the functional form for the dynamics as $R _ { t } = f ( A _ { t - 1 5 } , \cdot \cdot \cdot , A _ { t } , R _ { t - 1 5 } , \cdot \cdot \cdot , R _ { t - 1 } )$ .
+
+The test errors and log-likelihood are given in Table 2. We see that BNNs with $\alpha = 0 . 5$ and $\alpha = 1 . 0$ perform best here, whereas Gaussian processes or the MLP obtain rather poor results.
+
+Each row in Figure 4 visualizes long term predictions of the MLP and BNNs trained with VB and $\alpha = 0 . 5$ in two specific cases. In the top row we see that while all three methods produce wrong predictions in expectation (compare dark blue curve to red curve). However, BNNs trained with $V B$ and with $\alpha = 0 . 5$ exhibit a bi-modal distribution of predicted trajectories, with one mode following the ground-truth very closely. By contrast, the MLP misses the upper mode completely. The bottom row shows that the VB and $\alpha = 0 . 5$ also produce more tight confident bands in other settings.
+
+Next, we learn policies using the trained models. Here we use a relatively long horizon of $T = 7 5$ steps. Table 1 shows average rewards obtained when applying the policies to the real dynamics. Because both benchmark and models have an autoregressive component, we do an initial warm-up phase using random exploration before we apply the policies to the system and start to measure rewards.
+
+We observe that GPs perform very poorly in this benchmark. We believe the reason for this is the long search horizon, which makes the uncertainties in the predictive distributions of the GPs become very large. Tighter confidence bands, as illustrated in Figure 4 seem to be key for learning good policies. Overall, $\alpha = 1 . 0$ performs best with $\alpha = 0 . 5$ being very close.
+
+# 5 RELATED WORK
+
+There has been relatively little attention to using Bayesian neural networks for reinforcement learning. In Blundell et al. (2015) a Thompson sampling approach is used for a contextual bandits problem; the focus is tackling the exploration-exploitation trade-off, while the work in Watter et al. (2015) combines variational auto-encoder with stochastic optimal control for visual data. Compared to our approach the first of these contributions focusses on the exploration/exploitation dilemma, while the second one uses a stochastic optimal control approach to solve the learning problem. By contrast, our work seeks to find an optimal parameterized policy.
+
+Policy gradient techniques are a prominent class of policy search algorithms (Peters & Schaal, 2008). While model-based approaches were often used in discrete spaces (Wang & Dietterich, 2003), model-free approaches tended to be more popular in continuous spaces (e.g. Peters & Schaal (2006)).
+
+Our work can be seen as a Monte-Carlo model-based policy gradient technique in continuous stochastic systems. Similar work was done using Gaussian processes (Deisenroth & Rasmussen, 2011) and with recurrent neural networks (Schaefer et al., 2007) . The Gaussian process approach, while restricted to a Gaussian state distribution, allows propagating beliefs over the roll-out procedure. More recently Gu et al. (2016) augment a model-free learning procedure with data generated from model-based roll-outs.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+We have extended the standard Bayesian neural network (BNN) model with the addition of a random input noise source $z$ . This enables principled Bayesian inference over complex stochastic functions. We have shown that our BNNs with random inputs can be trained with high accuracy by minimizing $\alpha$ -divergences, with $\alpha = 0 . 5$ , which often produces better results than variational Bayes. We have also presented an algorithm that uses random roll-outs and stochastic optimization for learning a parameterized policy in a batch scenario. This algorithm particular suited for industry domains.
+
+Our BNNs with random inputs have allowed us to solve a challenging benchmark problem where model-based approaches usually fail. They have also shown promising results on industry benchmarks including real-world data from a gas turbine. In particular, our experiments indicate that a BNN trained with $\alpha = 0 . 5$ as divergence measure in conjunction with the presented algorithm for policy optimization is a powerful black-box tool for policy search.
+
+As future work we will consider safety and exploration. For safety, we believe having uncertainty over the underlaying stochastic functions will allows us to optimize policies by focusing on worst case results instead of on average performance. For exploration, having uncertainty on the stochastic functions will be useful for efficient data collection.
+
+# ACKNOWLEDGEMENTS
+
+José Miguel Hernández-Lobato acknowledges support from the Rafael del Pino Foundation. The authors would like to thank Ryan P. Adams, Hans-Georg Zimmermann, Matthew J. Johnson, David Duvenaud and Justin Bayer for helpful discussions.
+
+# REFERENCES
+
+D.P. Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific optimization and computation series. 2002. ISBN 9781886529083.
+
+Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural networks. In ICML, pp. 1613–1622, 2015.
+
+Thang D Bui, Daniel Hernández-Lobato, Yingzhen Li, José Miguel Hernández-Lobato, and Richard E Turner. Deep Gaussian processes for regression using approximate expectation propagation. In ICML, pp. 1472–1481, 2016.
+
+Marc Deisenroth and Carl E Rasmussen. PILCO: A model-based and data-efficient approach to policy search. In ICML, pp. 465–472, 2011.
+
+Marc Peter Deisenroth, Gerhard Neumann, and Jan Peters. A survey on policy search for robotics. Foundations and Trends in Robotics, 2:1–142, 2013.
+
+A. Draeger, S. Engell, and H. Ranke. Model predictive control using neural networks. IEEE Control Systems, 15:61–66, 1995.
+
+Yarin Gal, Rowan Mcallister, and Carl Rasmussen. Improving PILCO with Bayesian neural networks dynamics models. In Data-Efficient Machine Learning workshop, ICML, 2016.
+
+Shixiang Gu, Timothy Lillicrap, Ilya Sutskever, and Sergey Levine. Continuous deep q-learning with model-based acceleration. arXiv preprint arXiv:1603.00748, 2016.
+
+Alexander Hans and Steffen Udluft. Efficient uncertainty propagation for reinforcement learning with limited data. In ICANN, pp. 70–79. Springer, 2009.
+
+Daniel Hein, Alexander Hentschel, Thomas A Runkler, and Steffen Udluft. Reinforcement learning with particle swarm optimization policy (PSO-P) in continuous state and action spaces. IJSIR, 7: 23–42, 2016a.
+
+Daniel Hein, Alexander Hentschel, Volkmar Sterzing, Michel Tokic, and Steffen Udluft. Introduction to the" industrial benchmark". arXiv preprint arXiv:1610.03793, 2016b.
+
+José Miguel Hernández-Lobato, Matthew W Hoffman, and Zoubin Ghahramani. Predictive entropy search for efficient global optimization of black-box functions. In NIPS, pp. 918–926, 2014.
+
+José Miguel Hernández-Lobato, Yingzhen Li, Mark Rowland, Daniel Hernández-Lobato, Thang Bui, and Richard E Turner. Black-box $\alpha$ -divergence minimization. In ICML, pp. 1511–1520, 2016.
+
+Rein Houthooft, Xi Chen, Yan Duan, John Schulman, Filip De Turck, and Pieter Abbeel. VIME: Variational information maximizing exploration. In NIPS, pp. 1109–1117, 2016.
+
+Diederik P Kingma, Tim Salimans, and Max Welling. Variational dropout and the local reparameterization trick. In NIPS, pp. 2575–2583, 2015.
+
+Jonathan Ko, Daniel J Klein, Dieter Fox, and Dirk Haehnel. Gaussian processes and reinforcement learning for identification and control of an autonomous blimp. In IEEE Robotics and Automation, pp. 742–747, 2007.
+
+Tom Minka. Divergence measures and message passing. Technical report, Microsoft Research, 2005.
+
+Jan Peters and Stefan Schaal. Policy gradient methods for robotics. In IROS, pp. 2219–2225, 2006.
+
+Jan Peters and Stefan Schaal. Reinforcement learning of motor skills with policy gradients. Neural networks, 21:682–697, 2008.
+
+Carl Edward Rasmussen, Malte Kuss, et al. Gaussian processes in reinforcement learning. In NIPS, pp. 751–758, 2003.
+
+Anton Maximilian Schaefer, Steffen Udluft, and Hans-Georg Zimmermann. The recurrent control neural network. In ESANN, pp. 319–324, 2007.
+
+Edward Snelson and Zoubin Ghahramani. Sparse Gaussian processes using pseudo-inputs. In NIPS, pp. 1257–1264, 2005.
+
+Theano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv e-print abs/1605.02688.
+
+V. Tresp. The wet game of chicken. Technical report, 1994.
+
+Bo Wahlberg. System identification using Laguerre models. IEEE Transactions on Automatic Control, 36:551–562, 1991.
+
+M. J. Wainwright and M. I. Jordan. Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning, 1:1–305, 2008.
+
+Xin Wang and Thomas G Dietterich. Model-based policy gradient reinforcement learning. In ICML, pp. 776–783, 2003.
+
+Manuel Watter, Jost Springenberg, Joschka Boedecker, and Martin Riedmiller. Embed to control: A locally linear latent dynamics model for control from raw images. In NIPS, pp. 2728–2736, 2015.
+
+![](images/900ebf513e7d10564c74abf691b5b00c755842a97617ac6027dd8d673f447030.jpg)  
+Figure 5: Ground truth and predictive distributions for two toy problems introduced in main text. Top: bi-modal prediction problem, Bottom: heteroskedastic prediction problem. Left column: Training data (blue points) and ground truth functions (red). Columns 2-4: predictions generated with VB, $\alpha = 0 . 5$ and $\alpha = 1 . 0$ , respectively.
+
+# A ROBUSTNESS OF $\alpha = 0 . 5$ AND $\alpha = 1 . 0$ WHEN $q ( \mathbf { z } )$ IS NOT LEARNED
+
+We evaluate the accuracy of the predictive distributions generated by BNNs with stochastic inputs trained by minimizing (8) for different BNNs parameterized by $\alpha$ in two simple regression problems. The first one is characterized by a bimodal predictive distribution. The second is characterized by a heteroskedastic predictive distribution. In the latter case the magnitude of the noise in the targets changes as a function of the input features.
+
+In the first problem $x \in [ - 2 , 2 ]$ and $y$ is obtained as $y = 1 0 \sin ( x ) + \epsilon$ with probability 0.5 and $y = 1 0 \cos ( x ) + \epsilon$ , otherwise, where $\epsilon \sim \mathcal { N } ( 0 , 1 )$ and $\epsilon$ is independent of $x$ . The plot in the top of the 1st column in Figure 5 shows a training dataset obtained by sampling 2500 values of $x$ uniformly at random. The plot clearly shows that the distribution of $y$ for a particular $x$ is bimodal. In the second problem $x \in [ - 4 , 4 ]$ and $y$ is obtained as $y = 7 \sin ( x ) + 3 | \cos ( x / 2 ) | \epsilon$ . The plot in the bottom of the 1st column in Figure 5 shows a training dataset obtained with 1000 values of $x$ uniformly at random. The plot clearly shows that the distribution of $y$ is heteroskedastic, with a noise variance that is a function of $x$ .
+
+We evaluated the predictive performance obtained by minimizing (8) using $\alpha = 0 . 5$ and $\alpha = 1 . 0$ and also by running VB. However, we do not learn $q ( \mathbf { z } )$ and keeping it instead fixed to the prior $p ( \mathbf { z } )$ .
+
+We fitted a neural network with 2 hidden layers and 50 hidden units per layer using Adam with its default parameter values, with a learning rate of 0.01 in the first problem and 0.002 in the second problem. We used mini-batches of size 250 and 1000 training epochs. To approximate the expectations in 8, we draw $K = 5 0$ samples from $q$ .
+
+The plots in the 3rd and 4th columns of Figure 5 show the predictions obtained with $\alpha = 0 . 5$ and $\alpha = 1 . 0$ , respectively. In these cases, the predictive distribution is able to capture the bimodality in the first problem and the heteroskedasticity pattern in the second problem in both cases The plots in the 2nd column of Figure 5 show the predictions obtained with VB, which converges to suboptimal solutions in which the predictive distribution has a single mode (in the first problem) or is homoskedastic (in the second problem). Tables 3 and 4 show the average test RMSE and log-likelihood obtained by each method on each problem.
+
+These results show that Bayesian neural networks trained with $\alpha = 0 . 5$ or $\alpha = 1 . 0$ are more robust than VB and can still model complex predictive distributions, which may be multimodal and heteroskedastic, even when $q ( \mathbf { z } )$ is not learned and is instead kept fixed to the prior $p ( \mathbf { z } )$ . By contrast, VB fails to capture complex stochastic patterns in this setting.
+
+Table 3: Test error and log-likelihood for the bi-modal prediction problem.   
+
+<table><tr><td>Method</td><td>RMSE</td><td>Log-likelihood</td></tr><tr><td>VB</td><td>5.12</td><td>-3.05</td></tr><tr><td>α = 0.5</td><td>5.14</td><td>-2.10</td></tr><tr><td>α = 1.0</td><td>5.15</td><td>-2.11</td></tr></table>
+
+Table 4: Test error and log-likelihood for the heteroskedastic prediction problem.   
+
+<table><tr><td>Method</td><td>RMSE</td><td>Log-likelihood</td></tr><tr><td>VB</td><td>1.88</td><td>-2.05</td></tr><tr><td>α = 0.5</td><td>1.89</td><td>-1.78</td></tr><tr><td>α =1.0</td><td>1.94</td><td>-1.98</td></tr></table>
+
+# B METHODS
+
+In the experiments we compare to the following methods:
+
+Standard MLP. The standard multi-layer preceptron (MLP) is equivalent to our BNNs, but does not have uncertainty over the weight $\mathcal { W }$ and does not include any stochastic inputs. We train this method using early stopping on a subset of the training data. When we perform roll-outs using algorithm 1, the predictions of the MLP are made stochastic by adding Gaussian noise to its output. The noise variance is fixed by maximum likelihood on some validation data after model training.
+
+Variational Bayes (VB). The most prominent approach in training modern BNNs is to optimize the variational lower bound (Blundell et al., 2015; Houthooft et al., 2016; Gal et al., 2016). This is in practice equivalent to $\alpha$ -divergence minimization when $\alpha  0$ (Hernández-Lobato et al., 2016). In our experiments we use $\alpha$ -divergence minimization with $\alpha = 1 0 ^ { - 6 }$ to implement this method.
+
+Gaussian Processes (GPs). Gaussian Processes have recently been used for policy search under the name of PILCO (Deisenroth & Rasmussen, 2011). For each dimension of the target variables, we fit a different sparse GP using the FITC approximation (Snelson & Ghahramani, 2005). In particular, each sparse GP is trained using 150 inducing inputs by using the method stochastic expectation propagation (Bui et al., 2016). After this training process we approximate the sparse GP by using a feature expansion with random basis functions (see supplementary material of Hernández-Lobato et al. 2014). This allows us to draw samples from the GP posterior distribution over functions, enabling the use of Algorithm 1 for policy training. Note that PILCO will instead moment-match at every roll-out step as it works by propagating Gaussian distributions. However, in our experiments we obtained better performance by avoiding the moment matching step with the aforementioned approximation based on random basis functions.
+
+Particle Swarm Optimization Policy(PSO-P). We use this method to estimate an upper bound for reward performance. PSO-P is a model predictive control (MPC) method that uses the true dynamics when applicable (Hein et al., 2016a). For a given state $\mathbf { s } _ { t }$ , the best action is selected using the standard receding horizon approach on the real environment. Note that this is not a benchmark method to compare to, we use it instead as an indicator of what the best possible reward can be achieved for a fixed planning horizon $T$ .
+
+# C MODEL PARAMETERS
+
+For all tasks we will use a standard MLP with two hidden layer with 20 hidden units each as policy representation. The activation functions for the hidden units are rectifiers: $\varphi ( x ) = \operatorname* { m a x } ( x , 0 )$ . If present, bounding of the actions is realized using the tanh activation function on the outputs of the policy. All models based on neural network will share the same hyperparameter. We use ADAM as learning algorithm in all tasks.
+
+WetChicken The neural network models are set to 2 hidden layers and 20 hidden units per layer. We use 2500 random state transitions for training. We found that assuming no observation noise by setting $\Gamma$ to a constant of $1 0 ^ { - 5 }$ helped the models converge to lower energy values.
+
+For policy training we use a horizon of size $T = 5$ and optimize the policy network for 100 epochs, averaging over $K = 2 0$ samples in each gradient update, with mini-batches of size 10 and learning rate set to 10−5.
+
+Turbine The world model and the BNNs have two hidden layers with 50 hidden units each. For policy training and world-model evaluation we perform a roll-out with horizon $T = 2 0$ . For learning the policy we use minibaches of size 10 and draw $K = 1 0$ samples from $q$ .
+
+Industrial Benchmark For the neural network models we use two hidden layers with 75 hidden units.We use a horizon of $T = 7 5$ , training for 500 epochs with batches of size 50 and $K = 2 5$ samples for each rollout.
+
+# D COMPUTATIONAL COMPLEXITY
+
+MODEL TRAINING
+
+All models were trained using theano and a single GPU. Training the standard neural network is fast, the training time for this method was between 5 - 20 minutes, depending on data set size and dimensionality of the benchmark. In theano, the computational graph of the BNNs is similar to that of an ensemble of standard neural networks. The training time for the BNNs varied between 30 minutes to 5 hours depending on data size and dimensionality of benchmark. The sparse Gaussian Process was optimized using an expectation propagation algorithm and after training, it was approximated with a Bayesian linear model with fixed basis functions whose weights are initialized randomly (see Appendix B). We choose the inducing points in the GPs and the number of training epochs for these models so that the resulting training time was comparable to that of the BNNs.
+
+# POLICY SEARCH
+
+For policy training we used a single CPU. All methods are of similar complexity as they are all trained using Algorithm 1. Depending on the horizon, data set size and network topology, training took between 20 minutes (Wet-Chicken, $T = 5$ ), 3-4 hours (Turbine, $T = 2 0$ ) and 14-16 hours (industrial benchmark, $T = 7 5$ ).

parse/train/H1fl8S9ee/H1fl8S9ee_content_list.json

@@ -0,0 +1,1958 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING AND POLICY SEARCH IN STOCHASTIC DYNAMICAL SYSTEMS WITH BAYESIAN NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            769,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Stefan Depeweg Siemens AG and Technical University of Munich stefan.depeweg@siemens.com ",
+        "bbox": [
+            183,
+            194,
+            508,
+            237
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "José Miguel Hernández-Lobato University of Cambridge jmh233@cam.ac.uk ",
+        "bbox": [
+            593,
+            195,
+            813,
+            237
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Finale Doshi-Velez   \nHarvard University   \nfinale@seas.harvard.edu   \nSteffen Udluft   \nSiemens AG   \nsteffen.udluft@siemens.com ",
+        "bbox": [
+            184,
+            258,
+            411,
+            300
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            493,
+            258,
+            750,
+            300
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            337,
+            544,
+            352
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present an algorithm for policy search in stochastic dynamical systems using model-based reinforcement learning. The system dynamics are described with Bayesian neural networks (BNNs) that include stochastic input variables. These input variables allow us to capture complex statistical patterns in the transition dynamics (e.g. multi-modality and heteroskedasticity), which are usually missed by alternative modeling approaches. After learning the dynamics, our BNNs are then fed into an algorithm that performs random roll-outs and uses stochastic optimization for policy learning. We train our BNNs by minimizing $\\alpha$ -divergences with $\\alpha = 0 . 5$ , which usually produces better results than other techniques such as variational Bayes. We illustrate the performance of our method by solving a challenging problem where model-based approaches usually fail and by obtaining promising results in real-world scenarios including the control of a gas turbine and an industrial benchmark. ",
+        "bbox": [
+            233,
+            369,
+            766,
+            550
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            578,
+            336,
+            594
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In model-based reinforcement learning, an agent uses its experience to first learn a model of the environment and then uses that model to reason about what action to take next. We consider the case in which the agent observes the current state $\\mathbf { s } _ { t }$ , takes some action a, and then observes the next state $\\mathbf { s } _ { t + 1 }$ . The problem of learning the model corresponds then to learning a stochastic transition function $p ( \\mathbf { s } _ { t + 1 } | \\mathbf { s } _ { t } , \\mathbf { a } )$ specifying the conditional distribution of $\\mathbf { s } _ { t + 1 }$ given $\\mathbf { s } _ { t }$ and a. Most classic control theory texts, e.g. Bertsekas (2002), will start with the most general model of dynamical systems: ",
+        "bbox": [
+            173,
+            611,
+            825,
+            694
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6675445b118e1894828573db54721e2fa9be729a8577289b4d5822917a5c9577.jpg",
+        "text": "$$\n\\mathbf { s } _ { t + 1 } = f ( \\mathbf { s } _ { t } , \\mathbf { a } , z , \\mathcal { W } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            424,
+            703,
+            573,
+            719
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "where $f$ is some deterministic function parameterized by weights $\\mathcal { W }$ that takes as input the current state $\\mathbf { s } _ { t }$ , the control signal a, and some stochastic disturbance $z$ . ",
+        "bbox": [
+            171,
+            728,
+            823,
+            756
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, to date, we have not been able to robustly learn dynamical system models to such a level of generality. Popular modes for transition functions include Gaussian processes (Rasmussen et al., 2003; Ko et al., 2007; Deisenroth & Rasmussen, 2011), fixed bases such as Laguerre functions (Wahlberg, 1991), and adaptive basis functions or neural networks (Draeger et al., 1995). All of these methods assume deterministic transition functions, perhaps with some addition of Gaussian observation noise. Thus, they are severely limited in the kinds of stochasticity—or transition noise—they can express. In many real-world scenarios stochasticity may often arise due to some unobserved environmental feature that can affect the dynamics in complex ways (such as unmeasured gusts of wind on a boat). ",
+        "bbox": [
+            174,
+            762,
+            826,
+            875
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work we use Bayesian neural networks (BNNs) in conjunction with a random input noise source $z$ to express stochastic dynamics. We take advantage of a very recent inference advance based on $\\alpha$ -divergence minimization (Hernández-Lobato et al., 2016), with $\\alpha = 0 . 5$ , to learn with high accuracy BNN transition functions that are both scalable and expressive in terms of stochastic patterns. Previous work achieved one but not both of these two characteristics. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            821,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We focus our evaluation on the off-policy batch reinforcement learning scenario, in which we are given an initial batch of data from an already-running system and are asked to find a better (ideally near-optimal) policy. Such scenarios are common in real-world industry settings such as turbine control, where exploration is restricted to avoid possible damage to the system. We propose an algorithm that uses random roll-outs and stochastic optimization for learning an optimal policy from the predictions of BNNs. This method produces (to our knowledge) the first model-based solution of a 20-year-old benchmark problem: the Wet-Chicken (Tresp, 1994). We also obtain very promising results on a real-world application on controlling gas turbines and on an industrial benchmark. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            251
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            266,
+            326,
+            282
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 MODEL-BASED REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            296,
+            529,
+            311
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We consider reinforcement learning problems in which an agent acts in a stochastic environment by sequentially choosing actions over a sequence of time steps, in order to minimize a cumulative cost. We assume that our environment has some true dynamics $T _ { \\mathrm { t r u e } } ( \\mathbf { s } _ { t + 1 } | \\mathbf { s } , \\mathbf { a } )$ , and we are given a cost function $c ( \\mathbf { s } _ { t } )$ . In the model-based reinforcement learning setting, our goal is to learn an approximation $T _ { \\mathrm { a p p r o x } } ( \\mathbf { s } _ { t + 1 } | \\mathbf { s } , \\mathbf { a } )$ for the true dynamics based on collected samples $( \\mathbf { s } _ { t } , \\mathbf { a } , \\mathbf { s } _ { t + 1 } )$ . The agent then tries to solve the control problem in which $T _ { \\mathrm { a p p r o x } }$ is assumed to be the true dynamics. ",
+        "bbox": [
+            173,
+            323,
+            825,
+            407
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 BAYESIAN NEURAL NETWORKS WITH STOCHASTIC INPUTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            424,
+            620,
+            438
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Given data $\\mathcal { D } = \\{ \\mathbf { x } _ { n } , \\mathbf { y } _ { n } \\} _ { n = 1 } ^ { N }$ , formed by feature vectors $\\mathbf { x } _ { n } \\in \\mathbb { R } ^ { D }$ and targets $\\mathbf { y } _ { n } \\in \\mathbb { R } ^ { K }$ , we assume that ${ \\bf y } _ { n } = f ( { \\bf x } _ { n } , z _ { n } ; \\mathcal { W } ) + \\epsilon _ { n }$ , where $f ( \\cdot , \\cdot ; \\mathcal W )$ is the output of a neural network with weights The network receives as input the feature vector ${ \\bf x } _ { n }$ and the random disturbance $z _ { n } \\sim \\mathcal { N } ( 0 , \\gamma )$ . The activation functions for the hidden layers are rectifiers: $\\varphi ( x ) = \\operatorname* { m a x } ( x , 0 )$ . The activation functions for the output layers are the identity function: $\\varphi ( x ) = x$ . The network output is corrupted by the additive noise variable $\\epsilon _ { n } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { \\tilde { \\Sigma } } )$ with diagonal covariance matrix $\\pmb { \\Sigma }$ . The role of the noise disturbance $z _ { n }$ is to capture unobserved stochastic features that can affect the network’s output in complex ways. Without $z _ { n }$ , randomness is only given by the additive Gaussian observation noise $\\epsilon _ { n }$ , which can only describe limited stochastic patterns. The network has $L$ layers, with $V _ { l }$ hidden units in layer $l$ , and $\\mathcal { W } = \\{ \\mathbf { W } _ { l } \\} _ { l = 1 } ^ { L }$ is the collection of $V _ { l } \\times ( V _ { l - 1 } + 1 )$ weight matrices. The $+ 1$ is introduced here to account for the additional per-layer biases. ",
+        "bbox": [
+            173,
+            446,
+            825,
+            602
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One could argue why $\\epsilon _ { n }$ is needed at all when we are already using the more flexible stochastic model based on $z _ { n }$ . The reason for this is that, in practice, we make predictions with the above model by averaging over a finite number of samples of $z _ { n }$ and $\\mathcal { W }$ . By using $\\epsilon _ { n }$ , we obtain a predictive distribution whose density is well defined and given by a mixture of Gaussians. If we eliminate $\\epsilon _ { n }$ , the predictive density is degenerate and given by a mixture of delta functions. ",
+        "bbox": [
+            174,
+            608,
+            825,
+            679
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathbf { Y }$ be an $N \\times K$ matrix with the targets ${ \\bf y } _ { n }$ and $\\mathbf { X }$ be an $N \\times D$ matrix of feature vectors ${ \\bf x } _ { n }$ We denote by $\\mathbf { z }$ the $N$ -dimensional vector with the values of the random disturbances $z _ { 1 } , \\dots , z _ { N }$ that were used to generate the data. The likelihood function is ",
+        "bbox": [
+            174,
+            685,
+            825,
+            728
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/45f074be29f85244c79c31c7b7105f4271d4de5055722a82f5e0402161881ccb.jpg",
+        "text": "$$\np ( \\mathbf { Y } \\mid \\mathcal { W } , \\mathbf { z } , \\mathbf { X } ) = \\prod _ { n = 1 } ^ { N } p ( \\mathbf { y } _ { n } \\mid \\mathcal { W } , \\mathbf { z } , \\mathbf { x } _ { n } ) = \\prod _ { n = 1 } ^ { N } \\prod _ { k = 1 } ^ { K } \\mathcal { N } \\big ( y _ { n , k } \\mid f ( \\mathbf { x } _ { n } , z _ { n } ; \\mathcal { W } ) , \\pmb { \\Sigma } \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            729,
+            759,
+            773
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The prior for each entry in $\\mathbf { z }$ is $\\mathcal { N } ( 0 , \\gamma )$ . We also specify a Gaussian prior distribution for each entry in each of the weight matrices in $\\mathcal { W }$ . That is, ",
+        "bbox": [
+            173,
+            776,
+            825,
+            805
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/89cec8a0f30c547ca37fd2401ad65d984e03a1f338baebf52c20688a0ed6b17d.jpg",
+        "text": "$$\np ( \\mathbf { z } ) = \\prod _ { n = 1 } ^ { N } \\mathcal { N } ( z _ { n } | 0 , \\gamma ) , \\qquad p ( \\mathcal { W } ) = \\prod _ { l = 1 } ^ { L } \\prod _ { i = 1 } ^ { V _ { l } } \\prod _ { j = 1 } ^ { V _ { l - 1 } + 1 } \\mathcal { N } ( w _ { i j , l } | 0 , \\lambda ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            809,
+            745,
+            854
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { i j , l }$ is the entry in the $i$ -th row and $j$ -th column of $\\mathbf { W } _ { l }$ and $\\gamma$ and $\\lambda$ are a prior variances. The posterior distribution for the weights $\\mathcal { W }$ and the random disturbances $\\mathbf { z }$ is given by Bayes’ rule: ",
+        "bbox": [
+            173,
+            857,
+            823,
+            885
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ac2537d694267562ecbaa9b824b1b2e7c88d881ce2a99054f408b6a00742979.jpg",
+        "text": "$$\np ( \\mathcal { W } , \\mathbf { z } \\mid \\mathcal { D } ) = \\frac { p ( \\mathbf { Y } \\mid \\mathcal { W } , \\mathbf { z } , \\mathbf { X } ) p ( \\mathcal { W } ) p ( \\mathbf { z } ) } { p ( \\mathbf { Y } \\mid \\mathbf { X } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            887,
+            640,
+            922
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/62d41847667e5f61a0c16af5a6179e44cadeb706fbfc28d39e46d6482e8d2e90.jpg",
+        "image_caption": [
+            "Figure 1: Solution for the minimization of the $\\alpha$ -divergence between the posterior $p$ (in blue) and the Gaussian approximation $q$ (in red and unnormalized). Figure source Minka (2005). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            99,
+            761,
+            258
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a new input vector $\\mathbf { x } _ { \\star }$ , we can then make predictions for $\\mathbf { y } _ { \\star }$ using the predictive distribution ",
+        "bbox": [
+            173,
+            321,
+            820,
+            338
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e026789eb32d5befef91b9ff83b77e28f9be726938be5e696bb156f626b5898f.jpg",
+        "text": "$$\np ( \\mathbf { y } _ { \\star } | \\mathbf { x } _ { \\star } , \\mathcal { D } ) = \\int \\left[ \\int \\mathcal { N } ( y _ { \\star } | f ( \\mathbf { x } _ { \\star } , z _ { \\star } ; \\mathcal { W } ) , \\pm ) \\mathcal { N } ( z _ { \\star } | 0 , 1 ) d z _ { \\star } \\right] p ( \\mathcal { W } , \\mathbf { z } | \\mathcal { D } ) d \\mathcal { W } d \\mathbf { z } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            343,
+            777,
+            377
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, the exact computation of (4) is intractable and we have to use approximations. ",
+        "bbox": [
+            166,
+            381,
+            782,
+            397
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 $_ { \\pmb { \\alpha } }$ -DIVERGENCE MINIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            406,
+            431,
+            420
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We approximate the exact posterior distribution $p ( \\mathcal { W } , \\mathbf { z } \\mid \\mathcal { D } )$ with the factorized Gaussian distribution ",
+        "bbox": [
+            173,
+            431,
+            823,
+            446
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c4d0b215a362cf8cef47b9b2486e6a6ee8240537b3acd7fe17a856241c145018.jpg",
+        "text": "$$\nq ( \\mathcal W , \\mathbf z ) = \\left[ \\prod _ { l = 1 } ^ { L } \\prod _ { i = 1 } ^ { V _ { l } } \\prod _ { j = 1 } ^ { V _ { l - 1 } + 1 } \\mathcal N ( w _ { i j , l } | m _ { i j , l } ^ { w } , v _ { i j , l } ^ { w } ) \\right] \\left[ \\prod _ { n = 1 } ^ { N } \\mathcal N ( z _ { n } | m _ { n } ^ { z } , v _ { n } ^ { z } ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            454,
+            735,
+            506
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The parameters mwij,l, $v _ { i j , l } ^ { w }$ and $m _ { n } ^ { z } , \\ v _ { n } ^ { z }$ are determined by minimizing a divergence between $p ( \\mathcal { W } , \\mathbf { z } \\vert \\mathcal { D } )$ and the approximation $q$ . After fitting $q$ , we make predictions by replacing $p ( \\mathcal { W } , \\mathbf { z } \\mid \\mathcal { D } )$ with $q$ in (4) and approximating the integrals in (4) with empirical averages over samples of $\\mathcal { W } \\sim q$ . ",
+        "bbox": [
+            174,
+            506,
+            826,
+            551
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We aim to adjust the parameters of (5) by minimizing the $\\alpha$ -divergence between $p ( \\mathcal { W } , \\mathbf { z } \\vert \\mathcal { D } )$ and $q ( \\boldsymbol { \\mathcal { W } } , { \\mathbf z } )$ (Minka, 2005): ",
+        "bbox": [
+            173,
+            556,
+            825,
+            585
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec18d555332ccc442b40130d3552961f4d3f625103cc0a12ce7afe8ac700a34f.jpg",
+        "text": "$$\n\\mathrm { D } _ { \\alpha } [ p ( \\mathcal { W } , \\mathbf { z } | \\mathcal { D } ) | | q ( \\mathcal { W } , \\mathbf { z } ) ] = \\frac { 1 } { \\alpha ( \\alpha - 1 ) } \\left( 1 - \\int p ( \\mathcal { W } , \\mathbf { z } | \\mathcal { D } ) ^ { \\alpha } q ( \\mathcal { W } , \\mathbf { z } ) ^ { ( 1 - \\alpha ) } \\right) d \\mathcal { W } d \\mathbf { z } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            214,
+            592,
+            782,
+            626
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "which includes a parameter $\\alpha \\in \\mathbb { R }$ that controls the properties of the optimal $q$ . Figure 1 illustrates these properties for the one-dimensional case. When $\\alpha \\geq 1$ , $q$ tends to cover the whole posterior distribution $p$ . When $\\alpha \\leq 0$ , $q$ tends to fit a local mode in $p$ . The value $\\alpha = 0 . 5$ is expected to achieve a balance between these two tendencies. Importantly, when $\\alpha  0$ , the solution obtained is the same as with variational Bayes (VB) (Wainwright & Jordan, 2008). ",
+        "bbox": [
+            173,
+            630,
+            825,
+            702
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The direct minimization of (6) is infeasible in practice for arbitrary $\\alpha$ . Instead, we follow HernándezLobato et al. (2016) and optimize an energy function whose minimizer corresponds to a local minimization of $\\alpha$ -divergences, with one $\\alpha$ -divergence for each of the $N$ likelihood factors in (1). Since $q$ is Gaussian and the priors $p ( \\mathcal { W } )$ and $p ( \\mathbf { z } )$ are also Gaussian, we represent $q$ as ",
+        "bbox": [
+            173,
+            707,
+            826,
+            763
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b08444a945b9b2460a3d82b7860e9a83f50f22291b39d414ab36ecab4a6ed8a9.jpg",
+        "text": "$$\nq ( \\mathcal { W } , { \\mathbf z } ) \\propto \\left[ \\prod _ { n = 1 } ^ { N } f ( \\mathcal { W } ) f _ { n } ( z _ { n } ) \\right] p ( \\mathcal { W } ) p ( { \\mathbf z } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            768,
+            642,
+            813
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $f ( \\mathcal W )$ is a Gaussian factor that approximates the geometric mean of the $N$ likelihood factors in (1) as a function of $\\mathcal { W }$ . Each $f _ { n } ( z _ { n } )$ is also a Gaussian factor that approximates the $n$ -th likelihood factor in (1) as a function of $z _ { n }$ . We adjust $f ( \\mathcal W )$ and the $f _ { n } ( z _ { n } )$ by minimizing local $\\alpha$ -divergences. In particular, we minimize the energy function ",
+        "bbox": [
+            174,
+            816,
+            826,
+            875
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a2408d2089f5fc03b80899761acb7fe257e2506501400705c6cce150d67ce394.jpg",
+        "text": "$$\nE _ { \\alpha } ( q ) = - \\log Z _ { q } - \\frac { 1 } { \\alpha } \\sum _ { n = 1 } ^ { N } \\log \\mathbf { E } _ { \\mathcal { W } , z _ { n } \\sim q } \\left[ \\left( \\frac { p ( \\mathbf { y } _ { n } \\mid \\mathcal { W } , \\mathbf { x } _ { n } , z _ { n } , \\pmb { \\Sigma } ) } { f ( \\mathcal { W } ) f _ { n } ( z _ { n } ) } \\right) ^ { \\alpha } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            878,
+            738,
+            922
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "(Hernández-Lobato et al., 2016), where $f ( \\mathcal W )$ and $f _ { n } ( z _ { n } )$ are in exponential Gaussian form and parameterized in terms of the parameters of $q$ and the priors $p ( \\mathcal { W } )$ and $p ( z _ { n } )$ , that is, ",
+        "bbox": [
+            169,
+            102,
+            825,
+            132
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d02d640f3c15585ebc84319b468e8708a45a3f48c38d1f123f40be12655104a.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle f ( \\mathcal { W } ) = \\exp \\left\\{ \\sum _ { l = 1 } ^ { L } \\sum _ { i = 1 } ^ { V _ { l } } \\sum _ { j = 1 } ^ { V _ { l - 1 } + 1 } \\frac { 1 } { N } \\left( \\frac { \\lambda v _ { i , j , l } ^ { w } } { \\lambda - v _ { i , j , l } ^ { w } } w _ { i , j , l } ^ { 2 } + \\frac { m _ { i , j , l } ^ { w } } { v _ { i , j , l } ^ { w } } w _ { i , j , l } \\right) \\right\\} \\propto \\left[ \\frac { q ( \\mathcal { W } ) } { p ( \\mathcal { W } ) } \\right] ^ { \\frac { 1 } { N } } , } } \\\\ { { \\displaystyle f _ { n } ( z _ { n } ) = \\exp \\left\\{ \\frac { \\gamma v _ { n } ^ { z } } { \\gamma - v _ { n } ^ { z } } z _ { n } ^ { 2 } + \\frac { m _ { n } ^ { z } } { v _ { n } ^ { z } } z _ { n } \\right\\} \\propto \\frac { q ( z _ { n } ) } { p ( z _ { n } ) } } , }  \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            135,
+            776,
+            222
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and $\\log Z _ { q }$ is the logarithm of the normalization constant of the exponential Gaussian form of $q$ : ",
+        "bbox": [
+            173,
+            222,
+            802,
+            237
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e08789085601e9d420815a6d30a82817112494b0d6df19a7251a40c384eddb45.jpg",
+        "text": "$$\n\\log Z _ { q } = \\sum _ { l = 1 } ^ { L } \\sum _ { i = 1 } ^ { V _ { l } } \\sum _ { j = 1 } ^ { V _ { l - 1 } + 1 } \\left[ \\frac { 1 } { 2 } \\log \\left( 2 \\pi v _ { i , j , l } ^ { w } \\right) + \\frac { \\left( m _ { i , j , l } ^ { w } \\right) ^ { 2 } } { v _ { i , j , l } ^ { w } } \\right] + \\sum _ { n = 1 } ^ { N } \\left[ \\frac { 1 } { 2 } \\log \\left( 2 \\pi v _ { n } ^ { z } \\right) + \\frac { \\left( m _ { n } ^ { z } \\right) ^ { 2 } } { v _ { n } ^ { z } } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            239,
+            781,
+            299
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The scalable optimization of (8) is done in practice by using stochastic gradient descent. For this, we subsample the sums for $n = 1 , \\ldots , N$ in (8) and (11) using mini-batches and approximate the expectations over $q$ in (8) with an average over $K$ samples drawn from $q$ . We can then use the reparametrization trick (Kingma et al., 2015) to obtain gradients from the resulting stochastic approximator to (8). The hyper-parameters $\\pmb { \\Sigma }$ , $\\lambda$ and $\\gamma$ can also be tuned by minimizing (8). In practice we only tune $\\pmb { \\Sigma }$ and keep $\\lambda = 1$ and $\\gamma = d$ . The latter means that the prior scale of each $z _ { n }$ grows with the data dimensionality. This guarantees that, a priori, the effect of each $z _ { n }$ in the neural network’s output does not diminish when more and more features are available. ",
+        "bbox": [
+            173,
+            300,
+            825,
+            411
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Minimizing (8) when $\\alpha  0$ is equivalent to running the method VB (Hernández-Lobato et al., 2016), which has recently been used to train Bayesian neural networks in reinforcement learning problems (Blundell et al., 2015; Houthooft et al., 2016; Gal et al., 2016). However, we propose to minimize (8) using $\\alpha = 0 . 5$ , which often results in better test log-likelihood values. ",
+        "bbox": [
+            174,
+            417,
+            825,
+            474
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We have also observed $\\alpha = 0 . 5$ to be more robust than VB when $q ( \\mathbf { z } )$ is not fully optimized. In particular, $\\alpha = 0 . 5$ can still capture complex stochastic patterns even when we do not learn $q ( \\mathbf { z } )$ and instead keep it fixed to the prior $p ( \\mathbf { z } )$ . By contrast, VB fails completely in this case (see Appendix A). ",
+        "bbox": [
+            174,
+            481,
+            825,
+            523
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 POLICY SEARCH USING BNNS WITH STOCHASTIC INPUTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            535,
+            684,
+            553
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We now describe a gradient-based policy search algorithm that uses the BNNs with stochastic disturbances from the previous section. The motivation for our approach lies in its applicability to industrial systems: we wish to estimate a policy in parametric form, using only an available batch of state transitions obtained from an already-running system. We assume that the true dynamics present stochastic patterns that arise due to some unobserved process affecting the system in complex ways. ",
+        "bbox": [
+            174,
+            566,
+            825,
+            637
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Model-based policy search methods include two key parts (Deisenroth et al., 2013). The first part consists in learning a dynamics model from data in the form of state transitions $\\left( \\mathbf { s } _ { t } , \\mathbf { a } _ { t } , \\mathbf { s } _ { t + 1 } \\right)$ , where $\\mathbf { s } _ { t }$ denotes the current state, $\\mathbf { a } _ { t }$ is the action applied and $\\mathbf { s } _ { t + 1 }$ is the resulting state. The second part consists in learning the parameters $\\mathcal { W } _ { \\pi }$ of a deterministic policy function $\\pi$ that returns the optimal action $\\mathbf { a } _ { t } = \\pi ( \\mathbf { s } _ { t } ; \\mathcal { W } _ { \\pi } )$ as function of the current state $\\mathbf { s } _ { t }$ . The policy function can be a neural network with deterministic weights given by $\\mathcal { W } _ { \\pi }$ . ",
+        "bbox": [
+            173,
+            643,
+            825,
+            728
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The first part in the aforementioned procedure is a standard regression task, which we solve by using the modeling approach from the previous section. We assume the dynamics to be stochastic with the following true transition model: ",
+        "bbox": [
+            173,
+            734,
+            825,
+            776
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ba0fc1003a8afd6a4f228197a5b2e205b216d9ae382bd31249adadc4748d485.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { s } _ { t } = f _ { \\mathrm { t r u e } } \\left( \\mathbf { s } _ { t - 1 } , \\mathbf { a } _ { t - 1 } , z _ { t } ; \\mathcal { W } _ { \\mathrm { t r u e } } \\right) , \\quad z _ { t } \\sim \\mathcal { N } ( 0 , \\gamma ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            780,
+            665,
+            796
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the input disturbances $z _ { t } \\sim \\mathcal { N } ( 0 , \\gamma )$ account for the stochasticity in the dynamics. When the Markov state $\\mathbf { s } _ { t }$ is hidden and we are given only observations $\\mathbf { o } _ { t }$ , we can use the time embedding theorem using a suitable window of length $n$ and approximate: ",
+        "bbox": [
+            174,
+            800,
+            825,
+            842
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cb1d513b500e6931c81cb07b5bf58f6423301fe06c93d5e4f18377a012cac4ed.jpg",
+        "text": "$$\n\\hat { \\mathbf { s } } ( t ) = \\left[ \\mathbf { o } _ { t - n } , \\cdot \\cdot \\cdot \\mathbf { \\sigma } , \\mathbf { o } _ { t } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            844,
+            576,
+            862
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The transition model in equation 12 specifies a probability distribution $p ( \\mathbf { s } _ { t } | \\mathbf { s } _ { t - 1 } , \\mathbf { a } _ { t - 1 } )$ that we approximate using a BNN with stochastic inputs: ",
+        "bbox": [
+            171,
+            864,
+            823,
+            893
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/201a29352e89a75b523d4bce4257f45ecd70b06bf4dfea7fbcb72dc28ae2e4c5.jpg",
+        "text": "$$\np ( \\mathbf { s } _ { t } | \\mathbf { s } _ { t - 1 } , \\mathbf { a } _ { t - 1 } ) \\approx \\int \\mathcal { N } ( \\mathbf { s } _ { t } | f ( \\mathbf { s } _ { t - 1 } , \\mathbf { a } _ { t - 1 } , z _ { t } ; \\mathcal { W } ) , \\pmb { \\Sigma } ) q ( \\mathcal { W } ) \\mathcal { N } ( z _ { t } | 0 , \\gamma ) d \\mathcal { W } d z _ { t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
+            896,
+            761,
+            929
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/4bc543a8f922f587ceb4444e57b204340a26e49f322cbc44e5fe9419c07c04ed.jpg",
+        "table_caption": [
+            "Algorithm 1 Model-based policy search using Bayesian neural networks with stochastic inputs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>1: Input: D= {Sn,an,△n} for n ∈1..N 2: Fit q(W) and ∑ by optimizing (8). 3: function UNFOLD(So) 4: sample{W1,.,WK} from q(W) 5: C↑0 6: fork=1:Kdo 7: fort=O :Tdo 2+1~N(0,2） 8: 9: △t←f(st,π(st;Wπ),2t+1;Wk) e+1~N(0,∑) 10: 11:</td></tr></table>",
+        "bbox": [
+            179,
+            136,
+            503,
+            345
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/5585956c667bef16e2cc88bd3a8a1c016f2f1b236743a4501b1c456350bb7093.jpg",
+        "image_caption": [
+            "Figure 2: Predictive distribution of $y _ { t }$ given by different methods in four different scenarios. Ground truth (red) is obtained by sampling from the real dynamics. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            522,
+            101,
+            821,
+            280
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where the feature vectors in our BNN are now $\\mathbf { s } _ { t - 1 }$ and $\\mathbf { a } _ { t - 1 }$ and the targets are given by $\\mathbf { s } _ { t }$ . In this expression, the integration with respect to $\\mathcal { W }$ accounts for stochasticity arising from lack of knowledge of the model parameters, while the integration with respect to $z _ { t }$ accounts for stochasticity arising from unobserved processes that cannot be modeled. In practice, these integrals are approximated by an average over samples of $z _ { t } \\sim \\mathcal { N } ( 0 , \\gamma )$ and $\\mathcal { W } \\sim q$ . ",
+        "bbox": [
+            173,
+            372,
+            825,
+            443
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the second part of our model-based policy search algorithm, we optimize the parameters $\\mathcal { W } _ { \\pi }$ of a policy that minimizes the sum of expected cost over a finite horizon $T$ with respect to our belief $q ( \\mathcal { W } )$ . This expected cost is obtained by averaging over multiple virtual roll-outs. For each roll-out we sample $w _ { i } \\sim q$ and then simulate state trajectories using the model $\\mathbf { s } _ { t + 1 } = f ( \\mathbf { s } _ { t } , \\mathbf { a } _ { t } , z _ { t } ; \\mathcal { W } _ { i } ) + \\boldsymbol { \\epsilon } _ { t + 1 }$ with policy $\\mathbf { a } _ { t } = \\pi ( \\mathbf { s } _ { t } ; \\mathcal { W } _ { \\pi } )$ , input noise $z _ { t } \\sim \\mathcal { N } ( 0 , \\gamma )$ and additive noise $\\epsilon _ { t + 1 } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\Sigma )$ . This procedure allows us to obtain estimates of the policy’s expected cost for any particular cost function. If model, policy and cost function are differentiable, we are then able to tune $\\mathcal { W } _ { \\pi }$ by stochastic gradient descent over the roll-out average. ",
+        "bbox": [
+            173,
+            448,
+            826,
+            561
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given a cost function $c ( \\mathbf { s } _ { t } )$ , the objective to be optimized by our policy search algorithm is ",
+        "bbox": [
+            171,
+            566,
+            769,
+            583
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6bce70517a7dea64cbace7146c1340369c71663042c2ec9a31e8056b0ad2eac.jpg",
+        "text": "$$\n\\begin{array} { r } { J ( \\mathcal { W } _ { \\pi } ) = \\mathbf { E } \\left[ \\sum _ { t = 1 } ^ { T } c ( \\mathbf { s } _ { t } ) \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            587,
+            594,
+            614
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We approximate (15) by using (14), replacing $\\mathbf { a } _ { t }$ with $\\pi ( \\mathbf { s } _ { t } ; \\mathcal { W } _ { \\pi } )$ and using sampling to approximate the expectations: ",
+        "bbox": [
+            174,
+            618,
+            825,
+            646
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f841228e5d520d6c5cc3f529debbe3beb6addf973daa37a05087b6cad4846fbb.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\cal J } ( \\mathcal { W } _ { \\pi } ) = \\int \\left[ \\displaystyle { \\sum _ { t = 1 } ^ { T } c ( { \\bf s } _ { t } ) } \\right] \\left[ \\displaystyle { \\prod _ { t = 1 } ^ { T } \\int } \\mathcal { N } ( { \\bf s } _ { t } | f ( { \\bf s } _ { t - 1 } , \\pi ( { \\bf s } _ { t - 1 } ; \\mathcal { W } _ { \\pi } ) , z _ { t } ; \\mathcal { W } ) , { \\bf \\Sigma } ) q ( \\mathcal { W } ) \\mathcal { N } ( z _ { t } | 0 , \\gamma ) d \\mathcal { W } d z _ { t } \\right] } } \\\\ { { \\displaystyle p ( { \\bf s } _ { 0 } ) d { \\bf s } _ { 0 } \\cdot \\cdot \\cdot \\cdot d { \\bf s } _ { T } } } \\\\  { \\displaystyle ~ = \\int \\left[ \\displaystyle { \\sum _ { t = 1 } ^ { T } c ( { \\bf s } _ { t } ^ { \\mathcal { W } , \\{ z _ { 1 } , \\dots , z _ { t } \\} , \\{ { \\bf s } _ { t } \\} , \\dots , { \\bf s } _ { t } \\} , \\mathcal { W } _ { \\pi } ) }  q ( \\mathcal { W } ) d \\mathcal { W } \\left[ \\displaystyle { \\prod _ { t = 1 } ^ { T } \\mathcal { N } ( \\epsilon _ { t } | { \\bf 0 } , { \\bf \\Sigma } ) \\mathcal { N } ( z _ { t } | { \\bf 0 } , \\gamma ) d \\epsilon _ { t } d z _ { t } } \\right] p ( { \\bf s } _ { 0 } ) d { \\bf s } _ { 0 } } } \\\\  {\\right] \\displaystyle \\approx \\displaystyle \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\left[ \\displaystyle { \\sum _ { t = 1 } ^ { T } c ( { \\bf s } _ { t } ^ { \\mathcal { W } ^ { k } , \\{ z _ { 1 } ^ { k } , \\dots , z _ { t } ^ { k } \\} , \\{ { \\bf s } _ { t } ^ { k } , \\dots , { \\bf s } _ { t } ^ { k } \\} , \\mathcal { W } _ { \\pi } ) }  . } } \\end{\\right]array} \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            646,
+            828,
+            785
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The first line in (16) is obtained by using the assumption that the dynamics are Markovian with respect to the current state and the current action and by replacing $p ( \\mathbf { s } _ { t } | \\mathbf { s } _ { t - 1 } , \\mathbf { a } _ { t - 1 } )$ with the right-hand side of (14). In the second line, sW,{z1,...,zt},{\u000f1,...,\u000ft},Wπt is the state that is obtained at time t in a roll-out generated by using a policy with parameters $\\mathcal { W } _ { \\pi }$ , a transition function parameterized by $\\mathcal { W }$ and input noise $z _ { 1 } , \\dots , z _ { t }$ , with additive noise values $\\epsilon _ { 1 } , \\ldots , \\epsilon _ { t }$ . In the last line we have approximated the integration with respect to $\\mathcal { W } , z _ { 1 } , \\ldots , z _ { T } , \\epsilon _ { 1 } , \\ldots , \\epsilon _ { T }$ and $\\mathbf { s } _ { 0 }$ by averaging over $K$ samples of these variables. To sample $\\mathbf { s } _ { 0 }$ , we draw this variable uniformly from the available transitions $\\left( \\mathbf { s } _ { t } , \\mathbf { a } _ { t } , \\mathbf { s } _ { t + 1 } \\right)$ . ",
+        "bbox": [
+            173,
+            786,
+            825,
+            890
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The expected cost (15) can then be optimized by stochastic gradient descent using the gradients of the Monte Carlo approximation given by the last line of (16). Algorithm 1 computes this Monte ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/84707d6ab02c7350759998233bfcfcaf2a82dd4d7fcca965d2bb73ef7c1ee91e.jpg",
+        "image_caption": [
+            "Figure 3: Visualization of three policies in state space. Waterfall is indicated by top black bar. Left: policy $\\pi _ { V B }$ obtained with a BNN trained with VB. Avg. reward is $- 2 . 5 3$ . Middle: policy $\\pi _ { \\alpha = 0 . 5 }$ obtained with a BNN trained with $\\alpha = 0 . 5$ . Avg. reward is $- 2 . 3 1$ . Right: policy $\\pi _ { G P }$ obtained by using a Gaussian process model. Avg. reward is $- 2 . 9 4$ . Color and arrow indicate direction of paddling of policy when in state $\\mathbf { s } _ { t }$ , arrow length indicates action magnitude. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            98,
+            815,
+            241
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/eb954dace85d6ad50cc674045868588f229d20811f17cb9ed07b44fb7aa4836d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>MLP</td><td>VB</td><td>α=0.5</td><td>α=1.0</td><td>GP</td><td>PSO-P</td></tr><tr><td>Wetchicken</td><td>-2.71±0.09</td><td>-2.67±0.10</td><td>-2.37±0.01</td><td>-2.42±0.01</td><td>-3.05±0.06</td><td>-2.34</td></tr><tr><td>Turbine</td><td>-0.65±0.14</td><td>-0.45±0.02</td><td>-0.41±0.03</td><td>-0.55±0.08</td><td>-0.64±0.18</td><td>NA</td></tr><tr><td>Industrial</td><td>-183.5±4.1</td><td>-180.2±0.6</td><td>-174.2±1.1</td><td>-171.1±2.1</td><td>-285.2±20.5</td><td>-145.5</td></tr><tr><td>Avg. Rank</td><td>3.6±0.3</td><td>3.1±0.2</td><td>1.5±0.2</td><td>2.3±0.3</td><td>4.5±0.3</td><td></td></tr></table>",
+        "bbox": [
+            176,
+            349,
+            825,
+            420
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Policy performances over different benchmarks. Printed are average values over 5 runs with respective standard errors. Bottom row is the average rank over all $5 \\times 3$ runs. ",
+        "bbox": [
+            173,
+            430,
+            823,
+            458
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Carlo approximation. The gradients can then be obtained using automatic differentiation tools such as Theano (Theano Development Team). Note that Algorithm 1 uses the BNNs to make predictions for the change in the state $\\Delta _ { t } = { \\bf s } _ { t + 1 } - { \\bf s } _ { t }$ instead of for the next state $\\mathbf { s } _ { t + 1 }$ since this approach often performs better in practice (Deisenroth & Rasmussen, 2011). ",
+        "bbox": [
+            174,
+            487,
+            825,
+            542
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            560,
+            326,
+            575
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We now evaluate the performance of our algorithm for policy search in different benchmark problems. These problems are chosen based on two reasons. First, they contain complex stochastic dynamics and second, they represent real-world applications common in industrial settings. A theano implementation of algorithm 1 is available online1. See the appendix B for a short introduction to all methods we compare to and appendix C for the hyper-parameters used. ",
+        "bbox": [
+            174,
+            593,
+            825,
+            662
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 WET-CHICKEN BENCHMARK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            681,
+            413,
+            695
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The Wet-Chicken benchmark (Tresp, 1994) is a challenging problem for model-based policy search that presents both bi-modal and heteroskedastic transition dynamics. We use the two-dimensional version of the problem (Hans & Udluft, 2009) and extend it to the continuous case. ",
+        "bbox": [
+            174,
+            708,
+            825,
+            750
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this problem, a canoeist is paddling on a two-dimensional river. The canoeist’s position at time $t$ is $( x _ { t } , y _ { t } )$ . The river has width $w = 5$ and length $l = 5$ with a waterfall at the end, that is, at $y _ { t } = l$ . The canoeist wants to move as close to the waterfall as possible because at time $t$ he gets reward $r _ { t } = - ( l - y _ { t } )$ . However, going beyond the waterfall boundary makes the canoeist fall down, having to start back again at the origin $( 0 , 0 )$ . At time $t$ the canoeist can choose an action $( a _ { t , x } , a _ { t , y } ) \\in [ - 1 , 1 ] ^ { 2 }$ that represents the direction and magnitude of his paddling. The river dynamics have stochastic turbulences $s _ { t }$ and drift $v _ { t }$ that depend on the canoeist’s position on the $x$ axis. The larger $x _ { t }$ , the larger the drift and the smaller $x _ { t }$ , the larger the turbulences. The underlying dynamics are given by the following system of equations. The drift and the turbulence magnitude are given by $v _ { t } \\stackrel { - } { = } 3 x _ { t } w ^ { - 1 }$ and $s _ { t } = 3 . 5 - v _ { t }$ , respectively. The new location $( x _ { t + 1 } , y _ { t + 1 } )$ is given by the current ",
+        "bbox": [
+            173,
+            757,
+            825,
+            897
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/4e127e47145cae0eedbc5aefc38b8a5b2b0d86e24e0ffd48ee8c5016bbcf4a4e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>MLP</td><td>VB</td><td>α=0.5</td><td>α=1.0</td><td>GP</td></tr><tr><td colspan=\"6\">MSE</td></tr><tr><td>WetChicken</td><td>1.289±0.013</td><td>1.347±0.015</td><td>1.347±0.008</td><td>1.359±0.017</td><td>1.359±0.017</td></tr><tr><td>Turbine</td><td>0.16±0.001</td><td>0.21±0.003</td><td>0.192±0.002</td><td>0.237±0.004</td><td>0.492±0.026</td></tr><tr><td>Industrial</td><td>0.0186±0.0052</td><td>0.0182±0.0052</td><td>0.017±0.0046</td><td>0.0171±0.0047</td><td>0.0233±0.0049</td></tr><tr><td>Avg. Rank</td><td>2.0±0.34</td><td>3.1±0.24</td><td>2.4±0.23</td><td>2.9±0.36</td><td>4.6±0.23</td></tr><tr><td colspan=\"6\">Log-Likelihood</td></tr><tr><td>WetChicken</td><td>-1.755±0.003</td><td>-1.140±0.033</td><td>-1.057±0.014</td><td>-1.070±0.011</td><td>-1.722±0.011</td></tr><tr><td>Turbine</td><td>-0.868±0.007</td><td>-0.775±0.004</td><td>-0.746±0.013</td><td>-0.774±0.015</td><td>-2.663±0.131</td></tr><tr><td>Industrial</td><td>0.767±0.047</td><td>1.132±0.064</td><td>1.328±0.108</td><td>1.326±0.098</td><td>0.724±0.04</td></tr><tr><td>Avg. Rank</td><td>4.3±0.12</td><td>2.6±0.16</td><td>1.3±0.15</td><td>2.1±0.18</td><td>4.7±0.12</td></tr></table>",
+        "bbox": [
+            176,
+            99,
+            825,
+            238
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Model test error and test log-likelihood for different benchmarks. Printed are average values over 5 runs with respective standard errors. Bottom row is the average rank over all $5 \\times 3$ runs. ",
+        "bbox": [
+            173,
+            247,
+            825,
+            276
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "location $( x _ { t } , y _ { t } )$ and current action $( a _ { t , x } , a _ { t , y } )$ using ",
+        "bbox": [
+            173,
+            292,
+            517,
+            309
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d37bc8c960693d7fd49ff12ee63c79eb448ada2b9d0380956118e39bd4162023.jpg",
+        "text": "$$\nx _ { t + 1 } = \\left\\{ \\begin{array} { l l } { 0 } & { \\mathrm { i f } \\quad x _ { t } + a _ { t , x } < 0 } \\\\ { 0 } & { \\mathrm { i f } \\quad \\hat { y } _ { t + 1 } > l } \\\\ { w } & { \\mathrm { i f } \\quad x _ { t } + a _ { t , x } > w } \\\\ { x _ { t } + a _ { t , x } } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. , \\qquad y _ { t + 1 } = \\left\\{ \\begin{array} { l l } { 0 } & { \\mathrm { i f } \\quad \\hat { y } _ { t + 1 } < 0 } \\\\ { 0 } & { \\mathrm { i f } \\quad \\hat { y } _ { t + 1 } > l } \\\\ { \\hat { y } _ { t + 1 } } & { \\mathrm { o t h e r w i s e } } \\end{array} \\right. ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            315,
+            756,
+            383
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { y } _ { t + 1 } = y _ { t } + \\left( a _ { t , y } - 1 \\right) + v _ { t } + s _ { t } \\tau _ { t }$ and $\\tau _ { t } \\sim \\mathrm { U n i f } ( [ - 1 , 1 ] )$ is a random variable that represents the current turbulence. These dynamics result in rich transition distributions depending on the position as illustrated by the plots in Figure 2. As the canoeist moves closer to the waterfall, the distribution for the next state becomes increasingly bi-modal (see Figure 1c) because when he is close to the waterfall, the change in the current location can be large if the canoeist falls down the waterfall and starts again at $( 0 , 0 )$ . The distribution may also be truncated uniform for states close to the borders (see Figure 1d). Furthermore the system has heteroskedastic noise, the smaller the value of $x _ { t }$ the higher the noise variance (compare Figure 1a with 1b). Because of these properties, the Wet-Chicken problem is especially difficult for model-based reinforcement learning methods. To our knowledge it has only been solved using model-free approaches after a discretization of the state and action sets (Hans & Udluft, 2009). For model training we use a batch 2500 random state transitions. ",
+        "bbox": [
+            173,
+            390,
+            825,
+            544
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The predictive distributions of different models for $y _ { t + 1 }$ are shown in Figure 2 for specific choices of $( x _ { t } , y _ { t } )$ and $( a _ { x , t } , a _ { y , t } )$ . These plots show that BNNs with $\\alpha = 0 . 5$ are very close to the ground-truth. While it is expected that Gaussian processes fail to model multi-modalities in Figure 1c, the FTIC approximation allows them to model the heteroskedasticity to an extent. VB captures the stochastic patterns on a global level, but often under or over-estimates the true probability density in specific regions. The test-loglikelihood and test MSE in $y$ -dimension are reported in Table 2 for all methods. (the transitions for $x$ are deterministic given $y$ ). ",
+        "bbox": [
+            173,
+            550,
+            825,
+            648
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "After fitting the models, we train policies using Algorithm 1 with a horizon of size $T = 5$ . Table 1 shows the average reward obtained by each method. BNNs with $\\alpha = 0 . 5$ perform best and produce policies that are very close to the optimal upper bound, as indicated by the performance of the particle swarm optimization policy (PSO-P). In this problem VB seems to lack robustness and has much larger empirical variance across experiment repetitions than $\\alpha = 0 . 5$ or $\\alpha = 1 . 0$ . ",
+        "bbox": [
+            174,
+            655,
+            825,
+            724
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows three example policies, πVB , $\\pi _ { \\alpha = 0 . 5 }$ and $\\pi _ { \\mathrm { G P } }$ (Figure 3a,3b and 3c, respectively). The policies obtained by BNNs with random inputs (VB and $\\alpha = 0 . 5$ ) show a richer selection of actions. The biggest differences are in the middle-right regions of the plots, where the drift towards the waterfall is large and the bi-modal transition for $y$ (missed by the GP) is more important. ",
+        "bbox": [
+            174,
+            732,
+            825,
+            787
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 INDUSTRIAL APPLICATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            804,
+            406,
+            819
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now present results on two industrial cases. First, we focus on data generated by a real gas turbine and second, we consider a recently introduced simulator called the \"industrial benchmark\", with code publicly available2 (Hein et al., 2016b). According to the authors: \"The \"industrial benchmark\" aims at being realistic in the sense, that it includes a variety of aspects that we found to be vital in industrial applications.\" ",
+        "bbox": [
+            174,
+            830,
+            825,
+            900
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/73a893d718f52a13f31cd7f47999e4a4d6a4fe0125923c611a0ccda9a0afe6c5.jpg",
+        "image_caption": [
+            "Figure 4: Roll-outs of algorithm 1 for two starting states $\\mathbf { s } _ { 0 }$ (top/bottom) using different types of BNNs (left to right) with $K = 7 5$ samples for $T = 7 5$ steps. Action sequence $A _ { 0 } , \\cdots , A _ { T = 7 5 }$ given by dataset for each $\\mathbf { s } _ { 0 }$ . From left to right: model trained using VB, $\\alpha = 0 . 5$ and $\\alpha = 1 . 0$ respectively. Red: trajectory observed in dataset, blue: sample average, light blue: individual samples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            99,
+            807,
+            367
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.1 GAS TURBINE DATA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            469,
+            364,
+            483
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For the experiment with gas turbine data we simulate a task with partial observability. To that end we use 40,000 observations of a 30 dimensional time-series of sensor recordings from a real gas turbine. We are also given a cost function that evaluates the performance of the current state of the turbine. The features in the time-series are grouped into three different sets: a set of environmental variables $E _ { t }$ (e.g. temperature and measurements from sensors in the turbine) that cannot be influenced by the agent, a set of variables relevant for the cost function $N _ { t }$ (e.g. the turbines current pollutant emission) and a set of steering variables $A _ { t }$ that can be manipulated to control the turbine. ",
+        "bbox": [
+            174,
+            496,
+            825,
+            594
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We first train a world model as a reflection of the real turbine dynamics. To that end we define the world model’s transitions for $N _ { t }$ to have the functional form $N _ { t } = f ( E _ { t - 5 } , . . , E _ { t } , A _ { t - 5 } , . . A _ { t } ) .$ . The world model assumes constant transitions for the environmental variables: $E _ { t + 1 } = E _ { t }$ . To make fair comparisons, our world model is given by a non-Bayesian neural network with deterministic weights and with additive Gaussian output noise. ",
+        "bbox": [
+            174,
+            601,
+            825,
+            670
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We then use the world model to generate an artificial batch of data for training the different methods. The inputs in this batch are still the same as in the original turbine data, but the outputs are now sampled from the world model. After generating the artificial data, we only keep a small subset of the original inputs to the world model. The aim of this experiment is to learn policies that are robust to noise in the dynamics. This noise would originate from latent factors that cannot be controlled, such as the missing features that were originally used to generate the outputs by the world model but which are no longer available. After training the models for the dynamics, we use algorithm 1 for policy optimization. The resulting policies are then finally evaluated in the world model. ",
+        "bbox": [
+            174,
+            678,
+            825,
+            789
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Tables 2 and 1 show the respective model and policy performances for each method. The experiment was repeated 5 times and we report average results. We observe that $\\alpha = 0 . 5$ performs best in this scenario, having the highest test log-likelihood and best policy performance. ",
+        "bbox": [
+            176,
+            796,
+            825,
+            838
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 INDUSTRIAL BENCHMARK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
+            410,
+            868
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this benchmark the hidden Markov state space $\\mathbf { s } _ { t }$ consists of 27 variables, whereas the observable state $\\mathbf { o } _ { t }$ is only 5 dimensional. This observable state consists of 3 adjustable steering variables $A _ { t }$ : the velocity $v ( t )$ , the gain $g ( t )$ and the shift $s ( t )$ . We also observe the fatigue $f ( t )$ and consumption $c ( t )$ that together form the reward signal $R ( t ) = - ( 3 f ( t ) + c ( t ) )$ . Also visible is the setpoint $S$ , a constant hyper-parameter of the benchmark that indicates the complexity of the dynamics. ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For each setpoint $S \\in \\{ 1 0 , 2 0 , \\cdots , 1 0 0 \\}$ we generate 7 trajectories of length 1000 using random exploration. This batch with $7 0 , 0 0 0$ state transitions forms the training set. We use 30, 000 state transitions, consisting of 3 trajectories for each setpoint, as test set. ",
+        "bbox": [
+            176,
+            138,
+            821,
+            180
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For data preprocessing, in addition to the standard normalization process, we apply a log transformation to the reward variable. Because the reward is bounded in the interval $[ 0 , R _ { m a x } ]$ , we use a logit transformation to map this interval into the real line. We define the functional form for the dynamics as $R _ { t } = f ( A _ { t - 1 5 } , \\cdot \\cdot \\cdot , A _ { t } , R _ { t - 1 5 } , \\cdot \\cdot \\cdot , R _ { t - 1 } )$ . ",
+        "bbox": [
+            174,
+            188,
+            825,
+            244
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The test errors and log-likelihood are given in Table 2. We see that BNNs with $\\alpha = 0 . 5$ and $\\alpha = 1 . 0$ perform best here, whereas Gaussian processes or the MLP obtain rather poor results. ",
+        "bbox": [
+            174,
+            251,
+            823,
+            279
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Each row in Figure 4 visualizes long term predictions of the MLP and BNNs trained with VB and $\\alpha = 0 . 5$ in two specific cases. In the top row we see that while all three methods produce wrong predictions in expectation (compare dark blue curve to red curve). However, BNNs trained with $V B$ and with $\\alpha = 0 . 5$ exhibit a bi-modal distribution of predicted trajectories, with one mode following the ground-truth very closely. By contrast, the MLP misses the upper mode completely. The bottom row shows that the VB and $\\alpha = 0 . 5$ also produce more tight confident bands in other settings. ",
+        "bbox": [
+            174,
+            285,
+            825,
+            369
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Next, we learn policies using the trained models. Here we use a relatively long horizon of $T = 7 5$ steps. Table 1 shows average rewards obtained when applying the policies to the real dynamics. Because both benchmark and models have an autoregressive component, we do an initial warm-up phase using random exploration before we apply the policies to the system and start to measure rewards. ",
+        "bbox": [
+            174,
+            376,
+            825,
+            445
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We observe that GPs perform very poorly in this benchmark. We believe the reason for this is the long search horizon, which makes the uncertainties in the predictive distributions of the GPs become very large. Tighter confidence bands, as illustrated in Figure 4 seem to be key for learning good policies. Overall, $\\alpha = 1 . 0$ performs best with $\\alpha = 0 . 5$ being very close. ",
+        "bbox": [
+            174,
+            453,
+            825,
+            510
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            531,
+            339,
+            547
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "There has been relatively little attention to using Bayesian neural networks for reinforcement learning. In Blundell et al. (2015) a Thompson sampling approach is used for a contextual bandits problem; the focus is tackling the exploration-exploitation trade-off, while the work in Watter et al. (2015) combines variational auto-encoder with stochastic optimal control for visual data. Compared to our approach the first of these contributions focusses on the exploration/exploitation dilemma, while the second one uses a stochastic optimal control approach to solve the learning problem. By contrast, our work seeks to find an optimal parameterized policy. ",
+        "bbox": [
+            174,
+            566,
+            825,
+            665
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Policy gradient techniques are a prominent class of policy search algorithms (Peters & Schaal, 2008). While model-based approaches were often used in discrete spaces (Wang & Dietterich, 2003), model-free approaches tended to be more popular in continuous spaces (e.g. Peters & Schaal (2006)). ",
+        "bbox": [
+            174,
+            671,
+            825,
+            714
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our work can be seen as a Monte-Carlo model-based policy gradient technique in continuous stochastic systems. Similar work was done using Gaussian processes (Deisenroth & Rasmussen, 2011) and with recurrent neural networks (Schaefer et al., 2007) . The Gaussian process approach, while restricted to a Gaussian state distribution, allows propagating beliefs over the roll-out procedure. More recently Gu et al. (2016) augment a model-free learning procedure with data generated from model-based roll-outs. ",
+        "bbox": [
+            174,
+            720,
+            826,
+            804
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            833,
+            488,
+            848
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have extended the standard Bayesian neural network (BNN) model with the addition of a random input noise source $z$ . This enables principled Bayesian inference over complex stochastic functions. We have shown that our BNNs with random inputs can be trained with high accuracy by minimizing $\\alpha$ -divergences, with $\\alpha = 0 . 5$ , which often produces better results than variational Bayes. We have also presented an algorithm that uses random roll-outs and stochastic optimization for learning a parameterized policy in a batch scenario. This algorithm particular suited for industry domains. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our BNNs with random inputs have allowed us to solve a challenging benchmark problem where model-based approaches usually fail. They have also shown promising results on industry benchmarks including real-world data from a gas turbine. In particular, our experiments indicate that a BNN trained with $\\alpha = 0 . 5$ as divergence measure in conjunction with the presented algorithm for policy optimization is a powerful black-box tool for policy search. ",
+        "bbox": [
+            174,
+            138,
+            823,
+            208
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "As future work we will consider safety and exploration. For safety, we believe having uncertainty over the underlaying stochastic functions will allows us to optimize policies by focusing on worst case results instead of on average performance. For exploration, having uncertainty on the stochastic functions will be useful for efficient data collection. ",
+        "bbox": [
+            174,
+            215,
+            825,
+            271
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            290,
+            334,
+            303
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "José Miguel Hernández-Lobato acknowledges support from the Rafael del Pino Foundation. The authors would like to thank Ryan P. Adams, Hans-Georg Zimmermann, Matthew J. Johnson, David Duvenaud and Justin Bayer for helpful discussions. ",
+        "bbox": [
+            176,
+            315,
+            823,
+            357
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            378,
+            285,
+            393
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "D.P. Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific optimization and computation series. 2002. ISBN 9781886529083. ",
+        "bbox": [
+            174,
+            402,
+            823,
+            431
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural networks. In ICML, pp. 1613–1622, 2015. ",
+        "bbox": [
+            173,
+            440,
+            823,
+            470
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Thang D Bui, Daniel Hernández-Lobato, Yingzhen Li, José Miguel Hernández-Lobato, and Richard E Turner. Deep Gaussian processes for regression using approximate expectation propagation. In ICML, pp. 1472–1481, 2016. ",
+        "bbox": [
+            176,
+            479,
+            823,
+            522
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Marc Deisenroth and Carl E Rasmussen. PILCO: A model-based and data-efficient approach to policy search. In ICML, pp. 465–472, 2011. ",
+        "bbox": [
+            173,
+            531,
+            821,
+            561
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Marc Peter Deisenroth, Gerhard Neumann, and Jan Peters. A survey on policy search for robotics. Foundations and Trends in Robotics, 2:1–142, 2013. ",
+        "bbox": [
+            173,
+            571,
+            823,
+            599
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A. Draeger, S. Engell, and H. Ranke. Model predictive control using neural networks. IEEE Control Systems, 15:61–66, 1995. ",
+        "bbox": [
+            174,
+            609,
+            823,
+            638
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yarin Gal, Rowan Mcallister, and Carl Rasmussen. Improving PILCO with Bayesian neural networks dynamics models. In Data-Efficient Machine Learning workshop, ICML, 2016. ",
+        "bbox": [
+            174,
+            648,
+            823,
+            678
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Shixiang Gu, Timothy Lillicrap, Ilya Sutskever, and Sergey Levine. Continuous deep q-learning with model-based acceleration. arXiv preprint arXiv:1603.00748, 2016. ",
+        "bbox": [
+            174,
+            686,
+            823,
+            717
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alexander Hans and Steffen Udluft. Efficient uncertainty propagation for reinforcement learning with limited data. In ICANN, pp. 70–79. Springer, 2009. ",
+        "bbox": [
+            173,
+            726,
+            825,
+            756
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Daniel Hein, Alexander Hentschel, Thomas A Runkler, and Steffen Udluft. Reinforcement learning with particle swarm optimization policy (PSO-P) in continuous state and action spaces. IJSIR, 7: 23–42, 2016a. ",
+        "bbox": [
+            174,
+            765,
+            825,
+            808
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Daniel Hein, Alexander Hentschel, Volkmar Sterzing, Michel Tokic, and Steffen Udluft. Introduction to the\" industrial benchmark\". arXiv preprint arXiv:1610.03793, 2016b. ",
+        "bbox": [
+            171,
+            816,
+            823,
+            847
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "José Miguel Hernández-Lobato, Matthew W Hoffman, and Zoubin Ghahramani. Predictive entropy search for efficient global optimization of black-box functions. In NIPS, pp. 918–926, 2014. ",
+        "bbox": [
+            171,
+            856,
+            825,
+            886
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "José Miguel Hernández-Lobato, Yingzhen Li, Mark Rowland, Daniel Hernández-Lobato, Thang Bui, and Richard E Turner. Black-box $\\alpha$ -divergence minimization. In ICML, pp. 1511–1520, 2016. ",
+        "bbox": [
+            176,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Rein Houthooft, Xi Chen, Yan Duan, John Schulman, Filip De Turck, and Pieter Abbeel. VIME: Variational information maximizing exploration. In NIPS, pp. 1109–1117, 2016. ",
+        "bbox": [
+            171,
+            103,
+            825,
+            133
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma, Tim Salimans, and Max Welling. Variational dropout and the local reparameterization trick. In NIPS, pp. 2575–2583, 2015. ",
+        "bbox": [
+            173,
+            140,
+            825,
+            171
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jonathan Ko, Daniel J Klein, Dieter Fox, and Dirk Haehnel. Gaussian processes and reinforcement learning for identification and control of an autonomous blimp. In IEEE Robotics and Automation, pp. 742–747, 2007. ",
+        "bbox": [
+            176,
+            178,
+            823,
+            222
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Tom Minka. Divergence measures and message passing. Technical report, Microsoft Research, 2005. ",
+        "bbox": [
+            173,
+            229,
+            825,
+            247
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jan Peters and Stefan Schaal. Policy gradient methods for robotics. In IROS, pp. 2219–2225, 2006. ",
+        "bbox": [
+            173,
+            253,
+            825,
+            270
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jan Peters and Stefan Schaal. Reinforcement learning of motor skills with policy gradients. Neural networks, 21:682–697, 2008. ",
+        "bbox": [
+            171,
+            279,
+            825,
+            308
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Carl Edward Rasmussen, Malte Kuss, et al. Gaussian processes in reinforcement learning. In NIPS, pp. 751–758, 2003. ",
+        "bbox": [
+            173,
+            315,
+            825,
+            345
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Anton Maximilian Schaefer, Steffen Udluft, and Hans-Georg Zimmermann. The recurrent control neural network. In ESANN, pp. 319–324, 2007. ",
+        "bbox": [
+            171,
+            353,
+            826,
+            383
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Edward Snelson and Zoubin Ghahramani. Sparse Gaussian processes using pseudo-inputs. In NIPS, pp. 1257–1264, 2005. ",
+        "bbox": [
+            171,
+            391,
+            825,
+            420
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Theano Development Team. Theano: A Python framework for fast computation of mathematical expressions. arXiv e-print abs/1605.02688. ",
+        "bbox": [
+            171,
+            429,
+            826,
+            458
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "V. Tresp. The wet game of chicken. Technical report, 1994. ",
+        "bbox": [
+            173,
+            467,
+            566,
+            483
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Bo Wahlberg. System identification using Laguerre models. IEEE Transactions on Automatic Control, 36:551–562, 1991. ",
+        "bbox": [
+            171,
+            489,
+            825,
+            520
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "M. J. Wainwright and M. I. Jordan. Graphical models, exponential families, and variational inference. Foundations and Trends in Machine Learning, 1:1–305, 2008. ",
+        "bbox": [
+            171,
+            529,
+            825,
+            558
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Xin Wang and Thomas G Dietterich. Model-based policy gradient reinforcement learning. In ICML, pp. 776–783, 2003. ",
+        "bbox": [
+            173,
+            565,
+            825,
+            595
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Manuel Watter, Jost Springenberg, Joschka Boedecker, and Martin Riedmiller. Embed to control: A locally linear latent dynamics model for control from raw images. In NIPS, pp. 2728–2736, 2015. ",
+        "bbox": [
+            173,
+            603,
+            825,
+            633
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/900ebf513e7d10564c74abf691b5b00c755842a97617ac6027dd8d673f447030.jpg",
+        "image_caption": [
+            "Figure 5: Ground truth and predictive distributions for two toy problems introduced in main text. Top: bi-modal prediction problem, Bottom: heteroskedastic prediction problem. Left column: Training data (blue points) and ground truth functions (red). Columns 2-4: predictions generated with VB, $\\alpha = 0 . 5$ and $\\alpha = 1 . 0$ , respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            98,
+            810,
+            311
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A ROBUSTNESS OF $\\alpha = 0 . 5$ AND $\\alpha = 1 . 0$ WHEN $q ( \\mathbf { z } )$ IS NOT LEARNED ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            424,
+            764,
+            441
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the accuracy of the predictive distributions generated by BNNs with stochastic inputs trained by minimizing (8) for different BNNs parameterized by $\\alpha$ in two simple regression problems. The first one is characterized by a bimodal predictive distribution. The second is characterized by a heteroskedastic predictive distribution. In the latter case the magnitude of the noise in the targets changes as a function of the input features. ",
+        "bbox": [
+            174,
+            469,
+            825,
+            540
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In the first problem $x \\in [ - 2 , 2 ]$ and $y$ is obtained as $y = 1 0 \\sin ( x ) + \\epsilon$ with probability 0.5 and $y = 1 0 \\cos ( x ) + \\epsilon$ , otherwise, where $\\epsilon \\sim \\mathcal { N } ( 0 , 1 )$ and $\\epsilon$ is independent of $x$ . The plot in the top of the 1st column in Figure 5 shows a training dataset obtained by sampling 2500 values of $x$ uniformly at random. The plot clearly shows that the distribution of $y$ for a particular $x$ is bimodal. In the second problem $x \\in [ - 4 , 4 ]$ and $y$ is obtained as $y = 7 \\sin ( x ) + 3 | \\cos ( x / 2 ) | \\epsilon$ . The plot in the bottom of the 1st column in Figure 5 shows a training dataset obtained with 1000 values of $x$ uniformly at random. The plot clearly shows that the distribution of $y$ is heteroskedastic, with a noise variance that is a function of $x$ . ",
+        "bbox": [
+            173,
+            546,
+            825,
+            657
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We evaluated the predictive performance obtained by minimizing (8) using $\\alpha = 0 . 5$ and $\\alpha = 1 . 0$ and also by running VB. However, we do not learn $q ( \\mathbf { z } )$ and keeping it instead fixed to the prior $p ( \\mathbf { z } )$ . ",
+        "bbox": [
+            173,
+            665,
+            825,
+            694
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We fitted a neural network with 2 hidden layers and 50 hidden units per layer using Adam with its default parameter values, with a learning rate of 0.01 in the first problem and 0.002 in the second problem. We used mini-batches of size 250 and 1000 training epochs. To approximate the expectations in 8, we draw $K = 5 0$ samples from $q$ . ",
+        "bbox": [
+            174,
+            700,
+            825,
+            756
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The plots in the 3rd and 4th columns of Figure 5 show the predictions obtained with $\\alpha = 0 . 5$ and $\\alpha = 1 . 0$ , respectively. In these cases, the predictive distribution is able to capture the bimodality in the first problem and the heteroskedasticity pattern in the second problem in both cases The plots in the 2nd column of Figure 5 show the predictions obtained with VB, which converges to suboptimal solutions in which the predictive distribution has a single mode (in the first problem) or is homoskedastic (in the second problem). Tables 3 and 4 show the average test RMSE and log-likelihood obtained by each method on each problem. ",
+        "bbox": [
+            173,
+            762,
+            825,
+            861
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "These results show that Bayesian neural networks trained with $\\alpha = 0 . 5$ or $\\alpha = 1 . 0$ are more robust than VB and can still model complex predictive distributions, which may be multimodal and heteroskedastic, even when $q ( \\mathbf { z } )$ is not learned and is instead kept fixed to the prior $p ( \\mathbf { z } )$ . By contrast, VB fails to capture complex stochastic patterns in this setting. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/89d806a515235fe961334fdb19f11076bd32141b05a9a00eaf4598d7b755293f.jpg",
+        "table_caption": [
+            "Table 3: Test error and log-likelihood for the bi-modal prediction problem. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>RMSE</td><td>Log-likelihood</td></tr><tr><td>VB</td><td>5.12</td><td>-3.05</td></tr><tr><td>α = 0.5</td><td>5.14</td><td>-2.10</td></tr><tr><td>α = 1.0</td><td>5.15</td><td>-2.11</td></tr></table>",
+        "bbox": [
+            191,
+            101,
+            449,
+            170
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/afe890d5fbb76c24db5d97892a174c9578637c1a96f61328fde0287335315376.jpg",
+        "table_caption": [
+            "Table 4: Test error and log-likelihood for the heteroskedastic prediction problem. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>RMSE</td><td>Log-likelihood</td></tr><tr><td>VB</td><td>1.88</td><td>-2.05</td></tr><tr><td>α = 0.5</td><td>1.89</td><td>-1.78</td></tr><tr><td>α =1.0</td><td>1.94</td><td>-1.98</td></tr></table>",
+        "bbox": [
+            542,
+            101,
+            800,
+            170
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B METHODS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            226,
+            295,
+            242
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the experiments we compare to the following methods: ",
+        "bbox": [
+            174,
+            257,
+            550,
+            272
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Standard MLP. The standard multi-layer preceptron (MLP) is equivalent to our BNNs, but does not have uncertainty over the weight $\\mathcal { W }$ and does not include any stochastic inputs. We train this method using early stopping on a subset of the training data. When we perform roll-outs using algorithm 1, the predictions of the MLP are made stochastic by adding Gaussian noise to its output. The noise variance is fixed by maximum likelihood on some validation data after model training. ",
+        "bbox": [
+            174,
+            286,
+            825,
+            357
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Variational Bayes (VB). The most prominent approach in training modern BNNs is to optimize the variational lower bound (Blundell et al., 2015; Houthooft et al., 2016; Gal et al., 2016). This is in practice equivalent to $\\alpha$ -divergence minimization when $\\alpha  0$ (Hernández-Lobato et al., 2016). In our experiments we use $\\alpha$ -divergence minimization with $\\alpha = 1 0 ^ { - 6 }$ to implement this method. ",
+        "bbox": [
+            174,
+            372,
+            825,
+            428
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Gaussian Processes (GPs). Gaussian Processes have recently been used for policy search under the name of PILCO (Deisenroth & Rasmussen, 2011). For each dimension of the target variables, we fit a different sparse GP using the FITC approximation (Snelson & Ghahramani, 2005). In particular, each sparse GP is trained using 150 inducing inputs by using the method stochastic expectation propagation (Bui et al., 2016). After this training process we approximate the sparse GP by using a feature expansion with random basis functions (see supplementary material of Hernández-Lobato et al. 2014). This allows us to draw samples from the GP posterior distribution over functions, enabling the use of Algorithm 1 for policy training. Note that PILCO will instead moment-match at every roll-out step as it works by propagating Gaussian distributions. However, in our experiments we obtained better performance by avoiding the moment matching step with the aforementioned approximation based on random basis functions. ",
+        "bbox": [
+            174,
+            443,
+            825,
+            597
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Particle Swarm Optimization Policy(PSO-P). We use this method to estimate an upper bound for reward performance. PSO-P is a model predictive control (MPC) method that uses the true dynamics when applicable (Hein et al., 2016a). For a given state $\\mathbf { s } _ { t }$ , the best action is selected using the standard receding horizon approach on the real environment. Note that this is not a benchmark method to compare to, we use it instead as an indicator of what the best possible reward can be achieved for a fixed planning horizon $T$ . ",
+        "bbox": [
+            174,
+            612,
+            825,
+            695
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C MODEL PARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            715,
+            392,
+            732
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For all tasks we will use a standard MLP with two hidden layer with 20 hidden units each as policy representation. The activation functions for the hidden units are rectifiers: $\\varphi ( x ) = \\operatorname* { m a x } ( x , 0 )$ . If present, bounding of the actions is realized using the tanh activation function on the outputs of the policy. All models based on neural network will share the same hyperparameter. We use ADAM as learning algorithm in all tasks. ",
+        "bbox": [
+            174,
+            747,
+            825,
+            816
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "WetChicken The neural network models are set to 2 hidden layers and 20 hidden units per layer. We use 2500 random state transitions for training. We found that assuming no observation noise by setting $\\Gamma$ to a constant of $1 0 ^ { - 5 }$ helped the models converge to lower energy values. ",
+        "bbox": [
+            176,
+            833,
+            825,
+            875
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For policy training we use a horizon of size $T = 5$ and optimize the policy network for 100 epochs, averaging over $K = 2 0$ samples in each gradient update, with mini-batches of size 10 and learning rate set to 10−5. ",
+        "bbox": [
+            176,
+            881,
+            823,
+            922
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Turbine The world model and the BNNs have two hidden layers with 50 hidden units each. For policy training and world-model evaluation we perform a roll-out with horizon $T = 2 0$ . For learning the policy we use minibaches of size 10 and draw $K = 1 0$ samples from $q$ . ",
+        "bbox": [
+            173,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Industrial Benchmark For the neural network models we use two hidden layers with 75 hidden units.We use a horizon of $T = 7 5$ , training for 500 epochs with batches of size 50 and $K = 2 5$ samples for each rollout. ",
+        "bbox": [
+            174,
+            161,
+            825,
+            203
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D COMPUTATIONAL COMPLEXITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            223,
+            473,
+            239
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "MODEL TRAINING ",
+        "bbox": [
+            174,
+            256,
+            305,
+            270
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "All models were trained using theano and a single GPU. Training the standard neural network is fast, the training time for this method was between 5 - 20 minutes, depending on data set size and dimensionality of the benchmark. In theano, the computational graph of the BNNs is similar to that of an ensemble of standard neural networks. The training time for the BNNs varied between 30 minutes to 5 hours depending on data size and dimensionality of benchmark. The sparse Gaussian Process was optimized using an expectation propagation algorithm and after training, it was approximated with a Bayesian linear model with fixed basis functions whose weights are initialized randomly (see Appendix B). We choose the inducing points in the GPs and the number of training epochs for these models so that the resulting training time was comparable to that of the BNNs. ",
+        "bbox": [
+            174,
+            281,
+            825,
+            406
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "POLICY SEARCH ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            424,
+            290,
+            438
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For policy training we used a single CPU. All methods are of similar complexity as they are all trained using Algorithm 1. Depending on the horizon, data set size and network topology, training took between 20 minutes (Wet-Chicken, $T = 5$ ), 3-4 hours (Turbine, $T = 2 0$ ) and 14-16 hours (industrial benchmark, $T = 7 5$ ). ",
+        "bbox": [
+            174,
+            449,
+            823,
+            505
+        ],
+        "page_idx": 13
+    }
+]

parse/train/HyebplHYwB/HyebplHYwB.md

@@ -0,0 +1,487 @@
+# THE SHAPE OF DATA: INTRINSIC DISTANCE FOR DATA DISTRIBUTIONS
+
+Anton Tsitsulin∗† Marina Munkhoeva∗‡ Davide Mottin§ Panagiotis Karras§
+
+Alex Bronstein¶
+
+Ivan Oseledets‡
+
+Emmanuel Müller†
+
+# ABSTRACT
+
+The ability to represent and compare machine learning models is crucial in order to quantify subtle model changes, evaluate generative models, and gather insights on neural network architectures. Existing techniques for comparing data distributions focus on global data properties such as mean and covariance; in that sense, they are extrinsic and uni-scale. We develop a first-of-its-kind intrinsic and multi-scale method for characterizing and comparing data manifolds, using a lower-bound of the spectral Gromov-Wasserstein inter-manifold distance, which compares all data moments. In a thorough experimental study, we demonstrate that our method effectively discerns the structure of data manifolds even on unaligned data of different dimensionality, and showcase its efficacy in evaluating the quality of generative models.
+
+# 1 INTRODUCTION
+
+The geometric properties of neural networks provide insights about their internals (Morcos et al., 2018; Wang et al., 2018) and help researchers in the design of more robust models (Arjovsky et al., 2017; Binkowski et al. ´ , 2018). Generative models are a natural example of the need for geometric comparison of distributions. As generative models aim to reproduce the true data distribution $\mathbb { P } _ { d }$ by means of the model distribution $\mathbb { P } _ { g } ( \mathbf { z } ; \Theta )$ , more delicate evaluation procedures are needed. Oftentimes, we wish to compare data lying in entirely different spaces, for example to track model evolution or compare models having different representation space.
+
+In order to evaluate the performance of generative models, past research has proposed several extrinsic evaluation measures, most notably the Fréchet (Heusel et al., 2017) and Kernel (Binkowski et al. ´ , 2018) Inception Distances (FID and KID). Such measures only reflect the first two or three moments of distributions, meaning they can be insensitive to global structural problems. We showcase this inadvertence in Figure 1: here FID and KID are insensitive to the global structure of the data distribution. Besides, as FID and KID are based only on extrinsic properties they are unable to compare unaligned data manifolds.
+
+![](images/d1860f98e049f6c6274cae6e5dad00a92ab71810387053b7e054fe1b8fa8d575.jpg)  
+Figure 1: Two distributions having the same first 3 moments, meaning FID and KID scores are close to 0.
+
+In this paper, we start out from the observation that models capturing the multi-scale nature of the data manifold by utilizing higher distribution moment matching, such as MMD-GAN (Li et al., 2017) and Sphere-GAN (Park & Kwon, 2019), perform consistently better than their single-scale counterparts. On the other hand, using extrinsic information can be misleading, as it is dependent on factors external to the data, such as representation. To address this drawback, we propose IMD, an Intrinsic Multi-scale Distance, that is able to compare distributions using only intrinsic information about the data, and provide an efficient approximation thereof that renders computational complexity nearly linear. We demonstrate that IMD effectively quantifies difference in data distributions in three distinct application scenarios: comparing word vectors in languages with unaligned vocabularies, tracking dynamics of intermediate neural network representations, and evaluating generative models.
+
+# 2 RELATED WORK
+
+The geometric perspective on data is ubiquitous in machine learning. Geometric techniques enhance unsupervised and semi-supervised learning, generative and discriminative models (Belkin & Niyogi, 2002; Arjovsky et al., 2017; Mémoli, 2011). We outline the applications of the proposed manifold comparison technique and highlight the geometric intuition along the way.
+
+# 2.1 GENERATIVE MODEL EVALUATION
+
+Past research has explored many different directions for the evaluation of generative models. Setting aside models that ignore the true data distribution, such as the Inception Score (Salimans et al., 2016) and GILBO (Alemi & Fischer, 2018), we discuss most relevant geometric ideas below; we refer the reader to Borji (2019) for a comprehensive survey.
+
+Critic model-based metrics. Classifier two-sample tests (C2ST) (Lopez-Paz & Oquab, 2017) aim to assess whether two samples came from the same distribution by means of an auxiliary classifier. This idea is reminiscent of the GAN discriminator network (Goodfellow et al., 2014): if it is possible to train a model that distinguishes between samples from the model and the data distributions, it follows that these distributions are not entirely similar. The convergence process of the GAN-like discriminator (Arjovsky et al., 2017; Binkowski et al. ´ , 2018) lends itself to creating a family of metrics based on training a discriminative classifier (Im et al., 2018). Still, training a separate critic model is often computationally prohibitive and requires careful specification. Besides, if the critic model is a neural network, the resulting metric lacks interpretability and training stability.
+
+Many advanced GAN models such as Wasserstein, MMD, Sobolev and Spherical GANs impose different constraints on the function class so as to stabilize training (Arjovsky et al., 2017; Binkowski ´ et al., 2018; Mroueh et al., 2018; Park & Kwon, 2019). Higher-order moment matching (Binkowski ´ et al., 2018; Park & Kwon, 2019) enhances GAN performance, enabling GANs to capture multi-scale data properties, while multi-scale noise ameliorates GAN convergence problems (Jenni & Favaro, 2019). Still, no feasible multi-scale GAN evaluation metric has been proposed to date.
+
+Positional distribution comparison. In certain settings, it is acceptable to assign zero probability mass to the real data points (Odena et al., 2018). In effect, metrics that estimate a distribution’s location and dispersion provide useful input for generative model evaluations. For instance, the Fréchet Inception Distance (FID) (Heusel et al., 2017) computes the Wasserstein-2 (i.e., Fréchet) distance between distributions approximated with Gaussians, using only the estimated mean and covariance matrices; the Kernel Inception Distance (KID) (Binkowski et al. ´ , 2018) computes a polynomial kernel $\begin{array} { r } { \mathrm { k } ( x , y ) = ( \frac { 1 } { d } x ^ { \top } y + 1 ) ^ { 3 } } \end{array}$ and measures the associated Kernel Maximum Mean Discrepancy (kernel MMD). Unlike FID, KID has an unbiased estimator (Gretton et al., 2012; Binkowski et al. ´ , 2018). However, even while such methods, based on a limited number of moments, may be computationally inexpensive, they only provide a rudimentary characterization of distributions from a geometric viewpoint.
+
+Intrinsic geometric measures. The Geometry Score (Khrulkov & Oseledets, 2018) characterizes distributions in terms of their estimated persistent homology, which roughly corresponds to the number of holes in a manifold. Still, the Geometry Score assesses distributions merely in terms of their global geometry. In this work, we aim to provide a multi-scale geometric assessment.
+
+# 2.2 SIMILARITIES OF NEURAL NETWORK REPRESENTATIONS
+
+Learning how representations evolve during training or across initializations provides a pathway to the interpretability of neural networks (Raghu et al., 2017). Still, state-of-the-art methods for comparing representations of neural networks (Kornblith et al., 2019; Morcos et al., 2018; Wang et al., 2018) consider only linear projections. The intrinsic nature of IMD renders it appropriate for the task of comparing neural network representations, which can only rely on intrinsic information.
+
+Yin & Shen (2018) introduced the Pairwise Inner Product (PIP) loss, an unnormalized covariance error between sets, as a dissimilarity metric between word2vec embedding spaces with common vocabulary. We show in Section 4.2 how IMD is applicable to this comparison task too.
+
+# 3 MULTI-SCALE INTRINSIC DISTANCE
+
+At the core of deep learning lies the manifold hypothesis, which states that high-dimensional data, such as images or text, lie on a low-dimensional manifold (Narayanan & Mitter, 2010; Belkin & Niyogi, 2002; 2007). We aim to provide a theoretically motivated comparison of data manifolds based on rich intrinsic information. Our target measure should have the following properties:
+
+intrinsic – it is invariant to isometric transformations of the manifold, e.g. translations or rotations.
+
+multi-scale – it captures both local and global information.
+
+We expose our method starting out with heat kernels, which admit a notion of manifold metric and can be used to lower-bound the distance between manifolds.
+
+# 3.1 HEAT KERNELS ON MANIFOLDS AND GRAPHS
+
+Based on the heat equation, the heat kernel captures all the information about a manifold’s intrinsic geometry (Sun et al., 2009). Given the Laplace-Beltrami operator (LBO) $\Delta { _ { X } }$ on a manifold $\mathcal { X }$ , the heat equation is $\begin{array} { r } { \frac { \partial u } { \partial t } = \Delta _ { \mathcal { X } } u } \end{array}$ for $u : \mathbb { R } ^ { + } \times \bar { \mathcal { X } }  \mathbb { R } ^ { + }$ . A smooth function $u$ is a fundamental solution of the heat equation at point $x \in \mathcal { X }$ if $u$ satisfies both the heat equation and the Dirac condition $u ( t , x ^ { \prime } ) \to \delta ( \bar { x } ^ { \prime } - x )$ as $\hat { \ell } \to 0 ^ { + }$ . We assume the Dirichlet boundary condition $u ( t , x ) = 0$ for all $t$ and $x \in \partial \mathcal { X }$ . The heat kernel $\mathrm { k } \chi$ : $\mathcal { X } \times \mathcal { X } \times \mathbb { R } ^ { + }  \mathbb { R } _ { 0 } ^ { + }$ is the unique solution of the heat equation; while heat kernels can be defined on hyperbolic spaces and other exotic geometries, we restrict our exposition to Euclidean spaces $\chi = \mathbb { R } ^ { d }$ , on which the heat kernel is defined as:
+
+$$
+\operatorname { k } _ { \mathbb { R } ^ { d } } ( x , x ^ { \prime } , t ) = { \frac { 1 } { ( 4 \pi t ) ^ { d / 2 } } } \exp \left( - { \frac { \| x - x ^ { \prime } \| ^ { 2 } } { 4 t } } \right)
+$$
+
+$\mathcal { X }$ ing sub, where nifoand of ar $\mathbb { R } ^ { d }$ ,e he heat kernel admits the expansi-th eigenvalue and eigenvector of $\operatorname { k } _ { \mathcal { X } } ( x , x ^ { \prime } , t ) =$ $\textstyle \sum _ { i = 0 } ^ { \infty } e ^ { - \dot { \lambda _ { i } } t } \phi _ { i } ( x ) \phi _ { i } ( x ^ { \prime } )$ $\lambda _ { i }$ $\phi _ { i }$ $i$ $\Delta { _ { X } }$ $t \simeq 0 ^ { + }$ our purposes, the Heat kernel is multi-scale: for a local domain $\mathcal { D }$ with Dirichlet condition, the localized heat kernel $\mathrm { k } _ { \mathcal { D } } ( x , x ^ { \prime } , t )$ is a good approximation of $\mathrm { k } _ { \mathcal { X } } ( x , x ^ { \prime } , t )$ if either (i) $\mathcal { D }$ is arbitrarily small and $t$ is small enough, or (ii) $t$ is for arbitrarily large and $\mathcal { D }$ is big enough. Formally,
+
+Definition 1 Multi-scale property (Grigor’yan, 2006; Sun et al., 2009) (i) For any smooth and relatively compact domain $\begin{array} { r } { \mathcal { D } \subseteq \dot { \mathcal { X } } , \operatorname* { l i m } _ { t  0 } \mathrm { k } _ { \mathcal { D } } ( x , x ^ { \prime } , t ) = \mathrm { k } _ { \mathcal { X } } ( x , x ^ { \prime } , t ) } \end{array}$ (ii) For any $t \in \mathbb { R } ^ { + }$ and any $x , x ^ { \prime } \in \mathcal { D } _ { 1 }$ localized heat kernel $\mathrm { k } _ { \mathcal { D } _ { 1 } } ( x , x ^ { \prime } , t ) \leq \mathrm { k } _ { \mathcal { D } _ { 2 } } ( x , x ^ { \prime } , t )$ if $\mathcal { D } _ { 1 } \subseteq \mathcal { D } _ { 2 }$ . Moreover, if $\left\{ \mathcal { D } _ { n } \right\}$ is an expanding and exhausting sequence $\textstyle \bigcup _ { i = 1 } ^ { \infty } D _ { i } = \mathcal { X }$ and $\mathcal { D } _ { i - 1 } \subseteq \mathcal { D } _ { i }$ , then $\begin{array} { r } { \operatorname* { l i m } _ { i  \infty } \mathrm { k } _ { \mathcal { D } _ { i } } ( x , x ^ { \prime } , t ) = } \end{array}$ $\mathrm { k } _ { \mathcal { X } } ( x , x ^ { \prime } , t )$ for any $t$ .
+
+Heat kernels are also defined for graphs in terms of their Laplacian matrices. An undirected graph is a pair $G = ( V , E )$ , where $V = ( v _ { 1 } , \ldots , v _ { n } ) , n = | V |$ , is the set of vertices and $E \subseteq ( V \times V )$ the set of edges. The adjacency matrix of $G$ is a $n \times n$ matrix A having ${ \bf A } _ { i j } = 1$ if $( i , j ) \in E$ and $A _ { i j } = 0$ otherwise. The normalized graph Laplacian is the matrix $\pmb { \mathscr { L } } = \mathbf { I } - \mathbf { D } ^ { - \frac { 1 } { 2 } } \mathbf { A } \mathbf { D } ^ { - \frac { 1 } { 2 } }$ , where $\mathbf { D }$ is the diagonal matrix in which entry $\mathbf { D } _ { i i }$ holds the degree of node $i$ , i.e, $\begin{array} { r } { \mathbf { D } _ { i i } = \sum _ { j = 1 } ^ { n } \mathbf { A } _ { i j } } \end{array}$ . Since the Laplacian matrix is symmetric, its eigenvectors $\phi _ { 1 } , . . . , \phi _ { n }$ , are real and orthonormal. Thus, it is factorized as $\pmb { \mathcal { L } } = \pmb { \Phi } \pmb { \Lambda } \pmb { \Phi } ^ { \dagger }$ , where $\pmb { \Lambda }$ is a diagonal matrix with the sorted eigenvalues $\lambda _ { 1 } \leq . . . \leq \lambda _ { n }$ , and $\Phi$ is the orthonormal matrix $\Phi = \left( \phi _ { 1 } , \ldots , \phi _ { n } \right)$ having the eigenvectors of $\pmb { \mathcal { L } }$ as its columns. The heat kernel on a graph is also given by the solution to the heat equation on a graph, which requires an eigendecomposition of its Laplacian: $\begin{array} { r } { \mathbf { H } _ { t } = e ^ { - t \mathcal { L } } = \Phi e ^ { - t \Lambda } \Phi ^ { \dagger } = \sum _ { i } e ^ { - t \sum _ { i } } \dot { \phi _ { i } } \dot { \phi _ { i } ^ { \dagger } } } \end{array}$ .
+
+A useful invariant of the heat kernel is the heat kernel trace $\mathrm { h k t } _ { \mathcal X } : \mathcal X \times \mathbb R _ { 0 } ^ { + } \to \mathbb R _ { 0 } ^ { + }$ , defined by a diagonal restriction as $\begin{array} { r } { \mathrm { h k t } _ { \mathcal { X } } ( t ) = \int _ { \mathcal { X } } \mathrm { k } _ { \mathcal { X } } ( x , x , t ) d x = \sum _ { i = 0 } ^ { \infty } e ^ { - \lambda _ { i } t } } \end{array}$ 0  0or, in the discrete case, $\begin{array} { r } { \mathrm { h k t } \pmb { \mathscr { \mathbf { \rho } } } ( t ) = \mathrm { T r } ( \mathbf { H } _ { t } ) = \sum _ { i } e ^ { - t \lambda _ { i } } } \end{array}$ . Heat kernels traces (HKTs) have been successfully applied to the analysis of 3D shapes (Sun et al., 2009) and graphs (Tsitsulin et al., 2018). The HKT contains all the information in the graph’s spectrum, both local and global, as the eigenvalues $\lambda _ { i }$ can be inferred therefrom (Mémoli, 2011, Remark 4.8). For example, if there are $c$ connected components in the graph, then $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \mathrm { h k t } _ { \mathcal { L } } ( t ) = c } \end{array}$ .
+
+# 3.2 CONVERGENCE TO THE LAPLACE-BELTRAMI OPERATOR
+
+An important property of graph Laplacians is that it is possible to construct a graph among points sampled from a manifold $\mathcal { X }$ such that the spectral properties of its Laplacian resemble those of the Laplace-Beltrami operator on $\mathcal { X }$ . Belkin and Niyogi (Belkin & Niyogi, 2002) proposed such a construction, the point cloud Laplacian, which is used for dimensionality reduction in a technique called Laplacian eigenmaps. Convergence to the LBO has been proven for various definitions of the graph Laplacian, including the one we use (Belkin & Niyogi, 2007; Hein et al., 2007; Coifman & Lafon, 2006; Ting et al., 2010). We recite the convergence results for the point cloud Laplacian from Belkin & Niyogi (2007):
+
+Theorem 1 Let $\lambda _ { n , i } ^ { t _ { n } }$ and $\boldsymbol { \phi } _ { n , i } ^ { t _ { n } }$ be the $i ^ { \mathrm { t h } }$ eigenvalue and eigenvector, respectively, of the point cloud Laplacian $\pmb { \mathscr { L } } ^ { t _ { n } }$ ; let $\lambda _ { i }$ and $\phi _ { i }$ be the $i ^ { \mathrm { t h } }$ eigenvalue and eigenvector of the LBO $\Delta$ . Then, there exists $t _ { n } \to 0$ such that
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { l i m } _ { n \to \infty } \lambda _ { n , i } ^ { t _ { n } } = \lambda _ { i } } \\ { \displaystyle \operatorname* { l i m } _ { n \to \infty } \big \| \phi _ { n , i } ^ { t _ { n } } - \phi _ { i } \big \| _ { 2 } = 0 } \end{array}
+$$
+
+Still, the point cloud Laplacian involves the creation of an $\mathcal { O } ( n ^ { 2 } )$ matrix; for the sake of scalability, we use the $k$ -nearest-neighbours $( k \mathrm { N N } )$ graph by OR-construction (i.e., based on bidirectional $k \mathrm { N N }$ relationships among points), whose Laplacian converges to the LBO for data with sufficiently high intrinsic dimension (Ting et al., 2010). As for the choice of $k$ , a random geometric $k \mathrm { N N }$ graph is connected when $k \geq \log n / \log 7 \approx 0 . 5 1 3 9 \log n$ (Balister et al., 2005); $k = 5$ yields connected graphs for all sample sizes we tested.
+
+# 3.3 SPECTRAL GROMOV-WASSERSTEIN DISTANCE
+
+Even while it is a multi-scale metric on manifolds, the heat kernel can be spectrally approximated by finite graphs constructed from points sampled from these manifolds. In order to construct a metric between manifolds, Mémoli (2011) suggests an optimal-transport-theory-based “meta-distance”: a spectral definition of the Gromov-Wasserstein distance between Riemannian manifolds based on matching the heat kernels at all scales. The cost of matching a pair of points $( x , x ^ { \prime } )$ on manifold $\mathcal { M }$ to a pair of points $( y , y ^ { \prime } )$ on manifold $\mathcal { N }$ at scale $t$ is given by their heat kernels ${ \mathrm { k } } _ { \mathcal { M } } , { \mathrm { k } } _ { \mathcal { N } }$ :
+
+$$
+\Gamma ( x , y , x ^ { \prime } , y ^ { \prime } , t ) = \left| \mathrm { k } _ { \mathcal { M } } ( x , x ^ { \prime } , t ) - \mathrm { k } _ { \mathcal { N } } ( y , y ^ { \prime } , t ) \right| .
+$$
+
+The distance between the manifolds is then defined in terms of the infimal measure coupling
+
+$$
+d _ { \mathrm { G W } } ( \mathcal { M } , \mathcal { N } ) = \operatorname* { i n f } _ { \mu } \operatorname* { s u p } _ { t > 0 } e ^ { - 2 ( t + t ^ { - 1 } ) } \| \Gamma \| _ { L ^ { 2 } ( \mu \times \mu ) } ,
+$$
+
+where the infimum is sought over all measures $\mu$ on $\mathcal { M } \times \mathcal { N }$ marginalizing to the standard measures on $\mathcal { M }$ and $\mathcal { N }$ . For finite spaces, $\mu$ is a doubly-stochastic matrix. This distance is lower-bounded (Mémoli, 2011) in terms of the respective heat kernel traces as:
+
+$$
+d _ { \mathrm { G W } } ( \mathcal { M } , \mathcal { N } ) \geq \operatorname* { s u p } _ { t > 0 } e ^ { - 2 ( t + t ^ { - 1 } ) } \ \vert \mathrm { h k t } _ { \mathcal { M } } ( t ) - \mathrm { h k t } _ { \mathcal { N } } ( t ) \vert .
+$$
+
+This lower bound is the scaled $L _ { \infty }$ distance between the heat trace signatures $\mathrm { h k t } _ { \mathcal { M } }$ and $\mathrm { h k t } _ { \mathcal { N } }$ . The scaling factor $e ^ { - 2 ( t + t ^ { - 1 } ) }$ favors medium-scale differences, meaning that this lower bound is not sensitive to local perturbations. The maximum of the scaling factor occurs at $t = 1$ , and more than $1 - 1 0 ^ { - 8 }$ of the function mass lies between $t = 0 . 1$ and $t = 1 0$ .
+
+# 3.4 HEAT TRACE ESTIMATION
+
+Calculating the heat trace signature efficiently and accurately is a challenge on a large graph as it involves computing a trace of a large matrix exponential, i.e. $\operatorname { T r } ( e ^ { - t { \cal L } } )$ . A naive approach would be to use an eigendecomposition $\bar { \exp ( - t \pmb { \mathcal { L } } ) } = \bar { \Phi } \exp ( - t \pmb { \Lambda } ) \Phi ^ { \top }$ , which is infeasible for large $n$ . Recent work (Tsitsulin et al., 2018) suggested using either truncated Taylor expansion or linear interpolation of the interloping eigenvalues, however, both techniques are quite coarse. To combine accuracy and speed, we use the Stochastic Lanczos Quadrature (SLQ) (Ubaru et al., 2017; Golub & Meurant, 2009). This method combines the Hutchinson trace estimator (Hutchinson, 1989; Adams et al., 2018) and the Lanczos algorithm for eigenvalues. We aim to estimate the trace of a matrix function with a Hutchinson estimator:
+
+$$
+\mathrm { T r } ( \mathbf { \boldsymbol { f } } ( \mathbf { \mathcal { L } } ) ) = \mathbb { E } _ { p ( \mathbf { v } ) } ( \mathbf { v } ^ { \top } \mathbf { \boldsymbol { f } } ( \mathbf { \mathcal { L } } ) \mathbf { v } ) \approx \frac { n } { n _ { v } } \sum _ { i = 1 } ^ { n _ { v } } \mathbf { v } _ { i } ^ { \top } \mathbf { \boldsymbol { f } } ( \mathbf { \mathcal { L } } ) \mathbf { v } _ { i } ,
+$$
+
+where the function of interest $f ( \cdot ) = \exp ( \cdot )$ and $\mathbf { v } _ { i }$ are $n _ { v }$ random vectors drawn from a distribution $p ( \mathbf { v } )$ with zero mean and unit variance. A typical choice for $p ( \mathbf { v } )$ is Rademacher or a standard normal distribution. In practice, there is little difference, although in theory Rademacher has less variance, but Gaussian requires less random vectors (Avron & Toledo, 2011).
+
+To estimate the quadratic form ${ \bf v } _ { i } ^ { \top } f ( \pmb { \cal L } ) { \bf v } _ { i }$ in (3) with a symmetric real-valued matrix $\mathcal { L }$ and a smooth function $f$ , we plug in the eigendecomposition ${ \mathcal { L } } = \Phi \Lambda \Phi ^ { \top }$ , rewrite the outcome as a Riemann-Stieltjes integral and apply the $m$ -point Gauss quadrature rule (Golub & Welsch, 1969):
+
+$$
+\mathbf { v } _ { i } ^ { \top } f ( \mathcal { L } ) \mathbf { v } _ { i } = \mathbf { v } _ { i } ^ { \top } \Phi f ( \Lambda ) \Phi ^ { \top } \mathbf { v } _ { i } = \sum _ { j = 1 } ^ { n } f ( \lambda _ { j } ) \mu _ { j } ^ { 2 } = \int _ { a } ^ { b } f ( t ) d \mu ( t ) \approx \sum _ { k = 1 } ^ { m } \omega _ { k } f ( \theta _ { k } ) ,
+$$
+
+where $\mu _ { j } = [ \boldsymbol { \Phi } ^ { \top } \mathbf { v } _ { i } ] _ { j }$ and $\mu ( t )$ is a piecewise constant function defined as follows
+
+$$
+\mu ( t ) = { \left\{ \begin{array} { l l } { 0 , } & { { \mathrm { i f ~ } } t < a = \lambda _ { n } } \\ { \sum _ { j = 1 } ^ { i } \mu _ { j } ^ { 2 } , } & { { \mathrm { i f ~ } } \lambda _ { i } \leq t < \lambda _ { i - 1 } } \\ { \sum _ { j = 1 } ^ { n } \mu _ { j } ^ { 2 } , } & { { \mathrm { i f ~ } } b = \lambda _ { 1 } \leq t } \end{array} \right. }
+$$
+
+and $\theta _ { k }$ are the quadrature’s nodes and $\omega _ { k }$ are the corresponding weights. We obtain $\omega _ { k }$ and $\theta _ { k }$ with the $m$ -step Lanczos algorithm (Golub & Meurant, 2009), which we describe succinctly.
+
+Given the symmetric matrix $\mathcal { L }$ and an arbitrary starting unit-vector ${ \bf q } _ { 0 }$ , the $m$ -step Lanczos algorithm computes an $n \times m$ matrix $\mathbf { Q } = [ \mathbf { q } _ { 0 } , \mathbf { q } _ { 1 } , \dots , \mathbf { q } _ { m - 1 } ]$ with orthogonal columns and an $m \times m$ tridiagonal symmetric matrix $\mathbf { T }$ , such that $\mathbf { Q } ^ { \top } \mathcal { L } \mathbf { Q } = \mathbf { T }$ . The columns of $\mathbf { Q }$ constitute an orthonormal basis for the Krylov subspace $\kappa$ that spans vectors $\{ \mathbf { q } _ { 0 } , \pmb { \mathcal { L } } \mathbf { q } _ { 0 } , \dots , \pmb { \mathcal { L } } ^ { m - 1 } \mathbf { q } _ { 0 } \}$ ; each $\mathbf { q } _ { i }$ vector is given as a polynomial in $\mathcal { L }$ applied to the initial vector ${ \bf q } _ { 0 }$ : $\mathbf { q } _ { i } = p _ { i } ( \pmb { \mathscr { L } } ) \mathbf { q } _ { 0 }$ . These Lanczos polynomials are orthogonal with respect to the integral measure $\mu ( t )$ . As orthogonal polynomials satisfy the three term recurrence relation, we obtain $p _ { k + 1 }$ as a combination of $p _ { k }$ and $p _ { k - 1 }$ . The tridiagonal matrix storing the coefficients of such combinations, called the Jacobi matrix $\mathbf { J }$ , is exactly the tridiagonal symmetric matrix $\mathbf { T }$ . A classic result tells us that the nodes $\theta _ { k }$ and the weights $\omega _ { k }$ of the Gauss quadrature are the eigenvalues of $\mathbf { T }$ , $\lambda _ { k }$ , and the squared first components of its normalized eigenvectors, $\tau _ { k } ^ { 2 }$ , respectively (see Golub & Welsch (1969); Wilf (1962); Golub $\&$ Meurant (2009)). Thereby, setting $\mathbf { q } _ { 0 } = \mathbf { v } _ { i }$ , the estimate for the quadratic form becomes:
+
+$$
+\mathbf { v } _ { i } ^ { \top } f ( \mathcal { L } ) \mathbf { v } _ { i } \approx \sum _ { k = 1 } ^ { m } \tau _ { k } ^ { 2 } f ( \lambda _ { k } ) , \quad \tau _ { k } = \mathbf { U } _ { 0 , k } = \mathbf { e } _ { 1 } ^ { \top } \mathbf { u } _ { k } , \quad \lambda _ { k } = \Lambda _ { k , k } \quad \mathbf { T } = \mathbf { U } \Lambda \mathbf { U } ^ { \top } ,
+$$
+
+Applying (5) over $n _ { v }$ random vectors in the Hutchinson trace estimator (3) yields the SLQ estimate:
+
+$$
+\mathrm { T r } ( f ( \mathcal { L } ) ) \approx \frac { n } { n _ { v } } \sum _ { i = 1 } ^ { n _ { v } } \left( \sum _ { k = 0 } ^ { m } \left( \tau _ { k } ^ { i } \right) ^ { 2 } f \left( \lambda _ { k } ^ { i } \right) \right) = \Gamma .
+$$
+
+We derive error bounds for the estimator based on the Lanczos approximation of the matrix exponential, and show that even a few Lanczos steps, i.e., $m = 1 0$ , are sufficient for an accurate approximation of the quadratic form. However, the trace estimation error is theoretically dominated by the error of the Hutchinson estimator, e.g. for Gaussian $p ( \mathbf { v } )$ the bound on the number of samples to guarantee that the probability of the relative error exceeding $\epsilon$ is at most $\delta$ is $8 \epsilon ^ { - 2 } \ln ( 2 / \delta )$ (Roosta-Khorasani & Ascher, 2015). Although, in practice, we observe performance much better than the bound suggests. Hutchinson error implies nearing accuracy roughly $1 0 ^ { - 2 }$ with $n _ { v } \ge 1 0 \mathrm { k }$ random vectors, however, with as much as $n _ { v } = 1 0 0$ the error is already $\bar { 1 0 } ^ { - 3 }$ . Thus, we use default values of $m = 1 0$ and $n _ { v } = 1 0 0$ in all experiments in Section 4. Please see Appendix A for full derivations and figures.
+
+![](images/e4d7ebfbb912d062d5e1d08725ce648f18c54c58d7ee9dbf66ee461bb6d56877.jpg)  
+Figure 2: (a) IMD distances between language pairs for unaligned Wikipedia word embeddings and (b) distances from the simple English Wikipedia visualized for IMD, FID, and KID. We consider 16 languages: Polish, Russian, Greek, Hungarian, Turkish, Arabic, Hebrew, English, Simple English, Swedish, German, Spanish, Dutch, Portugese, Vietnamese, and Waray-Waray.
+
+# 3.5 PUTTING IMD TOGETHER
+
+We employ the heretofore described advances in differential geometry and numerical linear algebra to create IMD (Multi-Scale Intrinsic Distance), a fast, intrinsic method to lower-bound the spectral Gromov-Wasserstein distance between manifolds.
+
+We describe the overall computation of IMD in Algorithm 1. Given data samples in $\mathbb { R } ^ { d }$ , we build a $k \mathrm { N N }$ graph $G$ by OR-construction such that its Laplacian spectrum approximates the one of the Laplace-Beltrami operator of the underlying manifold (Ting et al., 2010), and then compute $\begin{array} { r c l } { \mathrm { h k t } _ { G } ( t ) } & { = } & { \sum _ { i } e ^ { - \lambda _ { i } t } ~ \approx ~ \Gamma } \end{array}$ . We compare heat traces in the spirit of Equation (2), i.e., $\left| \mathrm { h k t } _ { G _ { 1 } } ( t ) - \mathrm { h k t } _ { G _ { 2 } } ( t ) \right|$ for $t \in ( 0 . 1 , 1 0 )$ sampled from a logarithmically spaced grid.
+
+<table><tr><td>Algorithm1IMD algorithm.</td><td></td></tr><tr><td>function IMDESC(X)</td><td rowspan="3"></td></tr><tr><td>G ←kNN(X)</td></tr><tr><td>L ← Laplacian(G) return Γ = slq(L,s,nv)</td></tr><tr><td></td><td rowspan="3"></td></tr><tr><td>function IMDIsT(X,Y)</td></tr><tr><td>hktx ←IMDist(X)</td></tr><tr><td>hkty ← IMDist(Y)</td><td rowspan="2"></td></tr><tr><td>return supe-2(t+t-1)|nkt x - hktγl</td></tr></table>
+
+Constructing exact $k \mathrm { N N }$ graphs is an $\mathcal { O } ( d n ^ { 2 } )$ operation; however, approximation algorithms take near-linear time $\mathcal { O } ( d n ^ { 1 + \omega } )$ (Dong et al., 2011; Aumüller et al., 2019). In practice, with approximate $k \mathrm { N N }$ graph construction (Dong et al., 2011), computational time is low while result variance is similar to the exact case. The $m$ -step Lanczos algorithm on a sparse $n \times n$ $k \mathrm { N N }$ Laplacian $\pmb { \mathcal { L } }$ with one starting vector has $\mathcal { O } ( k n m )$ complexity, where $k n$ is the number of nonzero elements in $\mathcal { L }$ . The symmetric tridiagonal matrix eigendecomposition incurs an additional $\mathcal { O } ( m \log m )$ (Coakley & Rokhlin, 2013). We apply this algorithm over $n _ { v }$ starting vectors, yielding a complexity of $\mathcal { O } ( n _ { v } ( m \log m + k m n ) )$ ), with constant $k = 5$ and $m = 1 0$ by default. In effect, IMD’s time complexity stands between those of two common GAN evaluation methods: KID, which is $\mathcal { O } ( d n ^ { 2 } )$ and FID, which is $\mathcal { O } ( d ^ { 3 } + d n )$ . The time complexity of Geometry Score is unspecified in Khrulkov & Oseledets (2018), yet in Section 4.6 we show that its runtime grows exponentially in sample size.
+
+# 4 EXPERIMENTS
+
+We evaluate IMD on the ability to compare intermediate representations of machine learning models. For instance, in a recommender system we could detect whether a problem is related to the representation or the the classifier in the end of a pipeline. In this section, we show the effectiveness of our intrinsic measure on multiple tasks and show how our intrinsic distance can provide insights beyond previously proposed extrinsic measures.
+
+Summary of experiments. We examine the ability of $\mathrm { I M D } ^ { 1 }$ to measure several aspects of difference among data manifolds. We first consider a task from unsupervised machine translation with unaligned word embeddings and show that IMD captures correlations among language kinship (affinity or genealogical relationships). Second, we showcase how IMD handles data coming from data sources of unequal dimensionalities. Third, we study how IMD highlights differences among image data representations across initializations and through training process of neural networks.
+
+The problem of unaligned representations is particularly severe in the domain of natural language processing as the vocabulary is rarely comparable across different languages or even different documents. We employ IMD to measure the relative closeness of pairs of languages based on the word embeddings with different vocabularies. Figure 2 (a) shows a heatmap of pairwise IMD scores. IMD detects similar languages (Slavic, Semitic, Romanic, etc.) despite the lack of ground truth vocabulary alignment. On the other hand, Figure 11 in Appendix C shows that FID and KID, are not able to distinguish the intrinsic language-specific structure in word embeddings. Detailed description and setting of the experiment can be found in Appendix C.
+
+# 4.2 OPTIMIZING DIMENSIONALITY OF WORD EMBEDDINGS
+
+Comparing data having different dimensionality is cumbersome, even when representations are aligned. We juxtapose IMD by PIP loss (Yin & Shen, 2018) which allows the comparison of aligned representations for word embeddings. To this end, we measure IMD distance between English word embeddings of varying dimensions. Figure 3 shows the heatmap of the scores between sets of word vectors of different dimensionalities. Closer dimensionalities have lower distance scores for both metrics. However, IMD better highlights gradual change of the size of word
+
+vectors, e.g. word vectors of size 4 and 8 are clearly closer to each other than embeddings of size 4 and 16 in terms of IMD, which is not true for PIP.
+
+![](images/455c09e751b2500b5948333160b0de61d855227c8a4f2b738d7108e5ba699ec9.jpg)  
+Figure 3: Comparison of IMD and PIP loss on word embeddings of different dimension. IMD detects subtle changes in the dimensionality.
+
+# 4.3 TRACKING THE EVOLUTION OF IMAGE MANIFOLDS
+
+Next, we employ IMD to inspect the internal dynamics of neural networks. We investigate the stability of output layer manifolds across random initializations. We train 10 instances of the VGG-16 (Simonyan & Zisserman, 2015) network using different weight initializations on the CIFAR-10 and CIFAR-100 datasets. We compare the average IMD scores across representations in each network layer relative to the last layer. As Figure 4 (left) shows, for both CIFAR-10 and CIFAR-100, the convolutional layers exhibit similar behavior; IMD shows that consequent layers do not monotonically contribute to the separation of image representations, but start to do so after initial feature extraction stage comprised of 4 convolutional blocks. A low variance across the 10 networks trained from different random initializations indicates stability in the network structure.
+
+![](images/6c29d218e78ae1ac13c2f82082f72be477ed741c9cbed22488ed82bcdb206539.jpg)  
+Figure 4: (left) IMD score across convolutional layers of the VGG-16 network on CIFAR-10 and CIFAR-100 datasets; (right) training progression in terms of accuracy (dotted) and IMD (solid) on CIFAR-10 and CIFAR-100 datasets for VGG-16 and ResNet-20, with respect to VGG-16.
+
+Table 1: IMD agrees with KID and FID across varying datasets for GAN evaluation.   
+
+<table><tr><td></td><td colspan="2">MNIST</td><td colspan="2">FashionMNIST</td><td colspan="2">CIFAR10</td><td colspan="2">CelebA</td></tr><tr><td>Metric</td><td>WGAN</td><td>WGAN-GP</td><td>WGAN</td><td>WGAN-GP</td><td>WGAN</td><td>WGAN-GP</td><td>WGAN</td><td>WGAN-GP</td></tr><tr><td>IMD</td><td>57.74± 0.47</td><td>10.77± 0.42</td><td>118.14± 0.52</td><td>13.45± 0.54</td><td>18.10± 0.36</td><td>10.84± 0.42</td><td>10.11± 0.33</td><td>2.84± 0.31</td></tr><tr><td></td><td>KID ×10 47.26 ± 0.07</td><td>5.53± 0.03</td><td>119.93± 0.14</td><td>25.49± 0.07</td><td>93.89± 0.09</td><td>59.59 ± 0.09</td><td>217.28± 0.14</td><td>92.71± 0.08</td></tr><tr><td>FID</td><td>31.75 ± 0.07</td><td>8.95± 0.03</td><td>152.44± 0.12</td><td>35.31± 0.07</td><td>101.43 ± 0.09</td><td>80.65 ± 0.09</td><td>205.63± 0.09</td><td>85.55± 0.08</td></tr></table>
+
+We now examine the last network layers during training with different initializations. Figure 4 (right) plots the VGG16 validation errors and IMD scores relative to the final layer representations of two pretrained networks, VGG16 itself with last layer dimension $d = 5 1 2$ and ResNet-20 with $d = 6 4$ and ${ \sim } 5 0$ times less parameters. We observe that even in such unaligned spaces, IMD correctly identifies the convergence point of the networks. Surprisingly, we find that, in terms of IMD, VGG-16 representations progress towards not only the VGG-16 final layer, but the ResNet-20 final layer representation as well; this result suggests that these networks of distinct architectures share similar final structures.
+
+# 4.4 EVALUATING GENERATIVE MODELS
+
+We now move on to apply IMD to evaluation of generative models. First, we evaluate the sensitivity of IMD, FID, and KID to simple image transformations as a proxy to more intricate artifacts of modern generative models. We progressively blur images from the CIFAR-10 training set, and measure the distance to the original data manifold, averaging outcomes over 100 subsamples of 10k images each. To enable comparison across methods, we normalize each distance measure such that the distance between CIFAR-10 and MNIST is 1. Figure 5 reports the results at different levels $\sigma$ of Gaussian blur. We additionally report the normalized distance to the CIFAR-100 training set (dashed lines ). FID and KID quickly drift away from the original distribution and match MNIST, a dataset of a completely different nature. Contrariwise, IMD is more robust to noise and follows the datasets structure, as the relationships between objects remain mostly unaffected on low blur levels. Moreover, with both FID and KID, low noise $( \sigma = 1 )$ ) applied to CIFAR-10 suffices to exceed the distance of CIFAR-100, which is similar to CIFAR-10. IMD is much more robust, exceeding that distance only with $\sigma = 2$ .
+
+![](images/0c85003d35bd956205ddd365da52cfc894beee86b1e462604fc42ad3102b36ff.jpg)  
+Figure 5: FID, KID and IMD on the CIFAR-10 dataset with Gaussian blur.
+
+Next, we turn our attention to the sample-based evaluation of generative models. We then train the WGAN (Arjovsky et al., 2017) and WGAN-GP (Gulrajani et al., 2017) models on four datasets: MNIST, FashionMNIST, CIFAR10 and CelebA. We sample 10k samples, $\mathbf { Y }$ , from each GAN. We then uniformly subsample $1 0 k$ images from the corresponding original dataset, X, and compute the IMD, KID and FID scores between $\mathbf { X }$ and $\mathbf { Y }$ . Table 1 reports the average measure and its $9 9 \%$ confidence interval across 100 runs. IMD, as well as both FID and KID, reflect the fact that WGAN-GP is a more expressive model. We provide details on architecture, training, and generated samples in Appendix C. Additionally, in Appendix C we demonstrate superiority of IMD on synthetic data.
+
+![](images/99f37093d52640c71496dc7b6670f153aff34843393316867c6a85f50ce3e734.jpg)  
+Figure 6: Plotting the normalized heat trace allows interpretation of medium- and global-scale structure of datasets. Best viewed in color.
+
+![](images/d82cc328d51145b942a28d0e6ca3812063375a5955f61edeef6224a4197a7633.jpg)  
+Figure 7: Stability and scalability experiment: (left) stability of FID, KID and IMD wrt. sample size on CIFAR-10 and CIFAR-100 dataset; (right) scalability of FID, KID and IMD wrt. sample size on synthetic datasets.
+
+# 4.5 INTERPRETING IMD
+
+To understand how IMD operates, we investigate the behavior of heat kernel traces of different datasets that are normalized by a null model. Tsitsulin et al. (2018) proposed a normalization by the heat kernel trace of an empty graph, which amounts to taking the average, rather than the sum, of the original heat kernel diagonal. However, this normalization is not an appropriate null model as it ignores graph connectivity. We propose a heat kernel normalization by the expected heat kernel of an Erdos-Rényi graph (further details in the Appendix ˝ D).
+
+Figure 6 depicts the obtained normalized $\operatorname { h k t } _ { g }$ for all datasets we work with. We average results over 100 subsamples of $1 0 k$ images each. For $t = 1 0$ , i.e., at a medium scale, CelebA is most different from the random graph, while for large-scale $t$ values, which capture global community structure, $\frac { \mathrm { d h k t } _ { g } ( t ) } { \mathrm { d } t }$ reflects the approximate number of clusters in the data. Surprisingly, CIFAR-100 comes close to CIFAR-10 for large $t$ values; we have found that this is due to the fact that the pre-trained Inception network does not separate the CIFAR-100 data classes well enough. We conclude that the heat kernel trace is interpretable if we normalize it with an appropriate null model.
+
+# 4.6 VERIFYING STABILITY AND SCALABILITY OF IMD
+
+In addition to the complexity analysis in Section 3.5, we assess the scaling and sample stability of IMD. Since IMD, like FID, is a lower bound to an optimal transport-based metric, we cannot hope for an unbiased estimator. However, we empirically verify, in Figure 7 (left), that IMD does not diverge too much with increased sample size. Most remarkably, we observe that IMD with approximate $k \mathrm { N N }$ (Dong et al., 2011) does not induce additional variance, while it diverges slightly further than the exact version as the number of samples grows.
+
+In terms of scalability, Figure 7 (right) shows that the theoretical complexity is supported in practice. Using approximate $k \mathrm { N N }$ , we break the $\mathcal { O } ( n ^ { 2 } )$ performance of KID. FID’s time complexity appears constant, as its runtime is dominated by the $\mathcal { O } ( d ^ { 3 } )$ matrix square root operation. Geometry score (GS) fails to perform scalably, as its runtime grows exponentially. Due to this prohibitive computational cost, we eschew other comparison with GS. Furthermore, as IMD distance is computed through a low-dimensional heat trace representation of the manifold, we can store HKT for future comparisons, thereby enhancing performance in the case of many-to-many comparisons.
+
+# 5 DISCUSSION AND FUTURE WORK
+
+We introduced IMD, a geometry-grounded, first-of-its-kind intrinsic multi-scale method for comparing unaligned manifolds, which we approximate efficiently with guarantees, utilizing the Stochastic Lanczos Quadrature. We have shown the expressiveness of IMD in quantifying the change of data representations in NLP and image processing, evaluating generative models, and in the study of neural network representations. Since IMD allows comparing diverse manifolds, its applicability is not limited to the tasks we have evaluated, while it paves the way to the development of even more expressive techniques founded on geometric insights.
+
+# ACKNOWLEDGEMENTS
+
+This work was partially funded by the Ministry of Science and Education of Russian Federation as a part of Mega Grant Research Project 14.756.31.0001. Ivan Oseledets would like to thank Huawei for the support of his research.
+
+# REFERENCES
+
+Ryan P Adams, Jeffrey Pennington, Matthew J Johnson, Jamie Smith, Yaniv Ovadia, Brian Patton, and James Saunderson. Estimating the spectral density of large implicit matrices. arXiv preprint arXiv:1802.03451, 2018. 5
+
+Alexander A. Alemi and Ian Fischer. GILBO: One metric to measure them all. In NeurIPS, 2018. 2
+
+Martín Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In ICML, 2017. 1, 2, 8
+
+Martin Aumüller, Erik Bernhardsson, and Alexander Faithfull. ANN-benchmarks: A benchmarking tool for approximate nearest neighbor algorithms. Information Systems, 2019. 6
+
+Haim Avron and Sivan Toledo. Randomized algorithms for estimating the trace of an implicit symmetric positive semi-definite matrix. Journal of the ACM (JACM), 2011. 5
+
+Paul Balister, Béla Bollobás, Amites Sarkar, and Mark Walters. Connectivity of random k-nearest-neighbour graphs. Advances in Applied Probability, 2005. 4
+
+Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps and spectral techniques for embedding and clustering. In NIPS, 2002. 2, 3, 4
+
+Mikhail Belkin and Partha Niyogi. Convergence of laplacian eigenmaps. In NIPS, pp. 129–136, 2007. 3, 4
+
+Mikołaj Binkowski, Dougal J Sutherland, Michael Arbel, and Arthur Gretton. Demystifying MMD ´ gans. In ICLR, 2018. 1, 2
+
+Ali Borji. Pros and cons of gan evaluation measures. Computer Vision and Image Understanding, 179:41–65, 2019. 2
+
+Fan Chung, Linyuan Lu, and Van Vu. The spectra of random graphs with given expected degrees. Internet Mathematics, 2004. 16
+
+Ed S. Coakley and Vladimir Rokhlin. A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices. Applied and Computational Harmonic Analysis, 34(3): 379 – 414, 2013. 6
+
+Ronald R Coifman and Stéphane Lafon. Diffusion maps. Applied and computational harmonic analysis, 2006. 4
+
+Amin Coja-Oghlan. On the laplacian eigenvalues of $\mathtt { g } ( \mathtt { n } , \mathtt { p } )$ . Combinatorics, Probability and Computing, 2007. 15
+
+Wei Dong, Charikar Moses, and Kai Li. Efficient k-nearest neighbor graph construction for generic similarity measures. In WWW, 2011. 6, 9
+
+Gene H Golub and Gérard Meurant. Matrices, moments and quadrature with applications. Princeton University Press, 2009. 5
+
+Gene H Golub and John H Welsch. Calculation of gauss quadrature rules. Mathematics of computation, 1969. 5
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In NIPS, pp. 2672–2680, 2014. 2
+
+Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Schölkopf, and Alexander J. Smola. A kernel two-sample test. Journal of Machine Learning Research, 13:723–773, 2012. 2
+
+Alexander Grigor’yan. Heat kernels on weighted manifolds and applications. In Contemp. Math., pp. 93–191. AMS, 2006. 3
+
+Ishaan Gulrajani, Faruk Ahmed, Martín Arjovsky, Vincent Dumoulin, and Aaron C. Courville. Improved training of wasserstein GANs. In NIPS, 2017. 8
+
+Matthias Hein, Jean-Yves Audibert, and Ulrike von Luxburg. Graph laplacians and their convergence on random neighborhood graphs. Journal of Machine Learning Research, 2007. 4
+
+Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. GANs trained by a two time-scale update rule converge to a local nash equilibrium. In NIPS, 2017. 1, 2
+
+Marlis Hochbruck and Christian Lubich. On krylov subspace approximations to the matrix exponential operator. SIAM Journal on Numerical Analysis, 1997. 13
+
+MF Hutchinson. A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines. Communications in Statistics-Simulation and Computation, 18(3):1059–1076, 1989. 5
+
+Daniel Jiwoong Im, He Ma, Graham Taylor, and Kristin Branson. Quantitatively evaluating GANs with divergences proposed for training. In ICLR, 2018. 2
+
+Simon Jenni and Paolo Favaro. On stabilizing generative adversarial training with noise. In CVPR, 2019. 2
+
+Valentin Khrulkov and Ivan V. Oseledets. Geometry score: A method for comparing generative adversarial networks. In ICML, 2018. 2, 6
+
+Simon Kornblith, Mohammad Norouzi, Honglak Lee, and Geoffrey Hinton. Similarity of neural network representations revisited. In ICML, 2019. 2
+
+Chun-Liang Li, Wei-Cheng Chang, Yu Cheng, Yiming Yang, and Barnabás Póczos. MMD GAN: Towards deeper understanding of moment matching network. In NIPS, 2017. 1
+
+David Lopez-Paz and Maxime Oquab. Revisiting classifier two-sample tests. In ICLR, 2017. 2
+
+Facundo Mémoli. A spectral notion of gromov–wasserstein distance and related methods. Applied and Computational Harmonic Analysis, 2011. 2, 3, 4
+
+Ari Morcos, Maithra Raghu, and Samy Bengio. Insights on representational similarity in neural networks with canonical correlation. In NeurIPS, 2018. 1, 2
+
+Youssef Mroueh, Chun-Liang Li, Tom Sercu, Anant Raj, and Yu Cheng. Sobolev GAN. In ICLR, 2018. 2
+
+Hariharan Narayanan and Sanjoy Mitter. Sample complexity of testing the manifold hypothesis. In NIPS, 2010. 3
+
+Augustus Odena, Jacob Buckman, Catherine Olsson, Tom B. Brown, Christopher Olah, Colin A. Raffel, and Ian J. Goodfellow. Is generator conditioning causally related to GAN performance? In ICML, 2018. 2
+
+Sung Woo Park and Junseok Kwon. Sphere generative adversarial network based on geometric moment matching. In CVPR, 2019. 1, 2
+
+Maithra Raghu, Justin Gilmer, Jason Yosinski, and Jascha Sohl-Dickstein. SVCCA: Singular vector canonical correlation analysis for deep learning dynamics and interpretability. In NIPS, 2017. 2
+
+Radim Reh ˚u ˇ ˇrek and Petr Sojka. Software Framework for Topic Modelling with Large Corpora. In Proceedings of the LREC 2010 Workshop on New Challenges for NLP Frameworks, 2010. 15
+
+Farbod Roosta-Khorasani and Uri Ascher. Improved bounds on sample size for implicit matrix trace estimators. Foundations of Computational Mathematics, 2015. 5, 14
+
+Tim Salimans, Ian J. Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training GANs. In NIPS, 2016. 2
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. CoRR, abs/1409.1556, 2015. 7
+
+Jian Sun, Maks Ovsjanikov, and Leonidas Guibas. A concise and provably informative multi-scale signature based on heat diffusion. In Computer graphics forum. Wiley Online Library, 2009. 3
+
+Daniel Ting, Ling Huang, and Michael Jordan. An analysis of the convergence of graph laplacians. In ICML, 2010. 4, 6
+
+Anton Tsitsulin, Davide Mottin, Panagiotis Karras, Alexander M. Bronstein, and Emmanuel Müller. NetLSD: Hearing the shape of a graph. In KDD, 2018. 3, 4, 9, 16
+
+Shashanka Ubaru, Jie Chen, and Yousef Saad. Fast estimation of tr(f(a)) via stochastic lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 2017. 5, 13
+
+Liwei Wang, Lunjia Hu, Jiayuan Gu, Zhiqiang Hu, Yue Wu, Kun He, and John Hopcroft. Towards understanding learning representations: To what extent do different neural networks learn the same representation. In Advances in Neural Information Processing Systems, 2018. 1, 2
+
+Herbert S Wilf. Mathematics for the physical sciences. 1962. 5
+
+Zi Yin and Yuanyuan Shen. On the dimensionality of word embedding. In Advances in Neural Information Processing Systems, pp. 887–898, 2018. 2, 7
+
+# APPENDIX
+
+# A TRACE ESTIMATION ERROR BOUNDS
+
+We will use the error of the Lanczos approximation of the action of the matrix exponential $f ( \pmb { \mathcal { L } } ) \mathbf { v } = \exp ^ { - t \pmb { \mathcal { L } } } \mathbf { v }$ to estimate the error of the trace. We first rewrite quadratic form under summation in the trace approximation to a convenient form,
+
+$$
+\mathbf { v } ^ { \top } f ( \mathcal { L } ) \mathbf { v } \approx \sum _ { k = 0 } ^ { m } \tau _ { k } ^ { 2 } f ( \lambda _ { k } ) = \sum _ { k = 0 } ^ { m } [ \mathbf { e } _ { 1 } ^ { \top } \mathbf { u } _ { k } ] ^ { 2 } f ( \lambda _ { k } ) = \mathbf { e } _ { 1 } ^ { \top } \mathbf { U } f ( \Lambda ) \mathbf { U } ^ { \top } \mathbf { e } _ { 1 } = \mathbf { e } _ { 1 } ^ { \top } f ( \mathbf { T } ) \mathbf { e } _ { 1 } .
+$$
+
+Because the Krylov subspace ${ \cal { K } } _ { m } ( { \cal { L } } , { \bf { v } } )$ is built on top of vector $\mathbf { v }$ with $\mathbf { Q }$ as an orthogonal basis of ${ \cal { K } } _ { m } ( { \cal { L } } , { \bf { v } } )$ , i.e. $\mathbf { q } _ { 0 } = \mathbf { v }$ and $\mathbf { v } \perp \mathbf { q } _ { i }$ for $i \in ( 1 , \ldots , m - 1 )$ , the following holds
+
+$$
+\mathbf { v } ^ { \top } f ( \pmb { \mathscr { L } } ) \mathbf { v } \approx \mathbf { v } ^ { \top } \mathbf { Q } f ( \mathbf { T } ) \mathbf { e } _ { 1 } = \mathbf { e } _ { 1 } ^ { \top } f ( \mathbf { T } ) \mathbf { e } _ { 1 } .
+$$
+
+Thus, the error in quadratic form estimate ${ \mathbf { v } } ^ { \top } f ( \pmb { \mathscr { L } } ) { \mathbf { v } }$ is exactly the error of Lanczos approximation $f ( { \pmb { \mathscr { L } } } ) \mathbf { v } \approx \mathbf { Q } f ( \mathbf { T } ) { \dot { \mathbf { e } } } _ { 1 }$ . To obtain the error bounds, we use the Theorem 2 in Hochbruck & Lubich (1997), which we recite below.
+
+Theorem 2 Let $\pmb { \mathcal { L } }$ be a real symmetric positive semi-definite matrix with eigenvalues in the interval $[ 0 , 4 \rho ]$ . Then the error in the $m$ -step Lanczos approximation of $\exp ^ { - t \mathcal { L } } \mathbf { v }$ , i.e. $\epsilon _ { m } = \Vert \exp ^ { - t \mathcal { L } } \mathbf { v } - \mathbf { Q } _ { m } \exp ^ { - t \mathbf { T } _ { m } } \mathbf { e } _ { 1 } \Vert$ , is bounded in the following ways:
+
+$$
+\epsilon _ { m } \leq \left\{ \begin{array} { l l } { 1 0 e ^ { - m ^ { 2 } / ( 5 \rho t ) } , } & { \quad \sqrt { 4 \rho t } \leq m \leq 2 \rho t } \\ { 1 0 ( \rho t ) ^ { - 1 } e ^ { - \rho t } \Bigl ( \displaystyle \frac { e \rho t } { m } \Bigr ) ^ { m } , } & { \quad m \geq 2 \rho t } \end{array} \right.
+$$
+
+Since $\mathbf { v }$ is a unit vector, thanks to Cauchy–Bunyakovsky–Schwarz inequality, we can upper-bound the error of the quadratic form approximation by the error of the $\exp ^ { - t \pmb { c } } \mathbf { v }$ approximation, i.e. $\begin{array} { r } { | \mathbf { v } ^ { \top } f ( \pmb { \mathscr { L } } ) \mathbf { v } - \mathbf { e } _ { 1 } ^ { \top } \dot { \mathbf { U } } f ( \Lambda ) \mathbf { U } ^ { \top } \mathbf { e } _ { 1 } | \leq \dot { \| } \exp ^ { - t \pmb { \mathscr { L } } } \mathbf { v } - \dot { \mathbf { Q } } _ { m } \exp ^ { - t \pmb { \Upsilon } _ { m } } \mathbf { e } _ { 1 } \| = \dot { \epsilon _ { m } } . } \end{array}$ .
+
+Following the argumentation in Ubaru et al. (2017), we obtain a condition on the number of Lanczos steps $m$ by setting $\epsilon _ { m } \leq \frac { \epsilon } { 2 } f _ { m i n } ( \lambda )$ , where $f _ { m i n } ( \lambda )$ is the minimum value of $f$ on $[ \lambda _ { m i n } , \lambda _ { m a x } ]$ . We now derive the absolute error between the Hutchinson estimate of Equation (3) and the SLQ of Equation (6):
+
+$$
+\begin{array} { r l } & { \displaystyle \bigg | \operatorname { T r } _ { n _ { v } } \bigl ( f ( \mathscr { L } ) \bigr ) - \Gamma \bigg | = \frac { n } { n _ { v } } \bigg | \sum _ { i = 1 } ^ { n _ { v } } \mathbf { v } _ { i } ^ { \top } f ( \mathscr { L } ) \mathbf { v } _ { i } - \sum _ { i = 1 } ^ { n _ { v } } \mathbf { e } _ { 1 } ^ { \top } f ( \mathbf { T } ^ { ( i ) } ) \mathbf { e } _ { 1 } \bigg | } \\ & { \qquad \leq \frac { n } { n _ { v } } \displaystyle \sum _ { i = 1 } ^ { n _ { v } } \bigg | \mathbf { v } _ { i } ^ { \top } f ( \mathscr { L } ) \mathbf { v } _ { i } - \mathbf { e } _ { 1 } ^ { \top } f ( \mathbf { T } ^ { ( i ) } ) \mathbf { e } _ { 1 } \bigg | } \\ & { \qquad \leq \frac { n } { n _ { v } } \displaystyle \sum _ { i = 1 } ^ { n _ { v } } \epsilon _ { m } = n \epsilon _ { m } , } \end{array}
+$$
+
+where $\mathbf { T } ^ { ( i ) }$ is the tridiagonal matrix obtained with Lanczos algorithm with starting vector $\mathbf { v } _ { i }$
+
+$$
+\left| \operatorname { T r } _ { n _ { v } } f ( \pmb { \mathscr { L } } ) - \Gamma \right| \leq n \epsilon _ { m } \leq \frac { n \epsilon } { 2 } f _ { m i n } ( \lambda ) \leq \frac { \epsilon } { 2 } \operatorname { T r } ( f ( \pmb { \mathscr { L } } ) ) ,
+$$
+
+Finally, we formulate SLQ as an $( \epsilon , \delta )$ estimator,
+
+$$
+\begin{array} { r l } & { 1 - \delta \leq \operatorname* { P r } \Bigg [ \bigg | \operatorname { T r } ( f ( \mathcal { L } ) ) - \operatorname { T r } _ { n _ { v } } ( f ( \mathcal { L } ) ) \bigg | \leq \frac { \epsilon } { 2 } \bigg | \operatorname { T r } ( f ( \mathcal { L } ) ) \bigg | \Bigg ] } \\ & { \qquad \leq \operatorname* { P r } \Bigg [ \bigg | \operatorname { T r } ( f ( \mathcal { L } ) ) - \operatorname { T r } _ { n _ { v } } ( f ( \mathcal { L } ) ) \bigg | + \bigg | \operatorname { T r } _ { n _ { v } } ( f ( \mathcal { L } ) ) - \Gamma \bigg | \leq \frac { \epsilon } { 2 } \bigg | \operatorname { T r } ( f ( \mathcal { L } ) ) \bigg | + \frac { \epsilon } { 2 } \bigg | \operatorname { T r } ( f ( \mathcal { L } ) ) \bigg | \Bigg ] } \\ & { \qquad \leq \operatorname* { P r } \Bigg [ \bigg | \operatorname { T r } ( f ( \mathcal { L } ) ) - \Gamma \bigg | \leq \epsilon \bigg | \operatorname { T r } ( f ( \mathcal { L } ) ) \bigg | \Bigg ] , } \end{array}
+$$
+
+For the normalized Laplacian $\pmb { \mathcal { L } }$ , the minimum eigenvalue is 0 and $f _ { \mathrm { m i n } } ( 0 ) = \exp ( 0 ) = 1$ , hence $\epsilon _ { m } \leq \frac { \epsilon } { 2 }$ , and the eigenvalue interval has $\rho = 0 . 5$ . We can thus derive the appropriate number of Lanczos steps $m$ to achieve error $\epsilon$ ,
+
+$$
+\epsilon \leq \left\{ \begin{array} { l l } { 2 0 e ^ { - m ^ { 2 } / ( 2 . 5 t ) } , } & { \quad \sqrt { 2 t } \leq m \leq t } \\ { 4 0 t ^ { - 1 } e ^ { - 0 . 5 t } \Bigl ( \frac { 0 . 5 e t } { m } \Bigr ) ^ { m } , } & { \quad m \geq t } \end{array} \right.
+$$
+
+Figure 8 shows the tightness of the bound for the approximation of the matrix exponential action on vector $\mathbf { v }$ , $\epsilon _ { m } = \Vert \exp ( - t \pmb { \mathcal { L } } ) - \bar { \mathbf { Q } } _ { m } \exp ( - t \pmb { \mathrm { T } } _ { m } ) \mathbf { e } _ { 1 } \Vert$ . We can see that for most of the temperatures $t$ , very few Lanczos steps $m$ are sufficient, i.e. we can set $m \ = \ 1 0$ . However, the error from the Hutchinson estimator dominates the overall error. Figure 9 shows the error of trace estimation does not change with $m$ and for $t ~ = ~ 0 . 1$ is around $1 0 ^ { - 3 }$ . In case of a Rademacher $p ( \mathbf { v } )$ , the bound on the number of random samples is $\begin{array} { r } { \hat { n } _ { v } \ge \frac { 6 } { \epsilon ^ { 2 } } \log ( 2 / \delta ) } \end{array}$ (Roosta-Khorasani & Ascher, 2015). Employing 10k vectors results in the error bound of roughly $1 0 ^ { - 2 }$ . In practice, we observe the performance much better than given by the bound, see Figure 9.
+
+One particular benefit of small $m$ value is that we do not have to worry about the orthogonality loss in the Lanczos algorithm which often undermines its convergence. Since we do only a few Lanczos iterations, the rounding errors hardly accumulate causing little burden in terms of orthogonality loss between the basis vectors of the Krylov subspace.
+
+![](images/6b10904c09056ffcc85abdeb53d9e3c55826298c0940ab6834974fecc6cd8e7c.jpg)  
+Figure 8: Errors (solid) and error bounds (dotted) for the approximation of matrix exponential action with varying temperature $t$ .
+
+![](images/9141cae3567da944e5cf6fed8fa5486515b691e3a53f0cffd32618dbcc1ae092.jpg)  
+Figure 9: Trace estimation errors (solid) and error bounds (dotted) for: (left) the number of Lanczos steps $m$ with fixed number of random vectors $n _ { v } = 1 0 0$ ; (right) the number of random vectors $n _ { v }$ in Hutchinson estimator with fixed number of Lanczos steps $m = 1 0$ . Lines correspond to varying temperatures $t$ .
+
+# B VARIANCE REDUCTION
+
+We reduce variance of the randomized estimator through control variates. The idea is to use Taylor expansion to substitute a part of the trace estimate with its easily computed precise value,
+
+$$
+\begin{array} { l } { \displaystyle \mathrm { T r } ( \exp ( - t \mathcal { L } ) ) = \mathrm { s } \mathrm { 1 } \mathrm { q } \Big [ \exp ( - t \mathcal { L } ) - ( \mathbf { I } - t \mathcal { L } + \frac { t ^ { 2 } \mathcal { L } ^ { 2 } } { 2 } ) \Big ] + \mathrm { T r } ( \mathbf { I } - t \mathcal { L } + \frac { t ^ { 2 } \mathcal { L } ^ { 2 } } { 2 } ) } \\ { \displaystyle = \mathrm { s } \mathrm { 1 } \mathrm { q } \Big [ \exp ( - t \mathcal { L } ) - ( \mathbf { I } - t \mathcal { L } + \frac { t ^ { 2 } \mathcal { L } ^ { 2 } } { 2 } ) \Big ] + n + \mathrm { T r } ( - t \mathcal { L } ) + \frac { t ^ { 2 } \| \mathcal { L } \| _ { F } ^ { 2 } } { 2 } } \\ { \displaystyle = \mathrm { s } \mathrm { 1 } \mathrm { q } \Big [ \exp ( - t \mathcal { L } ) \Big ] + \mathrm { s } \mathrm { 1 } \mathrm { q } \Big [ t \mathcal { L } \Big ] - \mathrm { s } \mathrm { 1 } \mathrm { q } \Big [ \frac { t ^ { 2 } \mathcal { L } ^ { 2 } } { 2 } ) \Big ] - t n + \frac { t ^ { 2 } \| \mathcal { L } \| _ { F } ^ { 2 } } { 2 } , } \end{array}
+$$
+
+where we use the fact that $\| \pmb { \mathcal { L } } \| _ { F } = \sqrt { \mathrm { T r } ( \pmb { \mathcal { L } } ^ { \top } \pmb { \mathcal { L } } ) }$ and that the trace of normalized Laplacian is equal to $n$ . It does reduce the variance of the trace estimate for smaller temperatures $t \leq 1$ .
+
+To obtain this advantage over the whole range of $t$ , we utilize the following variance reduction form:
+
+$$
+\mathrm { T r } ( \exp ( - t \mathcal { L } ) ) = { \mathrm { s 1 } } \mathrm { q } \left[ \exp ( - t \mathcal { L } ) - \left( \mathbf { I } - \alpha t \mathcal { L } \right) \right] + n ( 1 - \alpha t ) ,
+$$
+
+where there exists an alpha that is optimal for every $t$ , namely setting $\alpha = 1 / \exp ( t )$ . We can see the variance reduction that comes from this procedure in the Figure 12.
+
+# C EXPERIMENTS DISCUSSION
+
+Here we include additional results that did not find their way to the main paper body.
+
+C.1 FID AND KID FAIL TO FIND STRUCTURE IN UNALIGNED CORPORA
+
+Figure 11 shows the matrix of distances for FID and KID aligned and colored in the same way as Figure 2 (a). FID and KID can not find meaningful structure in the data in the same way as IMD as they rely on extrinsic data properties.
+
+# C.2 WORD EMBEDDING EXPERIMENT DETAILS.
+
+We use gensim (Reh ˚u ˇ ˇrek & Sojka, 2010) to learn word vectors on the latest Wikipedia corpus snapshot on 16 languages: Polish, Russian, Greek, Hungarian, Turkish, Arabic, Hebrew, English, Simple English, Swedish, German, Spanish, Dutch, Portugese, Vietnamese, and Waray-Waray. We then compute FID, KID and IMD scores on all the pairs, we average 100 runs for the heatmap figures 2. For the different dimensionality experiment, we learn vectors on the English Wikipedia of sizes equal to the powers of 2 from 4 to 512. After that we compute IMD and covariance error, i.e. normalized PIP loss, between the pairs of sizes to generate the heatmap figure 3.
+
+![](images/52c846d8323449d18a990232232a9a23099fab6cb407661b19005403d6434b79.jpg)
+
+# C.3 VANILLA GAN ON TORUS
+
+We provide an additional experiment clearly showing the case where IMD is superior to its main competitors, FID and KID. We train two vanilla GANs on the points of a 3D torus. The bad
+
+Figure 10: Bad GAN produces samples inside the torus hole (red). FID and KID cannot detect such behaviour.   
+
+<table><tr><td>metric</td><td>good GAN</td><td>bad GAN</td></tr><tr><td>FID</td><td>0.00529± 0.00070</td><td>0.00627± 0.00076</td></tr><tr><td>KID</td><td>0.00172±</td><td>0.00259±</td></tr><tr><td></td><td>0.00073</td><td>0.00077</td></tr><tr><td>IMD</td><td>9.02059 ±</td><td>14.0732±</td></tr><tr><td></td><td>1.5195</td><td>2.1706</td></tr></table>
+
+GAN fails to learn the topology of the dataset it tries to mimic, yet previous metrics cannot detect this fact. IMD, on the contrary, can tell the difference. Figure 10 shows the points sampled from the GAN with some of the points inside the hole. KID and FID confidence intervals overlap for good and bad GANs, meanwhile IMD scores are clearly distinct from each other.
+
+# C.4 NORMALIZATION DETAILS
+
+For the purpose of normalizing IMD, we need to approximate that graph’s eigenvalues. Coja-Oghlan (2007) proved that $\lambda _ { 1 } \le 1 - c \bar { d } ^ { - 1 / 2 } \le \lambda _ { 2 } \le \lambda _ { n } \overset {  } { \le } 1 + c \bar { d } ^ { - 1 / 2 }$ for the core of the graph for some constant $c$ . We have empirically found that $c = 2$ provides a tight approximation for random graphs. That coincides with the analysis of Chung et al. (2004), who proved that √ $\lambda _ { n } = ( 1 + o ( 1 ) ) 2 \bar { d } ^ { - 1 / 2 }$ if $d _ { \operatorname* { m i n } } \gg \sqrt { \bar { d } } \log ^ { 3 } n$ even though in our case $d _ { \operatorname* { m i n } } = \bar { d } = k$ . We thus estimate the spectrum of a random Erdos-Rényi graph as growing linearly between ˝ $\lambda _ { 1 } = 1 - 2 \bar { d } ^ { - 1 / 2 }$ and $\lambda _ { n } = \bar { 1 } + 2 \bar { d } ^ { - 1 / 2 }$ , which corresponds to the underlying manifold being two-dimensional (Tsitsulin et al., 2018).
+
+![](images/01b747a53a256c570f4044e9e56cbd316761373846319656fc53dc904f356381.jpg)  
+Figure 11: FID and KID are not able to capture language affinity from unaligned word2vec embeddings.
+
+# D EXPERIMENTAL SETTINGS
+
+We train all our models on a single server with NVIDIA V100 GPU with 16Gb memory and $2 ~ \times ~ 2 0$ core Intel E5-2698 v4 CPU. For the experiment summarized in Table 1 in the Section 4.1 we train WGAN and WGAN-GP models on 4 datasets: MNIST, FashionMNIST, CIFAR10 and CelebA and sample 10k samples, Y, from each of the GANs. We uniformly subsample $1 0 \mathrm { k }$ images from the original datasets, $\mathbf { X }$ , and compute the IMD, KID and FID scores between $\mathbf { X }$ and $\mathbf { Y }$ . We report the mean as well as the $9 9 \%$ confidence interval across 100 runs.
+
+![](images/9872b645f540b370fa6c98948952a6c2a6dd312ead2b2bdd2ef3aa069b6fa5a4.jpg)  
+Figure 12: Variance of the trace estimate.
+
+Below we report the architectures, hyperparameters and generated samples of the models used for the experiments. We train each of the GANs for 200 epochs on MNIST, FMNIST and CIFAR-10, and for 50 epochs on CelebA dataset. For WGAN we use RMSprop optimizer with learning rate of $5 \times 1 0 ^ { - 5 }$ . For WGAN-GP we use Adam optimizer with learning rate of $1 0 ^ { - 4 }$ , $\beta _ { 1 } = 0 . 9 , \beta _ { 2 } = 0 . 9 9 9$ .
+
+# E GRAPH EXAMPLE
+
+Figure 13 provides visual proof that the 5NN graph reflects the underlying manifold structure of the CIFAR-10 dataset. Clusters in the graph exactly correspond to CIFAR-10 classes.
+
+![](images/fc47a809ab08d10fa61b155a7773599cd0c257c818f1fe6ca35aa13bdf7c1169.jpg)  
+Figure 13: CIFAR-10 graph colored with true class labels.
+
+![](images/c1277459146dacfa1a72cef4e2c38403311417d60ba3839c6d4eddc24c9a1cb2.jpg)  
+Figure 14: MNIST samples (left: WGAN, right: WGAN-GP)
+
+![](images/1de2b527ed4015c1a8452175d20ef1cc40a97ca03d2b0a6c9b57a35545266329.jpg)  
+Figure 15: FashionMNIST samples (left: WGAN, right: WGAN-GP)
+
+![](images/9173ee4effda75755a5bad03a4ed5244e7b640547d72f69bdb69749b8bb0fe6d.jpg)  
+Figure 16: CIFAR-10 samples (left: WGAN, right: WGAN-GP)
+
+![](images/78b5b485dc4d787ec349fb3944ad33946186ef2ce996c94c6aa1d075a5464c38.jpg)  
+Figure 17: CelebA samples (left: WGAN, right: WGAN-GP)
+
+# MNIST WGAN
+
+ConvGenerator( (latent_to_features): Sequential( (0): Linear(in_feature $\mathord { \mathrm { : } } = 1 0 0$ , out_feature $_ { \mathrm { S } } = 5 1 2$ , bias $=$ True) (1): ReLU() ) (features_to_image): Sequential( (0): ConvTranspose2d(128, 64, kernel_size $: =$ (4, 4), stride $=$ (2, 2), padding $=$ (1, 1)) (1): ReLU() (2): BatchNorm2d(64, $\mathtt { e p s } { = } 1 \mathtt { e } { - } 0 5$ , momentum $\phantom { - } 1 { = } 0 \ . \ 1$ , affine $=$ True) (3): ConvTranspose2d(64, 32, kernel_size $=$ (4, 4), stride $=$ (2, 2), padding $=$ (1, 1)) (4): ReLU() (5): BatchNorm2d(32, $\mathtt { e p s } { = } 1 \mathtt { e } { - } 0 5$ , momentum $\phantom { - } 1 { = } 0 \ . \ 1$ , affine $=$ True) (6): ConvTranspose2d(32, 16, kernel_size $=$ (4, 4), stride $=$ (2, 2), padding $=$ (1, 1)) (7): ReLU() (8): BatchNorm2d(16, $\mathtt { e p s } { = } 1 \mathtt { e } { - } 0 5$ , momentum $\phantom { - } 1 { = } 0 \ . \ 1$ , affine $=$ True) (9): ConvTranspose2d(16, 1, kernel_size $=$ (4, 4), stride $=$ (2, 2), padding $=$ (1, 1)) (10): Sigmoid() )   
+)   
+ConvDiscriminator( (image_to_features): Sequential( (0): Conv2d(1, 16, kernel_size $=$ (4, 4), stride $=$ (2, 2), padding $=$ (1, 1)) (1): LeakyReLU(negative_slope $= 0 . 2$ 2) (2): Conv2d(16, 32, kernel_size $=$ (4, 4), stride $=$ (2, 2), padding $=$ (1, 1)) (3): LeakyReLU(negative_slope ${ } = 0$ .2) (4): Conv2d(32, 64, kernel_size $=$ (4, 4), stride $=$ (2, 2), padding $=$ (1, 1)) (5): LeakyReLU(negative_slope ${ } = 0$ .2) (6): Conv2d(64, 128, kernel_size $=$ (4, 4), stride $=$ (2, 2), padding $=$ (1, 1)) (7): Sigmoid() ) (features_to_prob): Sequential( (0): Linear(in_features $_ { : = 5 1 2 }$ , out_feature $\mathsf { S } = 1$ , bias $=$ True) (1): Sigmoid() )   
+)
+
+# MNIST WGAN-GP, FMNIST (WGAN, WGAN-GP)
+
+MNISTGenerator( (block1): Sequential( (0): ConvTranspose2d(256, 128, kernel_size $=$ (5, 5), stride $=$ (1, 1)) (1): ReLU(inplace) ) (block2): Sequential( (0): ConvTranspose2d(128, 64, kernel_size $=$ (5, 5), stride $=$ (1, 1)) (1): ReLU(inplace) (deconv_out): ConvTranspose2d(64, 1, kernel_size $=$ (8, 8), stride $: =$ (2, 2)) (preprocess): Sequential( (0): Linear(in_features $_ { ; = 1 2 8 }$ , out_feature $_ { \mathsf { S } } = 4 0 9 6$ , bias $=$ True) (1): ReLU(inplace) ) (sigmoid): Sigmoid()   
+)   
+MNISTDiscriminator( (main): Sequential( (0): Conv2d(1, 64, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2)) (1): ReLU(inplace) (2): Conv2d(64, 128, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2)) (3): ReLU(inplace) (4): Conv2d(128, 256, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2)) (5): ReLU(inplace) ) (output): Linear(in_feature $s { = } 4 0 9 6$ , out_features $^ { , = 1 }$ , bias $=$ True)   
+)
+
+# CIFAR-10 (WGAN, WGAN-GP)
+
+CIFARGenerator( (preprocess): Sequential( (0): Linear(in_features $_ { \cdot = 1 2 8 }$ , out_feature $_ { \mathsf { S } } = 4 0 9 6$ , bias $=$ True) (1): BatchNorm1d(4096, $\mathtt { \mathtt { e p s } } { = } 1 \mathtt { e } { - } 0 5$ , momentum $\phantom { - } 1 { = } 0 \ . \ 1$ , affine $=$ True) (2): ReLU(inplace) ) (block1): Sequential( (0): ConvTranspose2d(256, 128, kernel_size $=$ (2, 2), stride $=$ (2, 2)) (1): BatchNorm2d(128, eps=1e-05, momentum $\phantom { - } 1 { = } 0 \ . \ 1$ , affine $=$ True) (2): ReLU(inplace) ) (block2): Sequential( (0): ConvTranspose2d(128, 64, kernel_size $: =$ (2, 2), stride $=$ (2, 2)) (1): BatchNorm2d(64, eps $=$ 1e-05, momentum $\phantom { - } 1 { = } 0 \ . \ 1$ , affine $=$ True) (2): ReLU(inplace) ) (deconv_out): ConvTranspose2d(64, 3, kernel_size $=$ (2, 2), stride $=$ (2, 2)) (tanh): Tanh()   
+)   
+CIFARDiscriminator( (main): Sequential( (0): Conv2d(3, 64, kernel_size $=$ (3, 3), stride $=$ (2, 2), padding $=$ (1, 1)) (1): LeakyReLU(negative_slope $= 0 . 0 1$ ) (2): Conv2d(64, 128, kernel_size $=$ (3, 3), stride $=$ (2, 2), padding $=$ (1, 1)) (3): LeakyReLU(negative_slope=0.01) (4): Conv2d(128, 256, kernel_size $=$ (3, 3), stride $=$ (2, 2), padding $=$ (1, 1)) (5): LeakyReLU(negative_slope=0.01) ) (linear): Linear(in_feature $s { = } 4 0 9 6$ , out_features $^ { = 1 }$ , bias $=$ True)   
+)
+
+# CelebA (WGAN, WGAN-GP)
+
+CelebaGenerator( (preprocess): Sequential( (0): Linear(in_feature $_ { \tt S } = 1 2 8$ , out_feature $_ { \mathrm { S } } = 8 1 9 2$ , bias $=$ True) (1): BatchNorm1d(8192, $\mathtt { \mathtt { P S } } \mathtt { = } 1 \mathtt { e } \mathrm { - } 0 5$ , momentum $\phantom { - } 1 { = } 0 \ . \ 1$ , affine $=$ True) (2): ReLU(inplace) ) (block1): Sequential( (0): ConvTranspose2d(512, 256, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2), output_padding $=$ (1, 1), bias $=$ False) (1): BatchNorm2d(256, e $_ { \mathsf { P S } } { = } 1 \mathsf { e } { - } 0 5$ , momentum $\scriptstyle 1 = 0$ .1, affine $=$ True) (2): ReLU(inplace) ) (block2): Sequential( (0): ConvTranspose2d(256, 128, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2), output_padding $=$ (1, 1), bias $=$ False) (1): BatchNorm2d(128, $\mathtt { P } S { = } 1 { \in } { - } 0 5$ , momentum $\scriptstyle 1 = 0$ .1, affine $=$ True) (2): ReLU(inplace) ) (block3): Sequential( (0): ConvTranspose2d(128, 64, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2), output_padding $=$ (1, 1), bias $=$ False) (1): BatchNorm2d(64, ep $\mathtt { s } = 1 \mathtt { e } - 0 5$ , momentum $= 0 \cdot 1$ , affine $=$ True) (2): ReLU(inplace) ) (deconv_out): ConvTranspose2d(64, 3, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2), output_padding $=$ (1, 1)) (tanh): Tanh()   
+)   
+CelebaDiscriminator( (main): Sequential( (0): Conv2d(3, 64, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2)) (1): LeakyReLU(negative_slope ${ } = 0$ .01) (2): Conv2d(64, 128, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2)) (3): LeakyReLU(negative_slope ${ } = 0$ .01) (4): Conv2d(128, 256, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2)) (5): LeakyReLU(negative_slope ${ } = 0$ .01) (6): Conv2d(256, 512, kernel_size $=$ (5, 5), stride $=$ (2, 2), padding $=$ (2, 2)) (7): LeakyReLU(negative_slope ${ } = 0$ .01) (8): Conv2d(512, 1, kernel_siz ${ \tt a } =$ (4, 4), stride $=$ (1, 1)) )   
+)

parse/train/S191YzbRZ/S191YzbRZ.md

@@ -0,0 +1,311 @@
+# PROTOTYPE MATCHING NETWORKS FOR LARGE-SCALE MULTI-LABEL GENOMIC SEQUENCE CLASSIFICATION
+
+# ABSTRACT
+
+One of the fundamental tasks in understanding genomics is the problem of predicting Transcription Factor Binding Sites (TFBSs). With more than hundreds of Transcription Factors (TFs) as labels, genomic-sequence based TFBS prediction is a challenging multi-label classification task. There are two major biological mechanisms for TF binding: (1) sequence-specific binding patterns on genomes known as “motifs” and (2) interactions among TFs known as co-binding effects. In this paper, we propose a novel deep architecture, the Prototype Matching Network (PMN) to mimic the TF binding mechanisms. Our PMN model automatically extracts prototypes (“motif”-like features) for each TF through a novel prototypematching loss. Borrowing ideas from few-shot matching models, we use the notion of support set of prototypes and an LSTM to learn how TFs interact and bind to genomic sequences. On a reference TFBS dataset with 2.1 million genomic sequences, the PMN significantly outperforms baselines and validates our design choices empirically. To our knowledge, this is the first deep learning architecture that introduces prototype learning and considers TF-TF interactions for large scale TFBS prediction. Not only is the proposed architecture accurate, but it also models the underlying biology.
+
+# 1 INTRODUCTION
+
+Genomic sequences build the basis of a large body of research on understanding the biological processes in living organisms. Enabling machines to read and comprehend genomes is a longstanding and unfulfilled goal of computational biology. One of the fundamental task to understand genomes is the problem of predicting Transcription Factor Binding Sites (TFBSs), attracting much attention over the years (Consortium et al., 2012). Transcription Factors (TFs) are proteins which bind (i.e., attach) to DNA and control whether a gene is expressed or not. Patterns of how different genes expressed or not expressed control many important biological phenomena, including diseases such as cancer. Therefore accurate models for identifying and describing the binding sites of TFs are essential in understanding cells.
+
+Owing to the development of chromatin immunoprecipitation and massively parallel DNA sequencing (ChIP-seq) technologies (Park, 2009), maps of genome-wide binding sites are currently available for multiple TFs in a few cell types across human and mouse genomes via the ENCODE (Consortium et al., 2012) database. However, ChIP-seq experiments are slow and expensive; they have not been performed for many important cell types or organisms. Therefore, computational methods to identify TFBS accurately remain essential for understanding the functioning and evolution of genomes.
+
+An important feature of TFs is that they typically bind to sequence-specific patterns on genomes, known as “motifs” (Mitchell, 1989). Motifs are essentially a blueprint, or a “prototype” which a TF searches for in order to bind. However, motifs are only one part in determining whether or not a TF will bind to specific locations. If a TF binds in the absence of its motif, or it does not bind in the presence of its motif, then it is likely there are some external causes such as an interaction with another TF, known as co-binding effects in biology (Wang et al., 2012). This indicates that when designing a genomic-sequence based TFBS predictor, we should consider two modeling challenges: (1) how to automatically extract “motifs”-like features and (2) how to model the co-binding patterns and consider such patterns in predicting TFBSs. In this paper, we address both proposing a novel deep-learning model: prototype matching network (PMN).
+
+To address the first challenge of motif learning and matching, many bioinformatics studies tried to predict TFBSs by constructing motifs using position weight matrices (PWMs) which best represented the positive binding sites. To test a sequence for binding, the sequence is compared against the PWMs to see if there is a close match (Stormo, 2000). PWM-matching was later outperformed by convolutional neural network (CNN) and CNN-variant models that can learn PWM-like filters Alipanahi et al. (2015a). Different from basic CNNs, our proposed PMN is inspired by the idea of “prototype-matching” (Wallis et al., 2008; Krotov & Hopfield, 2016). These studies refer to the CNN type of model as the “feature-matching” mode of pattern recognition. While pure feature matching has proven effective, studies have shown a “prototype effect” where objects are likely recognized as a whole using a similarity measure from a blurred prototype representation, and prototypes do not necessarily match the object precisely (Wallis et al., 2008). It is plausible that humans use a combination of feature matching and prototype matching where feature-matching is used to construct a prototype for testing unseen samples (Krotov & Hopfield, 2016). For TFBS prediction, the underlying biology evidently favors computation models that can learn “prototypes” (i.e. effective motifs). Although motifs are indirectly learned in convolutional layers, existing deep learning studies of TFBS (details in Section 3) have not considered the angle of “motif-matching” using a similarity measure. We, instead, propose a novel prototype-matching loss to learn prototype embedding automatically for each TF involved in the data.
+
+None of the previous deep-learning studies for TFBS predictions have considered tackling the second challenge of including the co-binding effects among TFs in data modeling. From a machine learning angle, the genomic sequence based TFBS prediction is a multi-label sequence classification task. Rather than learning a prediction model for each TF (i.e., each label) predicting if the TF will bind or not on input, a joint model is ideal for outputting how a genomic sequence input is attached by a set of TFs (i.e., labels). The so-called “co-binding effects” connect deeply to how to model the dependency and combinations of TFs (labels). Multi-label classification is receiving increasing attention in deep learning (Wang et al., 2016; Guo & Gu, 2011; Wei et al., 2014) (detailed review in Section 3). Modeling the multi-label formulation for TFBS is an extremely challenging task because the number of labels (TFs) is in hundreds to thousands (e.g. 1,391 TFs in Vaquerizas et al. (2009)). The classic solution for multi-label classification using the powerset idea (i.e., the set of all subsets of the label set) is clearly not feasible (Tsoumakas & Katakis, 2006). Possible prior information about TF-TF interactions is unknown or limited in the biology literature.
+
+To tackle these obstacles, our proposed model PMN borrows ideas from the memory network and attention literature. Vinyals et al. (2016) proposed a “matching network” model where they train a differentiable nearest neighbor model to find the closest matching image from a support set on a new unseen image. They use a CNN to extract features and then match those features against the support set images. We replace this support set of images with a learned support set of prototypes from the large-scale training set of TFBS prediction, and we use this support set to match against a new test sample. The key difference is that our PMN model is not for few-shot learning and we seek to learn the support set (prototypes). Vinyals et al. (2016) uses an attentionLSTM to model how a test sample matches to different items in the support set through softmax based attention. Differently, we use what we call a combinationLSTM to model how the embedding of a test sample matches to a combination of relevant prototypes. Using multiple “hops”, the combinationLSTM updates the embedding of the input sequence by searching for which TFs (prototypes) are more relevant in the label combination. Instead of explicitly modeling interactions among labels, we try to use the combinationLSTM to mimic the underlying biology. The combinationLSTM tries to learn prototype embedding and represent high-order label combinations through a weighted sum of prototype embedding. This weighted summation can model many “co-binding effects” reported in the biology literature (Wang et al., 2012) (details in Section 2).
+
+In summary, we propose a novel PMN model by combining few-shot matching and prototype feature learning. To our knowledge, this is the first deep learning architecture to model TF-TF interactions in an end-to-end model. In addition, this is also the first paper to introduce large scale prototype learning using a deep learning architecture. On a reference TFBS dataset with 2.1 million genomic sequences, PMN significantly outperforms the state-of-the-art TFBS prediction baselines. We validate the learned prototypes through an existing database about TF-TF interactions. The TF groups obtained by clustering prototype embedding evidently captures the “cooperative effects” that has not been modeled by previous TFBS prediction works.
+
+The main contributions of our model are:
+
+![](images/39459ef05d7b3d48b5c251d50864383255133254f7b81f5306d86a45890e2d68.jpg)  
+Figure 1: Prototype Matching Network (PMN) Model. On the left is an overview of the model. The input sequence $x$ is encoded as $\hat { x }$ using $f$ (3-layer CNN). $\hat { x }$ is then matched against the learned prototypes using the combinationLSTM for $K$ “hops” so that it can update its output based on TF interactions for this input sequence. The final output $\hat { y }$ is based on a concatenation of the final updated sequence vector $h ^ { K }$ from the LSTM, and final read vector $r ^ { K }$ from the matching. On the right is a closer look at the internal aspects of the combinationLSTM.
+
+• We propose a novel model by combining few-shot matching with large-scale prototype feature learning.   
+• We design a novel prototype-matching loss to learn “motif”-like features in deep learning, which is important for the TFBS prediction task.   
+• We extend matching models from the few-shot single-label task to a large-scale multi-label task for genomic sequence classification.   
+• We implement an attention LSTM module to model label interactions in a novel way.   
+• Our model favors design choices mimicking the underlying biological processes. We think such modeling strategies are more fundamental especially on datasets from biology.
+
+# 2 PROTOTYPE MATCHING NETWORKS
+
+# 2.1 MODEL OVERVIEW
+
+Given a DNA sequence $x$ (composed of characters A,C,G,T) of length $T$ , we want to classify $x$ as a positive or negative binding site for each transcription factor $T F _ { 1 } , T F _ { 2 } , . . . , T F _ { \ell }$ in our dataset (i.e. multi-label binary classification). To do this, we seek to match $x$ to a bank of $\ell$ learned TF prototype vectors, $\{ p _ { 1 } , . . . , \stackrel { . } { p _ { \ell } } \}$ , where each prototype is loosely representative of a motif. In addition, since TFs may bind or not bind based on other TFs, we model the interactions among TFs in order to make a prediction. An overview of our model can be seen in Figure 1.
+
+# 2.2 EMBEDDING THE SEQUENCE AND PROTOTYPES
+
+The input sequence $\boldsymbol { x } \in \mathbb { R } ^ { 4 \times t }$ is encoded using a function $f$ (3-layer CNN, which has shown to be sufficient for genomic feature extraction) to produce sequence embedding $\hat { x } \in \mathbb R ^ { d }$ :
+
+$$
+{ \hat { x } } = f ( x )
+$$
+
+Each prototype vector $p _ { i } \in \mathbb { R } ^ { d }$ is learned via a lookup table with a constant input integer at each position (e.g. 1 as input to the first position and $t$ as input to position $t$ ). I.e. the prototypes are produced by a multiplication of the identity matrix I and the learned lookup table matrix $W \in$ $\mathbb { R } ^ { | T F s | \times d }$ .
+
+$$
+P = \mathrm { I } W
+$$
+
+Since our learned prototypes, $p _ { i }$ are randomly initialized, we introduce a prototype matching loss ${ \mathcal { L } } _ { p }$ , which forces a prototype to correspond to a specific TF. Our prototype matching loss is explained in section 2.4.
+
+2.3 LSTM TO LEARN LABEL INTERACTIONS AND TO UPDATE THE SEQUENCE EMBEDDING
+
+Once we have the sequence and prototype embedding vectors, we want to compare the sequence to the prototypes to get binding site probabilities for each TF. The main idea is that we want to modify the sequence embedding $\hat { x }$ conditioned on matching against the prototypes. Since interactions among TFs influence binding, we cannot simply match the sequence to the prototypes. To obtain TF interactions, we use an LSTM (combinationLSTM), similar to the attention LSTM (attLSTM) in Vinyals et al. (2016). Our combinationLSTM is what does the actual matching for classification, whereas Vinyals et al. (2016) use the output of the attLSTM to make the final matching prediction. The combinationLSTM uses $K$ “hops” to process the prototypes $p _ { 1 } , p _ { 2 } , . . . , p _ { \ell }$ by matching against an updated sequence embedding $\hat { h } ^ { k }$ . The hops allow the combinationLSTM to update the output vector based on which TFs match simultaneously. At each hop, the LSTM accepts a constant $\hat { x }$ , a concatenation of the previous LSTM hidden state $\bar { h } ^ { k - 1 }$ and read vector r $r ^ { k - 1 }$ , as well as the previous LSTM cell output $c ^ { k - 1 }$ . $h ^ { 0 }$ and $c ^ { 0 }$ are initialized with zeros, and $r ^ { 0 }$ is initialized with the mean of all prototype vectors, ${ \frac { 1 } { | p | } } \sum _ { i } ^ { | p | } p _ { i }$ .
+
+The output hidden state $\hat { h } ^ { k }$ is matched against each prototype using cosine similarity, producing a similarity score. Since this similarity is in the range [-1,1], we feed this output through sigmoid function, weighted by hyperparameter $\epsilon$ (we use $\scriptstyle \epsilon = 2 0$ ) to produce the similarity score $w _ { i } ^ { k }$ at hop $k$ in [0,1]. The read vector $r$ is updated by a weighted sum of the prototype vectors using the matching scores. At each hop, $h ^ { k }$ is updated using the current LSTM output hidden state and the sequence embedding.
+
+$$
+\begin{array} { r l } & { \hat { h } ^ { k } , c ^ { k } = \mathrm { L S T M } ( \hat { x } , [ h ^ { k - 1 } ; r ^ { k - 1 } ] , c ^ { k - 1 } ) } \\ & { \quad \quad h ^ { k } = \hat { h } ^ { k } + \hat { x } } \\ & { \quad \quad r ^ { k } = \displaystyle \sum _ { i = 1 } ^ { \lfloor p \rfloor } w _ { i } ^ { k } p _ { i } } \\ & { \quad \quad w _ { i } ^ { k } = 1 / \big ( 1 + e ^ { - \epsilon c ( \hat { h } ^ { k } , p _ { i } ) } \big ) } \\ & { \quad \quad c ( u , v ) = \frac { u \cdot v } { | | u | | _ { 2 } | | v | | _ { 2 } } } \end{array}
+$$
+
+In the TFBS task, eq. 5 is the important factor for modelling TF combinations. The output vector $r$ can model multiple prototypes matching at once through a linear combination. Furthermore, the LSTM with $K$ hops is needed because of the fact that a TF binding may influence other TFs in a sequential manner. For example, if $T F _ { i }$ matches to $\hat { h } ^ { k }$ in the first hop, $r ^ { k }$ is then used to output $\hat { h } ^ { k + 1 }$ which can match to $T F _ { j }$ at the next hop. In this case, $\hat { h } ^ { k + 1 }$ is a joint representation of $\hat { x }$ and the current matched prototypes, represented by $r ^ { k }$ . At each hop, the LSTM fine-tunes $w ^ { k }$ in order to find $T F$ binding combinations.
+
+The final output $\hat { y } \in \mathbb { R } ^ { | T F s | } .$ is computed from a concatenation of the final hidden state and read vectors $[ h ^ { k } ; r ^ { K } ]$ after the $K ^ { t h }$ hop using a linear transform and an element-wise sigmoid function to get a probability of binding for each TF. :
+
+$$
+\begin{array} { l } { { o = W ( [ h ^ { K } ; r ^ { K } ] ) } } \\ { { \hat { y } = 1 / ( 1 + e ^ { - o } ) } } \end{array}
+$$
+
+# 2.4 CLASSIFICATION AND PROTOTYPE MATCHING LOSS FUNCTIONS
+
+To classify a sequence, we use a standard binary cross entropy loss between each label $y _ { i }$ for $T F _ { i }$ and the corresponding $T F _ { i }$ output $\hat { y } _ { i }$ , which we call the classification loss, $\mathcal { L } _ { c }$ , for each label.
+
+We also introduce a prototype matching loss ${ \mathcal { L } } _ { p }$ , which forces a prototype to correspond to a specific TF since prototypes are learned from random initializations. The prototype matching loss works by using an $\mathrm { L _ { 2 } }$ between the true label $y _ { i }$ for $T F _ { i }$ and the final matching weight $w _ { i } ^ { K }$ between updated sequence $h ^ { K }$ and prototype $p _ { i }$ . This loss forces a prototype to match to all of its positive binding sequences. Each $\dot { w } _ { i } ^ { K }$ is from the final prototype matching weights after the $K ^ { t h }$ hop from Eq (6).
+
+Table 1: Comparison of previous deep-learning studies for TFBS and three closely related deep learning papers in the recent literature. The columns indicate properties: (1) whether the study has a joint deep architecture for multi-label prediction or not, (2) if the study learns prototype features (”motifs” in the TFBS literature), (3) whether the study models how input samples match prototypes, (4) if it uses RNN to model high-order combinations of labels, and finally (5) if the method considers current sample inputs for modeling label combinations. All previous TFBS studies do not model label interactions. PMN combines several key strategies from deep learning literature including: (a) learning label-specific prototype embedding (Snell et al., 2017) through prototype-matching loss, (b) using RNN to model higher-order label combinations (Wang et al., 2016), and (c) using LSTM to model such combinations dynamically (conditioned on the current input) (Vinyals et al., 2016). PMN is the only model that exhibits all desirable properties.
+
+<table><tr><td>Method</td><td>Multi-Label Joint Model</td><td>Task</td><td>Prototypes (Motifs) perLabel</td><td>Prototype Matching Loss</td><td>RNN for Label Combinations (Co-Binding)</td><td>Dynamic Label Combinations</td></tr><tr><td rowspan="4">DeepBind (Alipanahi et al.,2015a) DeepSEA (Zhou &amp; Troyanskaya,2015) DanQ(Quang&amp; Xie,2016)</td><td>×</td><td>TFBS</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>√</td><td>TFBS</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>√</td><td>TFBS</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>X1</td><td>TFBS</td><td>√</td><td>×</td><td>×</td><td>×</td></tr><tr><td>TFImpute(Qin&amp; Feng,2017) CNN-RNN (Wang et al.,2016)</td><td>√</td><td>Image</td><td>√</td><td>×</td><td>√</td><td>×</td></tr><tr><td>Memory-Matching (Vinyals et al.,2016)</td><td>×</td><td>FewShot</td><td>×</td><td>√</td><td>x2</td><td>大</td></tr><tr><td>Prototypical-Network (Snell et al., 2017)</td><td>×</td><td>FewShot</td><td>√</td><td>×</td><td>×</td><td>×</td></tr><tr><td>Prototype Matching Network: (this paper)</td><td>√</td><td>TFBS</td><td>√</td><td>√</td><td>√</td><td>√</td></tr></table>
+
+The important thing is that the loss is computed from the final weights $w ^ { K }$ . This allows the LSTM to attend to certain TFs at different hops before making its final decision, modeling the co-binding of TFs. The hyperparameter $\lambda$ controls the amount that each prototype is mapped to a specific TF. $\lambda { = } 0$ corresponds to random prototypes since we are not forcing $p _ { i }$ to match to a specific sequence.
+
+Thus, the final loss $\mathcal { L }$ is a summation of both the classification loss and the prototype matching loss:
+
+$$
+\begin{array} { r l } & { \mathcal { L } = - \mathcal { L } _ { c } - \lambda \mathcal { L } _ { p } } \\ & { \mathcal { L } _ { c } = \displaystyle \sum _ { i } ^ { | p | } ( y _ { i } \log \hat { y } _ { i } + ( 1 - y _ { i } ) \log ( 1 - \hat { y } _ { i } ) ) } \\ & { \mathcal { L } _ { p } = \displaystyle \sum _ { i } ^ { | p | } ( y _ { i } - w _ { i } ^ { K } ) ^ { 2 } } \end{array}
+$$
+
+# 2.5 TRAINING
+
+We trained our model using Adam (Kingma & Ba, 2014) with a batch size of 512 sequences for 40 epochs. Our results were based on the test set results from the best performing validation epoch. We use dropout (Srivastava et al., 2014) for regularization.
+
+# 3 RELATED WORKS
+
+Deep learning in bioinformatics: Deep learning is steadily gaining popularity in the bioinformatics community. This trend is credited to their ability to extract meaningful representations from large datasets. For instance, multiple recent studies have successfully used deep learning for modeling protein sequences (Lin et al., 2016; Zhou & Troyanskaya, 2014), modeling DNA sequences (Alipanahi et al., 2015b; Lanchantin et al., 2016), predicting gene expression (Singh et al., 2016), as well as understanding the effects of non-coding variants (Zhou & Troyanskaya, 2015; Quang & Xie, 2016)).
+
+Previous Studies of TFBS and more: Previous techniques for predicting TFBS include many sequence-motif based computational approaches that typically use position-based sequence information (Stormo, 2000). Relying on a set of known transcription factor binding sites (TFBSs) for a given
+
+TF, the binding preference is generally represented in the form of a position weight matrix (PWM) (Stormo, 2013; Mathelier et al., 2013) (also called position-specific scoring matrix) derived from a position frequency matrix (PFM). Recently this technique was outperformed by different variations of deep convolutional models (Alipanahi et al., 2015b; Lanchantin et al., 2016; Quang & Xie, 2016; Shrikumar et al., 2017). While motif-based PWMs are compact and interpretable, they can under-fit ChIP-seq data by failing to capture subtle but detectable and important sequence signals, such as direct DNA-binding preferences of certain TFs, cofactor binding sequences, accessibility signals, or other discriminative sequence features (Arvey et al., 2012; Le et al., 2017).
+
+Table 1 summarizes four most relevant deep learning studies of TFBS in the literature. Alipanahi et al. (2015b) was the first to use a deep learning based approach to predict TFBSs. They showed that a 1-layer CNN could outperform baseline motif matching approaches which used position weight matrices Machanick & Bailey (2011). Zhou & Troyanskaya (2015) used a similar method to predict the effects of variants in noncoding regions of the genome, where TFBS prediction was an intermediate step to predict variant effects. They used a 3-layer CNN model to predict 919 chromatin labels (including 690 TFBS labels). Quang & Xie (2015) extended this model using a bidirectional LSTM on top of the CNN outputs to model interactions among motifs. Note that this is not modeling interactions among labels (TFs), but rather among sequence features. Qin & Feng (2017) uses a similar lookup table approach for learning a representation for a TF, but their implementation is for transferring between cell lines where the target cell line has no experimental data. However, ChIP-seq experiments are relatively cheap given a specific TF and sequence of interest. We are more interested in modeling the underlying biology.
+
+Aside from deep learning models, there have been other works to model joint dependencies among TFs Kazemian et al. (2011); He et al. (2009); Gautier et al. (2008); Sinha & He (2007). Our primary goal is to extend previous deep learning methods to better model co-binding, thus we do not compare against these approaches.
+
+The formulation of TFBS prediction belongs to a general category of “biological sequence classification”. Sequence analysis plays an important role in the field of bioinformatics. Various methods have previously been proposed, including generative (e.g., Hidden Markov Models-HMMs) and discriminative approaches. Among the discriminative approaches, string kernel methods provide some of the most accurate results, such as for remote protein fold and homology detections (Leslie & Kuang, 2004; Kuksa et al., 2008). We omit a full survey of this topic due to its vast body of previous literature and loose connection to our TFBS formulations.
+
+Memory Matching Network and Attention RNN: Attention in combination with RNNs has been successfully used for many tasks Bahdanau et al. (2014). Various methods have extended the single step attention approach to attention with multiple ‘hops’ Sukhbaatar et al. (2015); Kumar et al. (2016). For example, for the Question Answering task, Kumar et al. (2016) introduced the Dynamic Memory Network which uses an iterative attention process coupled with an RNN over multiple ‘episodes’. Kumar et al. (2016) show that the attention gets more focused over successive hops or ‘episodes’ over the input, reaffirming the need for recurrent episodes. Most of these models are for sequential inputs. For tasks that involve a set (i.e., no order) as input or/and as output, Vinyals et al. (2015) introduced a general framework employing a similar content based attention with associative memory. To deal with non-sequential inputs, the input elements are stored as a unordered external memory. It uses an LSTM coupled with a dynamic attention mechanism over the memory vectors. After ‘K’ hops of the LSTM, the final output/memory retrieved from the process block does not depend on the ordering of the input elements. Vinyals et al. (2016) leverage this set framework for few-shot learning by introduce the “matching network” (MN) model. The MN model learns nearest neighbor classifier to find the closest matching image from a support set on a new unseen image. To this end, they use the same ‘process’ block from Vinyals et al. (2015) with modifications to incorporate ‘matching’. In each hop over the support set, they ‘match’ or compare the hidden state of the attLSTM with each of the support set elements. They use the output of the attLSTM to do the final support set matching.
+
+In our model, we use a set of learned prototype vectors instead of the support set of images. Both our model and the MN model uses an LSTM to learn the interactions among the items in the support set. However, the MN model uses a softmax attention and we use a sigmoid attention (due to the multi-label output). We compare a baseline model using a softmax attention (details in section 4.1).
+
+Table 2: Comparison to similar matching models   
+
+<table><tr><td></td><td>Task</td><td>Output</td><td>Comparison</td><td>Support Set</td></tr><tr><td>Vinyals et al. 2015</td><td>Few shot</td><td>Single task</td><td>Cosine Similarity</td><td>Individual support set images</td></tr><tr><td>Snell et al. 2017</td><td>Few shot</td><td>Single task</td><td>Squared Euclidean</td><td>Mean of each class from support set</td></tr><tr><td>Ours</td><td>Large-scale</td><td>Multi-task</td><td>Cosine Similarity</td><td>Learned prototype for each class</td></tr></table>
+
+Prototype Features and Prototypical Networks: While standard deep learning architectures have proven to work in many tasks, most operate under the feature-matching theory of pattern recognition where an input is decomposed into a set of features, and then are compared with those stored in the memory (Krotov & Hopfield, 2016). Prototype theory, on the other hand, proposes that objects are recognized as a whole and prototypes do not necessarily match the object precisely (Wallis et al., 2008). In this sense, the prototypes are blurred abstract representations which include all of the object’s features. Krotov & Hopfield (2016) show that pattern recognition is likely a combination of both feature-matching and prototype-matching. Transcription factors bind to motifs on DNA sequences, which we view as prototypes, the blurred features are constructed from a CNN (featurematching). Our method is motivated by the prototype-matching theory, where instead of searching for exact features to match against, the model tests an unseen sample against a set of prototypes using a defined similarity metric to make a classification.
+
+Snell et al. (2017) introduces prototypical networks for zero and one-shot learning, which assumes that the data points belonging to a particular class cluster around a single prototype. This prototype is representative for its class. In this model, each prototype embedding is the average embedding of all examples belonging to a certain class. This method is equivalent to a linear classifier if the Euclidean distance metric is used. While this method successfully learns a single prototype embedding for each class, it does not utilize a recurrent-attention mechanism over the support set. The prototypes in their method are restricted to the average embedding of its class and do not consider interactions among classes. Instead, our prototypes are the learned embedding for each label and play important roles in modelling the interactions among labels. Rippel et al. (2015) proposed a method for learning $\mathbf { k }$ -prototypes within each class, instead of just one. This in turn requires a $\mathbf { k }$ -means classifier to first group the intra-class embeddings before separating inter-class embeddings.
+
+Multi-label Classification in Deep Learning: Multi-label classification is receiving increasing attention in image classification and object recognition. In a multi-label classification task, multiple labels may be assigned to each instance. A problem transformation method for multilabel classification considers each different set of labels as a single label. Thus, it learns a binary classifier for every element in the powerset of the labels (Tsoumakas & Katakis, 2006). However, this may not be feasible in the case of a large number of labels. The most common and a more feasible problem transformation method learns a binary classifier for each label(Tsoumakas & Katakis, 2006).Gong et al. (2013) use ranking to train deep convolutional neural networks for multi-label image classification. In Hypotheses-CNN-Pooling (Wei et al., 2014), the output results from different hypotheses are aggregated with max pooling. These models treat the labels independently from each other. However, modeling the dependencies or co-occurrences of the labels is essential to gain a complete understanding of the image. To this end, Read et al. (2009) propose a chaining method to model label correlations. Xue et al. (2011), Ghamrawi & McCallum (2005) and Guo & Gu (2011) use graphical models to capture these dependencies. However, these approaches only model low order label correlations and can be computationally expensive. Wang et al. (2016) is the state-of-the-art multi-label study for object recognition. To characterize the high-order label dependencies, this model leverages the ability of an LSTM to model long-term dependency in a sequence. Briefly, the label prediction is treated as an ‘ordered prediction path’. This prediction path is essentially modeled by the RNN. While the CNN is used to extract image features, the RNN model takes the current label prediction as input at each time step and generates an embedding for the next predicted label. Although the RNN utilizes its hidden state to model the label dependencies, it is not dynamically conditioned on input samples. StarSpace Wu et al. (2017) is a method to learn entity embeddings in the input space which can handle multi-label outputs. Our method is different in that it extracts relationships directly among the outputs rather than in the input space.
+
+Table 3: Dataset Overall Summary. In each split, about $40 \%$ of its samples have more than 1 TF binding (i.e. a combination of TFs binding together), and each sample has an average of about 5.7 TFs binding.   
+
+<table><tr><td>Split</td><td>Total Samples</td><td>Co-Binding Samples</td><td>Mean # of TFs Binding Per Sample</td></tr><tr><td>Train</td><td>1446320</td><td>797475</td><td>5.62</td></tr><tr><td>Valid</td><td>331884</td><td>186509</td><td>5.75</td></tr><tr><td>Test</td><td>306297</td><td>170329</td><td>5.85</td></tr></table>
+
+![](images/466ce9ebf1c09cc16db7f48dc8a3b1f45e6c2501a9bc60fb4a1acb6c05109d42.jpg)  
+Figure 2: Dataset Per-TF Summary. For each TF, the left y-axis shows the percentage of positive samples this TF has out of all possible samples. For each TF, the right y-axis shows the percentage of positive samples which also have another TF binding.
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASETS AND EXPERIMENTAL SETUP FOR TFBS PREDICTION
+
+Dataset: We constructed our own dataset from ChIP-seq experiments in the ENCODE project database (Consortium et al., 2012). ChIP-seq experiments give binding affinity p-values for certain locations in the human genome for a specific cell type. We divided the entire human genome up into 200-length sequences, using a sliding window of 50 basepairs. We then extracted the 200-length windows surrounding the peak locations for 86 transcription factors in the human lymphoblastoid cell line (GM12878). Any peak with a measured p-value of at least 1 was considered a positive binding peak (it is important to note that we could get better results by setting a higher treshold, but we were interested in modelling all potential peaks). For each window in the genome, if any of the TF windows have a $5 5 0 \%$ overlap, we consider this a positive binding site window. We discard all windows with no TFs binding, resulting in a total of 2,084,501 binding site windows, or about $14 \%$ of the human genome.
+
+Data Statistics: We use windows in chromosomes 1, 8, and 21 as a validation set (331,884 sequences), chromosomes 3, 12, and 17 as a test set (306,297 sequences), and the rest as training (1,446,320 sequences). The number of positive training windows for each TF ranges from 793 ( $1 \%$ of training samples) to 380,824 $23 \%$ of training samples). The validation and test splits have similar percentages. About $40 \%$ of the windows have more than 1 TF binding to it, and each window has about 5 TFs binding. A overview summary of the dataset is shown in Table 3 and a per-TF summary of the dataset is shown in Figure 2.
+
+Model Variations: To test the PMN model on our TFBS dataset, we constructed 3 model variations
+
+1. CNN As commonly used in previous models (Quang & Xie, 2015; Lanchantin et al., 2016), we use a baseline 3-layer CNN model. We use $\{ 5 1 2 , 2 5 6 , 1 2 8 \}$ kernels of widths $\{ 9 , 5 , 3 \}$ at the 3 layers, respectively. The output of the CNN is maxpooled across the length, resulting in a final output vector of size 128. This architecture is used for both the single-label and multi-label CNN models.
+
+Table 4: TFBS Prediction Across 86 TFs in GM12878 Cell Line. We compare our PMN model to two CNN baseline models (single-label and multi-label). We use the same CNN model and extend it using prototypes and the combinationLSTM to create our PMN model. $\lambda$ represents the weighting of the prototype loss in eq. 10. Results are shown using statistics across all 86 TFs, where our PMN model outperforms the CNN models based on all 3 metrics used. The PMN also outperforms both CNN models significantly using a pairwise t-test.
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">auROC</td><td colspan="3">auPR</td><td colspan="3">Recall at 50% FDR</td></tr><tr><td>Mean</td><td>Std.</td><td>%Increase over single</td><td>Mean</td><td>Std.</td><td>%Increase over single</td><td>Mean</td><td>Std.</td><td>%Increase over single</td></tr><tr><td>CNN (single-label)</td><td>0.820</td><td>0.072</td><td>-</td><td>0.263</td><td>0.123</td><td></td><td>0.224</td><td>0.198</td><td></td></tr><tr><td>CNN (multi-label)</td><td>0.831</td><td>0.055</td><td>1.37</td><td>0.257</td><td>0.113</td><td>-2.52</td><td>0.215</td><td>0.186</td><td>-4.00</td></tr><tr><td>PMN (λ=1), no LSTM</td><td>0.830</td><td>0.057</td><td>1.30</td><td>0.267</td><td>0.116</td><td>1.22</td><td>0.231</td><td>0.197</td><td>3.09</td></tr><tr><td>PMN (入=1), softmax att</td><td>0.834</td><td>0.057</td><td>1.70</td><td>0.272</td><td>0.115</td><td>3.36</td><td>0.243</td><td>0.194</td><td>8.48</td></tr><tr><td>PMN (入=0), sigmoid att</td><td>0.837</td><td>0.055</td><td>2.13</td><td>0.271</td><td>0.113</td><td>3.00</td><td>0.229</td><td>0.186</td><td>1.92</td></tr><tr><td>PMN (入=0.5), sigmoid att</td><td>0.839</td><td>0.055</td><td>2.38</td><td>0.272</td><td>0.113</td><td>3.36</td><td>0.235</td><td>0.187</td><td>4.73</td></tr><tr><td>PMN (入=1), sigmoid att</td><td>0.840</td><td>0.054</td><td>2.45</td><td>0.270</td><td>0.114</td><td>2.47</td><td>0.234</td><td>0.187</td><td>4.17</td></tr></table>
+
+2. PMN no LSTM To demonstrate the effectiveness of the combinationLSTM module, we implement a PMN with no LSTM to iteratively update its weightings. In this model, we use eq. 5-9, except that we replace $\hat { h } ^ { k }$ in eq. 9 with $\hat { x }$ since there is no LSTM. The output (eq. 11) is then a concatenation of $r$ and $\hat { x }$ . We still use the full prototype loss $\lambda = 1$ ) in this model.
+
+3. PMN, softmax att Since softmax attention is typically used in attention models, we explored using a softmax function to replace eq. 6 from $k = 0$ until $k = K - 1$ , and then an elementwise sigmoid function (i.e. eq. 6) for the final output since it is multi-label classification.
+
+4. PMN, sigmoid att The full PMN model utilizes the LSTM module in eq. 6 over $K$ hops. We implement 3 variations of the prototype loss $( \lambda = 0 , \lambda = 0 . 5 , \lambda = 1 )$ , where $\lambda = 0$ represents no prototype loss, or random prototypes. We observed that $\lambda > 1$ did not result in improved results.
+
+We then extended the baseline CNN to use the learned prototypes $p _ { i }$ , and prototype matching LSTM (combinationLSTM). We call the combination of the CNN, prototypes, and combinationLSTM a PMN. We use $K = 5$ hops for the combinationLSTM because each sample has on average 5 positive label outputs. We also compared against a baseline single-task model for each TF, which assumes no interactions among TFs.
+
+Metrics: We use three separate metrics which are commonly used in large scale TFBS prediction. Since our labels are very unbalanced, we use area under ROC curve (auROC). However, auROC may not give a fair evaluation in unbalanced datasets (Lever et al., 2016; Ching et al., 2017). So we also use area under precision-call curve (auPR), and recall at $50 \%$ false discovery rate.
+
+# 4.2 LARGE-SCALE TFBS CLASSIFICATION RESULTS
+
+Table 4 shows the results of our models across the 86 TF labels. The joint CNN (multi-label) model outperformed the single label CNN models in auROC and auPR. The main advantage of the joint model is that it is faster than an individual model for each TF. The joint model’s improvement over the single-task models was not significant (p-value $< 0 . 0 5$ ) based on a one-tailed pairwise t-test. This is presumably because the joint model finds motifs similar among all motifs, but it doesn’t model interactions among TF labels.
+
+The PMN model outperformed both baseline CNN models in all 3 metrics. In addition, the improvement of the PMN over both CNN models was significant using a one-tailed pairwise t-test. We hypothesize that the combinationLSTM module accurately models co-binding better, leading to an increase in performance.
+
+In Figure 3, we show the per-epoch mean auROC results of the PMN vs CNN model. There are two important factors to note from this plot. First, the PMN models all outperform the baseline CNN. Second, the PMN models converge faster than the CNN. We hypothesize that the prototypes and similarity measure help the model generalize quickly. We assume this is the case since prototype matching models have been shown to work in few-shot cases (Vinyals et al., 2016; Snell et al., 2017).
+
+![](images/237bd4e7b90bc36f96b4d0cd6873a6580a6030c69092ee25d9c05aa9bb779066.jpg)  
+Figure 3: TFBS Per-Epoch Mean auROC (Left: Train, Right: Test)
+
+Table 5: Column “TF-A” represents a TF in one of the clusters obtained from hierarchical clustering. The subsequent column contains TFs that belong to the same cluster and, according to TRRUST database (Han et al., 2015), share target genes with TF-A. Remaining columns show the number of overlapping target genes between each TF and TF-A pair as well as their p-values for TF-TF cooperativity obtained from TRRUST. An interesting thing to note here is that in Cluster 1 and 2, TF that is away from TF-A in the cluster has lower p-value than the TF that is closer.
+
+<table><tr><td rowspan=1 colspan=1>Cluster</td><td rowspan=1 colspan=1>TF-A</td><td rowspan=1 colspan=1>TFssharingtargets</td><td rowspan=1 colspan=1>No.oftargetgenes</td><td rowspan=1 colspan=1>P-value</td></tr><tr><td rowspan=3 colspan=1>1</td><td rowspan=3 colspan=1>MYC</td><td rowspan=1 colspan=1>EP300</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>1.08E-10</td></tr><tr><td rowspan=1 colspan=1>USF1</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>5.77E-04</td></tr><tr><td rowspan=1 colspan=1>E2F4</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>3.93E-04</td></tr><tr><td rowspan=3 colspan=1>2</td><td rowspan=3 colspan=1>RELA</td><td rowspan=1 colspan=1>PAX5</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>8.86E-10</td></tr><tr><td rowspan=1 colspan=1>ATF2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>6.10E-04</td></tr><tr><td rowspan=1 colspan=1>CREBP</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>5.04E-04</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>ATF2</td><td rowspan=1 colspan=1>CREB1</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>2.69E-12</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>YY1</td><td rowspan=1 colspan=1>MAF</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>7.03E-04</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>STAT5A</td><td rowspan=1 colspan=1>BRCA1</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>8.54E-04</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>MAX</td><td rowspan=1 colspan=1>STAT3</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>9.47E-04</td></tr></table>
+
+This is also validated in TFs with the smallest amount of samples. In the $1 0 \mathrm { T F s }$ with the smallest amount of samples, the PMN $\lambda = 1 \AA$ ) has a $1 . 8 6 \%$ increase in mean auROC over the CNN. In the 10 TFs with the largest amount of samples, however, the PMN only results in an increase of $0 . 9 4 \%$ .
+
+Biological Validation of Learned Prototype Embedding: In our PMN model, we are learning a prototype for each TF that represents “motif”-like features for that TF. Each prototype is processed by the combinationLSTM which is capturing the information about TFs that bind simultaneously (or TF-TF cooperativity). Thus, we hypothesize that our prototype not only learns its TF embedding but should also reflect the TF-TF cooperativity. This information is biologically relevant as it answers a critical question in the field - “What TFs work together (or cooperate) to regulate a particular gene of interest?”. To test this hypothesis, we performed hierarchical cluster analysis on the prototypes for the 86 TFs and obtained multiple clusters containing 2 or more TFs (Figure in Appendix). Next, we searched TFs from each cluster in a reference database of human transcriptional regulatory interactions called TRRUST (Han et al., 2015). For each TF, say “TF-A”, in its database, TRRUST displays a list of TFs that regulate the same target genes as TF-A. It also shows the measures of the significance of their cooperativity as p-values, using protein-protein interactions derived from major databases. Having no expert knowledge in biology, we found some interesting results that are summarized in Table 5. We found pairs of TFs (in Clusters 1-6), whose prototypes had been clustered together, to have significant (p-value $< 0 . 0 0 0 1$ ) cooperativity curated in TRRUST database. These observations indicate that each prototype is learning sufficient combinatorial information that allows the clustering algorithm to group the TFs that cooperate during gene regulation. Therefore, the learned prototypes are not only guiding the model to better predictions but are capturing an embedding that can provide insights into the TF-TF cooperativity in the actual biological scenario.
+
+# 5 CONCLUSION
+
+Sequence analysis plays an important role in the field of bioinformatics. A prominent task is to understand how Transcription Factor proteins (TFs) bind to DNA. Researchers in biology hypothesize that each TF searches for certain sequence patterns on genome to bind to, known as “motifs”. Accordingly we propose a novel prototype matching network (PMN) for learning motif-like prototype features. On a support set of learned prototypes, we use a combinationLSTM for modeling label dependencies. The combinationLSTM tries to learn and mimic the underlying biological effects among labels (e.g. co-binding). Our results on a dataset of 2.1 million genomic strings show that the prototype matching model outperforms baseline variations not having prototype-matching or not using the combinationLSTM. This empirically validates our design choices to favor those mimicking the underlying biological mechanisms.
+
+Our PMN model is a general classification approach and not tied to the TFBS applications. We show this generality by applying it on the MNIST dataset and obtain convincing results in Appendix Section 7.1. MNIST differs from TFBS prediction in its smaller training size as well as in its multi-class properties. We plan a few future directions to extend the PMN. First, TFBSs vary across different cell types, cell stages and genomes. Extending PMN for considering the knowledge transfer is especially important for unannotated cellular contexts (e.g., cell types of rare diseases or rare organisms). Another direction is to add more domain-specific features. While we show that using prototype matching and the combinationLSTM can help modelling TF combinations, there are additional raw feature extraction methods that we could add in order to obtain better representations of genomics sequences. These include reverse complement sequence inputs or convolutional parameter sharing (Shrikumar et al., 2017), or an RNN to model lower level spatial interactions (Quang & Xie, 2015; Lanchantin et al.) among motifs.
+
+# 6 ACKNOWLEDGEMENTS
+
+We would like to thank Dr. Chongzhi Zang from the University of Virginia Medical School for helping generate the datasets and for the helpful discussions.
+
+# REFERENCES
+
+Babak Alipanahi, Andrew Delong, Matthew T Weirauch, and Brendan J Frey. Predicting the sequence specificities of DNA-and RNA-binding proteins by deep learning. 33(8):831–838, 2015a.
+
+Babak Alipanahi, Andrew Delong, Matthew T Weirauch, and Brendan J Frey. Predicting the sequence specificities of dna-and rna-binding proteins by deep learning. Nature Publishing Group, 2015b.
+
+Aaron Arvey, Phaedra Agius, William Stafford Noble, and Christina Leslie. Sequence and chromatin determinants of cell-type–specific transcription factor binding. Genome research, 22(9):1723–1734, 2012.
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014.
+
+Travers Ching, Daniel S Himmelstein, Brett K Beaulieu-Jones, Alexandr A Kalinin, Brian T Do, Gregory P Way, Enrico Ferrero, Paul-Michael Agapow, Wei Xie, Gail L Rosen, et al. Opportunities and obstacles for deep learning in biology and medicine. bioRxiv, pp. 142760, 2017.
+
+ENCODE Project Consortium et al. An integrated encyclopedia of dna elements in the human genome. Nature, 489(7414):57–74, 2012.
+
+Arnaud Gautier, Alexandre Juillerat, Christian Heinis, Ivan Reis Correa, Maik Kindermann, Florent ˆ Beaufils, and Kai Johnsson. An engineered protein tag for multiprotein labeling in living cells. Chemistry & biology, 15(2):128–136, 2008.
+
+Nadia Ghamrawi and Andrew McCallum. Collective multi-label classification. In Proceedings of the 14th ACM international conference on Information and knowledge management, pp. 195–200. ACM, 2005.
+
+Yunchao Gong, Yangqing Jia, Thomas Leung, Alexander Toshev, and Sergey Ioffe. Deep convolutional ranking for multilabel image annotation. arXiv preprint arXiv:1312.4894, 2013.
+
+Yuhong Guo and Suicheng Gu. Multi-label classification using conditional dependency networks. In IJCAI Proceedings-International Joint Conference on Artificial Intelligence, volume 22, pp. 1300, 2011.
+
+Heonjong Han, Hongseok Shim, Donghyun Shin, Jung Eun Shim, Yunhee Ko, Junha Shin, Hanhae Kim, Ara Cho, Eiru Kim, Tak Lee, et al. Trrust: a reference database of human transcriptional regulatory interactions. Scientific reports, 5:11432, 2015.
+
+Min He, Yi Wang, and Wei Li. Ppi finder: a mining tool for human protein-protein interactions. PloS one, 4(2):e4554, 2009.
+
+Majid Kazemian, Michael H Brodsky, and Saurabh Sinha. Genome surveyor 2.0: cis-regulatory analysis in drosophila. Nucleic acids research, 39(suppl 2):W79–W85, 2011.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Dmitry Krotov and John J Hopfield. Dense associative memory for pattern recognition. In Advances in Neural Information Processing Systems, pp. 1172–1180, 2016.
+
+Pavel P Kuksa, Pai-Hsi Huang, and Vladimir Pavlovic. Scalable algorithms for string kernels with inexact matching. In NIPS, pp. 881–888, 2008.
+
+Ankit Kumar, Ozan Irsoy, Peter Ondruska, Mohit Iyyer, James Bradbury, Ishaan Gulrajani, Victor Zhong, Romain Paulus, and Richard Socher. Ask me anything: Dynamic memory networks for natural language processing. In International Conference on Machine Learning, pp. 1378–1387, 2016.
+
+Jack Lanchantin, Ritambhara Singh, Beilun Wang, and Yanjun Qi. Deep motif dashboard: Visualizing and understanding genomic sequences using deep neural networks.
+
+Jack Lanchantin, Ritambhara Singh, Beilun Wang, and Yanjun Qi. Deep motif dashboard: Visualizing and understanding genomic sequences using deep neural networks. arXiv preprint arXiv:1608.03644, 2016.
+
+Daniel D Le, Tyler C Shimko, Arjun K Aditham, Allison M Keys, Yaron Orenstein, and Polly Fordyce. Comprehensive, high-resolution binding energy landscapes reveal context dependencies of transcription factor binding. bioRxiv, 2017. doi: 10.1101/193904. URL https://www. biorxiv.org/content/early/2017/09/26/193904.
+
+Christina Leslie and Rui Kuang. Fast string kernels using inexact matching for protein sequences. The Journal of Machine Learning Research, 5:1435–1455, 2004.
+
+Jake Lever, Martin Krzywinski, and Naomi Altman. Points of significance: Model selection and overfitting. Nature Methods, 13(9):703��704, 2016.
+
+Zeming Lin, Jack Lanchantin, and Yanjun Qi. Must-cnn: A multilayer shift-and-stitch deep convolutional architecture for sequence-based protein structure prediction. In Thirtieth AAAI Conference on Artificial Intelligence, 2016.
+
+Philip Machanick and Timothy L Bailey. Meme-chip: motif analysis of large dna datasets. Bioinformatics, 27(12):1696–1697, 2011.
+
+Anthony Mathelier, Xiaobei Zhao, Allen W Zhang, Franc¸ois Parcy, Rebecca Worsley-Hunt, David J Arenillas, Sorana Buchman, Chih-yu Chen, Alice Chou, Hans Ienasescu, et al. Jaspar 2014: an extensively expanded and updated open-access database of transcription factor binding profiles. Nucleic acids research, pp. gkt997, 2013.
+
+Pamela J Mitchell. Transcriptional regulation in mammalian cells by sequence-specific dna binding proteins. 1989.
+
+Peter J Park. Chip–seq: advantages and challenges of a maturing technology. Nature Reviews Genetics, 10(10):669–680, 2009.
+
+Qian Qin and Jianxing Feng. Imputation for transcription factor binding predictions based on deep learning. PLoS computational biology, 13(2):e1005403, 2017.
+
+Daniel Quang and Xiaohui Xie. Danq: a hybrid convolutional and recurrent deep neural network for quantifying the function of dna sequences. pp. 032821. Cold Spring Harbor Labs Journals, 2015.
+
+Daniel Quang and Xiaohui Xie. Danq: a hybrid convolutional and recurrent deep neural network for quantifying the function of dna sequences. Nucleic acids research, 44(11):e107–e107, 2016.
+
+Jesse Read, Bernhard Pfahringer, Geoff Holmes, and Eibe Frank. Classifier chains for multi-label classification. Machine Learning and Knowledge Discovery in Databases, pp. 254–269, 2009.
+
+Oren Rippel, Manohar Paluri, Piotr Dollar, and Lubomir Bourdev. Metric learning with adaptive density discrimination. arXiv preprint arXiv:1511.05939, 2015.
+
+Avanti Shrikumar, Peyton Greenside, and Anshul Kundaje. Reverse-complement parameter sharing improves deep learning models for genomics. bioRxiv, pp. 103663, 2017.
+
+Ritambhara Singh, Jack Lanchantin, Gabriel Robins, and Yanjun Qi. Deepchrome: deep-learning for predicting gene expression from histone modifications. Bioinformatics, 32(17):i639–i648, 2016.
+
+Saurabh Sinha and Xin He. Morph: probabilistic alignment combined with hidden markov models of cis-regulatory modules. PLoS computational biology, 3(11):e216, 2007.
+
+Jake Snell, Kevin Swersky, and Richard S Zemel. Prototypical networks for few-shot learning. arXiv preprint arXiv:1703.05175, 2017.
+
+Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research, 15(1):1929–1958, 2014.
+
+Gary D Stormo. Dna binding sites: representation and discovery. Bioinformatics, 16(1):16–23, 2000.
+
+Gary D Stormo. Modeling the specificity of protein-dna interactions. Quantitative Biology, 1(2): 115–130, 2013.
+
+Sainbayar Sukhbaatar, Jason Weston, Rob Fergus, et al. End-to-end memory networks. In Advances in neural information processing systems, pp. 2440–2448, 2015.
+
+Grigorios Tsoumakas and Ioannis Katakis. Multi-label classification: An overview. International Journal of Data Warehousing and Mining, 3(3), 2006.
+
+Juan M Vaquerizas, Sarah K Kummerfeld, Sarah A Teichmann, and Nicholas M Luscombe. A census of human transcription factors: function, expression and evolution. Nature Reviews Genetics, 10 (4):252–263, 2009.
+
+Oriol Vinyals, Samy Bengio, and Manjunath Kudlur. Order matters: Sequence to sequence for sets. arXiv preprint arXiv:1511.06391, 2015.
+
+Oriol Vinyals, Charles Blundell, Tim Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In Advances in Neural Information Processing Systems, pp. 3630–3638, 2016.
+
+Guy Wallis, Ulrike E Siebeck, Kellie Swann, Volker Blanz, and Heinrich H Bulthoff. The prototype ¨ effect revisited: Evidence for an abstract feature model of face recognition. Journal of Vision, 8(3): 20–20, 2008.
+
+Jiang Wang, Yi Yang, Junhua Mao, Zhiheng Huang, Chang Huang, and Wei Xu. Cnn-rnn: A unified framework for multi-label image classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2285–2294, 2016.
+
+Jie Wang, Jiali Zhuang, Sowmya Iyer, XinYing Lin, Troy W Whitfield, Melissa C Greven, Brian G Pierce, Xianjun Dong, Anshul Kundaje, Yong Cheng, et al. Sequence features and chromatin structure around the genomic regions bound by 119 human transcription factors. Genome research, 22(9):1798–1812, 2012.
+
+Yunchao Wei, Wei Xia, Junshi Huang, Bingbing Ni, Jian Dong, Yao Zhao, and Shuicheng Yan. Cnn: Single-label to multi-label. arXiv preprint arXiv:1406.5726, 2014.
+
+Ledell Wu, Adam Fisch, Sumit Chopra, Keith Adams, Antoine Bordes, and Jason Weston. Starspace: Embed all the things! arXiv preprint arXiv:1709.03856, 2017.
+
+Xiangyang Xue, Wei Zhang, Jie Zhang, Bin Wu, Jianping Fan, and Yao Lu. Correlative multi-label multi-instance image annotation. In Computer Vision (ICCV), 2011 IEEE International Conference on, pp. 651–658. IEEE, 2011.
+
+Jian Zhou and Olga G Troyanskaya. Deep supervised and convolutional generative stochastic network for protein secondary structure prediction. arXiv preprint arXiv:1403.1347, 2014.
+
+Jian Zhou and Olga G Troyanskaya. Predicting effects of noncoding variants with deep learning-based sequence model. volume 12, pp. 931–934. Nature Publishing Group, 2015.
+
+# 7 APPENDIX
+
+# 7.1 MNIST
+
+To validate the learned prototype matching on another task, we compare our PMN model against a standard CNN model on the MNIST dataset. We use a 3-layer CNN with $\{ 3 2 , 3 2 , 3 2 \}$ kernels of sizes $\{ 5 , 5 , 5 \}$ at the 3 layers, respectively.
+
+For the MNIST experiments, the PMN models do not show a drastic improvement over the baseline CNN. The results are shown in Table 6. However, we do note that based on the per-epoch plots in Figure 4, the PMN models do converge faster than the baseline CNN. In addition, we show in figure 5 that the PMN embeddings are better separated than the CNN embeddings. This is likely due to the fact that the PMN uses a similarity metric, and the fact that it can update its embedding based on which number prototypes it matches to. In other words, if an image looks similar to several numbers (e.g. “5” and $\mathbf { \vec { \nabla } } ^ { 6 }$ ), the PMN can update its output based on which one it matches more to. Note that although we train using a prototype loss for each class, we do not constrain the matching to only match to one prototype.
+
+Table 6: MNIST using 3-Layer CNN   
+
+<table><tr><td>Model</td><td>Accuracy</td></tr><tr><td>CNN</td><td>99.37</td></tr><tr><td>PMN (λ = 0)</td><td>99.43</td></tr><tr><td>PMN (λ = 0.5)</td><td>99.39</td></tr><tr><td>PMN (λ= 1)</td><td>99.50</td></tr></table>
+
+![](images/2a3dd723e6806592b7a987d5524ab08fda441e9d58c2a1f8b6dafa03d79eb6d6.jpg)  
+Figure 4: MNIST Per-Epoch Accuracy (Left: Train, Right: Test)
+
+![](images/9131f7771aba6bc01a7f2afa16ef9f1c1fb98809f5e635ad6f29186ffbfad9b2.jpg)  
+Figure 5: 2d T-SNE Embedding of final output vector before classification. Left: CNN, Right: PMN
+
+![](images/6b299788e16075335ddf2e3fb435f13c9bf2fa9a191fe9b59e5e01c023b7e780.jpg)  
+Figure 6: Hierarchical clustering of prototypes of 86 TFs.

parse/train/S191YzbRZ/S191YzbRZ_content_list.json

@@ -0,0 +1,1661 @@
+[
+    {
+        "type": "text",
+        "text": "PROTOTYPE MATCHING NETWORKS FOR LARGE-SCALE MULTI-LABEL GENOMIC SEQUENCE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            823,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            238,
+            544,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One of the fundamental tasks in understanding genomics is the problem of predicting Transcription Factor Binding Sites (TFBSs). With more than hundreds of Transcription Factors (TFs) as labels, genomic-sequence based TFBS prediction is a challenging multi-label classification task. There are two major biological mechanisms for TF binding: (1) sequence-specific binding patterns on genomes known as “motifs” and (2) interactions among TFs known as co-binding effects. In this paper, we propose a novel deep architecture, the Prototype Matching Network (PMN) to mimic the TF binding mechanisms. Our PMN model automatically extracts prototypes (“motif”-like features) for each TF through a novel prototypematching loss. Borrowing ideas from few-shot matching models, we use the notion of support set of prototypes and an LSTM to learn how TFs interact and bind to genomic sequences. On a reference TFBS dataset with 2.1 million genomic sequences, the PMN significantly outperforms baselines and validates our design choices empirically. To our knowledge, this is the first deep learning architecture that introduces prototype learning and considers TF-TF interactions for large scale TFBS prediction. Not only is the proposed architecture accurate, but it also models the underlying biology. ",
+        "bbox": [
+            233,
+            270,
+            766,
+            506
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            534,
+            336,
+            549
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Genomic sequences build the basis of a large body of research on understanding the biological processes in living organisms. Enabling machines to read and comprehend genomes is a longstanding and unfulfilled goal of computational biology. One of the fundamental task to understand genomes is the problem of predicting Transcription Factor Binding Sites (TFBSs), attracting much attention over the years (Consortium et al., 2012). Transcription Factors (TFs) are proteins which bind (i.e., attach) to DNA and control whether a gene is expressed or not. Patterns of how different genes expressed or not expressed control many important biological phenomena, including diseases such as cancer. Therefore accurate models for identifying and describing the binding sites of TFs are essential in understanding cells. ",
+        "bbox": [
+            174,
+            565,
+            825,
+            690
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Owing to the development of chromatin immunoprecipitation and massively parallel DNA sequencing (ChIP-seq) technologies (Park, 2009), maps of genome-wide binding sites are currently available for multiple TFs in a few cell types across human and mouse genomes via the ENCODE (Consortium et al., 2012) database. However, ChIP-seq experiments are slow and expensive; they have not been performed for many important cell types or organisms. Therefore, computational methods to identify TFBS accurately remain essential for understanding the functioning and evolution of genomes. ",
+        "bbox": [
+            174,
+            695,
+            825,
+            780
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "An important feature of TFs is that they typically bind to sequence-specific patterns on genomes, known as “motifs” (Mitchell, 1989). Motifs are essentially a blueprint, or a “prototype” which a TF searches for in order to bind. However, motifs are only one part in determining whether or not a TF will bind to specific locations. If a TF binds in the absence of its motif, or it does not bind in the presence of its motif, then it is likely there are some external causes such as an interaction with another TF, known as co-binding effects in biology (Wang et al., 2012). This indicates that when designing a genomic-sequence based TFBS predictor, we should consider two modeling challenges: (1) how to automatically extract “motifs”-like features and (2) how to model the co-binding patterns and consider such patterns in predicting TFBSs. In this paper, we address both proposing a novel deep-learning model: prototype matching network (PMN). ",
+        "bbox": [
+            174,
+            785,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To address the first challenge of motif learning and matching, many bioinformatics studies tried to predict TFBSs by constructing motifs using position weight matrices (PWMs) which best represented the positive binding sites. To test a sequence for binding, the sequence is compared against the PWMs to see if there is a close match (Stormo, 2000). PWM-matching was later outperformed by convolutional neural network (CNN) and CNN-variant models that can learn PWM-like filters Alipanahi et al. (2015a). Different from basic CNNs, our proposed PMN is inspired by the idea of “prototype-matching” (Wallis et al., 2008; Krotov & Hopfield, 2016). These studies refer to the CNN type of model as the “feature-matching” mode of pattern recognition. While pure feature matching has proven effective, studies have shown a “prototype effect” where objects are likely recognized as a whole using a similarity measure from a blurred prototype representation, and prototypes do not necessarily match the object precisely (Wallis et al., 2008). It is plausible that humans use a combination of feature matching and prototype matching where feature-matching is used to construct a prototype for testing unseen samples (Krotov & Hopfield, 2016). For TFBS prediction, the underlying biology evidently favors computation models that can learn “prototypes” (i.e. effective motifs). Although motifs are indirectly learned in convolutional layers, existing deep learning studies of TFBS (details in Section 3) have not considered the angle of “motif-matching” using a similarity measure. We, instead, propose a novel prototype-matching loss to learn prototype embedding automatically for each TF involved in the data. ",
+        "bbox": [
+            174,
+            104,
+            825,
+            353
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "None of the previous deep-learning studies for TFBS predictions have considered tackling the second challenge of including the co-binding effects among TFs in data modeling. From a machine learning angle, the genomic sequence based TFBS prediction is a multi-label sequence classification task. Rather than learning a prediction model for each TF (i.e., each label) predicting if the TF will bind or not on input, a joint model is ideal for outputting how a genomic sequence input is attached by a set of TFs (i.e., labels). The so-called “co-binding effects” connect deeply to how to model the dependency and combinations of TFs (labels). Multi-label classification is receiving increasing attention in deep learning (Wang et al., 2016; Guo & Gu, 2011; Wei et al., 2014) (detailed review in Section 3). Modeling the multi-label formulation for TFBS is an extremely challenging task because the number of labels (TFs) is in hundreds to thousands (e.g. 1,391 TFs in Vaquerizas et al. (2009)). The classic solution for multi-label classification using the powerset idea (i.e., the set of all subsets of the label set) is clearly not feasible (Tsoumakas & Katakis, 2006). Possible prior information about TF-TF interactions is unknown or limited in the biology literature. ",
+        "bbox": [
+            174,
+            358,
+            825,
+            539
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To tackle these obstacles, our proposed model PMN borrows ideas from the memory network and attention literature. Vinyals et al. (2016) proposed a “matching network” model where they train a differentiable nearest neighbor model to find the closest matching image from a support set on a new unseen image. They use a CNN to extract features and then match those features against the support set images. We replace this support set of images with a learned support set of prototypes from the large-scale training set of TFBS prediction, and we use this support set to match against a new test sample. The key difference is that our PMN model is not for few-shot learning and we seek to learn the support set (prototypes). Vinyals et al. (2016) uses an attentionLSTM to model how a test sample matches to different items in the support set through softmax based attention. Differently, we use what we call a combinationLSTM to model how the embedding of a test sample matches to a combination of relevant prototypes. Using multiple “hops”, the combinationLSTM updates the embedding of the input sequence by searching for which TFs (prototypes) are more relevant in the label combination. Instead of explicitly modeling interactions among labels, we try to use the combinationLSTM to mimic the underlying biology. The combinationLSTM tries to learn prototype embedding and represent high-order label combinations through a weighted sum of prototype embedding. This weighted summation can model many “co-binding effects” reported in the biology literature (Wang et al., 2012) (details in Section 2). ",
+        "bbox": [
+            174,
+            544,
+            825,
+            780
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In summary, we propose a novel PMN model by combining few-shot matching and prototype feature learning. To our knowledge, this is the first deep learning architecture to model TF-TF interactions in an end-to-end model. In addition, this is also the first paper to introduce large scale prototype learning using a deep learning architecture. On a reference TFBS dataset with 2.1 million genomic sequences, PMN significantly outperforms the state-of-the-art TFBS prediction baselines. We validate the learned prototypes through an existing database about TF-TF interactions. The TF groups obtained by clustering prototype embedding evidently captures the “cooperative effects” that has not been modeled by previous TFBS prediction works. ",
+        "bbox": [
+            174,
+            785,
+            823,
+            896
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The main contributions of our model are: ",
+        "bbox": [
+            176,
+            901,
+            442,
+            915
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/39459ef05d7b3d48b5c251d50864383255133254f7b81f5306d86a45890e2d68.jpg",
+        "image_caption": [
+            "Figure 1: Prototype Matching Network (PMN) Model. On the left is an overview of the model. The input sequence $x$ is encoded as $\\hat { x }$ using $f$ (3-layer CNN). $\\hat { x }$ is then matched against the learned prototypes using the combinationLSTM for $K$ “hops” so that it can update its output based on TF interactions for this input sequence. The final output $\\hat { y }$ is based on a concatenation of the final updated sequence vector $h ^ { K }$ from the LSTM, and final read vector $r ^ { K }$ from the matching. On the right is a closer look at the internal aspects of the combinationLSTM. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            106,
+            818,
+            247
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• We propose a novel model by combining few-shot matching with large-scale prototype feature learning.   \n• We design a novel prototype-matching loss to learn “motif”-like features in deep learning, which is important for the TFBS prediction task.   \n• We extend matching models from the few-shot single-label task to a large-scale multi-label task for genomic sequence classification.   \n• We implement an attention LSTM module to model label interactions in a novel way.   \n• Our model favors design choices mimicking the underlying biological processes. We think such modeling strategies are more fundamental especially on datasets from biology. ",
+        "bbox": [
+            173,
+            373,
+            826,
+            501
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 PROTOTYPE MATCHING NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            521,
+            506,
+            537
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 MODEL OVERVIEW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            556,
+            348,
+            570
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a DNA sequence $x$ (composed of characters A,C,G,T) of length $T$ , we want to classify $x$ as a positive or negative binding site for each transcription factor $T F _ { 1 } , T F _ { 2 } , . . . , T F _ { \\ell }$ in our dataset (i.e. multi-label binary classification). To do this, we seek to match $x$ to a bank of $\\ell$ learned TF prototype vectors, $\\{ p _ { 1 } , . . . , \\stackrel { . } { p _ { \\ell } } \\}$ , where each prototype is loosely representative of a motif. In addition, since TFs may bind or not bind based on other TFs, we model the interactions among TFs in order to make a prediction. An overview of our model can be seen in Figure 1. ",
+        "bbox": [
+            174,
+            582,
+            826,
+            666
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 EMBEDDING THE SEQUENCE AND PROTOTYPES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            685,
+            545,
+            700
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The input sequence $\\boldsymbol { x } \\in \\mathbb { R } ^ { 4 \\times t }$ is encoded using a function $f$ (3-layer CNN, which has shown to be sufficient for genomic feature extraction) to produce sequence embedding $\\hat { x } \\in \\mathbb R ^ { d }$ : ",
+        "bbox": [
+            173,
+            713,
+            825,
+            742
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4c640251ebdb28a9f34c7f915991557858899a5e9cb58c5888374c33e0afdcab.jpg",
+        "text": "$$\n{ \\hat { x } } = f ( x )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            465,
+            753,
+            532,
+            771
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Each prototype vector $p _ { i } \\in \\mathbb { R } ^ { d }$ is learned via a lookup table with a constant input integer at each position (e.g. 1 as input to the first position and $t$ as input to position $t$ ). I.e. the prototypes are produced by a multiplication of the identity matrix I and the learned lookup table matrix $W \\in$ $\\mathbb { R } ^ { | T F s | \\times d }$ . ",
+        "bbox": [
+            174,
+            786,
+            823,
+            843
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a9dfcb7c45baf76f789980db4d590faaaada8f744a45669b1f38c1d53ee029fa.jpg",
+        "text": "$$\nP = \\mathrm { I } W\n$$",
+        "text_format": "latex",
+        "bbox": [
+            467,
+            849,
+            531,
+            864
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Since our learned prototypes, $p _ { i }$ are randomly initialized, we introduce a prototype matching loss ${ \\mathcal { L } } _ { p }$ , which forces a prototype to correspond to a specific TF. Our prototype matching loss is explained in section 2.4. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            922
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 LSTM TO LEARN LABEL INTERACTIONS AND TO UPDATE THE SEQUENCE EMBEDDING ",
+        "bbox": [
+            173,
+            103,
+            820,
+            118
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Once we have the sequence and prototype embedding vectors, we want to compare the sequence to the prototypes to get binding site probabilities for each TF. The main idea is that we want to modify the sequence embedding $\\hat { x }$ conditioned on matching against the prototypes. Since interactions among TFs influence binding, we cannot simply match the sequence to the prototypes. To obtain TF interactions, we use an LSTM (combinationLSTM), similar to the attention LSTM (attLSTM) in Vinyals et al. (2016). Our combinationLSTM is what does the actual matching for classification, whereas Vinyals et al. (2016) use the output of the attLSTM to make the final matching prediction. The combinationLSTM uses $K$ “hops” to process the prototypes $p _ { 1 } , p _ { 2 } , . . . , p _ { \\ell }$ by matching against an updated sequence embedding $\\hat { h } ^ { k }$ . The hops allow the combinationLSTM to update the output vector based on which TFs match simultaneously. At each hop, the LSTM accepts a constant $\\hat { x }$ , a concatenation of the previous LSTM hidden state $\\bar { h } ^ { k - 1 }$ and read vector r $r ^ { k - 1 }$ , as well as the previous LSTM cell output $c ^ { k - 1 }$ . $h ^ { 0 }$ and $c ^ { 0 }$ are initialized with zeros, and $r ^ { 0 }$ is initialized with the mean of all prototype vectors, ${ \\frac { 1 } { | p | } } \\sum _ { i } ^ { | p | } p _ { i }$ . ",
+        "bbox": [
+            173,
+            127,
+            826,
+            318
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The output hidden state $\\hat { h } ^ { k }$ is matched against each prototype using cosine similarity, producing a similarity score. Since this similarity is in the range [-1,1], we feed this output through sigmoid function, weighted by hyperparameter $\\epsilon$ (we use $\\scriptstyle \\epsilon = 2 0$ ) to produce the similarity score $w _ { i } ^ { k }$ at hop $k$ in [0,1]. The read vector $r$ is updated by a weighted sum of the prototype vectors using the matching scores. At each hop, $h ^ { k }$ is updated using the current LSTM output hidden state and the sequence embedding. ",
+        "bbox": [
+            173,
+            324,
+            825,
+            409
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/00bc2f9786f2f5b70ec7344b185eefcea8d5fe30e5cf7f04919debc09a71b4e4.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { h } ^ { k } , c ^ { k } = \\mathrm { L S T M } ( \\hat { x } , [ h ^ { k - 1 } ; r ^ { k - 1 } ] , c ^ { k - 1 } ) } \\\\ & { \\quad \\quad h ^ { k } = \\hat { h } ^ { k } + \\hat { x } } \\\\ & { \\quad \\quad r ^ { k } = \\displaystyle \\sum _ { i = 1 } ^ { \\lfloor p \\rfloor } w _ { i } ^ { k } p _ { i } } \\\\ & { \\quad \\quad w _ { i } ^ { k } = 1 / \\big ( 1 + e ^ { - \\epsilon c ( \\hat { h } ^ { k } , p _ { i } ) } \\big ) } \\\\ & { \\quad \\quad c ( u , v ) = \\frac { u \\cdot v } { | | u | | _ { 2 } | | v | | _ { 2 } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            430,
+            632,
+            577
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the TFBS task, eq. 5 is the important factor for modelling TF combinations. The output vector $r$ can model multiple prototypes matching at once through a linear combination. Furthermore, the LSTM with $K$ hops is needed because of the fact that a TF binding may influence other TFs in a sequential manner. For example, if $T F _ { i }$ matches to $\\hat { h } ^ { k }$ in the first hop, $r ^ { k }$ is then used to output $\\hat { h } ^ { k + 1 }$ which can match to $T F _ { j }$ at the next hop. In this case, $\\hat { h } ^ { k + 1 }$ is a joint representation of $\\hat { x }$ and the current matched prototypes, represented by $r ^ { k }$ . At each hop, the LSTM fine-tunes $w ^ { k }$ in order to find $T F$ binding combinations. ",
+        "bbox": [
+            173,
+            584,
+            825,
+            689
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The final output $\\hat { y } \\in \\mathbb { R } ^ { | T F s | } .$ is computed from a concatenation of the final hidden state and read vectors $[ h ^ { k } ; r ^ { K } ]$ after the $K ^ { t h }$ hop using a linear transform and an element-wise sigmoid function to get a probability of binding for each TF. : ",
+        "bbox": [
+            174,
+            694,
+            825,
+            737
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/537eeb2d40ddc967ca60769aab14850262921154fb373f09254417fa8e1d3e4d.jpg",
+        "text": "$$\n\\begin{array} { l } { { o = W ( [ h ^ { K } ; r ^ { K } ] ) } } \\\\ { { \\hat { y } = 1 / ( 1 + e ^ { - o } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            742,
+            560,
+            784
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 CLASSIFICATION AND PROTOTYPE MATCHING LOSS FUNCTIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            795,
+            663,
+            810
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To classify a sequence, we use a standard binary cross entropy loss between each label $y _ { i }$ for $T F _ { i }$ and the corresponding $T F _ { i }$ output $\\hat { y } _ { i }$ , which we call the classification loss, $\\mathcal { L } _ { c }$ , for each label. ",
+        "bbox": [
+            174,
+            819,
+            821,
+            848
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We also introduce a prototype matching loss ${ \\mathcal { L } } _ { p }$ , which forces a prototype to correspond to a specific TF since prototypes are learned from random initializations. The prototype matching loss works by using an $\\mathrm { L _ { 2 } }$ between the true label $y _ { i }$ for $T F _ { i }$ and the final matching weight $w _ { i } ^ { K }$ between updated sequence $h ^ { K }$ and prototype $p _ { i }$ . This loss forces a prototype to match to all of its positive binding sequences. Each $\\dot { w } _ { i } ^ { K }$ is from the final prototype matching weights after the $K ^ { t h }$ hop from Eq (6). ",
+        "bbox": [
+            174,
+            852,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Comparison of previous deep-learning studies for TFBS and three closely related deep learning papers in the recent literature. The columns indicate properties: (1) whether the study has a joint deep architecture for multi-label prediction or not, (2) if the study learns prototype features (”motifs” in the TFBS literature), (3) whether the study models how input samples match prototypes, (4) if it uses RNN to model high-order combinations of labels, and finally (5) if the method considers current sample inputs for modeling label combinations. All previous TFBS studies do not model label interactions. PMN combines several key strategies from deep learning literature including: (a) learning label-specific prototype embedding (Snell et al., 2017) through prototype-matching loss, (b) using RNN to model higher-order label combinations (Wang et al., 2016), and (c) using LSTM to model such combinations dynamically (conditioned on the current input) (Vinyals et al., 2016). PMN is the only model that exhibits all desirable properties. ",
+        "bbox": [
+            173,
+            99,
+            825,
+            253
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/2ea72f0c3626f5ff9b3d75edd08c1bf7a82df9705120d5f0a303f226e5080dbd.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Multi-Label Joint Model</td><td>Task</td><td>Prototypes (Motifs) perLabel</td><td>Prototype Matching Loss</td><td>RNN for Label Combinations (Co-Binding)</td><td>Dynamic Label Combinations</td></tr><tr><td rowspan=\"4\">DeepBind (Alipanahi et al.,2015a) DeepSEA (Zhou &amp; Troyanskaya,2015) DanQ(Quang&amp; Xie,2016)</td><td>×</td><td>TFBS</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>√</td><td>TFBS</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>√</td><td>TFBS</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>X1</td><td>TFBS</td><td>√</td><td>×</td><td>×</td><td>×</td></tr><tr><td>TFImpute(Qin&amp; Feng,2017) CNN-RNN (Wang et al.,2016)</td><td>√</td><td>Image</td><td>√</td><td>×</td><td>√</td><td>×</td></tr><tr><td>Memory-Matching (Vinyals et al.,2016)</td><td>×</td><td>FewShot</td><td>×</td><td>√</td><td>x2</td><td>大</td></tr><tr><td>Prototypical-Network (Snell et al., 2017)</td><td>×</td><td>FewShot</td><td>√</td><td>×</td><td>×</td><td>×</td></tr><tr><td>Prototype Matching Network: (this paper)</td><td>√</td><td>TFBS</td><td>√</td><td>√</td><td>√</td><td>√</td></tr></table>",
+        "bbox": [
+            173,
+            265,
+            825,
+            383
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The important thing is that the loss is computed from the final weights $w ^ { K }$ . This allows the LSTM to attend to certain TFs at different hops before making its final decision, modeling the co-binding of TFs. The hyperparameter $\\lambda$ controls the amount that each prototype is mapped to a specific TF. $\\lambda { = } 0$ corresponds to random prototypes since we are not forcing $p _ { i }$ to match to a specific sequence. ",
+        "bbox": [
+            174,
+            407,
+            825,
+            463
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Thus, the final loss $\\mathcal { L }$ is a summation of both the classification loss and the prototype matching loss: ",
+        "bbox": [
+            173,
+            468,
+            823,
+            483
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/db0ba4701fa617f395e57436a95036655f0461728cd819263c387aaabde6a6e9.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } = - \\mathcal { L } _ { c } - \\lambda \\mathcal { L } _ { p } } \\\\ & { \\mathcal { L } _ { c } = \\displaystyle \\sum _ { i } ^ { | p | } ( y _ { i } \\log \\hat { y } _ { i } + ( 1 - y _ { i } ) \\log ( 1 - \\hat { y } _ { i } ) ) } \\\\ & { \\mathcal { L } _ { p } = \\displaystyle \\sum _ { i } ^ { | p | } ( y _ { i } - w _ { i } ^ { K } ) ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            487,
+            637,
+            599
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.5 TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            611,
+            287,
+            626
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We trained our model using Adam (Kingma & Ba, 2014) with a batch size of 512 sequences for 40 epochs. Our results were based on the test set results from the best performing validation epoch. We use dropout (Srivastava et al., 2014) for regularization. ",
+        "bbox": [
+            174,
+            635,
+            825,
+            678
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            695,
+            354,
+            712
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Deep learning in bioinformatics: Deep learning is steadily gaining popularity in the bioinformatics community. This trend is credited to their ability to extract meaningful representations from large datasets. For instance, multiple recent studies have successfully used deep learning for modeling protein sequences (Lin et al., 2016; Zhou & Troyanskaya, 2014), modeling DNA sequences (Alipanahi et al., 2015b; Lanchantin et al., 2016), predicting gene expression (Singh et al., 2016), as well as understanding the effects of non-coding variants (Zhou & Troyanskaya, 2015; Quang & Xie, 2016)). ",
+        "bbox": [
+            173,
+            724,
+            825,
+            810
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Previous Studies of TFBS and more: Previous techniques for predicting TFBS include many sequence-motif based computational approaches that typically use position-based sequence information (Stormo, 2000). Relying on a set of known transcription factor binding sites (TFBSs) for a given ",
+        "bbox": [
+            174,
+            819,
+            825,
+            862
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "TF, the binding preference is generally represented in the form of a position weight matrix (PWM) (Stormo, 2013; Mathelier et al., 2013) (also called position-specific scoring matrix) derived from a position frequency matrix (PFM). Recently this technique was outperformed by different variations of deep convolutional models (Alipanahi et al., 2015b; Lanchantin et al., 2016; Quang & Xie, 2016; Shrikumar et al., 2017). While motif-based PWMs are compact and interpretable, they can under-fit ChIP-seq data by failing to capture subtle but detectable and important sequence signals, such as direct DNA-binding preferences of certain TFs, cofactor binding sequences, accessibility signals, or other discriminative sequence features (Arvey et al., 2012; Le et al., 2017). ",
+        "bbox": [
+            173,
+            103,
+            825,
+            215
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1 summarizes four most relevant deep learning studies of TFBS in the literature. Alipanahi et al. (2015b) was the first to use a deep learning based approach to predict TFBSs. They showed that a 1-layer CNN could outperform baseline motif matching approaches which used position weight matrices Machanick & Bailey (2011). Zhou & Troyanskaya (2015) used a similar method to predict the effects of variants in noncoding regions of the genome, where TFBS prediction was an intermediate step to predict variant effects. They used a 3-layer CNN model to predict 919 chromatin labels (including 690 TFBS labels). Quang & Xie (2015) extended this model using a bidirectional LSTM on top of the CNN outputs to model interactions among motifs. Note that this is not modeling interactions among labels (TFs), but rather among sequence features. Qin & Feng (2017) uses a similar lookup table approach for learning a representation for a TF, but their implementation is for transferring between cell lines where the target cell line has no experimental data. However, ChIP-seq experiments are relatively cheap given a specific TF and sequence of interest. We are more interested in modeling the underlying biology. ",
+        "bbox": [
+            174,
+            219,
+            825,
+            401
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Aside from deep learning models, there have been other works to model joint dependencies among TFs Kazemian et al. (2011); He et al. (2009); Gautier et al. (2008); Sinha & He (2007). Our primary goal is to extend previous deep learning methods to better model co-binding, thus we do not compare against these approaches. ",
+        "bbox": [
+            174,
+            406,
+            825,
+            462
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The formulation of TFBS prediction belongs to a general category of “biological sequence classification”. Sequence analysis plays an important role in the field of bioinformatics. Various methods have previously been proposed, including generative (e.g., Hidden Markov Models-HMMs) and discriminative approaches. Among the discriminative approaches, string kernel methods provide some of the most accurate results, such as for remote protein fold and homology detections (Leslie & Kuang, 2004; Kuksa et al., 2008). We omit a full survey of this topic due to its vast body of previous literature and loose connection to our TFBS formulations. ",
+        "bbox": [
+            174,
+            467,
+            825,
+            564
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Memory Matching Network and Attention RNN: Attention in combination with RNNs has been successfully used for many tasks Bahdanau et al. (2014). Various methods have extended the single step attention approach to attention with multiple ‘hops’ Sukhbaatar et al. (2015); Kumar et al. (2016). For example, for the Question Answering task, Kumar et al. (2016) introduced the Dynamic Memory Network which uses an iterative attention process coupled with an RNN over multiple ‘episodes’. Kumar et al. (2016) show that the attention gets more focused over successive hops or ‘episodes’ over the input, reaffirming the need for recurrent episodes. Most of these models are for sequential inputs. For tasks that involve a set (i.e., no order) as input or/and as output, Vinyals et al. (2015) introduced a general framework employing a similar content based attention with associative memory. To deal with non-sequential inputs, the input elements are stored as a unordered external memory. It uses an LSTM coupled with a dynamic attention mechanism over the memory vectors. After ‘K’ hops of the LSTM, the final output/memory retrieved from the process block does not depend on the ordering of the input elements. Vinyals et al. (2016) leverage this set framework for few-shot learning by introduce the “matching network” (MN) model. The MN model learns nearest neighbor classifier to find the closest matching image from a support set on a new unseen image. To this end, they use the same ‘process’ block from Vinyals et al. (2015) with modifications to incorporate ‘matching’. In each hop over the support set, they ‘match’ or compare the hidden state of the attLSTM with each of the support set elements. They use the output of the attLSTM to do the final support set matching. ",
+        "bbox": [
+            173,
+            574,
+            825,
+            824
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our model, we use a set of learned prototype vectors instead of the support set of images. Both our model and the MN model uses an LSTM to learn the interactions among the items in the support set. However, the MN model uses a softmax attention and we use a sigmoid attention (due to the multi-label output). We compare a baseline model using a softmax attention (details in section 4.1). ",
+        "bbox": [
+            176,
+            830,
+            823,
+            886
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/08a2d00d5dec466d2d959196f9f474f168b4bc17deef2c54416ffa477c095107.jpg",
+        "table_caption": [
+            "Table 2: Comparison to similar matching models "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Task</td><td>Output</td><td>Comparison</td><td>Support Set</td></tr><tr><td>Vinyals et al. 2015</td><td>Few shot</td><td>Single task</td><td>Cosine Similarity</td><td>Individual support set images</td></tr><tr><td>Snell et al. 2017</td><td>Few shot</td><td>Single task</td><td>Squared Euclidean</td><td>Mean of each class from support set</td></tr><tr><td>Ours</td><td>Large-scale</td><td>Multi-task</td><td>Cosine Similarity</td><td>Learned prototype for each class</td></tr></table>",
+        "bbox": [
+            173,
+            127,
+            906,
+            185
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Prototype Features and Prototypical Networks: While standard deep learning architectures have proven to work in many tasks, most operate under the feature-matching theory of pattern recognition where an input is decomposed into a set of features, and then are compared with those stored in the memory (Krotov & Hopfield, 2016). Prototype theory, on the other hand, proposes that objects are recognized as a whole and prototypes do not necessarily match the object precisely (Wallis et al., 2008). In this sense, the prototypes are blurred abstract representations which include all of the object’s features. Krotov & Hopfield (2016) show that pattern recognition is likely a combination of both feature-matching and prototype-matching. Transcription factors bind to motifs on DNA sequences, which we view as prototypes, the blurred features are constructed from a CNN (featurematching). Our method is motivated by the prototype-matching theory, where instead of searching for exact features to match against, the model tests an unseen sample against a set of prototypes using a defined similarity metric to make a classification. ",
+        "bbox": [
+            174,
+            256,
+            825,
+            422
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Snell et al. (2017) introduces prototypical networks for zero and one-shot learning, which assumes that the data points belonging to a particular class cluster around a single prototype. This prototype is representative for its class. In this model, each prototype embedding is the average embedding of all examples belonging to a certain class. This method is equivalent to a linear classifier if the Euclidean distance metric is used. While this method successfully learns a single prototype embedding for each class, it does not utilize a recurrent-attention mechanism over the support set. The prototypes in their method are restricted to the average embedding of its class and do not consider interactions among classes. Instead, our prototypes are the learned embedding for each label and play important roles in modelling the interactions among labels. Rippel et al. (2015) proposed a method for learning $\\mathbf { k }$ -prototypes within each class, instead of just one. This in turn requires a $\\mathbf { k }$ -means classifier to first group the intra-class embeddings before separating inter-class embeddings. ",
+        "bbox": [
+            174,
+            428,
+            825,
+            580
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Multi-label Classification in Deep Learning: Multi-label classification is receiving increasing attention in image classification and object recognition. In a multi-label classification task, multiple labels may be assigned to each instance. A problem transformation method for multilabel classification considers each different set of labels as a single label. Thus, it learns a binary classifier for every element in the powerset of the labels (Tsoumakas & Katakis, 2006). However, this may not be feasible in the case of a large number of labels. The most common and a more feasible problem transformation method learns a binary classifier for each label(Tsoumakas & Katakis, 2006).Gong et al. (2013) use ranking to train deep convolutional neural networks for multi-label image classification. In Hypotheses-CNN-Pooling (Wei et al., 2014), the output results from different hypotheses are aggregated with max pooling. These models treat the labels independently from each other. However, modeling the dependencies or co-occurrences of the labels is essential to gain a complete understanding of the image. To this end, Read et al. (2009) propose a chaining method to model label correlations. Xue et al. (2011), Ghamrawi & McCallum (2005) and Guo & Gu (2011) use graphical models to capture these dependencies. However, these approaches only model low order label correlations and can be computationally expensive. Wang et al. (2016) is the state-of-the-art multi-label study for object recognition. To characterize the high-order label dependencies, this model leverages the ability of an LSTM to model long-term dependency in a sequence. Briefly, the label prediction is treated as an ‘ordered prediction path’. This prediction path is essentially modeled by the RNN. While the CNN is used to extract image features, the RNN model takes the current label prediction as input at each time step and generates an embedding for the next predicted label. Although the RNN utilizes its hidden state to model the label dependencies, it is not dynamically conditioned on input samples. StarSpace Wu et al. (2017) is a method to learn entity embeddings in the input space which can handle multi-label outputs. Our method is different in that it extracts relationships directly among the outputs rather than in the input space. ",
+        "bbox": [
+            174,
+            590,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/82843c12c152ad18beedeff8a53ea3816c99b7248766e1b134b73aac14528f27.jpg",
+        "table_caption": [
+            "Table 3: Dataset Overall Summary. In each split, about $40 \\%$ of its samples have more than 1 TF binding (i.e. a combination of TFs binding together), and each sample has an average of about 5.7 TFs binding. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Split</td><td>Total Samples</td><td>Co-Binding Samples</td><td>Mean # of TFs Binding Per Sample</td></tr><tr><td>Train</td><td>1446320</td><td>797475</td><td>5.62</td></tr><tr><td>Valid</td><td>331884</td><td>186509</td><td>5.75</td></tr><tr><td>Test</td><td>306297</td><td>170329</td><td>5.85</td></tr></table>",
+        "bbox": [
+            272,
+            155,
+            720,
+            242
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/466ce9ebf1c09cc16db7f48dc8a3b1f45e6c2501a9bc60fb4a1acb6c05109d42.jpg",
+        "image_caption": [
+            "Figure 2: Dataset Per-TF Summary. For each TF, the left y-axis shows the percentage of positive samples this TF has out of all possible samples. For each TF, the right y-axis shows the percentage of positive samples which also have another TF binding. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            247,
+            825,
+            411
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            500,
+            326,
+            515
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1 DATASETS AND EXPERIMENTAL SETUP FOR TFBS PREDICTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            536,
+            655,
+            550
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Dataset: We constructed our own dataset from ChIP-seq experiments in the ENCODE project database (Consortium et al., 2012). ChIP-seq experiments give binding affinity p-values for certain locations in the human genome for a specific cell type. We divided the entire human genome up into 200-length sequences, using a sliding window of 50 basepairs. We then extracted the 200-length windows surrounding the peak locations for 86 transcription factors in the human lymphoblastoid cell line (GM12878). Any peak with a measured p-value of at least 1 was considered a positive binding peak (it is important to note that we could get better results by setting a higher treshold, but we were interested in modelling all potential peaks). For each window in the genome, if any of the TF windows have a $5 5 0 \\%$ overlap, we consider this a positive binding site window. We discard all windows with no TFs binding, resulting in a total of 2,084,501 binding site windows, or about $14 \\%$ of the human genome. ",
+        "bbox": [
+            174,
+            564,
+            825,
+            715
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Data Statistics: We use windows in chromosomes 1, 8, and 21 as a validation set (331,884 sequences), chromosomes 3, 12, and 17 as a test set (306,297 sequences), and the rest as training (1,446,320 sequences). The number of positive training windows for each TF ranges from 793 ( $1 \\%$ of training samples) to 380,824 $23 \\%$ of training samples). The validation and test splits have similar percentages. About $40 \\%$ of the windows have more than 1 TF binding to it, and each window has about 5 TFs binding. A overview summary of the dataset is shown in Table 3 and a per-TF summary of the dataset is shown in Figure 2. ",
+        "bbox": [
+            174,
+            727,
+            825,
+            824
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Model Variations: To test the PMN model on our TFBS dataset, we constructed 3 model variations ",
+        "bbox": [
+            179,
+            834,
+            823,
+            848
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "1. CNN As commonly used in previous models (Quang & Xie, 2015; Lanchantin et al., 2016), we use a baseline 3-layer CNN model. We use $\\{ 5 1 2 , 2 5 6 , 1 2 8 \\}$ kernels of widths $\\{ 9 , 5 , 3 \\}$ at the 3 layers, respectively. The output of the CNN is maxpooled across the length, resulting in a final output vector of size 128. This architecture is used for both the single-label and multi-label CNN models. ",
+        "bbox": [
+            215,
+            854,
+            823,
+            922
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 4: TFBS Prediction Across 86 TFs in GM12878 Cell Line. We compare our PMN model to two CNN baseline models (single-label and multi-label). We use the same CNN model and extend it using prototypes and the combinationLSTM to create our PMN model. $\\lambda$ represents the weighting of the prototype loss in eq. 10. Results are shown using statistics across all 86 TFs, where our PMN model outperforms the CNN models based on all 3 metrics used. The PMN also outperforms both CNN models significantly using a pairwise t-test. ",
+        "bbox": [
+            173,
+            99,
+            825,
+            185
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/70cf0b38cd40a9e2535fa6cb2c022144c3a7aa652fdb8505c6f629d75684d377.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"3\">auROC</td><td colspan=\"3\">auPR</td><td colspan=\"3\">Recall at 50% FDR</td></tr><tr><td>Mean</td><td>Std.</td><td>%Increase over single</td><td>Mean</td><td>Std.</td><td>%Increase over single</td><td>Mean</td><td>Std.</td><td>%Increase over single</td></tr><tr><td>CNN (single-label)</td><td>0.820</td><td>0.072</td><td>-</td><td>0.263</td><td>0.123</td><td></td><td>0.224</td><td>0.198</td><td></td></tr><tr><td>CNN (multi-label)</td><td>0.831</td><td>0.055</td><td>1.37</td><td>0.257</td><td>0.113</td><td>-2.52</td><td>0.215</td><td>0.186</td><td>-4.00</td></tr><tr><td>PMN (λ=1), no LSTM</td><td>0.830</td><td>0.057</td><td>1.30</td><td>0.267</td><td>0.116</td><td>1.22</td><td>0.231</td><td>0.197</td><td>3.09</td></tr><tr><td>PMN (入=1), softmax att</td><td>0.834</td><td>0.057</td><td>1.70</td><td>0.272</td><td>0.115</td><td>3.36</td><td>0.243</td><td>0.194</td><td>8.48</td></tr><tr><td>PMN (入=0), sigmoid att</td><td>0.837</td><td>0.055</td><td>2.13</td><td>0.271</td><td>0.113</td><td>3.00</td><td>0.229</td><td>0.186</td><td>1.92</td></tr><tr><td>PMN (入=0.5), sigmoid att</td><td>0.839</td><td>0.055</td><td>2.38</td><td>0.272</td><td>0.113</td><td>3.36</td><td>0.235</td><td>0.187</td><td>4.73</td></tr><tr><td>PMN (入=1), sigmoid att</td><td>0.840</td><td>0.054</td><td>2.45</td><td>0.270</td><td>0.114</td><td>2.47</td><td>0.234</td><td>0.187</td><td>4.17</td></tr></table>",
+        "bbox": [
+            173,
+            196,
+            825,
+            321
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "2. PMN no LSTM To demonstrate the effectiveness of the combinationLSTM module, we implement a PMN with no LSTM to iteratively update its weightings. In this model, we use eq. 5-9, except that we replace $\\hat { h } ^ { k }$ in eq. 9 with $\\hat { x }$ since there is no LSTM. The output (eq. 11) is then a concatenation of $r$ and $\\hat { x }$ . We still use the full prototype loss $\\lambda = 1$ ) in this model. ",
+        "bbox": [
+            212,
+            349,
+            825,
+            421
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "3. PMN, softmax att Since softmax attention is typically used in attention models, we explored using a softmax function to replace eq. 6 from $k = 0$ until $k = K - 1$ , and then an elementwise sigmoid function (i.e. eq. 6) for the final output since it is multi-label classification. ",
+        "bbox": [
+            210,
+            422,
+            825,
+            463
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4. PMN, sigmoid att The full PMN model utilizes the LSTM module in eq. 6 over $K$ hops. We implement 3 variations of the prototype loss $( \\lambda = 0 , \\lambda = 0 . 5 , \\lambda = 1 )$ , where $\\lambda = 0$ represents no prototype loss, or random prototypes. We observed that $\\lambda > 1$ did not result in improved results. ",
+        "bbox": [
+            212,
+            464,
+            825,
+            518
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We then extended the baseline CNN to use the learned prototypes $p _ { i }$ , and prototype matching LSTM (combinationLSTM). We call the combination of the CNN, prototypes, and combinationLSTM a PMN. We use $K = 5$ hops for the combinationLSTM because each sample has on average 5 positive label outputs. We also compared against a baseline single-task model for each TF, which assumes no interactions among TFs. ",
+        "bbox": [
+            174,
+            525,
+            825,
+            594
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Metrics: We use three separate metrics which are commonly used in large scale TFBS prediction. Since our labels are very unbalanced, we use area under ROC curve (auROC). However, auROC may not give a fair evaluation in unbalanced datasets (Lever et al., 2016; Ching et al., 2017). So we also use area under precision-call curve (auPR), and recall at $50 \\%$ false discovery rate. ",
+        "bbox": [
+            174,
+            604,
+            825,
+            660
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.2 LARGE-SCALE TFBS CLASSIFICATION RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            678,
+            552,
+            693
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 4 shows the results of our models across the 86 TF labels. The joint CNN (multi-label) model outperformed the single label CNN models in auROC and auPR. The main advantage of the joint model is that it is faster than an individual model for each TF. The joint model’s improvement over the single-task models was not significant (p-value $< 0 . 0 5$ ) based on a one-tailed pairwise t-test. This is presumably because the joint model finds motifs similar among all motifs, but it doesn’t model interactions among TF labels. ",
+        "bbox": [
+            174,
+            704,
+            825,
+            787
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The PMN model outperformed both baseline CNN models in all 3 metrics. In addition, the improvement of the PMN over both CNN models was significant using a one-tailed pairwise t-test. We hypothesize that the combinationLSTM module accurately models co-binding better, leading to an increase in performance. ",
+        "bbox": [
+            174,
+            792,
+            825,
+            849
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Figure 3, we show the per-epoch mean auROC results of the PMN vs CNN model. There are two important factors to note from this plot. First, the PMN models all outperform the baseline CNN. Second, the PMN models converge faster than the CNN. We hypothesize that the prototypes and similarity measure help the model generalize quickly. We assume this is the case since prototype matching models have been shown to work in few-shot cases (Vinyals et al., 2016; Snell et al., 2017). ",
+        "bbox": [
+            176,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/237bd4e7b90bc36f96b4d0cd6873a6580a6030c69092ee25d9c05aa9bb779066.jpg",
+        "image_caption": [
+            "Figure 3: TFBS Per-Epoch Mean auROC (Left: Train, Right: Test) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            113,
+            795,
+            287
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Column “TF-A” represents a TF in one of the clusters obtained from hierarchical clustering. The subsequent column contains TFs that belong to the same cluster and, according to TRRUST database (Han et al., 2015), share target genes with TF-A. Remaining columns show the number of overlapping target genes between each TF and TF-A pair as well as their p-values for TF-TF cooperativity obtained from TRRUST. An interesting thing to note here is that in Cluster 1 and 2, TF that is away from TF-A in the cluster has lower p-value than the TF that is closer. ",
+        "bbox": [
+            173,
+            332,
+            439,
+            511
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/368051c9f0d8aaf96c31e2439f462f76be4f321ad0eb6f3b2ef61666cc3f9417.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Cluster</td><td rowspan=1 colspan=1>TF-A</td><td rowspan=1 colspan=1>TFssharingtargets</td><td rowspan=1 colspan=1>No.oftargetgenes</td><td rowspan=1 colspan=1>P-value</td></tr><tr><td rowspan=3 colspan=1>1</td><td rowspan=3 colspan=1>MYC</td><td rowspan=1 colspan=1>EP300</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>1.08E-10</td></tr><tr><td rowspan=1 colspan=1>USF1</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>5.77E-04</td></tr><tr><td rowspan=1 colspan=1>E2F4</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>3.93E-04</td></tr><tr><td rowspan=3 colspan=1>2</td><td rowspan=3 colspan=1>RELA</td><td rowspan=1 colspan=1>PAX5</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>8.86E-10</td></tr><tr><td rowspan=1 colspan=1>ATF2</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>6.10E-04</td></tr><tr><td rowspan=1 colspan=1>CREBP</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>5.04E-04</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>ATF2</td><td rowspan=1 colspan=1>CREB1</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>2.69E-12</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>YY1</td><td rowspan=1 colspan=1>MAF</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>7.03E-04</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>STAT5A</td><td rowspan=1 colspan=1>BRCA1</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>8.54E-04</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>MAX</td><td rowspan=1 colspan=1>STAT3</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>9.47E-04</td></tr></table>",
+        "bbox": [
+            457,
+            332,
+            820,
+            501
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This is also validated in TFs with the smallest amount of samples. In the $1 0 \\mathrm { T F s }$ with the smallest amount of samples, the PMN $\\lambda = 1 \\AA$ ) has a $1 . 8 6 \\%$ increase in mean auROC over the CNN. In the 10 TFs with the largest amount of samples, however, the PMN only results in an increase of $0 . 9 4 \\%$ . ",
+        "bbox": [
+            176,
+            541,
+            825,
+            583
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Biological Validation of Learned Prototype Embedding: In our PMN model, we are learning a prototype for each TF that represents “motif”-like features for that TF. Each prototype is processed by the combinationLSTM which is capturing the information about TFs that bind simultaneously (or TF-TF cooperativity). Thus, we hypothesize that our prototype not only learns its TF embedding but should also reflect the TF-TF cooperativity. This information is biologically relevant as it answers a critical question in the field - “What TFs work together (or cooperate) to regulate a particular gene of interest?”. To test this hypothesis, we performed hierarchical cluster analysis on the prototypes for the 86 TFs and obtained multiple clusters containing 2 or more TFs (Figure in Appendix). Next, we searched TFs from each cluster in a reference database of human transcriptional regulatory interactions called TRRUST (Han et al., 2015). For each TF, say “TF-A”, in its database, TRRUST displays a list of TFs that regulate the same target genes as TF-A. It also shows the measures of the significance of their cooperativity as p-values, using protein-protein interactions derived from major databases. Having no expert knowledge in biology, we found some interesting results that are summarized in Table 5. We found pairs of TFs (in Clusters 1-6), whose prototypes had been clustered together, to have significant (p-value $< 0 . 0 0 0 1$ ) cooperativity curated in TRRUST database. These observations indicate that each prototype is learning sufficient combinatorial information that allows the clustering algorithm to group the TFs that cooperate during gene regulation. Therefore, the learned prototypes are not only guiding the model to better predictions but are capturing an embedding that can provide insights into the TF-TF cooperativity in the actual biological scenario. ",
+        "bbox": [
+            174,
+            593,
+            825,
+            857
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            864,
+            318,
+            880
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sequence analysis plays an important role in the field of bioinformatics. A prominent task is to understand how Transcription Factor proteins (TFs) bind to DNA. Researchers in biology hypothesize that each TF searches for certain sequence patterns on genome to bind to, known as “motifs”. Accordingly we propose a novel prototype matching network (PMN) for learning motif-like prototype features. On a support set of learned prototypes, we use a combinationLSTM for modeling label dependencies. The combinationLSTM tries to learn and mimic the underlying biological effects among labels (e.g. co-binding). Our results on a dataset of 2.1 million genomic strings show that the prototype matching model outperforms baseline variations not having prototype-matching or not using the combinationLSTM. This empirically validates our design choices to favor those mimicking the underlying biological mechanisms. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            214
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Our PMN model is a general classification approach and not tied to the TFBS applications. We show this generality by applying it on the MNIST dataset and obtain convincing results in Appendix Section 7.1. MNIST differs from TFBS prediction in its smaller training size as well as in its multi-class properties. We plan a few future directions to extend the PMN. First, TFBSs vary across different cell types, cell stages and genomes. Extending PMN for considering the knowledge transfer is especially important for unannotated cellular contexts (e.g., cell types of rare diseases or rare organisms). Another direction is to add more domain-specific features. While we show that using prototype matching and the combinationLSTM can help modelling TF combinations, there are additional raw feature extraction methods that we could add in order to obtain better representations of genomics sequences. These include reverse complement sequence inputs or convolutional parameter sharing (Shrikumar et al., 2017), or an RNN to model lower level spatial interactions (Quang & Xie, 2015; Lanchantin et al.) among motifs. ",
+        "bbox": [
+            174,
+            219,
+            825,
+            386
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "6 ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            406,
+            398,
+            421
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Dr. Chongzhi Zang from the University of Virginia Medical School for helping generate the datasets and for the helpful discussions. ",
+        "bbox": [
+            174,
+            438,
+            823,
+            465
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            486,
+            285,
+            501
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Babak Alipanahi, Andrew Delong, Matthew T Weirauch, and Brendan J Frey. Predicting the sequence specificities of DNA-and RNA-binding proteins by deep learning. 33(8):831–838, 2015a. ",
+        "bbox": [
+            174,
+            517,
+            823,
+            546
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Babak Alipanahi, Andrew Delong, Matthew T Weirauch, and Brendan J Frey. Predicting the sequence specificities of dna-and rna-binding proteins by deep learning. Nature Publishing Group, 2015b. ",
+        "bbox": [
+            174,
+            556,
+            820,
+            587
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Aaron Arvey, Phaedra Agius, William Stafford Noble, and Christina Leslie. Sequence and chromatin determinants of cell-type–specific transcription factor binding. Genome research, 22(9):1723–1734, 2012. ",
+        "bbox": [
+            174,
+            597,
+            825,
+            640
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014. ",
+        "bbox": [
+            171,
+            651,
+            825,
+            681
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Travers Ching, Daniel S Himmelstein, Brett K Beaulieu-Jones, Alexandr A Kalinin, Brian T Do, Gregory P Way, Enrico Ferrero, Paul-Michael Agapow, Wei Xie, Gail L Rosen, et al. Opportunities and obstacles for deep learning in biology and medicine. bioRxiv, pp. 142760, 2017. ",
+        "bbox": [
+            176,
+            691,
+            825,
+            736
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "ENCODE Project Consortium et al. An integrated encyclopedia of dna elements in the human genome. Nature, 489(7414):57–74, 2012. ",
+        "bbox": [
+            173,
+            746,
+            823,
+            775
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Arnaud Gautier, Alexandre Juillerat, Christian Heinis, Ivan Reis Correa, Maik Kindermann, Florent ˆ Beaufils, and Kai Johnsson. An engineered protein tag for multiprotein labeling in living cells. Chemistry & biology, 15(2):128–136, 2008. ",
+        "bbox": [
+            174,
+            786,
+            826,
+            829
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Nadia Ghamrawi and Andrew McCallum. Collective multi-label classification. In Proceedings of the 14th ACM international conference on Information and knowledge management, pp. 195–200. ACM, 2005. ",
+        "bbox": [
+            173,
+            840,
+            826,
+            883
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yunchao Gong, Yangqing Jia, Thomas Leung, Alexander Toshev, and Sergey Ioffe. Deep convolutional ranking for multilabel image annotation. arXiv preprint arXiv:1312.4894, 2013. ",
+        "bbox": [
+            174,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yuhong Guo and Suicheng Gu. Multi-label classification using conditional dependency networks. In IJCAI Proceedings-International Joint Conference on Artificial Intelligence, volume 22, pp. 1300, 2011. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Heonjong Han, Hongseok Shim, Donghyun Shin, Jung Eun Shim, Yunhee Ko, Junha Shin, Hanhae Kim, Ara Cho, Eiru Kim, Tak Lee, et al. Trrust: a reference database of human transcriptional regulatory interactions. Scientific reports, 5:11432, 2015. ",
+        "bbox": [
+            176,
+            155,
+            821,
+            199
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Min He, Yi Wang, and Wei Li. Ppi finder: a mining tool for human protein-protein interactions. PloS one, 4(2):e4554, 2009. ",
+        "bbox": [
+            173,
+            207,
+            823,
+            237
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Majid Kazemian, Michael H Brodsky, and Saurabh Sinha. Genome surveyor 2.0: cis-regulatory analysis in drosophila. Nucleic acids research, 39(suppl 2):W79–W85, 2011. ",
+        "bbox": [
+            173,
+            246,
+            821,
+            275
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. ",
+        "bbox": [
+            174,
+            285,
+            823,
+            314
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Dmitry Krotov and John J Hopfield. Dense associative memory for pattern recognition. In Advances in Neural Information Processing Systems, pp. 1172–1180, 2016. ",
+        "bbox": [
+            173,
+            323,
+            823,
+            352
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Pavel P Kuksa, Pai-Hsi Huang, and Vladimir Pavlovic. Scalable algorithms for string kernels with inexact matching. In NIPS, pp. 881–888, 2008. ",
+        "bbox": [
+            173,
+            361,
+            823,
+            391
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ankit Kumar, Ozan Irsoy, Peter Ondruska, Mohit Iyyer, James Bradbury, Ishaan Gulrajani, Victor Zhong, Romain Paulus, and Richard Socher. Ask me anything: Dynamic memory networks for natural language processing. In International Conference on Machine Learning, pp. 1378–1387, 2016. ",
+        "bbox": [
+            174,
+            400,
+            826,
+            457
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jack Lanchantin, Ritambhara Singh, Beilun Wang, and Yanjun Qi. Deep motif dashboard: Visualizing and understanding genomic sequences using deep neural networks. ",
+        "bbox": [
+            171,
+            465,
+            825,
+            496
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jack Lanchantin, Ritambhara Singh, Beilun Wang, and Yanjun Qi. Deep motif dashboard: Visualizing and understanding genomic sequences using deep neural networks. arXiv preprint arXiv:1608.03644, 2016. ",
+        "bbox": [
+            174,
+            505,
+            823,
+            547
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Daniel D Le, Tyler C Shimko, Arjun K Aditham, Allison M Keys, Yaron Orenstein, and Polly Fordyce. Comprehensive, high-resolution binding energy landscapes reveal context dependencies of transcription factor binding. bioRxiv, 2017. doi: 10.1101/193904. URL https://www. biorxiv.org/content/early/2017/09/26/193904. ",
+        "bbox": [
+            173,
+            556,
+            826,
+            613
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Christina Leslie and Rui Kuang. Fast string kernels using inexact matching for protein sequences. The Journal of Machine Learning Research, 5:1435–1455, 2004. ",
+        "bbox": [
+            176,
+            622,
+            823,
+            652
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jake Lever, Martin Krzywinski, and Naomi Altman. Points of significance: Model selection and overfitting. Nature Methods, 13(9):703–704, 2016. ",
+        "bbox": [
+            173,
+            661,
+            825,
+            690
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Zeming Lin, Jack Lanchantin, and Yanjun Qi. Must-cnn: A multilayer shift-and-stitch deep convolutional architecture for sequence-based protein structure prediction. In Thirtieth AAAI Conference on Artificial Intelligence, 2016. ",
+        "bbox": [
+            173,
+            699,
+            826,
+            742
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Philip Machanick and Timothy L Bailey. Meme-chip: motif analysis of large dna datasets. Bioinformatics, 27(12):1696–1697, 2011. ",
+        "bbox": [
+            169,
+            751,
+            825,
+            781
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Anthony Mathelier, Xiaobei Zhao, Allen W Zhang, Franc¸ois Parcy, Rebecca Worsley-Hunt, David J Arenillas, Sorana Buchman, Chih-yu Chen, Alice Chou, Hans Ienasescu, et al. Jaspar 2014: an extensively expanded and updated open-access database of transcription factor binding profiles. Nucleic acids research, pp. gkt997, 2013. ",
+        "bbox": [
+            174,
+            790,
+            826,
+            847
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Pamela J Mitchell. Transcriptional regulation in mammalian cells by sequence-specific dna binding proteins. 1989. ",
+        "bbox": [
+            173,
+            856,
+            823,
+            886
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Peter J Park. Chip–seq: advantages and challenges of a maturing technology. Nature Reviews Genetics, 10(10):669–680, 2009. ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Qian Qin and Jianxing Feng. Imputation for transcription factor binding predictions based on deep learning. PLoS computational biology, 13(2):e1005403, 2017. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            133
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Daniel Quang and Xiaohui Xie. Danq: a hybrid convolutional and recurrent deep neural network for quantifying the function of dna sequences. pp. 032821. Cold Spring Harbor Labs Journals, 2015. ",
+        "bbox": [
+            173,
+            141,
+            823,
+            171
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Daniel Quang and Xiaohui Xie. Danq: a hybrid convolutional and recurrent deep neural network for quantifying the function of dna sequences. Nucleic acids research, 44(11):e107–e107, 2016. ",
+        "bbox": [
+            173,
+            179,
+            825,
+            209
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jesse Read, Bernhard Pfahringer, Geoff Holmes, and Eibe Frank. Classifier chains for multi-label classification. Machine Learning and Knowledge Discovery in Databases, pp. 254–269, 2009. ",
+        "bbox": [
+            173,
+            217,
+            823,
+            247
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Oren Rippel, Manohar Paluri, Piotr Dollar, and Lubomir Bourdev. Metric learning with adaptive density discrimination. arXiv preprint arXiv:1511.05939, 2015. ",
+        "bbox": [
+            173,
+            255,
+            825,
+            285
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Avanti Shrikumar, Peyton Greenside, and Anshul Kundaje. Reverse-complement parameter sharing improves deep learning models for genomics. bioRxiv, pp. 103663, 2017. ",
+        "bbox": [
+            174,
+            292,
+            823,
+            324
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Ritambhara Singh, Jack Lanchantin, Gabriel Robins, and Yanjun Qi. Deepchrome: deep-learning for predicting gene expression from histone modifications. Bioinformatics, 32(17):i639–i648, 2016. ",
+        "bbox": [
+            176,
+            330,
+            823,
+            361
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Saurabh Sinha and Xin He. Morph: probabilistic alignment combined with hidden markov models of cis-regulatory modules. PLoS computational biology, 3(11):e216, 2007. ",
+        "bbox": [
+            174,
+            369,
+            823,
+            398
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jake Snell, Kevin Swersky, and Richard S Zemel. Prototypical networks for few-shot learning. arXiv preprint arXiv:1703.05175, 2017. ",
+        "bbox": [
+            174,
+            406,
+            821,
+            436
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Nitish Srivastava, Geoffrey E Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research, 15(1):1929–1958, 2014. ",
+        "bbox": [
+            173,
+            445,
+            823,
+            488
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Gary D Stormo. Dna binding sites: representation and discovery. Bioinformatics, 16(1):16–23, 2000. ",
+        "bbox": [
+            171,
+            497,
+            825,
+            513
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Gary D Stormo. Modeling the specificity of protein-dna interactions. Quantitative Biology, 1(2): 115–130, 2013. ",
+        "bbox": [
+            173,
+            521,
+            825,
+            551
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Sainbayar Sukhbaatar, Jason Weston, Rob Fergus, et al. End-to-end memory networks. In Advances in neural information processing systems, pp. 2440–2448, 2015. ",
+        "bbox": [
+            173,
+            559,
+            823,
+            589
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Grigorios Tsoumakas and Ioannis Katakis. Multi-label classification: An overview. International Journal of Data Warehousing and Mining, 3(3), 2006. ",
+        "bbox": [
+            173,
+            597,
+            823,
+            627
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Juan M Vaquerizas, Sarah K Kummerfeld, Sarah A Teichmann, and Nicholas M Luscombe. A census of human transcription factors: function, expression and evolution. Nature Reviews Genetics, 10 (4):252–263, 2009. ",
+        "bbox": [
+            171,
+            635,
+            825,
+            679
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Oriol Vinyals, Samy Bengio, and Manjunath Kudlur. Order matters: Sequence to sequence for sets. arXiv preprint arXiv:1511.06391, 2015. ",
+        "bbox": [
+            173,
+            688,
+            823,
+            717
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Oriol Vinyals, Charles Blundell, Tim Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In Advances in Neural Information Processing Systems, pp. 3630–3638, 2016. ",
+        "bbox": [
+            173,
+            724,
+            825,
+            756
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Guy Wallis, Ulrike E Siebeck, Kellie Swann, Volker Blanz, and Heinrich H Bulthoff. The prototype ¨ effect revisited: Evidence for an abstract feature model of face recognition. Journal of Vision, 8(3): 20–20, 2008. ",
+        "bbox": [
+            173,
+            763,
+            823,
+            806
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jiang Wang, Yi Yang, Junhua Mao, Zhiheng Huang, Chang Huang, and Wei Xu. Cnn-rnn: A unified framework for multi-label image classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2285–2294, 2016. ",
+        "bbox": [
+            171,
+            815,
+            823,
+            859
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jie Wang, Jiali Zhuang, Sowmya Iyer, XinYing Lin, Troy W Whitfield, Melissa C Greven, Brian G Pierce, Xianjun Dong, Anshul Kundaje, Yong Cheng, et al. Sequence features and chromatin structure around the genomic regions bound by 119 human transcription factors. Genome research, 22(9):1798–1812, 2012. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            922
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Yunchao Wei, Wei Xia, Junshi Huang, Bingbing Ni, Jian Dong, Yao Zhao, and Shuicheng Yan. Cnn: Single-label to multi-label. arXiv preprint arXiv:1406.5726, 2014. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Ledell Wu, Adam Fisch, Sumit Chopra, Keith Adams, Antoine Bordes, and Jason Weston. Starspace: Embed all the things! arXiv preprint arXiv:1709.03856, 2017. ",
+        "bbox": [
+            174,
+            140,
+            825,
+            170
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Xiangyang Xue, Wei Zhang, Jie Zhang, Bin Wu, Jianping Fan, and Yao Lu. Correlative multi-label multi-instance image annotation. In Computer Vision (ICCV), 2011 IEEE International Conference on, pp. 651–658. IEEE, 2011. ",
+        "bbox": [
+            176,
+            178,
+            823,
+            222
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Jian Zhou and Olga G Troyanskaya. Deep supervised and convolutional generative stochastic network for protein secondary structure prediction. arXiv preprint arXiv:1403.1347, 2014. ",
+        "bbox": [
+            171,
+            231,
+            823,
+            258
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Jian Zhou and Olga G Troyanskaya. Predicting effects of noncoding variants with deep learning-based sequence model. volume 12, pp. 931–934. Nature Publishing Group, 2015. ",
+        "bbox": [
+            169,
+            268,
+            823,
+            297
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "7 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            294,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "7.1 MNIST ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            269,
+            147
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To validate the learned prototype matching on another task, we compare our PMN model against a standard CNN model on the MNIST dataset. We use a 3-layer CNN with $\\{ 3 2 , 3 2 , 3 2 \\}$ kernels of sizes $\\{ 5 , 5 , 5 \\}$ at the 3 layers, respectively. ",
+        "bbox": [
+            174,
+            157,
+            825,
+            200
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For the MNIST experiments, the PMN models do not show a drastic improvement over the baseline CNN. The results are shown in Table 6. However, we do note that based on the per-epoch plots in Figure 4, the PMN models do converge faster than the baseline CNN. In addition, we show in figure 5 that the PMN embeddings are better separated than the CNN embeddings. This is likely due to the fact that the PMN uses a similarity metric, and the fact that it can update its embedding based on which number prototypes it matches to. In other words, if an image looks similar to several numbers (e.g. “5” and $\\mathbf { \\vec { \\nabla } } ^ { 6 }$ ), the PMN can update its output based on which one it matches more to. Note that although we train using a prototype loss for each class, we do not constrain the matching to only match to one prototype. ",
+        "bbox": [
+            173,
+            204,
+            825,
+            330
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/96e40a219f9b79475d7651cc857f726f9c93162bf9c6efae60e29f6943792200.jpg",
+        "table_caption": [
+            "Table 6: MNIST using 3-Layer CNN "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Accuracy</td></tr><tr><td>CNN</td><td>99.37</td></tr><tr><td>PMN (λ = 0)</td><td>99.43</td></tr><tr><td>PMN (λ = 0.5)</td><td>99.39</td></tr><tr><td>PMN (λ= 1)</td><td>99.50</td></tr></table>",
+        "bbox": [
+            392,
+            368,
+            604,
+            444
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/2a3dd723e6806592b7a987d5524ab08fda441e9d58c2a1f8b6dafa03d79eb6d6.jpg",
+        "image_caption": [
+            "Figure 4: MNIST Per-Epoch Accuracy (Left: Train, Right: Test) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            484,
+            785,
+            640
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/9131f7771aba6bc01a7f2afa16ef9f1c1fb98809f5e635ad6f29186ffbfad9b2.jpg",
+        "image_caption": [
+            "Figure 5: 2d T-SNE Embedding of final output vector before classification. Left: CNN, Right: PMN "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
+            751,
+            805,
+            893
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b299788e16075335ddf2e3fb435f13c9bf2fa9a191fe9b59e5e01c023b7e780.jpg",
+        "image_caption": [
+            "Figure 6: Hierarchical clustering of prototypes of 86 TFs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            364,
+            968,
+            618
+        ],
+        "page_idx": 15
+    }
+]

parse/train/Syx5eT4KDS/Syx5eT4KDS.md

@@ -0,0 +1,431 @@
+# DISCRETE INFOMAX CODES FOR META-LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+This paper analyzes how generalization works in meta-learning. Our core contribution is an information-theoretic generalization bound for meta-learning, which identifies the expressivity of the task-specific learner as the key factor that makes generalization to new datasets difficult. Taking inspiration from our bound, we present Discrete InfoMax Codes (DIMCO), a novel meta-learning model that trains a stochastic encoder to output discrete codes. Experiments show that DIMCO requires less memory and less time for similar performance to previous metric learning methods and that our method generalizes particularly well in a challenging small-data setting.
+
+# 1 INTRODUCTION
+
+Generalizing to unseen data is a problem of vital importance in machine learning. Many deep metalearning methods optimize for generalization by directly minimizing the loss of held-out validation data. Recent works have used this framework to achieve impressive feats such as learning to classify using one labeled image per class (Snell et al., 2017; Finn et al., 2017), learning unsupervised update rules that generalize to different domains (Metz et al., 2018), and accelerating training procedures that are millions of steps long (Flennerhag et al., 2018). However, the meta-learning setup introduces a new overfitting problem: the model may overfit to the distribution of tasks seen during training. In other words, the meta-learning framework decreases task-specific overfitting at the cost of introducing task-wise overfitting.
+
+The primary aim of this work is to elucidate this tradeoff between task-specific and task-wise overfitting in meta-learning. Specifically, we use tools from information theory to bound the overall generalization gap of a meta-learner. Analogously to how generalization bounds for standard learning algorithms reveal the role of dataset size in generalizing to new datapoints, our bound reveals the roles of both dataset size $( m )$ and number of datasets $( n )$ in generalizing to new tasks. The specific form of our generalization gap suggests that meta-learning models can be implicitly regularized by constructing them to express each datapoint with a small number of bits.
+
+To this end, we propose Discrete InfoMax COdes (DIMCO), a deep neural network model that outputs a discrete representation of each datapoint. Note that a continuous representation $\tilde { x } \in \mathbb { R } ^ { n }$ (e.g. Snell et al. (2017)) uses $3 2 n$ bits per datapoint. This is wildly inefficient in terms of bit-efficiency: our experiments in Section 5 show that DIMCO’s discrete representation requires roughly $1 0 \times$ less bits per datapoint to achieve similar performance compared to continuous methods. DIMCO generalizes well to novel datasets because its learning objective encourages the model to compactly use all of its degrees of freedom, thus enabling it to effectively compare datapoints while only requiring a small number of bits per datapoint.
+
+Our specific contributions are:
+
+1. Derive a generalization bound for meta-learning that makes the tradeoff between taskspecific and task-wise overfitting concrete.   
+2. Propose DIMCO, a neural network model that is designed to have a low value of a specific term in our generalization bound.   
+3. Empirically demonstrate that DIMCO generalizes better than previous meta-learning methods when trained with small datasets, and that it is more memory- and time-efficient compared to previous image retrieval methods.
+
+This paper is organized as follows. We detail our problem setup and derive a generalization bound for meta-learning in Section 2. Taking motivation from our bound, we propose our meta-learning model (DIMCO) in Section 3. We put our analysis in the context of previous work in Section 4. Notably, we suggest that certain previous meta-learning methods may have benefitted from implicit regularization. We present experiments in Section 5 and conclude the paper with a discussion about limitations and future directions of our approach in Section 6.
+
+# 2 A GENERALIZATION BOUND FOR META-LEARNING
+
+Throughout this section, we denote data, model outputs, and labels as $\mathbf { x } , \widetilde \mathbf { x }$ , and $\mathbf { y }$ , respectively. We use capital symbols $X$ , $\widetilde { X }$ , $Y$ to denote the random variables corresponding to $\mathbf { x } , { \widetilde { \mathbf { x } } } ,$ y .
+
+The (discrete) Mutual Information between two random variables $X _ { 1 } , X _ { 2 }$ is defined as
+
+$$
+I ( X _ { 1 } ; X _ { 2 } ) = H ( X _ { 1 } ) - H ( X _ { 1 } | X _ { 2 } ) = H ( X _ { 2 } ) - H ( X _ { 2 } | X _ { 1 } )
+$$
+
+$I ( X _ { 1 } ; X _ { 2 } )$ is a symmetric quantity which measures the amount of information shared between $X _ { 1 }$ and $X _ { 2 }$ . It has its lowest value 0 when $X _ { 1 }$ and $X _ { 2 }$ are independent and increases with the correlation between $X _ { 1 }$ and $X _ { 2 }$ . We refer the reader to (Cover & Thomas, 2012) for further exposition.
+
+# 2.1 PROBLEM SETUP
+
+We begin by describing our meta-learning problem setup. Define a task $T$ to be a distribution over ${ \mathcal { Z } } = { \mathcal { X } } \times { \mathcal { Y } }$ . Let tasks $T ^ { 1 } , \ldots , T ^ { n }$ be sampled i.i.d. from a distribution of tasks $\tau$ . Associated with each task $T$ , we define a dataset $D _ { T } = z _ { T } ^ { 1 ^ { \ast } } , \dots , z _ { T } ^ { m } = ( x _ { T } ^ { 1 } , y _ { T } ^ { 1 } ) , \dots , ( x _ { T } ^ { m } , y _ { T } ^ { m } )$ which is a set of $m$ i.i.d. samples from the data distribution $( z _ { T } ^ { j } \sim T )$ ).
+
+We consider models with parameters $\theta$ that map datapoints $X$ to representations $\widetilde { X } ( X , \theta )$ . Our objective is the expecteation of the negative mutual information between representations an labels across all tasks:
+
+$$
+\begin{array} { r } {  { \mathcal { L } } ( \tau , \theta ) = -  { \mathbb { E } } _ { T \sim \tau } \left[ I ( \widetilde { X } ( X _ { T } , \theta ) ; Y _ { T } ) \right] . } \end{array}
+$$
+
+This objective is closely related to both previous loss functions and evaluation metrics in setups that involve testing on unseen classes (e.g. few-shot classification, image retrieval). We show that previous loss functions can be seen as approximations to (2) in Appendix A, and that mutual information is strongly correlated with metrics such as few-shot accuracy and Recall@1 in Section 5.1.
+
+An important difference between (2) and previous objectives for few-shot classification is that we do not split each task into support/query (also called train/test) sets. This property bypasses the pesky issue of using batch normalization (BN) during meta-training. As Nichol & Schulman (2018) points out, BN can leak information from the support set to the query set. This issue is not a problem in our setup since we do not assume a seperate "query set". Additionally, not using support/query splits enables meta-learning with tasks consisting of one image per class, as we demonstrate in Section 5.4. Note that the standard meta-learning setup cannot learn from such tasks since it requires at least two images per class (one for support, one for query) to compute the loss function. Overall, our model demonstrates that the commonly used construction of a held-out test set within each task is not strictly necessary for meta-learning.
+
+# 2.2 GENERALIZATION BOUND
+
+We bound the difference between expected loss and empirical loss:
+
+Theorem 1. Let $\tau , n , m , X , \widetilde { X } , Y , \theta , { \mathcal { L } }$ be defined as above. Let $d _ { \Theta }$ be the VC dimension of the encoder $\widetilde { X } ( \cdot )$ . Let $\hat { I } ( \widetilde { X } ( X _ { T } , \theta ) ; Y _ { T } )$ be the empirical estimate of the mutual information using finite dataset $D _ { T }$ , and define empirical loss as
+
+$$
+\hat { \mathcal { L } } ( T ^ { 1 : n } , \theta ) = - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \hat { I } ( \widetilde { X } ( X _ { T ^ { i } } , \theta ) ; Y _ { T ^ { i } } ) .
+$$
+
+![](images/12e9ffefd873e73d60d077a6e59b12e0d4e86b87e574e42f8b209cd3081fec03.jpg)  
+Figure $1 \colon \mathsf { A }$ graphical overview of Discrete InfoMax COdes (DIMCO). A dataset $D$ consists of pairs of images $X$ and labels $Y$ . DIMCO is a stochastic encoder that maps each image $X$ to a distribution of discrete codes $p ( { \widetilde { X } } | X )$ . Each discrete code $\widetilde { \mathbf { x } } \sim p ( \widetilde { X } | X )$ is a $p$ -way code of length $d$ . If $p = 4 , d = 2$ e(as in the diagram), each code consists of 2 symbols and each symbol is $\in \{ 1 , 2 , 3 , 4 \}$ . Inside the $4 \times 4$ grid that represents the $p ^ { d } = 1 6$ possible codes, the most likely row and column are colored. The most likely code in the diagram is $( 1 , 2 )$ with probability $3 0 \%$ . DIMCO is optimized by maximizing the mutual information between the discrete code and the label within each batch.
+
+The following inequality holds with high probability:
+
+$$
+\mathcal { L } ( \tau , \theta ) - \hat { \mathcal { L } } ( T ^ { 1 : n } , \theta ) \leq O \left( \sqrt { \frac { d _ { \Theta } } { n } \log \frac { n } { d _ { \Theta } } } \right) + O \left( \frac { | \widetilde { X } | \log ( m ) } { \sqrt { m } } \right) + O \left( \frac { | \widetilde { X } | | Y | } { m } \right)
+$$
+
+Proof. We use standard Chernoff bounds along with a finite sample bound for mutual information from Shamir et al. (2010); see Appendix B.
+
+The generalization gap has three terms, two of which decrease as $m$ increases, and the other decreases as $n$ increases. Typically for few-shot learning, $n$ is very large while $m$ is small: the typical miniImagenet 5-way 1-shot setup has $n > 1 0 ^ { 1 0 }$ and $m = 5$ . We therefore claim that the latter two terms are the main difficulties for generalizing to new tasks. We see from Theorem 1 that these terms can be reduced by using small $| \widetilde { X } |$ . Therefore, in the context of meta-learning, using short representations (i.e. small $| \widetilde { X } | )$ can compensate for having a small train set (i.e. small $m$ ).
+
+Meta-learning is typically formulated as performing two levels of learning: task-general learning and task-specific learning. Theorem 1 implies that in the tasks considered in recent literature, task-general learning should have almost no generalization gap, making task-specific learning the main source of (meta-)overfitting. This can be problematic since in many works, the task-specific learning algorithm is usually just a byproduct of whatever clever meta-learning loss was proposed. Our theorem suggests that we should pay more attention to directly regularizing this task-specific learner, and our DIMCO model, described in the next section, can be seen as a minimal working example in this direction.
+
+# 3 DISCRETE INFOMAX CODES (DIMCO)
+
+We now present our model, Discrete InfoMax COdes (DIMCO). Motivated by Section 2, DIMCO produces a short discrete code $\widetilde { X }$ and is trained by maximizing mutual information $I ( \widetilde { X } ; Y )$ . Figure 1 graphically shows the overall structure of DIMCO.
+
+# 3.1 FACTORIZED DISCRETE CODES
+
+We propose a factorized discrete representation scheme which enables us to represent discrete distributions with exponentially fewer parameters compared to listing the probability of each event. We represent each event as the product of $d$ independent events, each of which consists of $p$ different possibilities. We thus have $p ^ { d }$ events in total, but only require $p d$ parameters to represent the probability of each event. Binary codes can be viewed as a special case of this scheme where $p = 2$ . This factorization trick allows us to consider representations of size $| \widetilde { X } | = 6 4 ^ { 2 5 6 }$ (Section 5). This representation has the advantage of requiring only $d \log _ { 2 } p$ bits per datapoint, whereas a $D$ - dimensional continuous vector embedding requires $3 2 d$ bits (assuming 32-bit floats).
+
+# 3.2 MODEL
+
+Recall that we represent a given image using $d$ independent discrete distributions, each of which has $p$ possibilities. First, a (convolutional) neural network $\operatorname { e n c } ( { \mathord { \cdot } } )$ takes image $X$ as input and outputs a vector of length $d p$ , which we reshape into a matrix of size $d \times p$ :
+
+$$
+\operatorname { e n c } ( X ) = { \left[ \begin{array} { l l l l l } { l _ { 1 1 } } & { l _ { 1 2 } } & { l _ { 1 3 } } & { \dots } & { l _ { 1 p } } \\ { l _ { 2 1 } } & { l _ { 2 2 } } & { l _ { 2 3 } } & { \dots } & { l _ { 2 p } } \\ { \vdots } & { \vdots } & { \vdots } & { \ddots } & { \vdots } \\ { l _ { d 1 } } & { l _ { d 2 } } & { l _ { d 3 } } & { \dots } & { l _ { d p } } \end{array} \right] }
+$$
+
+Each row of this matrix represents the logits of a discrete distribution. We apply the softmax function to each row to get probabilities.
+
+$$
+\mathrm { s o f t m a x } ( l _ { i 1 } , \dots , l _ { i p } ) = p _ { i 1 } , \dots , p _ { i p }
+$$
+
+The ith codeword is sampled according to the categorical distribution following these probabilites:
+
+$$
+\widetilde { x } _ { i } \sim \mathrm { C a t } ( p _ { i 1 } , \dots , p _ { i p } ) .
+$$
+
+We simply concatenate each $\widetilde { x } _ { i }$ to obtain the representation: $\widetilde { \mathbf { x } } = ( \widetilde { x } _ { 1 } , \dots , \widetilde { x } _ { d } )$
+
+# 3.3 TRAINING
+
+Recall that $\widetilde { \mathbf { x } }$ is a discrete random variable and $\widetilde { X }$ is its distribution. Instead of sampling $\widetilde { \mathbf { x } } \sim \widetilde { X }$ , we directly use $\widetilde { X }$ to compute the objective:
+
+$$
+I ( \widetilde { X } ; Y ) = H ( \widetilde { X } ) - H ( \widetilde { X } | Y ) .
+$$
+
+The first term, $H ( { \tilde { X } } )$ , can be calculated by taking the average of all probabilities and computing the entropy:
+
+$$
+H ( \widetilde X ) = \sum _ { i = 1 } ^ { d } H ( \widetilde X _ { i } ) = \sum _ { i = 1 } ^ { d } H \left( \mathrm { C a t } \left( \frac { \sum _ { j = 1 } ^ { m } p _ { i 1 } ^ { j } } { m } , \frac { \sum _ { j = 1 } ^ { m } p _ { i 2 } ^ { j } } { m } , \ldots , \frac { \sum _ { j = 1 } ^ { m } p _ { i p } ^ { j } } { m } \right) \right)
+$$
+
+The second term is $\begin{array} { r } { H ( \widetilde { X } | Y ) = \sum _ { k = 1 } ^ { c } p ( Y = k ) H ( \widetilde { X } | Y = k ) } \end{array}$ where $c$ is the number of classes. The marginal probability of $\mathrm { Y }$ $( p ( Y = k ) )$ is the frequency of $k$ in $\{ \mathbf { y } ^ { 1 } , \ldots , \mathbf { y } ^ { m } \}$ . $H ( \widetilde { X } | Y = k )$ can be obtained by computing (9) using only $\mathbf { x } ^ { j }$ for which $\mathbf { y } ^ { j } = k$ .
+
+Though we have motivated the use of $I ( \widetilde { X } ; Y )$ as a loss function throughout this paper, we provide yet another perspective using the decomposition in (8). Minimizing $H ( \widetilde X | Y )$ encourages discriminatory behavior. This term encourages the average embedding of each class to be as concentrated as possible. Maximizing $H ( { \widetilde { X } } )$ incentivizes the model to overall use all possible values of $\widetilde { X }$ .
+
+We emphasize that such closed-form computation of $I ( \widetilde { X } ; Y )$ is only possible because we are using discrete codes.
+
+# 3.4 EVALUATION
+
+We map all images to their probabilities $p _ { i j }$ via (5) and (6) for all $i = 1 , \ldots , d$ and $j = 1 , \dotsc , p$ . We map each training image to its most likely code:
+
+$$
+\widetilde { \mathbf { x } } = \Big ( \underset { j } { \overset { \widetilde { x } _ { 1 } } { \operatorname { a r g m a x } } } p _ { 1 j } , \ \overset { \widetilde { x } _ { 2 } } { \underset { j } { \operatorname { a r g m a x } } } p _ { 2 j } , \ \cdot \cdot \cdot , \ \overbrace { \mathrm { a r g } \underset { j } { \operatorname { m a x } } p _ { d j } } ^ { \widetilde { x } _ { d } } \Big )
+$$
+
+Fix a train image and a test image, and let $\widetilde { \mathbf { x } }$ be the most likely code for the train image. The similarity ebetween train image and test image is measured by the probability of the test image producing $\widetilde { \mathbf { x } }$ . This amounts to computing the product 1 of the test image’s probabilites using $\widetilde { \mathbf { x } } _ { i }$ for each $i = 1 , \ldots , d$ :
+
+$$
+\prod _ { i = 1 } ^ { d } p _ { \widetilde { \mathbf { x } } _ { i } i } .
+$$
+
+We use this as a similarity metric for both few-shot classification and image retrieval. We perform few-shot classification by computing the most likely code for each class via (10) and classifying each test image by choosing the class that has highest value of (11). We similarly perform image retrival by mapping each support image to its most likely code (10) and for each query image retrieving the support image that has highest (11).
+
+# 4 RELATED WORK
+
+Regularizing Meta-Learners The ability to generalize to novel datasets is critical in meta-learning benchmarks, and even more so in benchmarks such as Meta-Dataset (Triantafillou et al., 2019), where a model is tested on datasets from an unseen domain. To the best of our knowledge, no works have proposed an explicit regularizer for generalizing to new tasks.
+
+Our analysis suggests that the success of some previous meta-learning methods can be attributed to being implicitly regularized by reducing the expressive power reducing task-specific learning. The following works have reported benefits from reducing the number of such task-specific parameters: Lee & Choi (2018) learns a subset of the full network to alter during task-specific learning, Rusu et al. (2018) explicitly represents each task with a low-dimensional latent space, and Zintgraf et al. (2018) alters only a pre-specified subset of the full network during task-specific learning. It may be surprising at first that these methods achieve higher accuracy than vanila MAML (Finn et al., 2017), which is more expressive since it alters all parameters during task-specific learning. Kim et al. (2018) also reports better meta-generalization through approximate variational inference with respect to a learned prior, which can be seen as restricting the search space of the task-specific learner. We showed through Theorem 1 that restricting inner-loop expressivity reduces the generalization gap; this provides theoretical understanding to this consensus that meta-learning models with simple task-specific learners generalize to new tasks more easily.
+
+Information Bottleneck Theorem 1 is close in spirit to the information bottleneck principle (Tishby et al., 2000; Tishby & Zaslavsky, 2015; Shwartz-Ziv & Tishby, 2017). This principle states that to generalize, one should maximize $I ( \widetilde { X } ; Y )$ while simultaneously minimizing $I ( { \widetilde { X } } ; X )$ . Likewise, our objective (2) is $I ( \widetilde { X } ; Y )$ while our bound (22) suggests that the representation capacity $| \widetilde { X } |$ should be low for generalization. Also related is the deterministic information bottleneck (Strouse & Schwab, 2017) which extends the information bottleneck by minimizing $H ( { \tilde { X } } )$ rather than $I ( { \widetilde { X } } ; X )$ . These three approaches to generalization are related via the chain of inequalities $I ( \widetilde { X } ; X ) \le H ( \widetilde { X } ) \le$ $\log | \widetilde { X } |$ , which is tight when $\widetilde { X }$ is an efficient code.
+
+Representation Learning Previous works have applied information-theoretic principles to analyze the objective of VAEs (Alemi et al., 2017; Chen et al., 2018), derive an objective for GANs to learn disentangled features (Chen et al., 2016), and to directly learn representations (Alemi et al., 2016; Hjelm et al., 2018; Oord et al., 2018; Grover & Ermon, 2018; Choi et al., 2019). Our work can also be viewed as an information-theoretic representation learning method, but we assume a supervised meta-learning setup and our main focus is the meta-generalization problem.
+
+Discrete Representations Discrete representations have been thoroughly studied in the context of information theory (Shannon, 1948). Recent deep learning methods directly learn discrete representations, by learning variational autoencoders with discrete latent variables (Rolfe, 2016; van den Oord et al., 2017; Razavi et al., 2019) or maximizing the mutual information between representation and data (Hu et al., 2017). DIMCO is related to but differs from these works as it assumes a supervised meta-learning setting and performs infomax using labels instead of data.
+
+Similarly to our setup, Jeong & Song (2018) uses labels to learn a binary hash code. Their focus is on the speedup gained by using sparse codes, whereas DIMCO learns a dense discrete code to generalize better. Additionally, their method solves a minimum cost flow problem within each batch to find the locally optimal code, whereas DIMCO is able to directly compute its loss function.
+
+Factorized Representations The idea of using factorized representations has appeared the contexts of quantizing a continuous input (Jegou et al., 2011), memory-efficient clustering (Norouzi & Fleet, 2013), and constructing an expressive attention mechanism using few parameters (Vaswani et al., 2017). Likewise, DIMCO factorizes its discrete representatations to increase its representation power.
+
+Metric Learning The structure and loss function of DIMCO is closely related to embedding-based meta-learning Vinyals et al. (2016); Snell et al. (2017); Oreshkin et al. (2018) and image retrieval Hoffer & Ailon (2015); Sohn (2016); Wu et al. (2017); Duan et al. (2018) methods. We show in Appendix A that the loss functions of these methods can be seen as approximation to the mutual information $( I ( \widetilde { X } ; Y ) )$ . While all of these previous methods require a support/query split within each task, DIMCO simply optimizes an information-theoretic quantity of each batch, removing the need for such structured batch construction.
+
+# 5 EXPERIMENTS
+
+We use the miniImageNet (Ravi & Larochelle, 2016) and CUB200 (Wah et al., 2011) datasets with standard splits for both in our experiments. The miniImageNet dataset is a subset of the Imagenet (Krizhevsky et al., 2012) dataset that was made for few-shot classification. It consists of 100 classes each containing 600 images of size $8 4 \times 8 4$ . The classes are split into 64 training, 16 validation, and 24 test classes. The Caltech-UCSD Birds-200-2011 (CUB200) dataset consists of 11788 images of birds from 200 classes. The classes are split into 100 training and 100 test classes.
+
+We use two different CNN backbones for our experiments: the 4-layer convnet commonly used for meta-learning (Finn et al., 2017; Sung et al., 2018; Liu et al., 2018), and the Inception network (Szegedy et al., 2015) with batch normalization (Ioffe & Szegedy, 2015) which is commonly used for deep image retrieval (Sohn, 2016; Movshovitz-Attias et al., 2017; Wu et al., 2017). We randomly initialize weights for the 4-layer convnet and use pretrained weights for the Inception network
+
+# 5.1 CORRELATION OF METRICS
+
+This experiment empirically verifies whether mutual information $I ( \widetilde { X } ; Y )$ is a reasonable metric for quality of representation. We trained DIMCO on the miniImagenet dataset with $p = d = 6 4$ for 20 epochs, for 8 independent runs. We plot the pairwise correlations between five different metrics ( (5, 10, 20)-way 1-shot accuracy, Recall $@ 1$ , and $I ( \widetilde { X } ; Y )$ ) in Figure 2. We see that all five metrics are very strongly correlated. We observed similar trends when training with with previously proposed loss functions: we visualize these results in Figure 5 of the appendix due to space constraints. Alongside this empirical evidence, we prove in Appendix A that previously used loss functions for few-shot classification and image retrieval are approximations to $I ( \widetilde { X } ; Y )$ .
+
+# 5.2 WHAT DOES EACH CODE REPRESENT?
+
+We inspected what each code represents in a DIMCO model $ d  = 1 6$ , $p = 6 4 .$ ) trained on the miniImagenet dataset. Recall that each image produces a $d \times p$ probability matrix (5, 6). For each of these $d p$ entries, we plotted the top 10 images in the test set that assigned highest probability to that entry. We show images corresponding to four such codes in Figure 2 (right) and more in Figure 6 of the appendix.
+
+We see that DIMCO learns a distributed representation. For example, the top code in Figure 2 represents the high-level concept of a furry animal and the shown 10 images span 4 different classes. On the other hand, the bottom code seems to focus on the color of the background. By aggregating such complementary features in each of its $d$ codewords, DIMCO is able to classify images that belong to previously unseen classes.
+
+![](images/9dad9f4655d982f4fcff4b2c238b971b41e77a79c0d27076551b36b515297007.jpg)  
+Figure 2: Left: Pairwise correlation between $\mathbf { M } \mathbf { I } = I ( X ; { \widetilde { X } } )$ and previous metrics. Right: Visualization of codes of a small trained DIMCO model $\left( d = 1 6 \right.$ , $p = 6 4 ^ { \prime }$ ); we show the top 10 test set images that assign highest probability to a specific code, for 4 different codes.
+
+![](images/4358d89791c6cfc1b52cef4a6064cb652948c17f77299d65f2c0f8411f1bfef0.jpg)  
+Figure 3: Image retrieval performance of DIMCO and N-pair loss on the CUB-200 dataset. The y-axis for both figures are the Recall $@ 1$ metric, and error bars reflect standard deviation computed from $n = 5$ runs per configuration. The $\mathbf { X }$ -axes represent (left) bits required to store one representation and (right) seconds required to perform retrieval for one query. Both $\mathbf { X }$ -axes are log-scale.
+
+# 5.3 TIME- AND MEMORY-EFFICIENT IMAGE RETRIEVAL
+
+We conducted an image retrieval experiment using the CUB200 dataset, and used multiclass Npair loss (Sohn, 2016) as a baseline. As is standard for image retrieval, we use the Inception network as specified in the beginning of this section. Using the same backbone, we trained DIMCO with $( p , d ) \in \{ 6 4 , 1 2 8 , 2 5 6 \} \times \{ 1 2 \bar { 8 } , 2 5 6 , 5 1 2 \}$ and multiclass N-pair with embedding dimension $\in \{ 1 2 8 , 2 5 6 , 5 1 2 \}$ . We measured the time per query for each method on a single Tesla P40 GPU by averaging the time required for 10000 batches of queries of size 32.
+
+Results in Figure 3 show that the compact code of DIMCO takes roughly an order of magnitude less memory for similar performance to N-pair loss, and requires less query time as well. This experiment also demonstrates that discrete representations can match the performance of modern methods that use continuous embeddings on this relatively large-scale task. We additionally note that DIMCO is able to train using large backbones without significantly overfitting, whereas experiments reported in Mishra et al. (2017) indicate that MAML (Finn et al., 2017) overfits tremendously when using a deeper backbone.
+
+![](images/de8fef7019d5e73acdd2af9ee30318f253d3d0b71fa82bc7d7897d338ca38f25.jpg)  
+Figure 4: Performance of methods trained using subsets of miniImageNet of varying size. The lowermost y axis value for each metric corresponds to the expected performance of random guessing.
+
+# 5.4 GENERALIZATION TO NEW TASKS
+
+This experiment measures how well DIMCO can generalize to new datasets after training with a small number of datasets. This challenging experimental setup measures how much generalizable information the model can extract from a limited set of datasets; it can be seen as the meta-learning analogue of measuring the performance of a classifier trained with a small dataset.
+
+We trained each model using $\{ 1 , 4 , 1 6 , 6 4 \}$ samples from each training class in the miniImageNet dataset. For example, by using 4 samples, we are reducing the full train split of (64 classes $\times ~ 6 0 0$ images per class) into (64 classes $\times 4$ images per class). We compare against three baseline methods: Triplet Nets(Hoffer & Ailon, 2015), multiclass N-pair loss(Sohn, 2016), and ProtoNets(Snell et al., 2017). We report the average and standard deviation of the top 5 results of a random hyperparameter search (see appendix for details). We show $\{ 5 , 1 0 , 2 0 \}$ -way 1-shot accuracies and Recal $@ 1$ of the test set in Figure 4.
+
+First note that DIMCO is the only method that can train using a dataset consisting of 1 example per class. This is because other methods require at least one train and test example per class within each batch, while DIMCO requires no such train/test (also called support/query) split and simply maximizes the mutual information within a batch. Furthermore, Figure 4 shows that DIMCO learns much more effectively when the number of examples per class is low; this is because DIMCO uses fewer bits to describe each datapoint, which lowers its generalization gap (24) when applying to novel datasets.
+
+# 6 CONCLUSION AND DISCUSSION
+
+Summary We derived a generalization bound for meta-learning using information-theoretic principles. Building on our bound, we proposed DIMCO, a model that learns a discrete representation of data by maximizing its mutual information with the label. DIMCO had benefits in time, memory, and generalization in our experiments.
+
+Towards Explicit Meta-Regularization In Section 4, we have suggested with analogy to Theorem 1 that the benefits of some previous meta-learning methods can be attributed to implicitly being regularized by reducing the expressivity of their task-specific learners. While DIMCO has demonstrated better generalization by reducing this expressivity via hyperparameters $( d , p )$ , this is only applicable to DIMCO’s specific setup of few-shot classification through mutual information maximization. In future work, we would like to explore explicit meta-regularization schemes that can be applied to other problems (regression, reinforcement learning etc.) and algorithms (MAML, Neural Process etc.).
+
+Meta-Learning Without Splits Our meta-learning problem setup in Section 2.1 does not assume a support/query (also called train/test) split within each dataset. Along with showing that the traditional support/query split is not strictly necessary, we demonstrated in Section 5.4 that removing it has the benefit of enabling meta-learning in datasets having one image per class. We believe future work could benefit from further exploring this space of meta-learning algorithms that use datasets without splits.
+
+# REFERENCES
+
+Alexander A Alemi, Ian Fischer, Joshua V Dillon, and Kevin Murphy. Deep variational information bottleneck. arXiv preprint arXiv:1612.00410, 2016.
+
+Alexander A Alemi, Ben Poole, Ian Fischer, Joshua V Dillon, Rif A Saurous, and Kevin Murphy. Fixing a broken elbo. arXiv preprint arXiv:1711.00464, 2017.
+
+Ron Amit and Ron Meir. Meta-learning by adjusting priors based on extended pac-bayes theory. arXiv preprint arXiv:1711.01244, 2017.
+
+Tian Qi Chen, Xuechen Li, Roger B Grosse, and David K Duvenaud. Isolating sources of disentanglement in variational autoencoders. In Advances in Neural Information Processing Systems, pp. 2610–2620, 2018.
+
+Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In Advances in neural information processing systems, pp. 2172–2180, 2016.
+
+Kristy Choi, Kedar Tatwawadi, Aditya Grover, Tsachy Weissman, and Stefano Ermon. Neural joint source-channel coding. In International Conference on Machine Learning, pp. 1182–1192, 2019.
+
+Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012.
+
+Yueqi Duan, Wenzhao Zheng, Xudong Lin, Jiwen Lu, and Jie Zhou. Deep adversarial metric learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2780–2789, 2018.
+
+Chelsea Finn and Sergey Levine. Meta-learning and universality: Deep representations and gradient descent can approximate any learning algorithm. arXiv preprint arXiv:1710.11622, 2017.
+
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. arXiv preprint arXiv:1703.03400, 2017.
+
+Sebastian Flennerhag, Pablo G Moreno, Neil D Lawrence, and Andreas Damianou. Transferring knowledge across learning processes. arXiv preprint arXiv:1812.01054, 2018.
+
+Anirudh Goyal, Riashat Islam, Daniel Strouse, Zafarali Ahmed, Matthew Botvinick, Hugo Larochelle, Sergey Levine, and Yoshua Bengio. Infobot: Transfer and exploration via the information bottleneck. arXiv preprint arXiv:1901.10902, 2019.
+
+Aditya Grover and Stefano Ermon. Uncertainty autoencoders: Learning compressed representations via variational information maximization. arXiv preprint arXiv:1812.10539, 2018.
+
+R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. arXiv preprint arXiv:1808.06670, 2018.
+
+Elad Hoffer and Nir Ailon. Deep metric learning using triplet network. In International Workshop on Similarity-Based Pattern Recognition, pp. 84–92. Springer, 2015.
+
+Weihua Hu, Takeru Miyato, Seiya Tokui, Eiichi Matsumoto, and Masashi Sugiyama. Learning discrete representations via information maximizing self-augmented training. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1558–1567. JMLR. org, 2017.
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015.
+
+Herve Jegou, Matthijs Douze, and Cordelia Schmid. Product quantization for nearest neighbor search. IEEE transactions on pattern analysis and machine intelligence, 33(1):117–128, 2011.
+
+Yeonwoo Jeong and Hyun Oh Song. Efficient end-to-end learning for quantizable representations. arXiv preprint arXiv:1805.05809, 2018.
+
+Taesup Kim, Jaesik Yoon, Ousmane Dia, Sungwoong Kim, Yoshua Bengio, and Sungjin Ahn. Bayesian model-agnostic meta-learning. arXiv preprint arXiv:1806.03836, 2018.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012.
+
+Yoonho Lee and Seungjin Choi. Gradient-based meta-learning with learned layerwise metric and subspace. 2018.
+
+Yanbin Liu, Juho Lee, Minseop Park, Saehoon Kim, and Yi Yang. Transductive propagation network for few-shot learning. arXiv preprint arXiv:1805.10002, 2018.
+
+Luke Metz, Niru Maheswaranathan, Brian Cheung, and Jascha Sohl-Dickstein. Meta-learning update rules for unsupervised representation learning. arXiv preprint arXiv:1804.00222, 2018.
+
+Nikhil Mishra, Mostafa Rohaninejad, Xi Chen, and Pieter Abbeel. A simple neural attentive metalearner. arXiv preprint arXiv:1707.03141, 2017.
+
+Yair Movshovitz-Attias, Alexander Toshev, Thomas K Leung, Sergey Ioffe, and Saurabh Singh. No fuss distance metric learning using proxies. In Proceedings of the IEEE International Conference on Computer Vision, pp. 360–368, 2017.
+
+Alex Nichol and John Schulman. Reptile: a scalable metalearning algorithm. arXiv preprint arXiv:1803.02999, 2018.
+
+Mohammad Norouzi and David J Fleet. Cartesian k-means. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3017–3024, 2013.
+
+Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.
+
+Boris Oreshkin, Pau Rodríguez López, and Alexandre Lacoste. Tadam: Task dependent adaptive metric for improved few-shot learning. In Advances in Neural Information Processing Systems, pp. 721–731, 2018.
+
+Sachin Ravi and Hugo Larochelle. Optimization as a model for few-shot learning. 2016.
+
+Ali Razavi, Aaron van den Oord, and Oriol Vinyals. Generating diverse high-fidelity images with vq-vae-2. arXiv preprint arXiv:1906.00446, 2019.
+
+Jason Tyler Rolfe. Discrete variational autoencoders. arXiv preprint arXiv:1609.02200, 2016.
+
+Andrei A Rusu, Dushyant Rao, Jakub Sygnowski, Oriol Vinyals, Razvan Pascanu, Simon Osindero, and Raia Hadsell. Meta-learning with latent embedding optimization. arXiv preprint arXiv:1807.05960, 2018.
+
+Ohad Shamir, Sivan Sabato, and Naftali Tishby. Learning and generalization with the information bottleneck. Theoretical Computer Science, 411(29-30):2696–2711, 2010.
+
+Claude Elwood Shannon. A mathematical theory of communication. Bell system technical journal, 27(3):379–423, 1948.
+
+Ravid Shwartz-Ziv and Naftali Tishby. Opening the black box of deep neural networks via information. arXiv preprint arXiv:1703.00810, 2017.
+
+Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Advances in Neural Information Processing Systems, pp. 4077–4087, 2017.
+
+Kihyuk Sohn. Improved deep metric learning with multi-class n-pair loss objective. In Advances in Neural Information Processing Systems, pp. 1857–1865, 2016.
+
+DJ Strouse and David J Schwab. The deterministic information bottleneck. Neural computation, 29 (6):1611–1630, 2017.
+
+Flood Sung, Yongxin Yang, Li Zhang, Tao Xiang, Philip HS Torr, and Timothy M Hospedales. Learning to compare: Relation network for few-shot learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1199–1208, 2018.
+
+Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1–9, 2015.
+
+Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. In 2015 IEEE Information Theory Workshop (ITW), pp. 1–5. IEEE, 2015.
+
+Naftali Tishby, Fernando C Pereira, and William Bialek. The information bottleneck method. arXiv preprint physics/0004057, 2000.
+
+Eleni Triantafillou, Tyler Zhu, Vincent Dumoulin, Pascal Lamblin, Kelvin Xu, Ross Goroshin, Carles Gelada, Kevin Swersky, Pierre-Antoine Manzagol, and Hugo Larochelle. Meta-dataset: A dataset of datasets for learning to learn from few examples. arXiv preprint arXiv:1903.03096, 2019.
+
+Aaron van den Oord, Oriol Vinyals, et al. Neural discrete representation learning. In Advances in Neural Information Processing Systems, pp. 6306–6315, 2017.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pp. 5998–6008, 2017.
+
+Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In Advances in neural information processing systems, pp. 3630–3638, 2016.
+
+Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie. The caltech-ucsd birds-200-2011 dataset. 2011.
+
+Chao-Yuan Wu, R Manmatha, Alexander J Smola, and Philipp Krahenbuhl. Sampling matters in deep embedding learning. In Proceedings of the IEEE International Conference on Computer Vision, pp. 2840–2848, 2017.
+
+Luisa M Zintgraf, Kyriacos Shiarlis, Vitaly Kurin, Katja Hofmann, and Shimon Whiteson. Caml: Fast context adaptation via meta-learning. arXiv preprint arXiv:1810.03642, 2018.
+
+# A PREVIOUS LOSS FUNCTIONS ARE APPROXIMATIONS TO MUTUAL INFORMATION
+
+Cross-entropy Loss The cross-entropy loss has directly been used for few-shot classification (Vinyals et al., 2016; Snell et al., 2017).
+
+Let $q ( \mathbf { y } | \widetilde { \mathbf { x } } ; \boldsymbol { \phi } )$ be a parameterized prediction of $\mathbf { y }$ given $\widetilde { \mathbf { x } }$ , which tries to approximate the true conditional distribution $q ( \mathbf { y } | \widetilde { \mathbf { x } } )$ . Typically in a classification network, $\phi$ is the parameters of a learned projection matrix and $q ( \cdot )$ e is the final linear layer. The expected cross-entropy loss can be written as
+
+$$
+\mathrm { x e n t } ( Y , \widetilde { X } ) = \mathbb { E } _ { { \mathbf { y } } \sim Y , \widetilde { \mathbf { x } } \sim \widetilde { X } } \left[ - \log q ( \mathbf { y } | \widetilde { \mathbf { x } } , \phi ) \right] .
+$$
+
+Assuming that the approximate distribution $q ( \cdot )$ is sufficiently close to $p ( \mathbf { y } \vert \widetilde { \mathbf { x } } )$ , minimizing (12) can be seen as
+
+$$
+\begin{array} { l l l } { \arg \operatorname* { m i n } \mathbf { x e n t } ( Y , \widetilde { X } ) } & { \approx } & { \arg \operatorname* { m i n } \mathbb { E } _ { \mathbf { y } \sim Y , \widetilde { \mathbf { x } } \sim \widetilde { X } } \left[ - \log p ( \mathbf { y } | \widetilde { \mathbf { x } } ) \right] } \\ & { = } & { \arg \operatorname* { m i n } H ( Y | \widetilde { X } ) = \arg \operatorname* { m a x } I ( \widetilde { X } ; Y ) , } \end{array}
+$$
+
+where the last equality uses the fact that $H ( Y )$ is independent of model parameters. Therefore, crossentropy minimization is approximate maximization of the mutual information between representation $\widetilde { X }$ and labels $Y$ .
+
+The approximation is that we parameterized $q ( \mathbf { y } | \widetilde { \mathbf { x } } ; \boldsymbol { \phi } )$ as a linear projection. This structure cannot generalize to new classes because the parameters $\phi$ are specific to the labels $\mathbf { y }$ seen during training. For a model to generalize to unseen classes, one must amortize the learning of this approximate conditional distribution. (Vinyals et al., 2016; Snell et al., 2017) sidestepped this issue by using the embeddings for each class as $\phi$ .
+
+Triplet Loss The Triplet loss (Hoffer & Ailon, 2015) is defined as
+
+$$
+\mathcal { L } _ { \mathrm { t r i p l e t } } = \left\| \widetilde { \mathbf { x } } _ { q } - \widetilde { \mathbf { x } } _ { p } \right\| _ { 2 } ^ { 2 } - \left\| \widetilde { \mathbf { x } } _ { q } - \widetilde { \mathbf { x } } _ { n } \right\| _ { 2 } ^ { 2 } ,
+$$
+
+where $\widetilde { \mathbf { x } } _ { q } , \widetilde { \mathbf { x } } _ { p } , \widetilde { \mathbf { x } } _ { n } \in \mathbb { R } ^ { d }$ are the embedding vectors of query, positive, and negative images. Let $\mathbf { y } _ { q }$ e e edenote the label of the query data. Recall that the pdf function of a unit Gaussian is $\log N ( \widetilde { \mathbf { x } } | \mu , 1 ) \overset { \cdot } { = }$ $- c _ { 1 } - c _ { 2 } \big | \big | \widetilde { \mathbf { x } } - \boldsymbol { \mu } \big | \big | _ { 2 } ^ { 2 }$ , where $c _ { 1 } , c _ { 2 }$ are constants. Let $p _ { p } ( \widetilde { \mathbf { x } } ) = N ( \widetilde { \mathbf { x } } _ { p } , 1 )$ and $p _ { n } ( \widetilde { \mathbf { x } } ) = N ( \widetilde { \mathbf { x } } _ { n } , 1 )$ be eunit Gaussian distributions centered at $\widetilde { \mathbf { x } } _ { p } , \widetilde { \mathbf { x } } _ { n }$ e erespectively. We have
+
+$$
+\begin{array} { r l } { \mathbb { E } \left[ - \mathcal { L } _ { \mathrm { t r i p l e t } } \right] } & { \propto \quad \mathbb { E } \left[ \log p _ { p } ( \widetilde { \mathbf { x } } ) - \log p _ { n } ( \widetilde { \mathbf { x } } ) \right] } \\ & { \approx \quad \mathbb { E } \left[ \log p _ { p } ( \widetilde { \mathbf { x } } ) - \log p ( \widetilde { \mathbf { x } } ) \right] } \\ & { = \quad - H ( \widetilde { X } | Y ) + H ( \widetilde { X } ) = I ( \widetilde { X } ; Y ) . } \end{array}
+$$
+
+Two approximations were made in the process. We first assumed that the embedding distribution of images not in $\mathbf { y } _ { q }$ is equal to the distribution of all embeddings. This is reasonable when each class only represents a small fraction of the full data. We also approximated the embedding distributions $p ( \tilde { \mathbf { x } } | \mathbf { y } ) , p ( \tilde { \mathbf { x } } )$ with unit Gaussian distributions centered at single samples from each.
+
+N-pair Loss Multiclass $N$ -pair loss (Sohn, 2016) was proposed as an alternative to Triplet loss. This loss function requires one positive embedding $\widetilde { \mathbf { x } } ^ { + }$ and multiple negative embeddings $\widetilde { \mathbf { x } } _ { 1 } , \hdots$ , $\widetilde { \mathbf { x } } _ { N - 1 }$ , and takes the form
+
+$$
+- \log \frac { \exp ( \widetilde { \mathbf { x } } ^ { \top } \widetilde { \mathbf { x } } ^ { + } ) } { \exp ( \widetilde { \mathbf { x } } ^ { \top } \widetilde { \mathbf { x } } ^ { + } ) + \sum _ { i = 1 } ^ { N - 1 } \exp ( \widetilde { \mathbf { x } } ^ { \top } \widetilde { \mathbf { x } } _ { i } ) } .
+$$
+
+This can be seen as the cross-entropy loss applied to softmax $( \widetilde { \mathbf { x } } ^ { \top } \widetilde { \mathbf { x } } ^ { + } , \widetilde { \mathbf { x } } ^ { \top } \widetilde { \mathbf { x } } _ { 1 } , \dots , \widetilde { \mathbf { x } } ^ { \top } \widetilde { \mathbf { x } } _ { N - 1 } )$
+
+Following the same logic as the cross-entropy loss, this is also an approximation to $I ( \widetilde { X } ; Y )$ . This objective should have less variance than Triplet loss since it approximates $p ( \widetilde { \mathbf { x } } )$ using more examples.
+
+Adversarial Metric Learning Deep Adversarial Metric Learning (Duan et al., 2018) tackles the problem of most negative exmples being uninformative by directly generating meaningful negative embeddings. This model employs a generator which takes as input the embeddings of anchor, positive, and negative images. The generator then outputs a "synthetic negative" embedding that is hard to distinguish from a positive embedding while being close to the negative embedding.
+
+This can be seen as optimizing
+
+$$
+\mathbb { E } \left[ \log p _ { p } ( \widetilde { \mathbf { x } } ) - \log p ( \widetilde { \mathbf { x } } ) \right] = I ( \widetilde { X } ; Y )
+$$
+
+by estimating $p ( \widetilde { \mathbf { x } } )$ using a generative network rather than directly from samples. Rather than emodelling the marginal distribution $p ( \widetilde { \mathbf { x } } )$ , this method conditionally models $p ( \widetilde { \mathbf { x } } ; \bar { \widetilde { \mathbf { x } } } _ { q } , \widetilde { \mathbf { x } } _ { p } , \widetilde { \mathbf { x } } _ { n } )$ so that $\widetilde { \mathbf { x } }$ is hard to distinguish from $\widetilde { \mathbf { x } } _ { p }$ ewhile sufficiently close to both $\widetilde { \mathbf { x } } _ { q }$ and $\widetilde { \mathbf { x } } _ { n }$ .
+
+# B PROOF OF THEOREM 1
+
+We restate and prove our main theorem.
+
+Theorem 1. Let $d _ { \Theta }$ be the VC dimension of the encoder $\widetilde { X } ( \cdot )$ . Let $\hat { I } ( \widetilde { X } ( X _ { T } , \theta ) ; Y _ { T } )$ be the empirical estimate of the mutual information using finite dataset $D _ { T }$ , and define empirical loss as
+
+$$
+\hat { \mathcal { L } } ( T ^ { 1 : n } , \theta ) = - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \hat { I } ( \widetilde { X } ( X _ { T ^ { i } } , \theta ) ; Y _ { T ^ { i } } ) .
+$$
+
+The following inequality holds with high probability:
+
+$$
+\mathcal { L } ( \tau , \theta ) - \hat { \mathcal { L } } ( T ^ { 1 : n } , \theta ) \leq O \left( \sqrt { \frac { d _ { \Theta } } { n } \log \frac { n } { d _ { \Theta } } } \right) + O \left( \frac { | \widetilde { X } | \log ( m ) } { \sqrt { m } } \right) + O \left( \frac { | \widetilde { X } | | Y | } { m } \right)
+$$
+
+Proof. We use the following lemma from Shamir et al. (2010), which we restate using our notation.
+
+Lemma 1. Let $\widetilde { X }$ be a random mapping of $X$ . Let $D$ be a sample of size m drawn from the joint probability distribution $p ( X , Y )$ . Denote the empirical mutual information observed from $D$ between $\widetilde { X }$ and $Y$ as ${ \hat { I } } ( { \widetilde { X } } ; Y )$ . For any $\delta \in ( 0 , 1 )$ , the following holds with probability at least $1 - \delta$ :
+
+$$
+| I ( \widetilde { X } ; Y ) - \hat { I } ( \widetilde { X } ; Y ) | \leq \frac { ( 3 | \widetilde { X } | + 2 ) \log ( m ) \sqrt { \log ( 4 / \delta ) } } { \sqrt { 2 m } } + \frac { ( | Y | + 1 ) ( | \widetilde { X } | + 1 ) - 4 } { m }
+$$
+
+We simplify this and plug in our specific quantities of interest $( \widetilde { X } ( X _ { T } , \theta ) , Y _ { T } )$ :
+
+$$
+\left| I \left( \widetilde { X } ( X _ { T } , \theta ) ; Y _ { T } \right) - \hat { I } \left( \widetilde { X } ( X _ { T } , \theta ) ; Y _ { T } \right) \right| \le O \left( \frac { | \widetilde { X } | \log ( m ) } { \sqrt { m } } \right) + O \left( \frac { | \widetilde { X } | | Y | } { m } \right) .
+$$
+
+We similarly bound the error caused by estimating $\mathcal { L }$ with a finite number of tasks sampled from $\tau$ . Denote the finite sample estimate of $\mathcal { L }$ as
+
+$$
+\hat { \mathcal { L } } ( \tau , \theta ) = - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } I ( \widetilde { X } ( X _ { T ^ { i } } , \theta ) ; Y _ { T ^ { i } } ) .
+$$
+
+Let the mapping $X \mapsto { \widetilde { X } }$ be parameterized by $\theta \in \Theta$ and let this model have VC dimension $d _ { \Theta }$ Using $d _ { \Theta }$ , we can state that with high probability,
+
+$$
+\Big | \mathcal { L } ( \tau , \theta ) - \hat { \mathcal { L } } ( \tau , \theta ) \Big | \leq O \left( \sqrt { \frac { d _ { \Theta } } { n } \log { \frac { n } { d _ { \Theta } } } } \right) ,
+$$
+
+where $d _ { \Theta }$ is the VC dimension of hypothesis class $\Theta$ .
+
+Combining equations (26, 24), we have with high probability
+
+$$
+\begin{array} { r l r } {  { | \mathcal { L } ( \tau , \theta ) - ( - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \hat { I } ( \widetilde { X } ( X _ { T ^ { i } } , \theta ) ; Y _ { T ^ { i } } ) ) | } } \\ & { } & { \leq | \mathcal { L } ( \tau , \theta ) - \hat { \mathcal { L } } ( \tau , \theta ) | + O ( \frac { | \widetilde { X } | \log ( m ) } { \sqrt { m } } ) + O ( \frac { | \widetilde { X } | | Y | } { m } ) } \\ & { } & { \leq O ( \sqrt { \frac { d _ { \Theta } } { n } \log \frac { n } { d _ { \Theta } } } ) + O ( \frac { | \widetilde { X } | \log ( m ) } { \sqrt { m } } ) + O ( \frac { | \widetilde { X } | | Y | } { m } ) } \end{array}
+$$
+
+# C EXPERIMENTS AND IMPLEMENTATION DETAILS
+
+Hardware Every experiment was conducted on a single Nvidia V100 GPU with CUDA 9.2. We used PyTorch version 1.0.1. Each experiment was performed with different fixed initial seeds; we manually fix seeds with manual_seed() for python, pytorch, and numpy.
+
+Optimizer For experiments with the 4-layer convnet, we use the Adam optimizer (Kingma & Ba, 2014) with learning rate 3e-4. For the Inception network, we use SGD with learning rate 3e-5 and momentum 0.9.
+
+We report the average of 500 batches of 1-shot accuracies and mutual information. $I ( \widetilde { X } ; Y )$ was computed using balanced batches of 16 images each from 5 different classes. We additionally show in Figure 5 the correlation between 1-shot accuracies, Recall@1, and NMI using three previously proposed losses (triplet, npair, protonet).
+
+Small Train Set Experiment For this experiment, we used the Adam optimizer and performed a log-uniform hyperparameter sweep for learning rate $\in$ [1e-7, 1e-3] For DIMCO, we swept $p \in$ [32, 128] and $d \in [ 1 6 , 3 2 ]$ . For other methods, we made the embedding dimension $\in [ 1 6 , 3 2 ]$ . For each combination of loss and number of training examples per class, we ran the experiment 64 times and reported the mean and standard deviation of the top 5.
+
+![](images/59ce6fb78226694d9b06d1d8feadc237e49ee6c6764dd96151d7538f89de35de.jpg)  
+Figure 5: Correlation between few-shot accuracy and retrieval measures.
+
+![](images/5da775c280e79fd2f1c2352b0a6a98762a99b428364c5bbe4226b238ed116583.jpg)  
+Figure 6: Additional visualizations of codes.

parse/train/Syx5eT4KDS/Syx5eT4KDS_content_list.json

@@ -0,0 +1,2219 @@
+[
+    {
+        "type": "text",
+        "text": "DISCRETE INFOMAX CODES FOR META-LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            787,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            145,
+            400,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
+            544,
+            224
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper analyzes how generalization works in meta-learning. Our core contribution is an information-theoretic generalization bound for meta-learning, which identifies the expressivity of the task-specific learner as the key factor that makes generalization to new datasets difficult. Taking inspiration from our bound, we present Discrete InfoMax Codes (DIMCO), a novel meta-learning model that trains a stochastic encoder to output discrete codes. Experiments show that DIMCO requires less memory and less time for similar performance to previous metric learning methods and that our method generalizes particularly well in a challenging small-data setting. ",
+        "bbox": [
+            233,
+            242,
+            766,
+            367
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            397,
+            336,
+            414
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generalizing to unseen data is a problem of vital importance in machine learning. Many deep metalearning methods optimize for generalization by directly minimizing the loss of held-out validation data. Recent works have used this framework to achieve impressive feats such as learning to classify using one labeled image per class (Snell et al., 2017; Finn et al., 2017), learning unsupervised update rules that generalize to different domains (Metz et al., 2018), and accelerating training procedures that are millions of steps long (Flennerhag et al., 2018). However, the meta-learning setup introduces a new overfitting problem: the model may overfit to the distribution of tasks seen during training. In other words, the meta-learning framework decreases task-specific overfitting at the cost of introducing task-wise overfitting. ",
+        "bbox": [
+            174,
+            430,
+            825,
+            556
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The primary aim of this work is to elucidate this tradeoff between task-specific and task-wise overfitting in meta-learning. Specifically, we use tools from information theory to bound the overall generalization gap of a meta-learner. Analogously to how generalization bounds for standard learning algorithms reveal the role of dataset size in generalizing to new datapoints, our bound reveals the roles of both dataset size $( m )$ and number of datasets $( n )$ in generalizing to new tasks. The specific form of our generalization gap suggests that meta-learning models can be implicitly regularized by constructing them to express each datapoint with a small number of bits. ",
+        "bbox": [
+            174,
+            563,
+            825,
+            660
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To this end, we propose Discrete InfoMax COdes (DIMCO), a deep neural network model that outputs a discrete representation of each datapoint. Note that a continuous representation $\\tilde { x } \\in \\mathbb { R } ^ { n }$ (e.g. Snell et al. (2017)) uses $3 2 n$ bits per datapoint. This is wildly inefficient in terms of bit-efficiency: our experiments in Section 5 show that DIMCO’s discrete representation requires roughly $1 0 \\times$ less bits per datapoint to achieve similar performance compared to continuous methods. DIMCO generalizes well to novel datasets because its learning objective encourages the model to compactly use all of its degrees of freedom, thus enabling it to effectively compare datapoints while only requiring a small number of bits per datapoint. ",
+        "bbox": [
+            174,
+            667,
+            825,
+            779
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our specific contributions are: ",
+        "bbox": [
+            174,
+            785,
+            372,
+            799
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. Derive a generalization bound for meta-learning that makes the tradeoff between taskspecific and task-wise overfitting concrete.   \n2. Propose DIMCO, a neural network model that is designed to have a low value of a specific term in our generalization bound.   \n3. Empirically demonstrate that DIMCO generalizes better than previous meta-learning methods when trained with small datasets, and that it is more memory- and time-efficient compared to previous image retrieval methods. ",
+        "bbox": [
+            212,
+            813,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper is organized as follows. We detail our problem setup and derive a generalization bound for meta-learning in Section 2. Taking motivation from our bound, we propose our meta-learning model (DIMCO) in Section 3. We put our analysis in the context of previous work in Section 4. Notably, we suggest that certain previous meta-learning methods may have benefitted from implicit regularization. We present experiments in Section 5 and conclude the paper with a discussion about limitations and future directions of our approach in Section 6. ",
+        "bbox": [
+            173,
+            103,
+            826,
+            188
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 A GENERALIZATION BOUND FOR META-LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            208,
+            635,
+            226
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Throughout this section, we denote data, model outputs, and labels as $\\mathbf { x } , \\widetilde \\mathbf { x }$ , and $\\mathbf { y }$ , respectively. We use capital symbols $X$ , $\\widetilde { X }$ , $Y$ to denote the random variables corresponding to $\\mathbf { x } , { \\widetilde { \\mathbf { x } } } ,$ y . ",
+        "bbox": [
+            173,
+            241,
+            825,
+            272
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The (discrete) Mutual Information between two random variables $X _ { 1 } , X _ { 2 }$ is defined as ",
+        "bbox": [
+            176,
+            279,
+            741,
+            295
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a36c21304609ab8a9c0083b400c5d1d38efaa721089f488e3dbd504d39c28476.jpg",
+        "text": "$$\nI ( X _ { 1 } ; X _ { 2 } ) = H ( X _ { 1 } ) - H ( X _ { 1 } | X _ { 2 } ) = H ( X _ { 2 } ) - H ( X _ { 2 } | X _ { 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            301,
+            684,
+            319
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "$I ( X _ { 1 } ; X _ { 2 } )$ is a symmetric quantity which measures the amount of information shared between $X _ { 1 }$ and $X _ { 2 }$ . It has its lowest value 0 when $X _ { 1 }$ and $X _ { 2 }$ are independent and increases with the correlation between $X _ { 1 }$ and $X _ { 2 }$ . We refer the reader to (Cover & Thomas, 2012) for further exposition. ",
+        "bbox": [
+            173,
+            327,
+            828,
+            369
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 PROBLEM SETUP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            387,
+            333,
+            402
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We begin by describing our meta-learning problem setup. Define a task $T$ to be a distribution over ${ \\mathcal { Z } } = { \\mathcal { X } } \\times { \\mathcal { Y } }$ . Let tasks $T ^ { 1 } , \\ldots , T ^ { n }$ be sampled i.i.d. from a distribution of tasks $\\tau$ . Associated with each task $T$ , we define a dataset $D _ { T } = z _ { T } ^ { 1 ^ { \\ast } } , \\dots , z _ { T } ^ { m } = ( x _ { T } ^ { 1 } , y _ { T } ^ { 1 } ) , \\dots , ( x _ { T } ^ { m } , y _ { T } ^ { m } )$ which is a set of $m$ i.i.d. samples from the data distribution $( z _ { T } ^ { j } \\sim T )$ ). ",
+        "bbox": [
+            173,
+            414,
+            825,
+            474
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We consider models with parameters $\\theta$ that map datapoints $X$ to representations $\\widetilde { X } ( X , \\theta )$ . Our objective is the expecteation of the negative mutual information between representations an labels across all tasks: ",
+        "bbox": [
+            174,
+            483,
+            825,
+            526
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7b4b1ed24afa5879b8258edda5f297fd4caf201db44be6f01cfa8b08a52ff093.jpg",
+        "text": "$$\n\\begin{array} { r } {  { \\mathcal { L } } ( \\tau , \\theta ) = -  { \\mathbb { E } } _ { T \\sim \\tau } \\left[ I ( \\widetilde { X } ( X _ { T } , \\theta ) ; Y _ { T } ) \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            531,
+            632,
+            559
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This objective is closely related to both previous loss functions and evaluation metrics in setups that involve testing on unseen classes (e.g. few-shot classification, image retrieval). We show that previous loss functions can be seen as approximations to (2) in Appendix A, and that mutual information is strongly correlated with metrics such as few-shot accuracy and Recall@1 in Section 5.1. ",
+        "bbox": [
+            173,
+            565,
+            825,
+            622
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "An important difference between (2) and previous objectives for few-shot classification is that we do not split each task into support/query (also called train/test) sets. This property bypasses the pesky issue of using batch normalization (BN) during meta-training. As Nichol & Schulman (2018) points out, BN can leak information from the support set to the query set. This issue is not a problem in our setup since we do not assume a seperate \"query set\". Additionally, not using support/query splits enables meta-learning with tasks consisting of one image per class, as we demonstrate in Section 5.4. Note that the standard meta-learning setup cannot learn from such tasks since it requires at least two images per class (one for support, one for query) to compute the loss function. Overall, our model demonstrates that the commonly used construction of a held-out test set within each task is not strictly necessary for meta-learning. ",
+        "bbox": [
+            173,
+            628,
+            826,
+            768
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 GENERALIZATION BOUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            786,
+            398,
+            801
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We bound the difference between expected loss and empirical loss: ",
+        "bbox": [
+            173,
+            813,
+            612,
+            828
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Let $\\tau , n , m , X , \\widetilde { X } , Y , \\theta , { \\mathcal { L } }$ be defined as above. Let $d _ { \\Theta }$ be the VC dimension of the encoder $\\widetilde { X } ( \\cdot )$ . Let $\\hat { I } ( \\widetilde { X } ( X _ { T } , \\theta ) ; Y _ { T } )$ be the empirical estimate of the mutual information using finite dataset $D _ { T }$ , and define empirical loss as ",
+        "bbox": [
+            174,
+            833,
+            825,
+            880
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/83b4b5b746a16d1066edc2244804f7cc00f60280553b80409b114035c75f5cce.jpg",
+        "text": "$$\n\\hat { \\mathcal { L } } ( T ^ { 1 : n } , \\theta ) = - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\hat { I } ( \\widetilde { X } ( X _ { T ^ { i } } , \\theta ) ; Y _ { T ^ { i } } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            887,
+            622,
+            928
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/12e9ffefd873e73d60d077a6e59b12e0d4e86b87e574e42f8b209cd3081fec03.jpg",
+        "image_caption": [
+            "Figure $1 \\colon \\mathsf { A }$ graphical overview of Discrete InfoMax COdes (DIMCO). A dataset $D$ consists of pairs of images $X$ and labels $Y$ . DIMCO is a stochastic encoder that maps each image $X$ to a distribution of discrete codes $p ( { \\widetilde { X } } | X )$ . Each discrete code $\\widetilde { \\mathbf { x } } \\sim p ( \\widetilde { X } | X )$ is a $p$ -way code of length $d$ . If $p = 4 , d = 2$ e(as in the diagram), each code consists of 2 symbols and each symbol is $\\in \\{ 1 , 2 , 3 , 4 \\}$ . Inside the $4 \\times 4$ grid that represents the $p ^ { d } = 1 6$ possible codes, the most likely row and column are colored. The most likely code in the diagram is $( 1 , 2 )$ with probability $3 0 \\%$ . DIMCO is optimized by maximizing the mutual information between the discrete code and the label within each batch. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            102,
+            816,
+            266
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The following inequality holds with high probability: ",
+        "bbox": [
+            173,
+            410,
+            519,
+            425
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/65ed082e341bac29ba220fe723d6c9a4631fbe5347dfa294ce541b84d975b75b.jpg",
+        "text": "$$\n\\mathcal { L } ( \\tau , \\theta ) - \\hat { \\mathcal { L } } ( T ^ { 1 : n } , \\theta ) \\leq O \\left( \\sqrt { \\frac { d _ { \\Theta } } { n } \\log \\frac { n } { d _ { \\Theta } } } \\right) + O \\left( \\frac { | \\widetilde { X } | \\log ( m ) } { \\sqrt { m } } \\right) + O \\left( \\frac { | \\widetilde { X } | | Y | } { m } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            431,
+            754,
+            474
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proof. We use standard Chernoff bounds along with a finite sample bound for mutual information from Shamir et al. (2010); see Appendix B. ",
+        "bbox": [
+            174,
+            488,
+            823,
+            517
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The generalization gap has three terms, two of which decrease as $m$ increases, and the other decreases as $n$ increases. Typically for few-shot learning, $n$ is very large while $m$ is small: the typical miniImagenet 5-way 1-shot setup has $n > 1 0 ^ { 1 0 }$ and $m = 5$ . We therefore claim that the latter two terms are the main difficulties for generalizing to new tasks. We see from Theorem 1 that these terms can be reduced by using small $| \\widetilde { X } |$ . Therefore, in the context of meta-learning, using short representations (i.e. small $| \\widetilde { X } | )$ can compensate for having a small train set (i.e. small $m$ ). ",
+        "bbox": [
+            173,
+            534,
+            825,
+            623
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Meta-learning is typically formulated as performing two levels of learning: task-general learning and task-specific learning. Theorem 1 implies that in the tasks considered in recent literature, task-general learning should have almost no generalization gap, making task-specific learning the main source of (meta-)overfitting. This can be problematic since in many works, the task-specific learning algorithm is usually just a byproduct of whatever clever meta-learning loss was proposed. Our theorem suggests that we should pay more attention to directly regularizing this task-specific learner, and our DIMCO model, described in the next section, can be seen as a minimal working example in this direction. ",
+        "bbox": [
+            173,
+            628,
+            825,
+            728
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 DISCRETE INFOMAX CODES (DIMCO) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            747,
+            531,
+            763
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We now present our model, Discrete InfoMax COdes (DIMCO). Motivated by Section 2, DIMCO produces a short discrete code $\\widetilde { X }$ and is trained by maximizing mutual information $I ( \\widetilde { X } ; Y )$ . Figure 1 graphically shows the overall structure of DIMCO. ",
+        "bbox": [
+            174,
+            779,
+            825,
+            824
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 FACTORIZED DISCRETE CODES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            842,
+            431,
+            856
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose a factorized discrete representation scheme which enables us to represent discrete distributions with exponentially fewer parameters compared to listing the probability of each event. We represent each event as the product of $d$ independent events, each of which consists of $p$ different possibilities. We thus have $p ^ { d }$ events in total, but only require $p d$ parameters to represent the probability of each event. Binary codes can be viewed as a special case of this scheme where $p = 2$ . This factorization trick allows us to consider representations of size $| \\widetilde { X } | = 6 4 ^ { 2 5 6 }$ (Section 5). This representation has the advantage of requiring only $d \\log _ { 2 } p$ bits per datapoint, whereas a $D$ - dimensional continuous vector embedding requires $3 2 d$ bits (assuming 32-bit floats). ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            102,
+            826,
+            162
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 MODEL ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            179,
+            267,
+            193
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Recall that we represent a given image using $d$ independent discrete distributions, each of which has $p$ possibilities. First, a (convolutional) neural network $\\operatorname { e n c } ( { \\mathord { \\cdot } } )$ takes image $X$ as input and outputs a vector of length $d p$ , which we reshape into a matrix of size $d \\times p$ : ",
+        "bbox": [
+            174,
+            204,
+            825,
+            247
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d197c820997942f3fb8660fffd6f8fcd0ecd2b424baa214a800a6a6a7fe52089.jpg",
+        "text": "$$\n\\operatorname { e n c } ( X ) = { \\left[ \\begin{array} { l l l l l } { l _ { 1 1 } } & { l _ { 1 2 } } & { l _ { 1 3 } } & { \\dots } & { l _ { 1 p } } \\\\ { l _ { 2 1 } } & { l _ { 2 2 } } & { l _ { 2 3 } } & { \\dots } & { l _ { 2 p } } \\\\ { \\vdots } & { \\vdots } & { \\vdots } & { \\ddots } & { \\vdots } \\\\ { l _ { d 1 } } & { l _ { d 2 } } & { l _ { d 3 } } & { \\dots } & { l _ { d p } } \\end{array} \\right] }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            252,
+            611,
+            327
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Each row of this matrix represents the logits of a discrete distribution. We apply the softmax function to each row to get probabilities. ",
+        "bbox": [
+            171,
+            330,
+            825,
+            359
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/998e5ea719943c5a4760f0cd3a2569f19744729d6863004d473febc7c4c229e1.jpg",
+        "text": "$$\n\\mathrm { s o f t m a x } ( l _ { i 1 } , \\dots , l _ { i p } ) = p _ { i 1 } , \\dots , p _ { i p }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            364,
+            604,
+            382
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The ith codeword is sampled according to the categorical distribution following these probabilites: ",
+        "bbox": [
+            171,
+            387,
+            818,
+            402
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/74b21412fe0ae3822739285a5fbb66ad7f19e58218125dc92ec60e1ee598ea33.jpg",
+        "text": "$$\n\\widetilde { x } _ { i } \\sim \\mathrm { C a t } ( p _ { i 1 } , \\dots , p _ { i p } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            409,
+            562,
+            426
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We simply concatenate each $\\widetilde { x } _ { i }$ to obtain the representation: $\\widetilde { \\mathbf { x } } = ( \\widetilde { x } _ { 1 } , \\dots , \\widetilde { x } _ { d } )$ ",
+        "bbox": [
+            173,
+            431,
+            687,
+            446
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            462,
+            287,
+            477
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Recall that $\\widetilde { \\mathbf { x } }$ is a discrete random variable and $\\widetilde { X }$ is its distribution. Instead of sampling $\\widetilde { \\mathbf { x } } \\sim \\widetilde { X }$ , we directly use $\\widetilde { X }$ to compute the objective: ",
+        "bbox": [
+            174,
+            486,
+            825,
+            520
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/05d885264523bba660645974f418eda89a8b6d91e9fdf703463cde8c3129e6a2.jpg",
+        "text": "$$\nI ( \\widetilde { X } ; Y ) = H ( \\widetilde { X } ) - H ( \\widetilde { X } | Y ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            523,
+            617,
+            547
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The first term, $H ( { \\tilde { X } } )$ , can be calculated by taking the average of all probabilities and computing the entropy: ",
+        "bbox": [
+            173,
+            560,
+            826,
+            590
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f951e807cf4d12e236a9e16073be85fe93d4649a473b4c60a0234fd856153eb0.jpg",
+        "text": "$$\nH ( \\widetilde X ) = \\sum _ { i = 1 } ^ { d } H ( \\widetilde X _ { i } ) = \\sum _ { i = 1 } ^ { d } H \\left( \\mathrm { C a t } \\left( \\frac { \\sum _ { j = 1 } ^ { m } p _ { i 1 } ^ { j } } { m } , \\frac { \\sum _ { j = 1 } ^ { m } p _ { i 2 } ^ { j } } { m } , \\ldots , \\frac { \\sum _ { j = 1 } ^ { m } p _ { i p } ^ { j } } { m } \\right) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            594,
+            745,
+            640
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The second term is $\\begin{array} { r } { H ( \\widetilde { X } | Y ) = \\sum _ { k = 1 } ^ { c } p ( Y = k ) H ( \\widetilde { X } | Y = k ) } \\end{array}$ where $c$ is the number of classes. The marginal probability of $\\mathrm { Y }$ $( p ( Y = k ) )$ is the frequency of $k$ in $\\{ \\mathbf { y } ^ { 1 } , \\ldots , \\mathbf { y } ^ { m } \\}$ . $H ( \\widetilde { X } | Y = k )$ can be obtained by computing (9) using only $\\mathbf { x } ^ { j }$ for which $\\mathbf { y } ^ { j } = k$ . ",
+        "bbox": [
+            173,
+            652,
+            825,
+            702
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Though we have motivated the use of $I ( \\widetilde { X } ; Y )$ as a loss function throughout this paper, we provide yet another perspective using the decomposition in (8). Minimizing $H ( \\widetilde X | Y )$ encourages discriminatory behavior. This term encourages the average embedding of each class to be as concentrated as possible. Maximizing $H ( { \\widetilde { X } } )$ incentivizes the model to overall use all possible values of $\\widetilde { X }$ . ",
+        "bbox": [
+            173,
+            708,
+            826,
+            772
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We emphasize that such closed-form computation of $I ( \\widetilde { X } ; Y )$ is only possible because we are using discrete codes. ",
+        "bbox": [
+            174,
+            779,
+            823,
+            810
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4 EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            827,
+            307,
+            840
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We map all images to their probabilities $p _ { i j }$ via (5) and (6) for all $i = 1 , \\ldots , d$ and $j = 1 , \\dotsc , p$ . We map each training image to its most likely code: ",
+        "bbox": [
+            173,
+            852,
+            825,
+            882
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/12bd1ce455f9bbc9d7e187f318593c008521280ef16e6b0fbeb86bb07cf815b4.jpg",
+        "text": "$$\n\\widetilde { \\mathbf { x } } = \\Big ( \\underset { j } { \\overset { \\widetilde { x } _ { 1 } } { \\operatorname { a r g m a x } } } p _ { 1 j } , \\ \\overset { \\widetilde { x } _ { 2 } } { \\underset { j } { \\operatorname { a r g m a x } } } p _ { 2 j } , \\ \\cdot \\cdot \\cdot , \\ \\overbrace { \\mathrm { a r g } \\underset { j } { \\operatorname { m a x } } p _ { d j } } ^ { \\widetilde { x } _ { d } } \\Big )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            887,
+            668,
+            928
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Fix a train image and a test image, and let $\\widetilde { \\mathbf { x } }$ be the most likely code for the train image. The similarity ebetween train image and test image is measured by the probability of the test image producing $\\widetilde { \\mathbf { x } }$ . This amounts to computing the product 1 of the test image’s probabilites using $\\widetilde { \\mathbf { x } } _ { i }$ for each $i = 1 , \\ldots , d$ : ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4bf6565a405970cc1dd53e0c89149226fccf262f3e1c6eca7bdead0445909618.jpg",
+        "text": "$$\n\\prod _ { i = 1 } ^ { d } p _ { \\widetilde { \\mathbf { x } } _ { i } i } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            450,
+            154,
+            513,
+            198
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We use this as a similarity metric for both few-shot classification and image retrieval. We perform few-shot classification by computing the most likely code for each class via (10) and classifying each test image by choosing the class that has highest value of (11). We similarly perform image retrival by mapping each support image to its most likely code (10) and for each query image retrieving the support image that has highest (11). ",
+        "bbox": [
+            174,
+            212,
+            825,
+            284
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            304,
+            344,
+            320
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Regularizing Meta-Learners The ability to generalize to novel datasets is critical in meta-learning benchmarks, and even more so in benchmarks such as Meta-Dataset (Triantafillou et al., 2019), where a model is tested on datasets from an unseen domain. To the best of our knowledge, no works have proposed an explicit regularizer for generalizing to new tasks. ",
+        "bbox": [
+            174,
+            337,
+            825,
+            392
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our analysis suggests that the success of some previous meta-learning methods can be attributed to being implicitly regularized by reducing the expressive power reducing task-specific learning. The following works have reported benefits from reducing the number of such task-specific parameters: Lee & Choi (2018) learns a subset of the full network to alter during task-specific learning, Rusu et al. (2018) explicitly represents each task with a low-dimensional latent space, and Zintgraf et al. (2018) alters only a pre-specified subset of the full network during task-specific learning. It may be surprising at first that these methods achieve higher accuracy than vanila MAML (Finn et al., 2017), which is more expressive since it alters all parameters during task-specific learning. Kim et al. (2018) also reports better meta-generalization through approximate variational inference with respect to a learned prior, which can be seen as restricting the search space of the task-specific learner. We showed through Theorem 1 that restricting inner-loop expressivity reduces the generalization gap; this provides theoretical understanding to this consensus that meta-learning models with simple task-specific learners generalize to new tasks more easily. ",
+        "bbox": [
+            173,
+            400,
+            826,
+            580
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Information Bottleneck Theorem 1 is close in spirit to the information bottleneck principle (Tishby et al., 2000; Tishby & Zaslavsky, 2015; Shwartz-Ziv & Tishby, 2017). This principle states that to generalize, one should maximize $I ( \\widetilde { X } ; Y )$ while simultaneously minimizing $I ( { \\widetilde { X } } ; X )$ . Likewise, our objective (2) is $I ( \\widetilde { X } ; Y )$ while our bound (22) suggests that the representation capacity $| \\widetilde { X } |$ should be low for generalization. Also related is the deterministic information bottleneck (Strouse & Schwab, 2017) which extends the information bottleneck by minimizing $H ( { \\tilde { X } } )$ rather than $I ( { \\widetilde { X } } ; X )$ . These three approaches to generalization are related via the chain of inequalities $I ( \\widetilde { X } ; X ) \\le H ( \\widetilde { X } ) \\le$ $\\log | \\widetilde { X } |$ , which is tight when $\\widetilde { X }$ is an efficient code. ",
+        "bbox": [
+            173,
+            597,
+            825,
+            723
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Representation Learning Previous works have applied information-theoretic principles to analyze the objective of VAEs (Alemi et al., 2017; Chen et al., 2018), derive an objective for GANs to learn disentangled features (Chen et al., 2016), and to directly learn representations (Alemi et al., 2016; Hjelm et al., 2018; Oord et al., 2018; Grover & Ermon, 2018; Choi et al., 2019). Our work can also be viewed as an information-theoretic representation learning method, but we assume a supervised meta-learning setup and our main focus is the meta-generalization problem. ",
+        "bbox": [
+            173,
+            739,
+            825,
+            824
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Discrete Representations Discrete representations have been thoroughly studied in the context of information theory (Shannon, 1948). Recent deep learning methods directly learn discrete representations, by learning variational autoencoders with discrete latent variables (Rolfe, 2016; van den Oord et al., 2017; Razavi et al., 2019) or maximizing the mutual information between representation and data (Hu et al., 2017). DIMCO is related to but differs from these works as it assumes a supervised meta-learning setting and performs infomax using labels instead of data. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            897
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similarly to our setup, Jeong & Song (2018) uses labels to learn a binary hash code. Their focus is on the speedup gained by using sparse codes, whereas DIMCO learns a dense discrete code to generalize better. Additionally, their method solves a minimum cost flow problem within each batch to find the locally optimal code, whereas DIMCO is able to directly compute its loss function. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            194
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Factorized Representations The idea of using factorized representations has appeared the contexts of quantizing a continuous input (Jegou et al., 2011), memory-efficient clustering (Norouzi & Fleet, 2013), and constructing an expressive attention mechanism using few parameters (Vaswani et al., 2017). Likewise, DIMCO factorizes its discrete representatations to increase its representation power. ",
+        "bbox": [
+            174,
+            212,
+            825,
+            267
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Metric Learning The structure and loss function of DIMCO is closely related to embedding-based meta-learning Vinyals et al. (2016); Snell et al. (2017); Oreshkin et al. (2018) and image retrieval Hoffer & Ailon (2015); Sohn (2016); Wu et al. (2017); Duan et al. (2018) methods. We show in Appendix A that the loss functions of these methods can be seen as approximation to the mutual information $( I ( \\widetilde { X } ; Y ) )$ . While all of these previous methods require a support/query split within each task, DIMCO simply optimizes an information-theoretic quantity of each batch, removing the need for such structured batch construction. ",
+        "bbox": [
+            174,
+            285,
+            825,
+            383
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            406,
+            326,
+            421
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We use the miniImageNet (Ravi & Larochelle, 2016) and CUB200 (Wah et al., 2011) datasets with standard splits for both in our experiments. The miniImageNet dataset is a subset of the Imagenet (Krizhevsky et al., 2012) dataset that was made for few-shot classification. It consists of 100 classes each containing 600 images of size $8 4 \\times 8 4$ . The classes are split into 64 training, 16 validation, and 24 test classes. The Caltech-UCSD Birds-200-2011 (CUB200) dataset consists of 11788 images of birds from 200 classes. The classes are split into 100 training and 100 test classes. ",
+        "bbox": [
+            174,
+            438,
+            825,
+            522
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We use two different CNN backbones for our experiments: the 4-layer convnet commonly used for meta-learning (Finn et al., 2017; Sung et al., 2018; Liu et al., 2018), and the Inception network (Szegedy et al., 2015) with batch normalization (Ioffe & Szegedy, 2015) which is commonly used for deep image retrieval (Sohn, 2016; Movshovitz-Attias et al., 2017; Wu et al., 2017). We randomly initialize weights for the 4-layer convnet and use pretrained weights for the Inception network ",
+        "bbox": [
+            174,
+            529,
+            825,
+            599
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 CORRELATION OF METRICS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            617,
+            406,
+            631
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This experiment empirically verifies whether mutual information $I ( \\widetilde { X } ; Y )$ is a reasonable metric for quality of representation. We trained DIMCO on the miniImagenet dataset with $p = d = 6 4$ for 20 epochs, for 8 independent runs. We plot the pairwise correlations between five different metrics ( (5, 10, 20)-way 1-shot accuracy, Recall $@ 1$ , and $I ( \\widetilde { X } ; Y )$ ) in Figure 2. We see that all five metrics are very strongly correlated. We observed similar trends when training with with previously proposed loss functions: we visualize these results in Figure 5 of the appendix due to space constraints. Alongside this empirical evidence, we prove in Appendix A that previously used loss functions for few-shot classification and image retrieval are approximations to $I ( \\widetilde { X } ; Y )$ . ",
+        "bbox": [
+            173,
+            642,
+            826,
+            761
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2 WHAT DOES EACH CODE REPRESENT? ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            779,
+            477,
+            792
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We inspected what each code represents in a DIMCO model $ d  = 1 6$ , $p = 6 4 .$ ) trained on the miniImagenet dataset. Recall that each image produces a $d \\times p$ probability matrix (5, 6). For each of these $d p$ entries, we plotted the top 10 images in the test set that assigned highest probability to that entry. We show images corresponding to four such codes in Figure 2 (right) and more in Figure 6 of the appendix. ",
+        "bbox": [
+            174,
+            804,
+            825,
+            875
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We see that DIMCO learns a distributed representation. For example, the top code in Figure 2 represents the high-level concept of a furry animal and the shown 10 images span 4 different classes. On the other hand, the bottom code seems to focus on the color of the background. By aggregating such complementary features in each of its $d$ codewords, DIMCO is able to classify images that belong to previously unseen classes. ",
+        "bbox": [
+            176,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/9dad9f4655d982f4fcff4b2c238b971b41e77a79c0d27076551b36b515297007.jpg",
+        "image_caption": [
+            "Figure 2: Left: Pairwise correlation between $\\mathbf { M } \\mathbf { I } = I ( X ; { \\widetilde { X } } )$ and previous metrics. Right: Visualization of codes of a small trained DIMCO model $\\left( d = 1 6 \\right.$ , $p = 6 4 ^ { \\prime }$ ); we show the top 10 test set images that assign highest probability to a specific code, for 4 different codes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            99,
+            812,
+            392
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/4358d89791c6cfc1b52cef4a6064cb652948c17f77299d65f2c0f8411f1bfef0.jpg",
+        "image_caption": [
+            "Figure 3: Image retrieval performance of DIMCO and N-pair loss on the CUB-200 dataset. The y-axis for both figures are the Recall $@ 1$ metric, and error bars reflect standard deviation computed from $n = 5$ runs per configuration. The $\\mathbf { X }$ -axes represent (left) bits required to store one representation and (right) seconds required to perform retrieval for one query. Both $\\mathbf { X }$ -axes are log-scale. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            473,
+            815,
+            604
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            703,
+            823,
+            731
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 TIME- AND MEMORY-EFFICIENT IMAGE RETRIEVAL ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            750,
+            575,
+            765
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We conducted an image retrieval experiment using the CUB200 dataset, and used multiclass Npair loss (Sohn, 2016) as a baseline. As is standard for image retrieval, we use the Inception network as specified in the beginning of this section. Using the same backbone, we trained DIMCO with $( p , d ) \\in \\{ 6 4 , 1 2 8 , 2 5 6 \\} \\times \\{ 1 2 \\bar { 8 } , 2 5 6 , 5 1 2 \\}$ and multiclass N-pair with embedding dimension $\\in \\{ 1 2 8 , 2 5 6 , 5 1 2 \\}$ . We measured the time per query for each method on a single Tesla P40 GPU by averaging the time required for 10000 batches of queries of size 32. ",
+        "bbox": [
+            174,
+            776,
+            825,
+            861
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results in Figure 3 show that the compact code of DIMCO takes roughly an order of magnitude less memory for similar performance to N-pair loss, and requires less query time as well. This experiment also demonstrates that discrete representations can match the performance of modern methods that use continuous embeddings on this relatively large-scale task. We additionally note that DIMCO is able to train using large backbones without significantly overfitting, whereas experiments reported in Mishra et al. (2017) indicate that MAML (Finn et al., 2017) overfits tremendously when using a deeper backbone. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/de8fef7019d5e73acdd2af9ee30318f253d3d0b71fa82bc7d7897d338ca38f25.jpg",
+        "image_caption": [
+            "Figure 4: Performance of methods trained using subsets of miniImageNet of varying size. The lowermost y axis value for each metric corresponds to the expected performance of random guessing. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            101,
+            820,
+            295
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            363,
+            825,
+            405
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 GENERALIZATION TO NEW TASKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            422,
+            450,
+            438
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This experiment measures how well DIMCO can generalize to new datasets after training with a small number of datasets. This challenging experimental setup measures how much generalizable information the model can extract from a limited set of datasets; it can be seen as the meta-learning analogue of measuring the performance of a classifier trained with a small dataset. ",
+        "bbox": [
+            174,
+            449,
+            825,
+            506
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We trained each model using $\\{ 1 , 4 , 1 6 , 6 4 \\}$ samples from each training class in the miniImageNet dataset. For example, by using 4 samples, we are reducing the full train split of (64 classes $\\times ~ 6 0 0$ images per class) into (64 classes $\\times 4$ images per class). We compare against three baseline methods: Triplet Nets(Hoffer & Ailon, 2015), multiclass N-pair loss(Sohn, 2016), and ProtoNets(Snell et al., 2017). We report the average and standard deviation of the top 5 results of a random hyperparameter search (see appendix for details). We show $\\{ 5 , 1 0 , 2 0 \\}$ -way 1-shot accuracies and Recal $@ 1$ of the test set in Figure 4. ",
+        "bbox": [
+            174,
+            512,
+            825,
+            609
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "First note that DIMCO is the only method that can train using a dataset consisting of 1 example per class. This is because other methods require at least one train and test example per class within each batch, while DIMCO requires no such train/test (also called support/query) split and simply maximizes the mutual information within a batch. Furthermore, Figure 4 shows that DIMCO learns much more effectively when the number of examples per class is low; this is because DIMCO uses fewer bits to describe each datapoint, which lowers its generalization gap (24) when applying to novel datasets. ",
+        "bbox": [
+            174,
+            617,
+            825,
+            714
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION AND DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            736,
+            468,
+            751
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Summary We derived a generalization bound for meta-learning using information-theoretic principles. Building on our bound, we proposed DIMCO, a model that learns a discrete representation of data by maximizing its mutual information with the label. DIMCO had benefits in time, memory, and generalization in our experiments. ",
+        "bbox": [
+            174,
+            767,
+            825,
+            824
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Towards Explicit Meta-Regularization In Section 4, we have suggested with analogy to Theorem 1 that the benefits of some previous meta-learning methods can be attributed to implicitly being regularized by reducing the expressivity of their task-specific learners. While DIMCO has demonstrated better generalization by reducing this expressivity via hyperparameters $( d , p )$ , this is only applicable to DIMCO’s specific setup of few-shot classification through mutual information maximization. In future work, we would like to explore explicit meta-regularization schemes that can be applied to other problems (regression, reinforcement learning etc.) and algorithms (MAML, Neural Process etc.). ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Meta-Learning Without Splits Our meta-learning problem setup in Section 2.1 does not assume a support/query (also called train/test) split within each dataset. Along with showing that the traditional support/query split is not strictly necessary, we demonstrated in Section 5.4 that removing it has the benefit of enabling meta-learning in datasets having one image per class. We believe future work could benefit from further exploring this space of meta-learning algorithms that use datasets without splits. ",
+        "bbox": [
+            174,
+            148,
+            825,
+            233
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            255,
+            285,
+            270
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alexander A Alemi, Ian Fischer, Joshua V Dillon, and Kevin Murphy. Deep variational information bottleneck. arXiv preprint arXiv:1612.00410, 2016. ",
+        "bbox": [
+            174,
+            279,
+            825,
+            308
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alexander A Alemi, Ben Poole, Ian Fischer, Joshua V Dillon, Rif A Saurous, and Kevin Murphy. Fixing a broken elbo. arXiv preprint arXiv:1711.00464, 2017. ",
+        "bbox": [
+            171,
+            318,
+            825,
+            348
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ron Amit and Ron Meir. Meta-learning by adjusting priors based on extended pac-bayes theory. arXiv preprint arXiv:1711.01244, 2017. ",
+        "bbox": [
+            174,
+            358,
+            823,
+            387
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Tian Qi Chen, Xuechen Li, Roger B Grosse, and David K Duvenaud. Isolating sources of disentanglement in variational autoencoders. In Advances in Neural Information Processing Systems, pp. 2610–2620, 2018. ",
+        "bbox": [
+            174,
+            398,
+            825,
+            441
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In Advances in neural information processing systems, pp. 2172–2180, 2016. ",
+        "bbox": [
+            176,
+            452,
+            823,
+            496
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Kristy Choi, Kedar Tatwawadi, Aditya Grover, Tsachy Weissman, and Stefano Ermon. Neural joint source-channel coding. In International Conference on Machine Learning, pp. 1182–1192, 2019. ",
+        "bbox": [
+            174,
+            506,
+            823,
+            536
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012. ",
+        "bbox": [
+            176,
+            545,
+            810,
+            563
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yueqi Duan, Wenzhao Zheng, Xudong Lin, Jiwen Lu, and Jie Zhou. Deep adversarial metric learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2780–2789, 2018. ",
+        "bbox": [
+            176,
+            571,
+            823,
+            616
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Chelsea Finn and Sergey Levine. Meta-learning and universality: Deep representations and gradient descent can approximate any learning algorithm. arXiv preprint arXiv:1710.11622, 2017. ",
+        "bbox": [
+            173,
+            626,
+            823,
+            656
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. arXiv preprint arXiv:1703.03400, 2017. ",
+        "bbox": [
+            173,
+            666,
+            823,
+            696
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Sebastian Flennerhag, Pablo G Moreno, Neil D Lawrence, and Andreas Damianou. Transferring knowledge across learning processes. arXiv preprint arXiv:1812.01054, 2018. ",
+        "bbox": [
+            173,
+            705,
+            823,
+            736
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Anirudh Goyal, Riashat Islam, Daniel Strouse, Zafarali Ahmed, Matthew Botvinick, Hugo Larochelle, Sergey Levine, and Yoshua Bengio. Infobot: Transfer and exploration via the information bottleneck. arXiv preprint arXiv:1901.10902, 2019. ",
+        "bbox": [
+            174,
+            746,
+            823,
+            790
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Aditya Grover and Stefano Ermon. Uncertainty autoencoders: Learning compressed representations via variational information maximization. arXiv preprint arXiv:1812.10539, 2018. ",
+        "bbox": [
+            173,
+            801,
+            823,
+            830
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. arXiv preprint arXiv:1808.06670, 2018. ",
+        "bbox": [
+            176,
+            840,
+            825,
+            883
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Elad Hoffer and Nir Ailon. Deep metric learning using triplet network. In International Workshop on Similarity-Based Pattern Recognition, pp. 84–92. Springer, 2015. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Weihua Hu, Takeru Miyato, Seiya Tokui, Eiichi Matsumoto, and Masashi Sugiyama. Learning discrete representations via information maximizing self-augmented training. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1558–1567. JMLR. org, 2017. ",
+        "bbox": [
+            174,
+            103,
+            826,
+            159
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015. ",
+        "bbox": [
+            176,
+            170,
+            823,
+            199
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Herve Jegou, Matthijs Douze, and Cordelia Schmid. Product quantization for nearest neighbor search. IEEE transactions on pattern analysis and machine intelligence, 33(1):117–128, 2011. ",
+        "bbox": [
+            174,
+            208,
+            823,
+            238
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yeonwoo Jeong and Hyun Oh Song. Efficient end-to-end learning for quantizable representations. arXiv preprint arXiv:1805.05809, 2018. ",
+        "bbox": [
+            176,
+            247,
+            825,
+            276
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Taesup Kim, Jaesik Yoon, Ousmane Dia, Sungwoong Kim, Yoshua Bengio, and Sungjin Ahn. Bayesian model-agnostic meta-learning. arXiv preprint arXiv:1806.03836, 2018. ",
+        "bbox": [
+            174,
+            285,
+            823,
+            315
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. ",
+        "bbox": [
+            173,
+            324,
+            823,
+            353
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012. ",
+        "bbox": [
+            176,
+            363,
+            825,
+            405
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yoonho Lee and Seungjin Choi. Gradient-based meta-learning with learned layerwise metric and subspace. 2018. ",
+        "bbox": [
+            174,
+            415,
+            823,
+            445
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yanbin Liu, Juho Lee, Minseop Park, Saehoon Kim, and Yi Yang. Transductive propagation network for few-shot learning. arXiv preprint arXiv:1805.10002, 2018. ",
+        "bbox": [
+            171,
+            454,
+            823,
+            484
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Luke Metz, Niru Maheswaranathan, Brian Cheung, and Jascha Sohl-Dickstein. Meta-learning update rules for unsupervised representation learning. arXiv preprint arXiv:1804.00222, 2018. ",
+        "bbox": [
+            171,
+            493,
+            823,
+            522
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Nikhil Mishra, Mostafa Rohaninejad, Xi Chen, and Pieter Abbeel. A simple neural attentive metalearner. arXiv preprint arXiv:1707.03141, 2017. ",
+        "bbox": [
+            174,
+            531,
+            823,
+            561
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yair Movshovitz-Attias, Alexander Toshev, Thomas K Leung, Sergey Ioffe, and Saurabh Singh. No fuss distance metric learning using proxies. In Proceedings of the IEEE International Conference on Computer Vision, pp. 360–368, 2017. ",
+        "bbox": [
+            174,
+            570,
+            826,
+            614
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Nichol and John Schulman. Reptile: a scalable metalearning algorithm. arXiv preprint arXiv:1803.02999, 2018. ",
+        "bbox": [
+            173,
+            623,
+            823,
+            652
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mohammad Norouzi and David J Fleet. Cartesian k-means. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3017–3024, 2013. ",
+        "bbox": [
+            173,
+            661,
+            825,
+            693
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018. ",
+        "bbox": [
+            169,
+            700,
+            825,
+            731
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Boris Oreshkin, Pau Rodríguez López, and Alexandre Lacoste. Tadam: Task dependent adaptive metric for improved few-shot learning. In Advances in Neural Information Processing Systems, pp. 721–731, 2018. ",
+        "bbox": [
+            174,
+            739,
+            826,
+            782
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sachin Ravi and Hugo Larochelle. Optimization as a model for few-shot learning. 2016. ",
+        "bbox": [
+            174,
+            792,
+            754,
+            808
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ali Razavi, Aaron van den Oord, and Oriol Vinyals. Generating diverse high-fidelity images with vq-vae-2. arXiv preprint arXiv:1906.00446, 2019. ",
+        "bbox": [
+            171,
+            816,
+            825,
+            847
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jason Tyler Rolfe. Discrete variational autoencoders. arXiv preprint arXiv:1609.02200, 2016. ",
+        "bbox": [
+            169,
+            856,
+            790,
+            872
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Andrei A Rusu, Dushyant Rao, Jakub Sygnowski, Oriol Vinyals, Razvan Pascanu, Simon Osindero, and Raia Hadsell. Meta-learning with latent embedding optimization. arXiv preprint arXiv:1807.05960, 2018. ",
+        "bbox": [
+            176,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ohad Shamir, Sivan Sabato, and Naftali Tishby. Learning and generalization with the information bottleneck. Theoretical Computer Science, 411(29-30):2696–2711, 2010. ",
+        "bbox": [
+            171,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Claude Elwood Shannon. A mathematical theory of communication. Bell system technical journal, 27(3):379–423, 1948. ",
+        "bbox": [
+            173,
+            147,
+            825,
+            178
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ravid Shwartz-Ziv and Naftali Tishby. Opening the black box of deep neural networks via information. arXiv preprint arXiv:1703.00810, 2017. ",
+        "bbox": [
+            169,
+            191,
+            826,
+            222
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Advances in Neural Information Processing Systems, pp. 4077–4087, 2017. ",
+        "bbox": [
+            173,
+            236,
+            825,
+            267
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Kihyuk Sohn. Improved deep metric learning with multi-class n-pair loss objective. In Advances in Neural Information Processing Systems, pp. 1857–1865, 2016. ",
+        "bbox": [
+            174,
+            281,
+            823,
+            311
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "DJ Strouse and David J Schwab. The deterministic information bottleneck. Neural computation, 29 (6):1611–1630, 2017. ",
+        "bbox": [
+            171,
+            325,
+            825,
+            356
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Flood Sung, Yongxin Yang, Li Zhang, Tao Xiang, Philip HS Torr, and Timothy M Hospedales. Learning to compare: Relation network for few-shot learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1199–1208, 2018. ",
+        "bbox": [
+            173,
+            369,
+            826,
+            414
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 1–9, 2015. ",
+        "bbox": [
+            176,
+            429,
+            825,
+            473
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. In 2015 IEEE Information Theory Workshop (ITW), pp. 1–5. IEEE, 2015. ",
+        "bbox": [
+            173,
+            488,
+            823,
+            517
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Naftali Tishby, Fernando C Pereira, and William Bialek. The information bottleneck method. arXiv preprint physics/0004057, 2000. ",
+        "bbox": [
+            173,
+            531,
+            823,
+            561
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Eleni Triantafillou, Tyler Zhu, Vincent Dumoulin, Pascal Lamblin, Kelvin Xu, Ross Goroshin, Carles Gelada, Kevin Swersky, Pierre-Antoine Manzagol, and Hugo Larochelle. Meta-dataset: A dataset of datasets for learning to learn from few examples. arXiv preprint arXiv:1903.03096, 2019. ",
+        "bbox": [
+            176,
+            577,
+            823,
+            621
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Aaron van den Oord, Oriol Vinyals, et al. Neural discrete representation learning. In Advances in Neural Information Processing Systems, pp. 6306–6315, 2017. ",
+        "bbox": [
+            173,
+            636,
+            823,
+            665
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pp. 5998–6008, 2017. ",
+        "bbox": [
+            176,
+            679,
+            825,
+            723
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In Advances in neural information processing systems, pp. 3630–3638, 2016. ",
+        "bbox": [
+            174,
+            738,
+            823,
+            768
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Catherine Wah, Steve Branson, Peter Welinder, Pietro Perona, and Serge Belongie. The caltech-ucsd birds-200-2011 dataset. 2011. ",
+        "bbox": [
+            173,
+            782,
+            823,
+            811
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chao-Yuan Wu, R Manmatha, Alexander J Smola, and Philipp Krahenbuhl. Sampling matters in deep embedding learning. In Proceedings of the IEEE International Conference on Computer Vision, pp. 2840–2848, 2017. ",
+        "bbox": [
+            174,
+            827,
+            823,
+            871
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Luisa M Zintgraf, Kyriacos Shiarlis, Vitaly Kurin, Katja Hofmann, and Shimon Whiteson. Caml: Fast context adaptation via meta-learning. arXiv preprint arXiv:1810.03642, 2018. ",
+        "bbox": [
+            178,
+            886,
+            823,
+            916
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A PREVIOUS LOSS FUNCTIONS ARE APPROXIMATIONS TO MUTUAL INFORMATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            753,
+            136
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Cross-entropy Loss The cross-entropy loss has directly been used for few-shot classification (Vinyals et al., 2016; Snell et al., 2017). ",
+        "bbox": [
+            173,
+            152,
+            823,
+            181
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Let $q ( \\mathbf { y } | \\widetilde { \\mathbf { x } } ; \\boldsymbol { \\phi } )$ be a parameterized prediction of $\\mathbf { y }$ given $\\widetilde { \\mathbf { x } }$ , which tries to approximate the true conditional distribution $q ( \\mathbf { y } | \\widetilde { \\mathbf { x } } )$ . Typically in a classification network, $\\phi$ is the parameters of a learned projection matrix and $q ( \\cdot )$ e is the final linear layer. The expected cross-entropy loss can be written as ",
+        "bbox": [
+            173,
+            188,
+            825,
+            231
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fcbc1beda5e5202843efa3de5740531778e26e7b9032b88e08f5499102c8e894.jpg",
+        "text": "$$\n\\mathrm { x e n t } ( Y , \\widetilde { X } ) = \\mathbb { E } _ { { \\mathbf { y } } \\sim Y , \\widetilde { \\mathbf { x } } \\sim \\widetilde { X } } \\left[ - \\log q ( \\mathbf { y } | \\widetilde { \\mathbf { x } } , \\phi ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            239,
+            632,
+            262
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Assuming that the approximate distribution $q ( \\cdot )$ is sufficiently close to $p ( \\mathbf { y } \\vert \\widetilde { \\mathbf { x } } )$ , minimizing (12) can be seen as ",
+        "bbox": [
+            174,
+            271,
+            825,
+            301
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8909565f52a459ba46acdf2b2cf7652f7db1f107a482487492cc2b97049a6a05.jpg",
+        "text": "$$\n\\begin{array} { l l l } { \\arg \\operatorname* { m i n } \\mathbf { x e n t } ( Y , \\widetilde { X } ) } & { \\approx } & { \\arg \\operatorname* { m i n } \\mathbb { E } _ { \\mathbf { y } \\sim Y , \\widetilde { \\mathbf { x } } \\sim \\widetilde { X } } \\left[ - \\log p ( \\mathbf { y } | \\widetilde { \\mathbf { x } } ) \\right] } \\\\ & { = } & { \\arg \\operatorname* { m i n } H ( Y | \\widetilde { X } ) = \\arg \\operatorname* { m a x } I ( \\widetilde { X } ; Y ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
+            306,
+            717,
+            353
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where the last equality uses the fact that $H ( Y )$ is independent of model parameters. Therefore, crossentropy minimization is approximate maximization of the mutual information between representation $\\widetilde { X }$ and labels $Y$ . ",
+        "bbox": [
+            176,
+            359,
+            823,
+            405
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The approximation is that we parameterized $q ( \\mathbf { y } | \\widetilde { \\mathbf { x } } ; \\boldsymbol { \\phi } )$ as a linear projection. This structure cannot generalize to new classes because the parameters $\\phi$ are specific to the labels $\\mathbf { y }$ seen during training. For a model to generalize to unseen classes, one must amortize the learning of this approximate conditional distribution. (Vinyals et al., 2016; Snell et al., 2017) sidestepped this issue by using the embeddings for each class as $\\phi$ . ",
+        "bbox": [
+            173,
+            411,
+            825,
+            482
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Triplet Loss The Triplet loss (Hoffer & Ailon, 2015) is defined as ",
+        "bbox": [
+            173,
+            500,
+            620,
+            515
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5d6c674cf896b9484f4dde500016b2f55cb6224d5dc0d64e64bda5e30a544ab7.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { t r i p l e t } } = \\left\\| \\widetilde { \\mathbf { x } } _ { q } - \\widetilde { \\mathbf { x } } _ { p } \\right\\| _ { 2 } ^ { 2 } - \\left\\| \\widetilde { \\mathbf { x } } _ { q } - \\widetilde { \\mathbf { x } } _ { n } \\right\\| _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            523,
+            606,
+            547
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\widetilde { \\mathbf { x } } _ { q } , \\widetilde { \\mathbf { x } } _ { p } , \\widetilde { \\mathbf { x } } _ { n } \\in \\mathbb { R } ^ { d }$ are the embedding vectors of query, positive, and negative images. Let $\\mathbf { y } _ { q }$ e e edenote the label of the query data. Recall that the pdf function of a unit Gaussian is $\\log N ( \\widetilde { \\mathbf { x } } | \\mu , 1 ) \\overset { \\cdot } { = }$ $- c _ { 1 } - c _ { 2 } \\big | \\big | \\widetilde { \\mathbf { x } } - \\boldsymbol { \\mu } \\big | \\big | _ { 2 } ^ { 2 }$ , where $c _ { 1 } , c _ { 2 }$ are constants. Let $p _ { p } ( \\widetilde { \\mathbf { x } } ) = N ( \\widetilde { \\mathbf { x } } _ { p } , 1 )$ and $p _ { n } ( \\widetilde { \\mathbf { x } } ) = N ( \\widetilde { \\mathbf { x } } _ { n } , 1 )$ be eunit Gaussian distributions centered at $\\widetilde { \\mathbf { x } } _ { p } , \\widetilde { \\mathbf { x } } _ { n }$ e erespectively. We have ",
+        "bbox": [
+            174,
+            556,
+            823,
+            616
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f95c4dee5fc4ffd418ab3311a1c2f20e2dc0051c7d1da8512a5cca5d5661db39.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathbb { E } \\left[ - \\mathcal { L } _ { \\mathrm { t r i p l e t } } \\right] } & { \\propto \\quad \\mathbb { E } \\left[ \\log p _ { p } ( \\widetilde { \\mathbf { x } } ) - \\log p _ { n } ( \\widetilde { \\mathbf { x } } ) \\right] } \\\\ & { \\approx \\quad \\mathbb { E } \\left[ \\log p _ { p } ( \\widetilde { \\mathbf { x } } ) - \\log p ( \\widetilde { \\mathbf { x } } ) \\right] } \\\\ & { = \\quad - H ( \\widetilde { X } | Y ) + H ( \\widetilde { X } ) = I ( \\widetilde { X } ; Y ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            625,
+            673,
+            689
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Two approximations were made in the process. We first assumed that the embedding distribution of images not in $\\mathbf { y } _ { q }$ is equal to the distribution of all embeddings. This is reasonable when each class only represents a small fraction of the full data. We also approximated the embedding distributions $p ( \\tilde { \\mathbf { x } } | \\mathbf { y } ) , p ( \\tilde { \\mathbf { x } } )$ with unit Gaussian distributions centered at single samples from each. ",
+        "bbox": [
+            173,
+            695,
+            826,
+            752
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "N-pair Loss Multiclass $N$ -pair loss (Sohn, 2016) was proposed as an alternative to Triplet loss. This loss function requires one positive embedding $\\widetilde { \\mathbf { x } } ^ { + }$ and multiple negative embeddings $\\widetilde { \\mathbf { x } } _ { 1 } , \\hdots$ , $\\widetilde { \\mathbf { x } } _ { N - 1 }$ , and takes the form ",
+        "bbox": [
+            173,
+            770,
+            825,
+            813
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f8fae4d797d2d84e08a92982052123c748371f257082de55e172929f5bc3c8ee.jpg",
+        "text": "$$\n- \\log \\frac { \\exp ( \\widetilde { \\mathbf { x } } ^ { \\top } \\widetilde { \\mathbf { x } } ^ { + } ) } { \\exp ( \\widetilde { \\mathbf { x } } ^ { \\top } \\widetilde { \\mathbf { x } } ^ { + } ) + \\sum _ { i = 1 } ^ { N - 1 } \\exp ( \\widetilde { \\mathbf { x } } ^ { \\top } \\widetilde { \\mathbf { x } } _ { i } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            821,
+            619,
+            861
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This can be seen as the cross-entropy loss applied to softmax $( \\widetilde { \\mathbf { x } } ^ { \\top } \\widetilde { \\mathbf { x } } ^ { + } , \\widetilde { \\mathbf { x } } ^ { \\top } \\widetilde { \\mathbf { x } } _ { 1 } , \\dots , \\widetilde { \\mathbf { x } } ^ { \\top } \\widetilde { \\mathbf { x } } _ { N - 1 } )$ ",
+        "bbox": [
+            171,
+            869,
+            769,
+            887
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Following the same logic as the cross-entropy loss, this is also an approximation to $I ( \\widetilde { X } ; Y )$ . This objective should have less variance than Triplet loss since it approximates $p ( \\widetilde { \\mathbf { x } } )$ using more examples. ",
+        "bbox": [
+            171,
+            895,
+            831,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Adversarial Metric Learning Deep Adversarial Metric Learning (Duan et al., 2018) tackles the problem of most negative exmples being uninformative by directly generating meaningful negative embeddings. This model employs a generator which takes as input the embeddings of anchor, positive, and negative images. The generator then outputs a \"synthetic negative\" embedding that is hard to distinguish from a positive embedding while being close to the negative embedding. ",
+        "bbox": [
+            173,
+            102,
+            826,
+            174
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This can be seen as optimizing ",
+        "bbox": [
+            174,
+            180,
+            377,
+            195
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e5c3a55422cc355c051cfa0a8db7455f9258d8493507145e295f04cc9463c78.jpg",
+        "text": "$$\n\\mathbb { E } \\left[ \\log p _ { p } ( \\widetilde { \\mathbf { x } } ) - \\log p ( \\widetilde { \\mathbf { x } } ) \\right] = I ( \\widetilde { X } ; Y )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            202,
+            606,
+            223
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "by estimating $p ( \\widetilde { \\mathbf { x } } )$ using a generative network rather than directly from samples. Rather than emodelling the marginal distribution $p ( \\widetilde { \\mathbf { x } } )$ , this method conditionally models $p ( \\widetilde { \\mathbf { x } } ; \\bar { \\widetilde { \\mathbf { x } } } _ { q } , \\widetilde { \\mathbf { x } } _ { p } , \\widetilde { \\mathbf { x } } _ { n } )$ so that $\\widetilde { \\mathbf { x } }$ is hard to distinguish from $\\widetilde { \\mathbf { x } } _ { p }$ ewhile sufficiently close to both $\\widetilde { \\mathbf { x } } _ { q }$ and $\\widetilde { \\mathbf { x } } _ { n }$ . ",
+        "bbox": [
+            173,
+            231,
+            825,
+            275
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B PROOF OF THEOREM 1 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            295,
+            397,
+            311
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We restate and prove our main theorem. ",
+        "bbox": [
+            174,
+            327,
+            434,
+            342
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Let $d _ { \\Theta }$ be the VC dimension of the encoder $\\widetilde { X } ( \\cdot )$ . Let $\\hat { I } ( \\widetilde { X } ( X _ { T } , \\theta ) ; Y _ { T } )$ be the empirical estimate of the mutual information using finite dataset $D _ { T }$ , and define empirical loss as ",
+        "bbox": [
+            174,
+            347,
+            823,
+            378
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/61bda3f977d6fee79ff6b7fe2596bee6ef02d9752640b4c3d954f1d68da7f2a8.jpg",
+        "text": "$$\n\\hat { \\mathcal { L } } ( T ^ { 1 : n } , \\theta ) = - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\hat { I } ( \\widetilde { X } ( X _ { T ^ { i } } , \\theta ) ; Y _ { T ^ { i } } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            386,
+            622,
+            426
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The following inequality holds with high probability: ",
+        "bbox": [
+            173,
+            433,
+            521,
+            449
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ebed0e4c8dfb46d1cf4c2f5a14b1628ca3d7726e26347a5af1f5c58b3014b65e.jpg",
+        "text": "$$\n\\mathcal { L } ( \\tau , \\theta ) - \\hat { \\mathcal { L } } ( T ^ { 1 : n } , \\theta ) \\leq O \\left( \\sqrt { \\frac { d _ { \\Theta } } { n } \\log \\frac { n } { d _ { \\Theta } } } \\right) + O \\left( \\frac { | \\widetilde { X } | \\log ( m ) } { \\sqrt { m } } \\right) + O \\left( \\frac { | \\widetilde { X } | | Y | } { m } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            455,
+            754,
+            497
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof. We use the following lemma from Shamir et al. (2010), which we restate using our notation. ",
+        "bbox": [
+            171,
+            513,
+            825,
+            529
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1. Let $\\widetilde { X }$ be a random mapping of $X$ . Let $D$ be a sample of size m drawn from the joint probability distribution $p ( X , Y )$ . Denote the empirical mutual information observed from $D$ between $\\widetilde { X }$ and $Y$ as ${ \\hat { I } } ( { \\widetilde { X } } ; Y )$ . For any $\\delta \\in ( 0 , 1 )$ , the following holds with probability at least $1 - \\delta$ : ",
+        "bbox": [
+            173,
+            537,
+            825,
+            587
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6fb4dbbdaa935cceefe72205d114690d8c85996ef0e619afabc7a62d220aa6e5.jpg",
+        "text": "$$\n| I ( \\widetilde { X } ; Y ) - \\hat { I } ( \\widetilde { X } ; Y ) | \\leq \\frac { ( 3 | \\widetilde { X } | + 2 ) \\log ( m ) \\sqrt { \\log ( 4 / \\delta ) } } { \\sqrt { 2 m } } + \\frac { ( | Y | + 1 ) ( | \\widetilde { X } | + 1 ) - 4 } { m }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            593,
+            758,
+            631
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We simplify this and plug in our specific quantities of interest $( \\widetilde { X } ( X _ { T } , \\theta ) , Y _ { T } )$ : ",
+        "bbox": [
+            173,
+            647,
+            692,
+            665
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3c098b21aa1ce0d78a4e2ce44f8dd5cb32b4864b94f4978d28ed5c056edafcda.jpg",
+        "text": "$$\n\\left| I \\left( \\widetilde { X } ( X _ { T } , \\theta ) ; Y _ { T } \\right) - \\hat { I } \\left( \\widetilde { X } ( X _ { T } , \\theta ) ; Y _ { T } \\right) \\right| \\le O \\left( \\frac { | \\widetilde { X } | \\log ( m ) } { \\sqrt { m } } \\right) + O \\left( \\frac { | \\widetilde { X } | | Y | } { m } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            207,
+            671,
+            753,
+            714
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We similarly bound the error caused by estimating $\\mathcal { L }$ with a finite number of tasks sampled from $\\tau$ . Denote the finite sample estimate of $\\mathcal { L }$ as ",
+        "bbox": [
+            173,
+            728,
+            825,
+            757
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e3ef74d953a25e52d0b063e1607be83ca1017368e2ce22f464728410ab87c39f.jpg",
+        "text": "$$\n\\hat { \\mathcal { L } } ( \\tau , \\theta ) = - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } I ( \\widetilde { X } ( X _ { T ^ { i } } , \\theta ) ; Y _ { T ^ { i } } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            765,
+            612,
+            806
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Let the mapping $X \\mapsto { \\widetilde { X } }$ be parameterized by $\\theta \\in \\Theta$ and let this model have VC dimension $d _ { \\Theta }$ Using $d _ { \\Theta }$ , we can state that with high probability, ",
+        "bbox": [
+            176,
+            821,
+            825,
+            853
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d3ebee489d9ced6cc1b0ee30c4bb448229323a993f5cbe967305374cb710ac58.jpg",
+        "text": "$$\n\\Big | \\mathcal { L } ( \\tau , \\theta ) - \\hat { \\mathcal { L } } ( \\tau , \\theta ) \\Big | \\leq O \\left( \\sqrt { \\frac { d _ { \\Theta } } { n } \\log { \\frac { n } { d _ { \\Theta } } } } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            859,
+            625,
+            902
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $d _ { \\Theta }$ is the VC dimension of hypothesis class $\\Theta$ . ",
+        "bbox": [
+            173,
+            909,
+            526,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Combining equations (26, 24), we have with high probability ",
+        "bbox": [
+            173,
+            102,
+            575,
+            119
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a2702f78dcd615feac2416f287bbc29e1718189e5ddb20531ea09400cd7b0212.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { | \\mathcal { L } ( \\tau , \\theta ) - ( - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\hat { I } ( \\widetilde { X } ( X _ { T ^ { i } } , \\theta ) ; Y _ { T ^ { i } } ) ) | } } \\\\ & { } & { \\leq | \\mathcal { L } ( \\tau , \\theta ) - \\hat { \\mathcal { L } } ( \\tau , \\theta ) | + O ( \\frac { | \\widetilde { X } | \\log ( m ) } { \\sqrt { m } } ) + O ( \\frac { | \\widetilde { X } | | Y | } { m } ) } \\\\ & { } & { \\leq O ( \\sqrt { \\frac { d _ { \\Theta } } { n } \\log \\frac { n } { d _ { \\Theta } } } ) + O ( \\frac { | \\widetilde { X } | \\log ( m ) } { \\sqrt { m } } ) + O ( \\frac { | \\widetilde { X } | | Y | } { m } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            125,
+            718,
+            262
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C EXPERIMENTS AND IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            300,
+            606,
+            318
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Hardware Every experiment was conducted on a single Nvidia V100 GPU with CUDA 9.2. We used PyTorch version 1.0.1. Each experiment was performed with different fixed initial seeds; we manually fix seeds with manual_seed() for python, pytorch, and numpy. ",
+        "bbox": [
+            173,
+            332,
+            825,
+            376
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Optimizer For experiments with the 4-layer convnet, we use the Adam optimizer (Kingma & Ba, 2014) with learning rate 3e-4. For the Inception network, we use SGD with learning rate 3e-5 and momentum 0.9. ",
+        "bbox": [
+            174,
+            388,
+            823,
+            431
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We report the average of 500 batches of 1-shot accuracies and mutual information. $I ( \\widetilde { X } ; Y )$ was computed using balanced batches of 16 images each from 5 different classes. We additionally show in Figure 5 the correlation between 1-shot accuracies, Recall@1, and NMI using three previously proposed losses (triplet, npair, protonet). ",
+        "bbox": [
+            174,
+            438,
+            825,
+            494
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Small Train Set Experiment For this experiment, we used the Adam optimizer and performed a log-uniform hyperparameter sweep for learning rate $\\in$ [1e-7, 1e-3] For DIMCO, we swept $p \\in$ [32, 128] and $d \\in [ 1 6 , 3 2 ]$ . For other methods, we made the embedding dimension $\\in [ 1 6 , 3 2 ]$ . For each combination of loss and number of training examples per class, we ran the experiment 64 times and reported the mean and standard deviation of the top 5. ",
+        "bbox": [
+            174,
+            510,
+            825,
+            580
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/59ce6fb78226694d9b06d1d8feadc237e49ee6c6764dd96151d7538f89de35de.jpg",
+        "image_caption": [
+            "Figure 5: Correlation between few-shot accuracy and retrieval measures. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            276,
+            816,
+            719
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/5da775c280e79fd2f1c2352b0a6a98762a99b428364c5bbe4226b238ed116583.jpg",
+        "image_caption": [
+            "Figure 6: Additional visualizations of codes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            189,
+            823,
+            808
+        ],
+        "page_idx": 15
+    }
+]

parse/train/WZnVnlFBKFj/WZnVnlFBKFj.md

@@ -0,0 +1,607 @@
+# Federated Learning With Quantized Global Model Updates
+
+# Abstract
+
+We study federated learning (FL), which enables mobile devices to utilize their local datasets to collaboratively train a global model with the help of a central server, while keeping data localized. At each iteration, the server broadcasts the current global model to the devices for local training, and aggregates the local model updates from the devices to update the global model. Previous work on the communication efficiency of FL has mainly focused on the aggregation of model updates from the devices, assuming perfect broadcasting of the global model. In this paper, we instead consider broadcasting a compressed version of the global model. This is to further reduce the communication cost of FL, which can be particularly limited when the global model is to be transmitted over a wireless medium. We introduce a lossy FL (LFL) algorithm, in which both the global model and the local model updates are quantized before being transmitted. We analyze the convergence behavior of the proposed LFL algorithm assuming the availability of accurate local model updates at the server. Numerical experiments show that the proposed LFL scheme, which quantizes the global model update (with respect to the global model estimate at the devices) rather than the global model itself, significantly outperforms other existing schemes studying quantization of the global model at the PS-to-device direction. Also, the performance loss of the proposed scheme is marginal compared to the fully lossless approach, where the PS and the devices transmit their messages entirely without any quantization.
+
+# 1 Introduction
+
+Federated learning (FL) enables wireless devices to collaboratively train a global model by utilizing locally available data and computational capabilities under the coordination of a parameter server (PS) while the data never leaves the devices McMahan $\&$ Ramage (2017).
+
+respect to the global model In FL with devices the goal is to minimize a loss function $\pmb { \theta } \in \mathbb { R } ^ { d }$ , where $\begin{array} { r } { F _ { m } \left( \pmb { \theta } \right) = \frac { 1 } { B _ { m } } \sum _ { \pmb { u } \in \mathcal { B } _ { m } } f \left( \pmb { \theta } , \pmb { u } \right) } \end{array}$ $\begin{array} { r } { F ( \pmb { \theta } ) = \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } F _ { m } \left( \pmb { \theta } \right) } \end{array}$ m=1 B  is the loss function with at device $m$ , with $B _ { m }$ representing device $m$ ’s local dataset of size $B _ { m }$ , $\begin{array} { r } { B \triangleq \sum _ { m = 1 } ^ { M } B _ { m } } \end{array}$ , and $f ( \cdot , \cdot )$ is an empirical loss function. Having access to the global model $\pmb { \theta }$ , device $m$ utilizes its local dataset and performs multiple iterations of stochastic gradient descent (SGD) in order to minimize the local loss function $F _ { m } \left( \pmb { \theta } \right)$ . It then sends the local model update to the server, which aggregates the local updates from all the devices to update the global model.
+
+FL mainly targets mobile applications at the network edge, and the wireless communication links connecting these devices to the network are typically limited in bandwidth and power, and suffer from various channel impairments such as fading, shadowing, or interference; hence the need to develop an FL framework with limited communication requirements becomes more vital. While communication-efficient FL has been widely studied, prior works mainly focused on the devices-to-PS links, assuming perfect broadcasting of the global model to the devices at each iteration. In this paper, we design an FL algorithm aiming to reduce the cost of both PS-to-device and devices-to-PS communications. To address the importance of quantization at the PS-to-device direction, we highlight that some devices simply may not have the sufficient bandwidth to receive the global model update when the model size is relatively large, particularly in the wireless setting, where the devices are away from the base station. This would result in consistent exclusion of these devices, resulting in significant performance loss. Moreover, the impact of quantization in the device-to-PS direction is less severe due to the impact of averaging local updates at the PS.
+
+Related work There is a fast-growing body of literature on the communication efficiency of FL targeting restricted bandwidth devices. Several studies address this issue by considering communications with rate limitations, and propose different compression and quantization techniques Konecny et al. (2016); McMahan et al. (2017); Konecny & Richtarik (2018); Dowlin et al. (2016); Konecny et al. (2015); Lin et al. (2018b); He et al. (2018); M. M. Amiri & G¨und¨uz (2020), as well as performing local updates to reduce the frequency of communications from the devices to the PS Lin et al. (2018a); Stich (2019). Statistical challenges arise in FL since the data samples may not be independent and identically distributed (iid) across devices. The common sources of the dependence or bias in data distribution are the participating devices being located in a particular geographic region, and/or at a particular time window P. Kairouz et al. (2019). Different approaches have been studied to mitigate the effect of non-iid data in FL McMahan et al. (2017); Hsieh et al. (2019); Li et al. (2020a); Wang et al. (2020); Eichner et al. (2019); Zhao et al. (2018). Also, FL suffers from a significant variability in the system, which is mainly due to the hardware, network connectivity, and available power associated with different devices Li et al. (2019). Active device selection schemes have been introduced to alleviate significant variability in FL systems, where a subset of devices share the resources and participate at each iteration of training Kang et al. (2019); Nishio & Yonetani (2019); Amiri et al. (2020b); Yang et al. (2020; 2019). There have also been efforts in developing convergence guarantees for FL under various scenarios, considering iid data across the devices Stich (2019); Wang & Joshi (2019); Woodworth et al. (2019); Zhou & Cong (2018); Koloskova et al. (2020), non-iid data Koloskova et al. (2020); Li et al. (2020a); Haddadpour & Mahdavi (2019); Li et al. (2020c), participation of all the devices Khaled et al. (2020); Wang et al. (2019); Yu et al. (2018); Huo et al. (2020), or only a subset of devices at each iteration Li et al. (2020b); Karimireddy et al. (2020); Rizk et al. (2020); Li et al. (2020c); Amiri et al. (2020a), and FL under limited communication constraints Amiri et al. (2020a); Recht et al. (2011); Alistarh et al. (2018).
+
+FL with compressed global model transmission has been studied recently in Caldas et al. (2019); Tang et al. (2019) aiming to alleviate the communication footprint from the PS to the devices. The global model parameters are relatively skewed/diverse and the efficiency of quantization diminishes significantly when the peak-to-average ratio of the parameters is large. To overcome this, in Caldas et al. (2019) the PS first employs a linear transform in order to spread the information of the global model vector more evenly among its dimensions, and broadcasts a quantized version of the resultant vector, and the devices apply the inverse linear transform to estimate the global model. We highlight that this approach requires a relatively high computational overhead due to employing the linear transform at the PS and its inverse at the devices, where this overhead grows with the size of the model parameters. Furthermore, the performance evaluation in Caldas et al. (2019) is limited to the experimental results On the other hand, in Tang et al. (2019) the PS broadcasts quantized global model with error accumulation to compensate the quantization error.
+
+Our contributions With the exception of Caldas et al. (2019); Tang et al. (2019), the literature on FL considers perfect broadcasting of the global model from the PS to the devices. With this assumption, no matter what type of local update or device-to-PS communication strategy is used, all the devices are synchronized with the same global model at each iteration. In this paper, we instead consider broadcasting a quantized version of the global model update by the PS, which provides the devices with a lossy estimate of the global model (rather than its accurate estimate) with which to perform local training. This further reduces the communication cost of FL, which can be particularly limited for transmission over a wireless medium while serving a massive number of devices. Also, it is interesting to investigate the impact of various hyperparameters on the performance of FL with lossy broadcasting of the global model since FL involves transmission over wireless networks with limited bandwidth. We introduce a lossy FL (LFL) algorithm, where at each iteration the PS broadcasts a compressed version of the global model update to all the devices through quantization. To be precise, the PS exploits the knowledge of the last global model estimate available at the devices as side information to quantize the global model update. The devices recover an estimate of the current global model by combining the received quantized global model update with their previous estimate, and perform local training using their estimate, and return the local model updates, again employing quantization. The PS updates the global model after receiving the quantized local model updates from the devices. We provide convergence analysis of the LFL algorithm investigating the impact of lossy broadcasting on the performance of FL. Numerical experiments on the MNIST and CIFAR-10 datasets illustrate the efficiency of the proposed LFL algorithm. We observe that the proposed LFL scheme, which leads to a significant communication cost saving, provides a promising performance with no visible gap to the performance of the fully lossless scenario where the communication from both PS-to-device and device-to-PS directions is assumed to be perfect. Also, it is illustrated that the proposed LFL scheme significantly outperforms the schemes introduced in Caldas et al. (2019) and Tang et al. (2019) considering compression from the PS to devices.
+
+The proposed LFL algorithm differs from the approaches in Caldas et al. (2019); Tang et al. (2019), since we propose broadcasting the global model update, with respect to the previous estimate at the devices, rather than the global model itself. We remark that the global model update has less variability/variance and peak-to-average ratio than the global model (see Figure 2), and hence, for the same communication load, the devices can have a more accurate estimate of the global model. However, this would require all the devices to track the global model at each iteration, even if they do not participate in the learning process by sending their local update. We argue that broadcasting the global model update to the whole set of devices, rather than a randomly chosen subset, would introduce limited additional communication cost as broadcasting is typically more efficient than sending independent information to devices. Moreover, in practice, the subset of participating devices remain the same for a number of iterations, until a device leaves or joins. Our algorithm can easily be adopted to such scenarios by sending the global model, rather than the model update, every time the subset of devices changes. Also, compared to the approach in Caldas et al. (2019), the LFL algorithm requires a significantly smaller computational overhead. Furthermore, unlike Caldas et al. (2019), we provide an in-depth convergence analysis of the proposed LFL algorithm. The advantage of the proposed LFL algorithm over the approaches introduced in Caldas et al. (2019); Tang et al. (2019) is shown numerically, where, despite its significantly smaller communication load, it provides considerably higher accuracy.
+
+Notation The set of real numbers is denoted by $\mathbb { R }$ . For $x \in \mathbb { R }$ , $| x |$ returns the absolute value of $x$ . For a vector of real numbers $_ { x }$ , the largest and the smallest absolute values among all the entries of $_ { x }$ are represented by max $\{ | { \pmb x } | \}$ and $\operatorname* { m i n } { \left\{ \left| x \right| \right\} }$ , respectively. For an integer $_ i$ , we let $[ i ] \triangleq \{ 1 , 2 , \dots , i \}$ . The $l _ { 2 }$ -norm of vector $_ { x }$ is denoted by $\lVert \mathbf { x } \rVert _ { 2 }$ .
+
+# 2 Lossy Federated Learning (LFL) Algorithm
+
+We consider a lossy PS-to-device transmission, in which the PS sends a compressed version of the global model to the devices. This reduces the communication cost, and can be particularly beneficial when the PS resources are limited, and/or communication takes place over a constrained bandwidth medium. We denote the estimate of the global model $\pmb \theta ( t )$ at the devices by $\widehat { \pmb \theta } ( t )$ , where $t$ represents the global iteration count. Having recovered $\widehat { \pmb \theta } ( t )$ , the devices perform a $\tau$ -step SGD with respect to their local datasets, and transmit their local model updates to the PS using quantization while accumulating the quantization error.
+
+# 2.1 Global Model Broadcasting
+
+In the proposed LFL algorithm, the PS performs stochastic quantization similarly to the QSGD algorithm introduced in Alistarh et al. (2017) with a slight modification to broadcast the information about the global model to the devices. In particular, at global iteration $t$ , the PS aims to broadcast the global model update $\pmb \theta ( t ) - \widehat \pmb \theta ( t - 1 )$ to the devices. We present the stochastic quantization technique we use, denoted by $Q ( \cdot , \cdot )$ , in Appendix A.
+
+Lemma 1. For the quantization function $\varphi \left( x , q \right)$ and vector $Q ( x , q )$ given in (21b) and (22), respectively, we have
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { \varphi } \left[ \varphi ( x , q ) \right] = x , \quad \mathbb { E } _ { \varphi } \left[ \varphi ^ { 2 } ( x , q ) \right] \leq x ^ { 2 } + 1 / ( 4 q ^ { 2 } ) , } \\ & { \mathbb { E } _ { \varphi } \left[ \pmb { Q } ( \pmb { x } , q ) \right] = \pmb { x } , \quad \mathbb { E } _ { \varphi } \left[ \left\| \pmb { Q } ( \pmb { x } , q ) \right\| _ { 2 } ^ { 2 } \right] \leq \left\| \pmb { x } \right\| _ { 2 } ^ { 2 } + \varepsilon d \left\| \pmb { x } \right\| _ { 2 } ^ { 2 } / ( 4 q ^ { 2 } ) , } \end{array}
+$$
+
+# Algorithm 1 LFL
+
+1: for $t = 0 , \ldots , T - 1$ do • Global model broadcasting   
+2: PS broadcasts $Q \big ( \pmb \theta ( t ) - \widehat \pmb \theta ( t - 1 ) , q _ { 1 } \big )$   
+3: ${ \widehat { \pmb \theta } } ( t ) = { \widehat { \pmb \theta } } ( t - 1 ) + { \pmb Q } { \big ( } { \pmb \theta } ( t ) - { \widehat { \pmb \theta } } ( t - 1 ) , q _ { 1 } { \big ) }$ • Local update aggregation   
+4: for $m = 1 , \ldots , M$ in parallel do   
+5: Device $m$ transmits $\begin{array} { r } { Q \big ( \Delta \theta _ { m } ( t ) + \delta _ { m } ( t ) , q _ { 2 } \big ) = Q \big ( \theta _ { m } ^ { \tau + 1 } ( t ) - \widehat \theta ( t ) + \delta _ { m } ( t ) , q _ { 2 } \big ) } \end{array}$   
+6: 7: $\begin{array} { r } { \pmb { \theta } ( t + 1 ) = \pmb { \widehat { \theta } } ( t ) + \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \pmb { Q } \big ( \Delta \pmb { \theta } _ { m } ( t ) + \pmb { \delta } _ { m } ( t ) , q _ { 2 } \big ) } \end{array}$   
+8: end for
+
+where $\mathbb { E } _ { \varphi }$ represents expectation with respect to the quatization function $\varphi \left( \cdot , \cdot \right)$ , and $0 \leq \varepsilon \leq$ 1 is defined as $\varepsilon \triangleq \left( \operatorname* { m a x } \left\{ \left| \pmb { x } \right| \right\} - \operatorname* { m i n } \left\{ \left| \pmb { x } \right| \right\} \right) ^ { 2 } / \left\| \pmb { x } \right\| _ { 2 } ^ { 2 }$ .
+
+The proof of Lemma 1 is provided in Appendix B. We highlight that the value of $\varepsilon$ depends on the skewness of the magnitudes of the entries of $_ { x }$ , where it increases for a more skewed entries with a higher variance. We have $\varepsilon = 0$ , if and only if all the entries of $_ { x }$ have the same magnitude, and $\varepsilon = 1$ , if and only if $_ { x }$ has only one non-zero entry.
+
+Given a quantization level $q _ { 1 }$ , the PS broadcasts $Q \big ( \pmb \theta ( t ) - \widehat \theta ( t - 1 ) , q _ { 1 } \big )$ to the devices at global iteration $t$ . Then the devices obtain the following estimate of $\pmb \theta ( t )$ :
+
+$$
+\widehat { \pmb { \theta } } ( t ) = \widehat { \pmb { \theta } } ( t - 1 ) + \pmb { Q } \big ( \pmb { \theta } ( t ) - \widehat { \pmb { \theta } } ( t - 1 ) , q _ { 1 } \big ) ,
+$$
+
+which is equivalent to $\begin{array} { r } { \widehat { \pmb { \theta } } ( t ) = \pmb { \theta } ( 0 ) + \sum _ { i = 1 } ^ { t } \pmb { Q } \big ( \pmb { \theta } ( i ) - \widehat { \pmb { \theta } } ( i - 1 ) , q _ { 1 } \big ) } \end{array}$ , where we assumed that $\widehat { \pmb { \theta } } ( 0 ) = \pmb { \theta } ( 0 )$ . We note that, having the knowledge of the compressed vector $Q \big ( \theta ( i ) - \widehat { \theta } ( i -$ $1 ) , q _ { 1 } )$ , $\forall i \in [ t ]$ , the PS can also track $\widehat { \pmb \theta } ( t )$ at each iteration.
+
+# 2.2 Local Update Aggregation
+
+After recovering $\widehat { \pmb \theta } ( t )$ , device $m$ performs a $\tau$ -step local SGD, where the $i$ -th step corresponds to $\pmb { \theta } _ { m } ^ { i + 1 } ( t ) = \pmb { \theta } _ { m } ^ { i } ( t ) - \eta _ { m } ^ { i } ( t ) \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) .$ , $i \in [ \tau ]$ , where $\pmb { \theta } _ { m } ^ { 1 } ( t ) = \widehat { \pmb { \theta } } ( t )$ , and $\xi _ { m } ^ { i } ( t )$ denotes the local mini-batch chosen uniformly at random from the local dataset $B _ { m }$ . It then aims to transmit local model update $\Delta \pmb { \theta } _ { m } ( t ) = \pmb { \theta } _ { m } ^ { \tau + 1 } ( t ) - \widehat { \pmb { \theta } } ( t )$ through quantization with error compensation and transmits $Q ( \Delta \theta _ { m } ( t ) + \delta _ { m } ( t ) , q _ { 2 } )$ using a quantization level $q _ { 2 }$ , where $\delta _ { m } ( t )$ retains the quantization error, and is updated as
+
+$$
+\delta _ { m } ( t + 1 ) = \Delta \theta _ { m } ( t ) + \delta _ { m } ( t ) - Q \big ( \Delta \theta _ { m } ( t ) + \delta _ { m } ( t ) , q _ { 2 } \big ) ,
+$$
+
+where we set $\pmb { \delta } _ { m } ( 0 ) = \mathbf { 0 }$ . Having received $Q ( \Delta \theta _ { m } ( t ) + \delta _ { m } ( t ) , q _ { 2 } )$ from device $m$ , $\forall m \in [ M ]$ the PS updates the global model as
+
+$$
+\pmb { \theta } ( t + 1 ) = \widehat { \pmb { \theta } } ( t ) + \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \pmb { Q } \big ( \Delta \pmb { \theta } _ { m } ( t ) + \pmb { \delta } _ { m } ( t ) , q _ { 2 } \big ) .
+$$
+
+Algorithm 1 summarizes the proposed LFL algorithm.
+
+Remark 1. We do not consider error compensation at the $P S$ with LFL since we have observed performance degradation numerically when compensating the quantization error at the $P S$ . We argue that LFL naturally accumulates the quantization error at the PS since it sends the quatized global model update with respect to the last global model estimate at the devices. We further highlight that the proposed approach is not limited to any specific quantization technique, and any compression technique can be used within the proposed framework.
+
+# 3 Convergence Analysis of LFL Algorithm
+
+Here we analyze the convergence behaviour of LFL, where for simplicity of the analysis, we assume that the devices can transmit their local updates, $\Delta \pmb \theta _ { m } ( t )$ , $\forall m$ , accurately/in a lossless fashion to the PS, and focus on the impact of lossy broadcasting on the convergence.
+
+# 3.1 Preliminaries
+
+We denote the optimal solution minimizing loss function $F ( \pmb \theta )$ by $\theta ^ { * }$ , and the minimum loss as $F ^ { * }$ , i.e., $\pmb { \theta } ^ { * } \triangleq \arg \operatorname* { m i n } _ { \pmb { \theta } } \boldsymbol { F } ( \pmb { \theta } )$ , and $F ^ { * } \triangleq F ( \theta ^ { * } )$ . We also denote the minimum value of the local loss function at device m by F ∗m. We further define Γ , F ∗ − PMm=1 BmB F Fm , where $\Gamma \geq 0$ , and its magnitude indicates the bias in the data distribution across devices.
+
+For ease of analysis, we set $\eta _ { m } ^ { \ i } ( t ) = \eta ( t )$ . Thus, the $_ i$ -th step SGD at device $m$ is given by
+
+$$
+{ { \pmb \theta } _ { m } ^ { i + 1 } } ( t ) = { \pmb \theta } _ { m } ^ { i } ( t ) - \eta ( t ) \nabla F _ { m } \left( { \pmb \theta } _ { m } ^ { i } ( t ) , { \pmb \xi } _ { m } ^ { i } ( t ) \right) , \quad i \in [ \tau ] , m \in [ M ] ,
+$$
+
+here $\pmb { \theta } _ { m } ^ { 1 } ( t ) = \widehat { \pmb { \theta } } ( t )$ , given in (2). Device $m$ transmits the local model update
+
+$$
+\Delta \pmb { \theta } _ { m } ( t ) = \pmb { \theta } _ { m } ^ { \tau + 1 } ( t ) - \widehat { \pmb { \theta } } ( t ) = - \eta ( t ) \sum _ { i = 1 } ^ { \tau } \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) , \quad m \in [ M ] ,
+$$
+
+and the PS updates the global model as
+
+$$
+\begin{array} { r } { \pmb { \theta } ( t + 1 ) = \widehat { \pmb { \theta } } ( t ) - \eta ( t ) \sum _ { m = 1 } ^ { M } \sum _ { i = 1 } ^ { \tau } \frac { B _ { m } } { B } \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) . } \end{array}
+$$
+
+Assumption 1. The expected squared $l _ { 2 }$ -norm of the stochastic gradients are bounded, i.e.,
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \boldsymbol \xi } \left[ \left. \nabla F _ { m } \left( \theta _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \right. _ { 2 } ^ { 2 } \right] \leq G ^ { 2 } , \quad \forall i \in [ \tau ] , \forall m \in [ M ] , \ \forall t . } \end{array}
+$$
+
+Assumption 2. The loss functions $F _ { 1 } , \ldots , F _ { M }$ are $L$ -smooth; that is, $\forall \pmb { v } , \pmb { w } \in \mathbb { R } ^ { d }$
+
+$$
+2 \big ( F _ { m } ( \pmb { v } ) - F _ { m } ( \pmb { w } ) \big ) \leq 2 \langle \pmb { v } - \pmb { w } , \nabla F _ { m } ( \pmb { w } ) \rangle + L \| \pmb { v } - \pmb { w } \| _ { 2 } ^ { 2 } , \quad \forall m \in [ M ] .
+$$
+
+# 3.2 Strongly Convex Loss Function
+
+Here we provide convergence analysis assuming that the loss functions $F _ { 1 } , \ldots , F _ { M }$ are $\mu$ - strongly convex; that is, $\forall \pmb { v } , \pmb { w } \in \mathbb { R } ^ { d }$ ,
+
+$$
+2 \big ( F _ { m } ( \pmb { v } ) - F _ { m } ( \pmb { w } ) \big ) \geq 2 \big \langle \pmb { v } - \pmb { w } , \nabla F _ { m } ( \pmb { w } ) \big \rangle + \mu \left\| \pmb { v } - \pmb { w } \right\| _ { 2 } ^ { 2 } , \quad \forall m \in [ M ] .
+$$
+
+In the following theorem, whose proof is provided in Appendix C, we present the convergence rate of the LFL algorithm assuming that the devices can send their local updates accurately. Theorem 1. Let $0 < \eta ( t ) \leq \operatorname* { m i n } \left\{ 1 , \frac { 1 } { \mu \tau } \right\}$ , $\forall t$ . We have
+
+$\begin{array} { r } { \mathbb { E } \big [ \| \pmb { \theta } ( t ) - \pmb { \theta } ^ { * } \| _ { 2 } ^ { 2 } \big ] \leq \left( \prod _ { i = 0 } ^ { t - 1 } A ( i ) \right) \| \pmb { \theta } ( 0 ) - \pmb { \theta } ^ { * } \| _ { 2 } ^ { 2 } + \sum _ { j = 0 } ^ { t - 1 } B ( j ) \prod _ { i = j + 1 } ^ { t - 1 } A ( i ) , } \end{array}$ where
+
+$$
+\begin{array} { r l } & { A ( i ) \triangleq 1 - \mu \eta ( i ) \left( \tau - \eta ( i ) ( \tau - 1 ) \right) , } \\ & { B ( i ) \triangleq ( 1 - \mu \eta ( i ) \left( \tau - \eta ( i ) ( \tau - 1 ) \right) ) \left( \frac { \eta ( i - 1 ) \tau G } { 2 q _ { 1 } } \right) ^ { 2 } \varepsilon d + \eta ^ { 2 } ( i ) ( \tau ^ { 2 } + \tau - 1 ) G ^ { 2 } } \\ & { \qquad + \left( 1 + \mu ( 1 - \eta ( i ) ) \right) \eta ^ { 2 } ( i ) G ^ { 2 } \frac { \tau ( \tau - 1 ) ( 2 \tau - 1 ) } { 6 } + 2 \eta ( i ) ( \tau - 1 ) \Gamma , } \end{array}
+$$
+
+for some $0 \leq \varepsilon \leq 1$ , and the expectation is with respect to the stochastic gradient function and stochastic quantization.
+
+Corollary 1. From the $L$ -smoothness of the loss function, for $0 < \eta ( t ) \leq \operatorname* { m i n } \left\{ 1 , \frac { 1 } { \mu \tau } \right\}$ , $\forall t$ , and a total of $T$ global iterations, it follows that
+
+$$
+\begin{array} { r l r } {  { \mathbb { E } [ F ( \pmb { \theta } ( T ) ) ] - F ^ { * } \le \frac { L } { 2 } \mathbb { E } \big [ \| \pmb { \theta } ( T ) - \pmb { \theta } ^ { * } \| _ { 2 } ^ { 2 } \big ] } } \\ & { } & { \le \frac { L } { 2 } \Big ( \prod _ { i = 0 } ^ { T - 1 } A ( i ) \Big ) \| \pmb { \theta } ( 0 ) - \pmb { \theta } ^ { * } \| _ { 2 } ^ { 2 } + \frac { L } { 2 } \sum _ { j = 0 } ^ { T - 1 } B ( j ) \prod _ { i = j + 1 } ^ { T - 1 } A ( i ) , } \end{array}
+$$
+
+where the last inequality follows from (11a). Considering $\eta ( t ) = \eta$ and $\tau = 1$ , we have
+
+$$
+\begin{array} { r l } & { \mathbb { E } \left[ F ( \pmb { \theta } ( T ) ) \right] - F ^ { * } \le \displaystyle \frac { L } { 2 } ( 1 - \mu \eta ) ^ { T } \left. \pmb { \theta } ( 0 ) - \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } } \\ & { \qquad + \displaystyle \frac { L } { 2 } \Big ( ( 1 - \mu \eta ) \Big ( \frac { \varepsilon d } { 4 q _ { 1 } ^ { 2 } } \Big ) + 1 \Big ) \left( 1 - ( 1 - \mu \eta ) ^ { T } \right) \Big ( \frac { \eta G ^ { 2 } } { \mu } \Big ) . } \end{array}
+$$
+
+Asymptotic convergence analysis Here we show that, for a decreasing learning rate over time, such that $0$ . For $0 < \eta ( t ) \leq \operatorname* { m i n } \{ 1 , \frac { 1 } { \mu \tau } \}$ $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \eta ( t ) = 0 } \end{array}$ , we have , and given small enough $0 \leq A ( t ) < 1$ , and $\varepsilon$ $\begin{array} { r } { \operatorname* { l i m } _ { T \to \infty } \prod _ { i = 0 } ^ { T - 1 } A ( i ) = 0 } \end{array}$ , $\begin{array} { r } { \operatorname* { l i m } _ { T \to \infty } \mathbb { E } \left[ F ( \pmb { \theta } ( T ) ) \right] - F ^ { * } = } \end{array}$ . For simplicity, assume $\begin{array} { r } { \eta ( t ) = \frac { \alpha } { t + \beta } } \end{array}$ αt+β , for constant values α and β. For j  0, B(j) → 0, and for limited $j$ values, $\begin{array} { r } { \prod _ { i = j + 1 } ^ { T - 1 } A ( i ) \to 0 } \end{array}$ , and so, according to (12), $\begin{array} { r } { \operatorname* { l i m } _ { T \to \infty } \mathbb { E } \left[ F ( \pmb { \theta } ( T ) ) \right] - F ^ { * } = 0 } \end{array}$ .
+
+# 3.3 Non-Convex Loss Function
+
+Next, we provide convergence guarantees of the proposed LFL scheme for $L$ -smooth and nonconvex loss functions $F _ { 1 } , \ldots , F _ { M }$ . For the non-convex case, we provide a weaker notion of convergence Liu $\&$ Wright (2015) $\begin{array} { r } { \operatorname* { l i m } _ { T \to \infty } \mathbb { E } \big [ \| \nabla F ( \pmb { \theta } ( T ) ) \| _ { 2 } ^ { 2 } \big ]  0 } \end{array}$ . In the following theorem, we bound $\begin{array} { r } { \frac { 1 } { \sum _ { t = 0 } ^ { T - 1 } \eta ( t ) } \sum _ { t = 0 } ^ { T - 1 } \eta ( t ) \mathbb { E } \big [ \left. \nabla F ( \pmb { \theta } ( t ) ) \right. _ { 2 } ^ { 2 } \big ] } \end{array}$ with the proof provided in Appendix F.
+
+Theorem 2. Performing the LFL algorithm for $T \geq 1$ global iterations assuming that the $P S$ receives the local model updates accurately leads to
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { 1 } { \sum _ { t = 0 } ^ { T - 1 } \eta ( t ) } \sum _ { t = 0 } ^ { T - 1 } \eta ( t ) \mathbb { E } \big [ \| \nabla F ( \pmb \theta ( t ) ) \| _ { 2 } ^ { 2 } \big ] \leq \frac { 2 \big ( F ( \pmb \theta ( 0 ) ) - F ^ { * } \big ) } { \tau \sum _ { t = 0 } ^ { T - 1 } \eta ( t ) } + \frac { 2 \Gamma } { \tau \sum _ { t = 0 } ^ { T - 1 } \eta ( t ) } } \\ & { \qquad + \frac { 1 } { \sum _ { t = 0 } ^ { T - 1 } \eta ( t ) } \displaystyle \sum _ { t = 0 } ^ { T - 1 } \Big ( \frac { \eta ( t - 1 ) G } { 2 q _ { 1 } } \Big ) ^ { 2 } \big ( \eta ( t ) ( 2 \tau - 1 ) L + 2 \big ) \varepsilon d \tau L } \\ & { \qquad + 2 G ^ { 2 } \tau L \frac { \sum _ { t = 0 } ^ { T - 1 } \eta ^ { 2 } ( t ) } { \sum _ { t = 0 } ^ { T - 1 } \eta ( t ) } + L ^ { 2 } G ^ { 2 } ( \tau - 1 ) ( 2 \tau - 1 ) \frac { \sum _ { t = 0 } ^ { T - 1 } \eta ^ { 3 } ( t ) } { 3 \sum _ { t = 0 } ^ { T - 1 } \eta ( t ) } . } \end{array}
+$$
+
+Choice of $\varepsilon$ We highlight that $\varepsilon$ appears in the convergence analysis of the LFL algorithm in inequalities (45), (63), in which we have
+
+$$
+\begin{array} { r l } & { \mathbb { E } [ \Big ( \operatorname* { m a x } \Big \{ \Big | \sum _ { m = 1 } ^ { M } \sum _ { i = 1 } ^ { \tau } \frac { B _ { m } } { B } \nabla F _ { m } ( \theta _ { m } ^ { i } ( t - 1 ) , \xi _ { m } ^ { i } ( t - 1 ) ) \Big | \Big \}  } \\ & { \quad  - \operatorname* { m i n } \Big \{ \Big | \sum _ { m = 1 } ^ { M } \sum _ { i = 1 } ^ { \tau } \frac { B _ { m } } { B } \nabla F _ { m } ( \theta _ { m } ^ { i } ( t - 1 ) , \xi _ { m } ^ { i } ( t - 1 ) ) \Big | \Big \} ) ^ { 2 } ] } \\ & { \leq \varepsilon \mathbb { E } [ \Big \| \sum _ { m = 1 } ^ { M } \sum _ { i = 1 } ^ { \tau } \frac { B _ { m } } { B } \nabla F _ { m } ( \theta _ { m } ^ { i } ( t - 1 ) , \xi _ { m } ^ { i } ( t - 1 ) ) \Big \| _ { 2 } ^ { 2 } ] , } \end{array}
+$$
+
+which follows from (26b), where we note that
+
+$$
+\pmb { \theta } ( t ) - \widehat { \pmb { \theta } } ( t - 1 ) = - \eta ( t - 1 ) \sum _ { m = 1 } ^ { M } \sum _ { i = 1 } ^ { r } \frac { B _ { m } } { B } \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t - 1 ) , \xi _ { m } ^ { i } ( t - 1 ) \right) .
+$$
+
+On average the entries of $\pmb \theta ( t ) - \widehat \pmb \theta ( t - 1 )$ , given in (16), are not expected to have very diverse magnitudes. Thus, the inequality in (15) should hold for a relatively small value of $\varepsilon$ . We have observed numerically that $\varepsilon \approx 1 0 ^ { - 3 }$ satisfies inequality (15) for the LFL algorithm.
+
+Impact of number of local SGD steps $\forall t$ , it is easy to verify that the upper bound is simplified as follows: $\tau$ For the non-convex case, assuming n $\begin{array} { r l } {  { \frac { 1 } { T } \sum _ { t = 0 } ^ { T - 1 } \mathbb { E } \bigl [ \bigl \| \nabla F ( \pmb { \theta } ( t ) ) \bigr \| _ { 2 } ^ { 2 } \bigr ] } \qquad } & { { } } \end{array}$ , given in Theorem $\eta ( t ) = \eta$ ,
+
+$$
+h ( \tau ) = \frac { a _ { - 1 } } { \tau } + a _ { 0 } + a _ { 1 } \tau + a _ { 2 } \tau ^ { 2 } ,
+$$
+
+where
+
+$$
+\begin{array} { c } { { a _ { - 1 } \triangleq \displaystyle \frac { 2 \left( F ( 0 ) - F ^ { * } + \Gamma \right) } { \eta T } , \quad a _ { 0 } \triangleq \displaystyle \frac { \eta ^ { 2 } L ^ { 2 } G ^ { 2 } } { 3 } , } } \\ { { a _ { 1 } \triangleq ( 2 - \eta L ) \eta L G ^ { 2 } \Big ( 1 + \displaystyle \frac { \varepsilon d } { 4 q _ { 1 } ^ { 2 } } \Big ) , \quad a _ { 2 } \triangleq \eta ^ { 2 } L ^ { 2 } G ^ { 2 } \Big ( \displaystyle \frac { 2 } { 3 } + \displaystyle \frac { \varepsilon d } { 2 q _ { 1 } ^ { 2 } } \Big ) . } } \end{array}
+$$
+
+We have $d h ( \tau ) / d \tau = - a _ { - 1 } / \tau ^ { 2 } + a _ { 1 } + 2 a _ { 2 } \tau$ , where we note that $a _ { 1 } + 2 a _ { 2 } \geq 0$ . For relatively small $a _ { - 1 }$ values, particularly $a _ { - 1 } \leq a _ { 1 } + 2 a _ { 2 }$ , $h ( \tau )$ increases with $\tau$ , and $\tau = 1$ minimizes $h ( \tau )$ ; that is, when the training is started close to the optimal solution ( $F ( 0 ) { - } F ^ { * }$ is relatively small), and/or $\eta$ is relatively large, $\tau = 1$ may be the best choice. On the other hand, for relatively large $a _ { - 1 }$ values, the best $\tau$ can be the nearest integer to the positive solution of $( a _ { 1 } + 2 a _ { 2 } \tau ) \tau ^ { 2 } - a _ { - 1 } = 0$ .
+
+Table 1: CNN architecture for image classification on MNIST and CIFAR-10.   
+
+<table><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>CIFAR-10</td></tr><tr><td rowspan=2 colspan=1>3 × 3 convolutional layer,32 channels,ReLU activation, same padding</td><td rowspan=1 colspan=1>3 × 3 convolutional layer,32 channels,ReLU activation,same padding</td></tr><tr><td rowspan=1 colspan=1>3 × 3 convolutional layer,32 channels,ReLU activation,same padding</td></tr><tr><td rowspan=1 colspan=2>2 × 2 max pooling</td></tr><tr><td rowspan=3 colspan=1>3 × 3 convolutional layer,64 channels,ReLU activation,same padding</td><td rowspan=1 colspan=1>dropout with probability 0.2</td></tr><tr><td rowspan=1 colspan=1>3 × 3 convolutional layer, 64 channels,ReLU activation, same padding</td></tr><tr><td rowspan=1 colspan=1>3 × 3 convolutional layer, 64 channels,ReLU activation,same padding</td></tr><tr><td rowspan=1 colspan=2>2 × 2 max pooling</td></tr><tr><td rowspan=3 colspan=1>3 × 3 convolutional layer,64 channels,ReLU activation, same padding</td><td rowspan=1 colspan=1>dropout with probability 0.3</td></tr><tr><td rowspan=1 colspan=1>3 × 3 convolutional layer,128 channels,ReLU activation,same padding</td></tr><tr><td rowspan=1 colspan=1>3 × 3 convolutional layer,128 channels,ReLU activation, same padding</td></tr><tr><td rowspan=1 colspan=2>2 ×2 max pooling</td></tr><tr><td rowspan=1 colspan=1>fully connected layer with 128 units,ReLUactivation</td><td rowspan=1 colspan=1>dropout with probability 0.4</td></tr><tr><td rowspan=1 colspan=2>softmax output layer with10 units</td></tr></table>
+
+# 4 Numerical Experiments
+
+Here we investigate the performance of the proposed LFL algorithm for image classification on both MNIST LeCun et al. (1998) and CIFAR-10 Krizhevsky $\&$ Hinton (2009) datasets utilizing ADAM optimizer Kingma $\&$ Ba (2017). We consider $M = 4 0$ devices, and we measure the performance as the accuracy with respect to the test samples, called test accuracy.
+
+Network architecture We train different convolutional neural networks (CNNs) with MNIST and CIFAR-10 datasets. The architectures of these CNNs are described in Table 1.
+
+Data distribution We consider two data distribution scenarios. In the non-iid scenario, we split the training data samples with the same label (from the same class) to $M / 1 0$ disjoint subsets (assume that $M$ is divisible by 10). We then assign each subset of data samples, selected at random, to a different device. In the iid scenario, we randomly split the training data samples to $M$ disjoint subsets, and assign each subset to a distinct device. We consider non-iid and iid data distributions while training using MNIST and CIFAR-10, respectively.
+
+State-of-the-art approaches We consider two approaches with lossy broadcasting introduced in Caldas et al. (2019) and Tang et al. (2019) as the state-of-the-art approaches. With the scheme in Caldas et al. (2019), referred to as lossy transformed global model (LTGM), the PS first employs a linear transform to project the global model. It then quantizes the resultant vector after the linear transform, and sends the quantized vector to the devices. The devices employ the inverse of the linear transform and use the recovered vector for local training. As suggested in Caldas et al. (2019), we consider Walsh-Hadamrd transform and employ the stochastic quantization scheme presented in Appendix A at the PS. On the other hand, with the approach studied in Tang et al. (2019), referred to as lossy global model (LGM), the PS directly quantizes the global model plus the quantization error accumulated from the previous iterations and shares the quantized global model with the devices, while updating the qunatization error. For fairness, we consider the quantization scheme presented in Appendix A with the LGM scheme, and assume the same technique for transmission in the device-to-PS direction introduced in Section 2.2.
+
+![](images/548a22435ea30820701de024a68bfa8f885796e5d29d61f38714eab827c7628b.jpg)  
+Figure 1: Test accuracy using MNIST and CIFAR-10 for training with local mini-batch size $\vert \xi _ { m } ^ { i } ( t ) \vert = 5 0 0$ and $| \xi _ { m } ^ { i } ( t ) | = 2 5 0$ , respectively.
+
+Benchmark approaches We consider the performance of the lossless broadcasting (LB) scenario, where the devices receive the current global model accurately, and perform the quantization with error compensation approach as described in Section 2.2. We highlight that this approach requires transmission of $R _ { \mathrm { L B } } = 3 3 d$ bits from the PS, where we assume that each entry of the global model is represented by 33 bits. Thus, the saving ratio in the communication bits of broadcasting from the PS using LFL versus LB is
+
+$$
+\frac { R _ { \mathrm { L B } } } { R _ { \mathrm { Q } } } = \frac { 3 3 d } { 6 4 + d \left( 1 + \log _ { 2 } ( q _ { 1 } + 1 ) \right) } \stackrel { ( \mathrm { a } ) } { \approx } \frac { 3 3 } { 1 + \log _ { 2 } ( q _ { 1 } + 1 ) } ,
+$$
+
+where (a) follows assuming that $d \gg 1$ . We further consider the performance of the fully lossless approach, where in addition to having the accurate global model at the devices, we assume that the PS receives the local model updates from the devices accurately.
+
+In Figure 1 we illustrate the performance of different approaches for non-iid and iid scenarios using MNIST and CIFAR-10, respectively, for training with $M = 4 0$ devices. Figure 1a demonstrates test accuracy of different approaches for non-iid data using MNIST with local mini-batch size $\vert \xi _ { m } ^ { i } ( t ) \vert = 5 0 0$ and number of local iterations $\tau = 4$ . We set $q _ { 2 } = 2$ for all the approaches where the devices perform quantization, and $q _ { 1 } = 2$ for the LFL and LGM schemes. We observe that the proposed LFL algorithm with $( q _ { 1 } , q _ { 2 } ) = ( 2 , 2 )$ performs as well as the fully lossless and LB approaches, despite a factor of 12.77 savings in the number of bits that need to be broadcast compared to the LB approach. This illustrates the efficiency of the LFL algorithm for the non-iid scenario providing significant communication cost savings without any visible performance degradation. On the other hand, the performance of the LGM algorithm drops after an intermediate number of training iterations, which shows that the quantization level $q _ { 1 } = 2$ does not provide the devices with an accurate estimate of the global model to rely on for local training. This is particularly more harmful in later iterations as the algorithm approaches the optimal point where a more accurate estimate of the global model is required for training. We highlight that the proposed LFL algorithm resolves this deficiency with the LGM algorithm through quantizating the global model update rather than the global model providing a more accurate estimate of the global model to the devices even with a relatively small quantization level $q _ { 1 } = 2$ . Throughout our experiments, we found that the random linear transform with the LTGM scheme is not highly efficient in providing a transformed vector with a relatively small peak-to-average ratio, and the quantization level $q _ { 1 }$ should be relatively large to guarantee that the algorithm succeeds in learning. Therefore, we set $q _ { 1 } = 5 0$ for the LTGM scheme, which is a relatively large quantization value. The advantage of the proposed LFL algorithm over the LTGM and LGM algorithms for the non-iid scenario can be clearly seen in the figure.
+
+A similar observation is made in Figure 1b illustrating the perforance of different approaches for iid data using CIFAR-10 with local mini-batch size $| \xi _ { m } ^ { i } ( t ) | = 2 5 0$ and number of local iterations $\tau = 5$ . The the LFL algorithm with $( q _ { 1 } , q _ { 2 } ) = ( 5 , 3 )$ provides $\times$ 9.2 smaller communication load compared to LB with $q _ { 2 } = 3$ without any visible performance degradation with respect to the fully lossless and LB approaches. It also significantly outperforms the LGM algorithm with $( q _ { 1 } , q _ { 2 } ) = ( 5 , 3 )$ , which shows the advantage of quantizing the global model update rather than the global model for iid data. We also observe that the accuracy level of the LTGM algorithm drops significantly after around 200 global iterations even for a large quantization level $q _ { 1 } = 1 0 0 0$ , which shows the deficiency of the linear transform to provide a relatively small peak-to-average ratio for the transformed vector.
+
+![](images/776ebc75fab6db5b9d298ab85ecf3006585dbdbdc310407ad35f250d3fc99f71.jpg)  
+Figure 2: Empirical variance and peak-to-average ratio of the vector quantized at the PS.
+
+In Figure 2, we investigate the empirical variance and the peak-to-average ratio of the vector (considering absolute values of its entries) to be quantized at the PS with different schemes for the experimental settings used in Figure 1. This result is provided to better justify the benefits of the proposed LFL scheme over LGM and LTGM shown in Figure 1. We observe that the global model update, which is quantized at the PS with LFL, has significantly smaller empirical variance than the global model, which is quantized at the PS with LGM. This justifies the improvement of LFL over LGM reflecting smaller quantization error when quantizing the global model update rather than the global model, particularly towards the end of training, where the empirical variance of the global model with LGM has an increasing trend over time. Also, both the empirical variance and the peak-to-average ratio of the transformed vector with LTGM increases over time, particularly for training on CIFAR-10. This illustrates that the quantization error increases with time, which may be more harmful towards the end of training while approaching the optimal solution. We note that the relatively small empirical variance of the transformed vector with LTGM is due to the linear transform applied at the PS which scales down the entries of the global model vector. The relatively large peak-to-average ratio indicates that the quantized vector with LTGM may not provide an accurate estimate of the actual transformed vector at the PS.
+
+# 5 Conclusion
+
+FL is demanding in terms of bandwidth, particularly when deep networks with huge numbers of parameters are trained across a large number of devices. Communication is typically the major bottleneck, since it involves iterative transmission over a bandwidth-limited wireless medium between the PS and a massive number of devices at the edge. With the goal of reducing the communication cost, we have studied FL with lossy broadcasting, where, in contrast to most of the existing work in the literature, the PS broadcasts a compressed version of the global model to the devices. We have considered broadcasting quantized global model updates from the PS, which can be used to estimate the current global model at the devices for local SGD iterations. The PS aggregates the quantized local model updates from the devices, according to which it updates the global model. We have derived convergence guarantees for the proposed LFL algorithm to analyze the impact of lossy broadcasting on the FL performance assuming accurate local model updates at the PS. Numerical experiments have shown the efficiency of the proposed LFL algorithm in providing an accurate estimate of the global model to the devices, where it performs as well as the fully lossless and LB approaches for both non-iid and iid data despite the significant reduction in the communication load. It also significantly outperforms the LTGM Caldas et al. (2019) and LGM Tang et al. (2019) algorithms studying compression in the PS-to-device direction thanks to quantizing the global model update rather than the global model at the PS.
+
+References   
+D. Alistarh, D. Grubic, J. Z. Li, R. Tomioka, and M. Vojnovic. QSGD: Communicationefficient SGD via randomized quantization and encoding. In Advances in Neural Information Processing Systems (NIPS), pp. 1709–1720, Long Beach, CA, December 2017.   
+D. Alistarh, T. Hoefler, M. Johansson, S. Khirirat, N. Konstantinov, and C. Renggli. The convergence of sparsified gradient methods. [Online]. https://arxiv.org/pdf/1809.10505.pdf, September 2018.   
+M. M. Amiri, D. G¨und¨uz, S. R. Kulkarni, and H. V. Poor. Convergence of update aware device scheduling for federated learning at the wireless edge. [Online]. https://arxiv.org/pdf/2001.10402.pdf, May 2020a.   
+M. M. Amiri, D. G¨und¨uz, S. R. Kulkarni, and H. V. Poor. Update aware device scheduling for federated learning at the wireless edge. In Proc. IEEE International Symposium on Information Theory (ISIT), Los Angeles, CA, USA, June 2020b.   
+S. Caldas, J. Konecny, H. B. McMahan, and A. Talwalkar. Expanding the reach of federated learning by reducing client resource requirements. [Online]. https://arxiv.org/pdf/1812.07210.pdf, January 2019.   
+N. Dowlin, R. Gilad-Bachrach, K. Laine, K. Lauter, M. Naehrig, and J. Wernsing. CryptoNets: Applying neural networks to encrypted data with high throughput and accuracy. In Proc. International Conference on Machine Learning (ICML), pp. 201–210, New York, NY, USA, 2016.   
+H. Eichner, T. Koren, H. B. McMahan, N. Srebro, and K. Talwar. Semi-cyclic stochastic gradient descent. [Online]. https://arxiv.org/pdf/1904.10120.pdf, April 2019.   
+F. Haddadpour and M. Mahdavi. On the convergence of local descent methods in federated learning. [Online]. https://arxiv.org/pdf/1910.14425.pdf, December 2019.   
+Lie He, An Bian, and Martin Jaggi. COLA: Decentralized linear learning. In Proc. Conference on Neural Information Processing Systems (NeurIPS), Montreal, Canada, 2018.   
+K. Hsieh, A. Phanishayee, O. Mutlu, and P. B. Gibbons. The non-IID data quagmire of decentralized machine learning. [Online]. https://arxiv.org/pdf/1910.00189.pdf, December 2019.   
+Z. Huo, Q. Yang, B. Gu, L. Carin, and H. Huang. Faster on-device training using new federated momentum algorithm. [Online]. https://arxiv.org/pdf/2002.02090.pdf, February 2020.   
+J. Kang, Z. Xiong, D. Niyato, H. Yu, Y.-C. Liang, and D. I. Kim. Incentive design for efficient federated learning in mobile networks: A contract theory approach. In Proc. IEEE VTS Asia Pacific Wireless Communications Symposium (APWCS), pp. 1–5, Singapore, August 2019.   
+S. P. Karimireddy, S. Kale, M. Mohri, S. J. Reddi, S. U. Stich, and A. T. Suresh. SCAFFOLD: Stochastic controlled averaging for federated learning. [Online]. https://arxiv.org/pdf/1910.06378.pdf, February 2020.   
+A. Khaled, K. Mishchenko, and P. Richtarik. First analysis of local GD on heterogeneous data. [Online]. https://arxiv.org/pdf/1909.04715.pdf, March 2020.   
+D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv:1412.6980v9 [cs.LG], January 2017.   
+A. Koloskova, N. Loizou, S. Boreiri, M. Jaggi, and S. U. Stich. A unified theory of decentralized sgd with changing topology and local updates. [Online]. https://arxiv.org/pdf/2003.10422.pdf, March 2020.   
+J. Konecny and P. Richtarik. Randomized distributed mean estimation: Accuracy vs communication. Frontiers in Applied Mathematics and Statistics, 4, December 2018.   
+J. Konecny, B. McMahan, and D. Ramage. Federated optimization: Distributed optimization beyond the datacenter. arXiv:1511.03575 [cs.LG], November 2015.   
+J. Konecny, H. B. McMahan, F. X. Yu, P. Richtarik, A. T. Suresh, and D. Bacon. Federated learning: Strategies for improving communication efficiency. In Proc. NIPS Workshop on Private Multi-Party Machine Learning, Barcelona, Spain, 2016.   
+A. Krizhevsky and G. Hinton. Learning multiple layers of features from tiny images. In Technical Report, University of Toronto, 2009.   
+Y. LeCun, C. Cortes, and C. Burges. The MNIST database of handwritten digits. http://yann.lecun.com/exdb/mnist/, 1998.   
+T. Li, A. K. Sahu, A. Talwalkar, and V. Smith. Federated learning: Challenges, methods, and future directions. [Online]. https://arxiv.org/pdf/1908.07873.pdf, August 2019.   
+T. Li, A. K. Sahu, M. Zaheer, M. Sanjabi, A. Talwalkar, and V. Smith. Federated optimization in heterogeneous networks. [Online]. https://arxiv.org/pdf/1812.06127.pdf, April 2020a.   
+T. Li, A. K. Sahu, M. Zaheer, M. Sanjabi, A. Talwalkar, and V. Smith. FedDANE: A federated newton-type method. [Online]. https://arxiv.org/pdf/2001.01920.pdf, January 2020b.   
+X. Li, K. Huang, W. Yang, S. Wang, and Z. Zhang. On the convergence of FedAvg on non-IID data. [Online]. https://arxiv.org/pdf/1907.02189.pdf, February 2020c.   
+T. Lin, S. U. Stich, and M. Jaggi. Don’t use large mini-batches, use local SGD. [Online]. https://arxiv.org/pdf/1808.07217.pdf, October 2018a.   
+Y. Lin, S. Han, H. Mao, Y. Wang, and W. J. Dally. Deep gradient compression: Reducing the communication bandwidth for distributed training. arXiv:1712.01887v2 [cs.CV], February 2018b.   
+J. Liu and S. J Wright. Asynchronous stochastic coordinate descent: Parallelism and convergence properties. SIAM Journal on Optimization, 25(1):351–376, 2015.   
+M. M. Amiri and D. G¨und¨uz. Machine learning at the wireless edge: Distributed stochastic gradient descent over-the-air. IEEE Transactions on Signal Processing, 68:2155 – 2169, April 2020.   
+H. B. McMahan and D. Ramage. Federated learning: Collaborative machine learning without centralized training data. https://www.googblogs.com/federated-learning-collaborativemachine-learning-without-centralized-training-data/, April 2017.   
+H. B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas. Communicationefficient learning of deep networks from decentralized data. In Proc. AISTATS, 2017.   
+T. Nishio and R. Yonetani. Client selection for federated learning with heterogeneous resources in mobile edge. In IEEE International Conference on Communications (ICC), pp. 1–7, Shanghai, China, May 2019.   
+P. Kairouz et al. Advances and open problems in federated learning. [Online]. https://arxiv.org/pdf/1912.04977.pdf, December 2019.   
+B. Recht, C. Re, S. Wright, and F. Niu. Hogwild: A lock-free approach to parallelizing stochastic gradient descent. In Advances in Neural Information Processing Systems (NIPS), pp. 693–701, 2011.   
+E. Rizk, S. Vlaski, and A. H. Sayed. Dynamic federated learning. [Online]. https://arxiv.org/pdf/2002.08782.pdf, May 2020.   
+S. U. Stich. Local sgd converges fast and communicates little. [Online]. https://arxiv.org/pdf/1805.09767.pdf, May 2019.   
+H. Tang, X. Lian, C. Yu, T. Zhang, and J. Liu. DoubleSqueeze: Parallel stochastic gradient descent with double-pass error-compensated compression. In Proc. International Conference on Machine Learning (PMLR), Long Beach, CA, 2019.   
+J. Wang and G. Joshi. Cooperative SGD: A unified framework for the design and analysis of communication-efficient sgd algorithms. [Online]. https://arxiv.org/pdf/1808.07576.pdf, January 2019.   
+S. Wang, T. Tuor, T. Salonidis, K. K. Leung, C. Makaya, T. He, and K. Chan. Adaptive federated learning in resource constrained edge computing systems. IEEE Journal on Selected Areas in Communications, 47(6):1205–1221, June 2019.   
+T. Wang, J.-Y. Zhu, A. Torralba, and A. A. Efros. Dataset distillation. [Online]. https://arxiv.org/pdf/1811.10959.pdf, February 2020.   
+B. Woodworth, J. Wang, A. Smith, H. B. McMahan, and N. Srebro. Graph oracle models, lower bounds, and gaps for parallel stochastic optimization. [Online]. https://arxiv.org/pdf/1805.10222.pdf, February 2019.   
+H. H. Yang, A. Arafa, T. Q. S. Quek, and H. V. Poor. Age-based scheduling policy for federated learning in mobile edge networks. [Online]. https://arxiv.org/pdf/1910.14648.pdf, October 2019.   
+H. H. Yang, Z. Liu, T. Q. S. Quek, and H. V. Poor. Scheduling policies for federated learning in wireless networks. IEEE Transactions on Communications, 68(1):317–333, January 2020.   
+H. Yu, S. Yang, and S. Zhu. Parallel restarted sgd with faster convergence and less communication: Demystifying why model averaging works for deep learning. [Online]. https://arxiv.org/pdf/1807.06629.pdf], November 2018.   
+Yue Zhao, Meng Li, Liangzhen Lai, Naveen Suda, Damon Civin, and Vikas Chandra. Federated learning with non-IID data. arXiv:1806.00582 [cs.LG], June 2018.   
+F. Zhou and G. Cong. On the convergence properties of a K-step averaging stochastic gradient descent algorithm for nonconvex optimization. In Proc. International Joint Conference on Artificial Intelligence (IJCAI), pp. 3219–3227, Stockholm, Sweden, July 2018.
+
+# A Stochastic quantization
+
+Given $\pmb { x } \in \mathbb { R } ^ { d }$ , with the $i$ -th entry denoted by $x _ { i }$ , we define
+
+$$
+\begin{array} { r l } & { x _ { \mathrm { m a x } } \triangleq \operatorname* { m a x } \left\{ | \pmb { x } | \right\} , } \\ & { x _ { \mathrm { m i n } } \triangleq \operatorname* { m i n } \left\{ | \pmb { x } | \right\} . } \end{array}
+$$
+
+Given a quantization level $q \geq 1$ , we have
+
+$$
+Q \left( x _ { i } , q \right) \triangleq \operatorname { s i g n } \left( x _ { i } \right) \cdot \left( x _ { \operatorname* { m i n } } + \left( x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } \right) \cdot \varphi \Big ( \frac { \lvert x _ { i } \rvert - x _ { \operatorname* { m i n } } } { x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } } , q \Big ) \right) , \quad \mathrm { f o r } \ i \in [ d ] ,
+$$
+
+where $\varphi ( \cdot , \cdot )$ is a quantization function defined in the following. For $0 \leq x \leq 1$ and $q \geq 1$ , let $l \in \{ 0 , 1 , \ldots , q - 1 \}$ be an integer such that $x \in [ l / q , ( l + 1 ) / q )$ . We then define
+
+$$
+\varphi \left( x , q \right) \triangleq \left\{ \begin{array} { l l } { l / q , } & { \mathrm { w i t h ~ p r o b a b i l i t y ~ } 1 - ( x q - l ) , } \\ { ( l + 1 ) / q , } & { \mathrm { w i t h ~ p r o b a b i l i t y ~ } x q - l . } \end{array} \right.
+$$
+
+We define
+
+$$
+\begin{array} { r } { \pmb { Q } ( \pmb { x } , \pmb { q } ) \triangleq [ \pmb { Q } ( \pmb { x } _ { 1 } , \pmb { q } ) , \cdots , \pmb { Q } ( \pmb { x } _ { d } , \pmb { q } ) ] ^ { T } , } \end{array}
+$$
+
+and we highlight that it is represented by
+
+$$
+R _ { \mathrm { Q } } = 6 4 + d \left( 1 + \log _ { 2 } ( q + 1 ) \right) { \mathrm { ~ b i t s } } ,
+$$
+
+where 64 bits are used to represent $x _ { \mathrm { m a x } }$ and $x _ { \mathrm { m i n } }$ , $d$ bits are used for $\mathrm { s i g n } ( x _ { i } )$ , $\forall i \in [ d ]$ , and $d \log _ { 2 } ( q + 1 )$ bits represent $\varphi ( ( | x _ { i } | - x _ { \operatorname* { m i n } } ) / ( x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } ) , q )$ , $\forall i \in [ d ]$ . We note that we have modified the QSGD scheme proposed in Alistarh et al. (2017) by normalizing the entries of vector $_ { x }$ with $x _ { \mathrm { m a x } } - x _ { \mathrm { m i n } }$ rather than $\lVert \pmb { x } \rVert _ { 2 }$ .
+
+# B Proof of Lemma 1
+
+Given $\varphi \left( x , q \right)$ in (21b), we have
+
+$$
+\mathbb { E } _ { \varphi } \left[ \varphi ( x , q ) \right] = \left( { \frac { l } { q } } \right) ( 1 + l - x q ) + \left( { \frac { l + 1 } { q } } \right) ( x q - l ) = x .
+$$
+
+Also, we have
+
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } _ { \varphi } \left[ \varphi ^ { 2 } ( x , q ) \right] = \left( \displaystyle \frac { l } { q } \right) ^ { 2 } \left( 1 + l - x q \right) + \left( \displaystyle \frac { l + 1 } { q } \right) ^ { 2 } \left( x q - l \right) = \displaystyle \frac { 1 } { q ^ { 2 } } \left( - l ^ { 2 } + 2 l x q + x q - l \right) } \\ { \displaystyle = x ^ { 2 } + \displaystyle \frac { 1 } { q ^ { 2 } } \left( x q - l \right) \left( 1 - x q + l \right) \stackrel { \mathrm { ( a ) } } { \leq } x ^ { 2 } + \displaystyle \frac { 1 } { 4 q ^ { 2 } } , } \end{array}
+$$
+
+where (a) follows since $( x q - l ) \left( 1 - x q + l \right) \leq 1 / 4$ . According to (24), (25) and the definition of $Q ( x , q )$ given in (22), it follows that
+
+$$
+\begin{array} { r l r } & { \mathbb { E } _ { \varphi } [ \pmb { Q } ( \pmb { x } , q ) ] = \pmb { x } , } & { ( 2 6 \mathrm { a } \pmb { \mathrm { a } }  } \\ & { \mathbb { E } _ { \varphi } [ \| \pmb { Q } ( \pmb { x } , q ) \| _ { 2 } ^ { 2 } ] = \sum _ { i = 1 } ^ { d } \mathbb { E } _ { \varphi } [ | \pmb { Q } ( \pmb { x } _ { i } , q ) | _ { 2 } ^ { 2 } ] = ( x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } ) ^ { 2 } \sum _ { i = 1 } ^ { d } \mathbb { E } _ { \varphi } \bigg [ \varphi ^ { 2 } \bigg ( \frac { | x _ { i } | - x _ { \operatorname* { m i n } } } { x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } } , q \bigg ) } \\ & { \quad \quad \quad \quad + d x _ { \operatorname* { m i n } } ^ { 2 } + 2 x _ { \operatorname* { m i n } } ( x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } ) \sum _ { i = 1 } ^ { d } \mathbb { E } _ { \varphi } \bigg [ \varphi \bigg ( \frac { | x _ { i } | - x _ { \operatorname* { m i n } } } { x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } } , q \bigg ) \bigg ] } \\ & { \overset { ( ) } { \le } ( x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } ) ^ { 2 } \sum _ { i = 1 } ^ { d } \bigg ( ( \frac { | x _ { i } | - x _ { \operatorname* { m i n } } } { x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } } ) ^ { 2 } + \frac { 1 } { 4 q ^ { 2 } } \bigg ) + d x _ { \operatorname* { m i n } } ^ { 2 } + 2 x _ { \operatorname* { m i n } } \sum _ { i = 1 } ^ { d } ( | x _ { i } | - x _ { \operatorname* { m i n } } } \\ & { \quad \quad \quad = \| x \| _ { 2 } ^ { 2 } + d \frac { ( x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } ) ^ { 2 } } { 4 q ^ { 2 } } \overset { ( ) } { \le } \| x \| _ { 2 } ^ { 2 } + \frac { \varepsilon d } { 4 q ^ { 2 } } , } &  ( 2 6 \mathrm { b } \| x \| _ { 2 } ^  2  \end{array}
+$$
+
+where (b) follows from (24) and (25), and (c) follows since $\varepsilon = \left( x _ { \operatorname* { m a x } } - x _ { \operatorname* { m i n } } \right) ^ { 2 } / \left\| x \right\| _ { 2 } ^ { 2 }$
+
+# C Proof of Theorem 1
+
+We have
+
+$$
+\begin{array} { r l } & { \mathbb { E } \left[ \left. \pmb { \theta } ( t + 1 ) - \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right] = \mathbb { E } \left[ \left. \widehat { \pmb { \theta } } ( t ) - \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right] + \mathbb { E } \left[ \left. \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \Delta \pmb { \theta } _ { m } ( t ) \right. _ { 2 } ^ { 2 } \right] } \\ & { \qquad + \ 2 \mathbb { E } \left[ \langle \widehat { \pmb { \theta } } ( t ) - \pmb { \theta } ^ { * } , \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \Delta \pmb { \theta } _ { m } ( t ) \rangle \right] . } \end{array}
+$$
+
+In the following, we bound the last two terms on the right hand side (RHS) of (27). From the convexity of $\left\| \cdot \right\| _ { 2 } ^ { 2 }$ , it follows that
+
+$$
+\begin{array} { r l } & { \mathbb { E } \bigg [ \bigg \| \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \Delta \theta _ { m } ( t ) \bigg \| _ { 2 } ^ { 2 } \bigg ] \leq \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \mathbb { E } \Big [ \| \Delta \theta _ { m } ( t ) \| _ { 2 } ^ { 2 } \Big ] } \\ & { \qquad = \eta ^ { 2 } ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \mathbb { E } \bigg [ \big \| \sum _ { i = 1 } ^ { \tau } \nabla F _ { m } \left( \theta _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \big \| _ { 2 } ^ { 2 } \bigg ] } \\ & { \qquad \leq \eta ^ { 2 } ( t ) \tau \sum _ { m = 1 } ^ { M } \sum _ { i = 1 } ^ { \tau } \frac { B _ { m } } { B } \mathbb { E } \left[ \big \| \nabla F _ { m } \left( \theta _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \big \| _ { 2 } ^ { 2 } \right] \overset { ( a ) } { \leq } \eta ^ { 2 } ( t ) \tau ^ { 2 } G ^ { 2 } , } \end{array}
+$$
+
+where (a) follows from Assumption 1.
+
+We rewrite the third term on the RHS of (27) as follows:
+
+$$
+\begin{array} { r l } {  { 2 \mathbb { E } [  \widehat { \theta } ( t ) - \theta ^ { * } , \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \Delta \theta _ { m } ( t )  ] } \quad } & { } \\ & { = 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \mathbb { E } [ \langle \theta ^ { * } - \widehat { \theta } ( t ) , \sum _ { i = 1 } ^ { \tau } \nabla F _ { m } ( \theta _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) ) \rangle ] } \\ & { = 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \mathbb { E } [ \langle \theta ^ { * } - \widehat { \theta } ( t ) , \nabla F _ { m } \big ( \widehat { \theta } ( t ) , \xi _ { m } ^ { 1 } ( t ) \big ) \rangle ] } \\ & { \quad + 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \mathbb { E } [ \langle \theta ^ { * } - \widehat { \theta } ( t ) , \sum _ { i = 2 } ^ { \tau } \nabla F _ { m } ( \theta _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) ) \rangle ] . } \end{array}
+$$
+
+We have
+
+$$
+\begin{array} { r l } & { 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \mathbb { E } \left[ \langle \theta ^ { * } - \widehat \theta ( t ) , \nabla F _ { m } \widehat ( \widehat \theta ( t ) , \xi _ { m } ^ { 1 } ( t ) ) \rangle \right] } \\ & { \qquad \stackrel { ( \sharp ) } { \le } 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \mathbb { E } \left[ \langle \theta ^ { * } - \widehat \theta ( t ) , \nabla F _ { m } \widehat ( \widehat \theta ( t ) ) \rangle \right] } \\ & { \qquad \stackrel { ( \sharp ) } { \le } 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \mathbb { E } \left[ F _ { m } ( \theta ^ { * } ) - F _ { m } \widehat ( \widehat \theta ( t ) ) - \frac { \mu } { 2 } \left. \widehat \theta ( t ) - \theta ^ { * } \right. _ { 2 } ^ { 2 } \right] } \\ & { \qquad = 2 \eta ( t ) \Big ( F ^ { * } - \mathbb { E } \left[ F ( \widehat \theta ( t ) ) \right] - \frac { \mu } { 2 } \mathbb { E } \left[ \left. \widehat \theta ( t ) - \theta ^ { * } \right. _ { 2 } ^ { 2 } \right] \Big ) } \\ & { \qquad \stackrel { ( \sharp ) } { \le } - \mu \eta ( t ) \mathbb { E } \left[ \left. \widehat \theta ( t ) - \theta ^ { * } \right. _ { 2 } ^ { 2 } \right] , } \end{array}
+$$
+
+where (a) follows since $\mathbb { E } _ { \xi } \left[ \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \right] = \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) \right)$ , $\forall i , m$ , (b) is the result of assuming $\mu$ -strongly loss functions, and (c) follows since $F ^ { * } \leq F ( { \widehat { \pmb { \theta } } } ( t ) )$ , $\forall t$ .
+
+Lemma 2. For $0 < \eta ( t ) \leq 1$ , we have
+
+$$
+\begin{array} { r l } {  { 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \mathbb { E } [ \langle \pmb { \theta } ^ { * } - \widehat { \pmb { \theta } } ( t ) , \sum _ { i = 2 } ^ { \tau } \nabla F _ { m } ( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) ) \rangle ] } \quad } & { } \\ & { \leq - \mu \eta ( t ) ( 1 - \eta ( t ) ) ( \tau - 1 ) \mathbb { E } [ \| \widehat { \pmb { \theta } } ( t ) - \pmb { \theta } ^ { * } \| _ { 2 } ^ { 2 } ] + \eta ^ { 2 } ( t ) ( \tau - 1 ) G ^ { 2 } } \\ & { \quad + ( 1 + \mu ( 1 - \eta ( t ) ) ) \eta ^ { 2 } ( t ) G ^ { 2 } \frac { \tau ( \tau - 1 ) ( 2 \tau - 1 ) } { 6 } + 2 \eta ( t ) ( \tau - 1 ) \Gamma . } \end{array}
+$$
+
+Proof. See Appendix D.
+
+By substituting (30) and (31) in (29), it follows that
+
+$$
+\begin{array} { r l } {  { 2 \mathbb { E } [ \langle \widehat { \pmb { \theta } } ( t ) - \pmb { \theta } ^ { * } , \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \Delta \pmb { \theta } _ { m } ( t ) \rangle ] } \qquad } & { } \\ & { \leq - \mu \eta ( t ) ( \tau - \eta ( t ) ( \tau - 1 ) ) \mathbb { E } [ \| \widehat { \pmb { \theta } } ( t ) - \pmb { \theta } ^ { * } \| _ { 2 } ^ { 2 } ] + \eta ^ { 2 } ( t ) ( \tau - 1 ) G ^ { 2 } } \\ & { \qquad + ( 1 + \mu ( 1 - \eta ( t ) ) ) \eta ^ { 2 } ( t ) G ^ { 2 } \frac { \tau ( \tau - 1 ) ( 2 \tau - 1 ) } { 6 } + 2 \eta ( t ) ( \tau - 1 ) \Gamma , } \end{array}
+$$
+
+which, together with the inequality in (28), leads to the following upper bound on $\mathbb { E } \left[ \left. \pmb { \theta } ( t + 1 ) - \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right]$ , when substituted into (27):
+
+$$
+\begin{array} { r l } & { \mathbb { E } \left[ \Vert \pmb { \theta } ( t + 1 ) - \pmb { \theta } ^ { * } \Vert _ { 2 } ^ { 2 } \right] \leq ( 1 - \mu \eta ( t ) ( \tau - \eta ( t ) ( \tau - 1 ) ) ) \mathbb { E } \left[ \left. \hat { \pmb { \theta } } ( t ) - \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right] + \eta ^ { 2 } ( t ) \left( \tau ^ { 2 } + \tau - 1 \right) G ^ { 2 } } \\ & { \qquad + \left( 1 + \mu ( 1 - \eta ( t ) ) \right) \eta ^ { 2 } ( t ) G ^ { 2 } \frac { \tau ( \tau - 1 ) ( 2 \tau - 1 ) } { 6 } + 2 \eta ( t ) ( \tau - 1 ) \Gamma . \ ( 3 3 ) } \end{array}
+$$
+
+Lemma 3. For $\widehat { \pmb \theta } ( t )$ given in (2), we have
+
+$$
+\mathbb { E } \left[ \left. \widehat { \pmb { \theta } } ( t ) - { \pmb { \theta } } ^ { * } \right. _ { 2 } ^ { 2 } \right] \leq \mathbb { E } \left[ \left. \pmb { \theta } ( t ) - \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right] + \left( \frac { \eta ( t - 1 ) \tau G } { 2 q _ { 1 } } \right) ^ { 2 } \varepsilon d .
+$$
+
+for some $0 \leq \varepsilon \leq 1$
+
+Proof. See Appendix E.
+
+According to Lemma 3, the inequality in (34) can be rewritten as follows:
+
+$$
+\begin{array} { r l } & { \mathbb { E } \left[ \| \pmb { \theta } ( t + 1 ) - \pmb { \theta } ^ { * } \| _ { 2 } ^ { 2 } \right] \leq ( 1 - \mu \eta ( t ) ( \tau - \eta ( t ) ( \tau - 1 ) ) ) \mathbb { E } \left[ \| \pmb { \theta } ( t ) - \pmb { \theta } ^ { * } \| _ { 2 } ^ { 2 } \right] } \\ & { \qquad + \left( 1 - \mu \eta ( t ) \left( \tau - \eta ( t ) ( \tau - 1 ) \right) \right) \left( \frac { \eta ( t - 1 ) \tau G } { 2 q _ { 1 } ( t ) } \right) ^ { 2 } \varepsilon d + \eta ^ { 2 } ( t ) \left( \tau ^ { 2 } + \tau - 1 \right) G ^ { 2 } } \\ & { \qquad + \left( 1 + \mu ( 1 - \eta ( t ) ) \right) \eta ^ { 2 } ( t ) G ^ { 2 } \frac { \tau ( \tau - 1 ) ( 2 \tau - 1 ) } { 6 } + 2 \eta ( t ) ( \tau - 1 ) \Gamma . } \end{array}
+$$
+
+Theorem 1 follows from the inequality in (35) having $0 < \eta ( t ) \leq \operatorname* { m i n } \left\{ 1 , \frac { 1 } { \mu \tau } \right\}$ , $\forall t$ .
+
+# D Proof of Lemma 2
+
+We have
+
+$$
+\begin{array} { r l } & { 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \cfrac { B _ { m } } { B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } \left[ \langle \pmb { \theta } ^ { * } - \widehat { \pmb { \theta } } ( t ) , \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \rangle \right] } \\ & { \quad = 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \cfrac { B _ { m } } { B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } \left[ \langle \pmb { \theta } _ { m } ^ { i } ( t ) - \widehat { \pmb { \theta } } ( t ) , \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \rangle \right] } \\ & { \quad \quad + 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \cfrac { B _ { m } } { B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } \left[ \langle \pmb { \theta } ^ { * } - \pmb { \theta } _ { m } ^ { i } ( t ) , \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \rangle \right] . } \end{array}
+$$
+
+We first bound the first term on the RHS of (36). We have
+
+$$
+\begin{array} { r l r } {  { 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } [ \langle \theta _ { m } ^ { i } ( t ) - \widehat \theta ( t ) , \nabla F _ { m } ( \theta _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) ) \rangle ] } } \\ & { } & { \leq \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } [ \frac { 1 } { \eta ( t ) } \| \theta _ { m } ^ { i } ( t ) - \widehat \theta ( t ) \| _ { 2 } ^ { 2 } + \eta ( t ) \| \nabla F _ { m } ( \theta _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) ) \| _ { 2 } ^ { 2 } ] } \\ & { } & { \overset { \mathrm { ( a ) } } { \leq } \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } [ \| \theta _ { m } ^ { i } ( t ) - \widehat \theta ( t ) \| _ { 2 } ^ { 2 } ] + \eta ^ { 2 } ( t ) ( \tau - 1 ) G ^ { 2 } , \quad \quad \quad \quad ( \widehat { \mathsf { L } } \eta ) \| \theta _ { m } ^ { i } ( t ) - \widehat \theta ( t ) \| _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+where (a) follows from Assumption 1. We have
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } \left[ \left\| \theta _ { m } ^ { i } ( t ) - \widehat { \theta } ( t ) \right\| _ { 2 } ^ { 2 } \right] } \\ & { = \eta ^ { 2 } ( t ) \displaystyle \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } \left[ \left\| \sum _ { j = 1 } ^ { i } \nabla F _ { m } \left( \theta _ { m } ^ { j } ( t ) , \xi _ { m } ^ { j } ( t ) \right) \right\| _ { 2 } ^ { 2 } \right] \stackrel { ( \mathrm { b } ) } { \leq } \eta ^ { 2 } ( t ) G ^ { 2 } \frac { \tau ( \tau - 1 ) ( 2 \tau - 1 ) } { 6 } , } \end{array}
+$$
+
+where (b) follows from the convexity of $\left\| \cdot \right\| _ { 2 } ^ { 2 }$ and Assumption 1. Plugging (38) into (37) yields
+
+$$
+\begin{array} { r l } & { 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } \left[ \langle \pmb { \theta } _ { m } ^ { i } ( t ) - \widehat { \pmb { \theta } } ( t ) , \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \rangle \right] } \\ & { \qquad \leq \eta ^ { 2 } ( t ) G ^ { 2 } \frac { \tau \left( \tau - 1 \right) \left( 2 \tau - 1 \right) } { 6 } + \eta ^ { 2 } ( t ) \left( \tau - 1 \right) G ^ { 2 } . } \end{array}
+$$
+
+For the second term on the RHS of (36), we have
+
+$$
+\begin{array} { r l } & { 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { \displaystyle B _ { m } } { \displaystyle B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } [ ( \theta ^ { * } - \theta _ { m } ^ { * } ( t ) , \nabla F _ { m } ( \theta _ { m } ^ { * } ( t ) , \xi _ { m } ^ { * } ( t ) ) ) ] } \\ & { ~ \stackrel { \mathrm { ( i ) } } { = } 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { \displaystyle B _ { m } } { \displaystyle B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } [ ( \theta ^ { * } - \theta _ { m } ^ { * } ( t ) , \nabla F _ { m } ( \theta _ { m } ^ { * } ( t ) ) ) ] } \\ & { \stackrel { \mathrm { ( B ) } } { \leq } 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { \displaystyle B _ { m } } { \displaystyle B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } [ F _ { m } ( \theta ^ { * } ) - F _ { m } ( \theta _ { m } ^ { * } ( t ) ) - \frac { \displaystyle \theta } { \displaystyle B } \| \theta _ { m } ^ { * } ( t ) - \theta ^ { * } \| _ { 2 } ^ { 2 } ] } \\ & { ~ = 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { \displaystyle B _ { m } } { \displaystyle B } \sum _ { i = 2 } ^ { \tau } \mathbb { E } [ F _ { m } ( \theta ^ { * } ) - F _ { m } ^ { * } + F _ { m } ^ { * } - F _ { m } ( \theta _ { m } ^ { * } ( t ) ) - \frac { \displaystyle \theta } { \displaystyle B } \| \theta _ { m } ^ { * } ( t ) - \theta ^ { * } \| _ { 2 } ^ { 2 } ] } \\ & { ~ = 2 \eta ( t ) ( \tau - 1 ) \Gamma + 2 \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { \displaystyle B _ { m } } { \displaystyle B } \sum _ { i = 2 } ^ { \tau } ( F _ { m } ^ { * } - \sum _ { i = 2 } ^ { \tau } ( F _ { m } ^ { * } - \mathbb { E } [ F _ { m } ( \theta _ { m } ^ { * } ( t ) ) ] ) } \\ &  ~ - \mu \eta ( t ) \sum _ { m = 1 } ^ { M } \frac { \displaystyle B _ { m } } { \displaystyle B } \sum _ { i = 2 } ^ { \tau } \mathbb  E  \end{array}
+$$
+
+where (a) follows since $\mathbb { E } _ { \xi } \left[ \nabla F _ { m } \left( \pmb { \theta } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \right] = \nabla F _ { m } \left( \pmb { \theta } ( t ) \right) , \forall i , m , t$ ; (b) follows from assuming that the loss functions are $\mu$ -strongly convex; and (c) follows since $F _ { m } ^ { * } \leq F _ { m } ( \pmb { \theta } _ { m } ^ { \imath } ( t ) )$ , $\forall m$ . We have
+
+$$
+\begin{array} { r l r } {  { - \big \| \theta _ { m } ^ { i } ( t ) - \theta ^ { * } \big \| _ { 2 } ^ { 2 } = - \big \| \theta _ { m } ^ { i } ( t ) - \widehat \theta ( t ) \big \| _ { 2 } ^ { 2 } - \big \| \widehat \theta ( t ) - \theta ^ { * } \big \| _ { 2 } ^ { 2 } - 2 \langle \theta _ { m } ^ { i } ( t ) - \widehat \theta ( t ) , \widehat \theta ( t ) - \theta ^ { * } \rangle } } \\ & { \overset { ( ) } { \leq } - \Big \| \theta _ { m } ^ { i } ( t ) - \widehat \theta ( t ) \Big \| _ { 2 } ^ { 2 } - \Big \| \widehat \theta ( t ) - \theta ^ { * } \Big \| _ { 2 } ^ { 2 } + \frac { 1 } { \eta ( t ) } \| \theta _ { m } ^ { i } ( t ) - \widehat \theta ( t ) \| _ { 2 } ^ { 2 } + \eta ( t ) \| \widehat \theta ( t ) - \theta ^ { * } \| _ { 2 } ^ { 2 } } \\ & { } & { = - ( 1 - \eta ( t ) ) \| \widehat \theta ( t ) - \theta ^ { * } \| _ { 2 } ^ { 2 } + \Big ( \frac { 1 } { \eta ( t ) } - 1 \Big ) \| \theta _ { m } ^ { i } ( t ) - \widehat \theta ( t ) \| _ { 2 } ^ { 2 } , \quad \quad \quad ( 4 ) } \end{array}
+$$
+
+where (a) follows from Cauchy-Schwarz inequality. Plugging (41) into (40) yields
+
+$$
+\begin{array} { r l } & { \frac { 2 \eta ( t ) } { M } \sum _ { m = 1 } ^ { M } \sum _ { i = 2 } ^ { \tau } \mathbb { E } \left[ \langle \pmb { \theta } ^ { * } - \pmb { \theta } _ { m } ^ { i } ( t ) , \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \rangle \right] } \\ & { \leq - \mu \eta ( t ) ( 1 - \eta ( t ) ) ( \tau - 1 ) \left\| \widehat { \theta } ( t ) - \pmb { \theta } ^ { * } \right\| _ { 2 } ^ { 2 } + \mu ( 1 - \eta ( t ) ) \eta ^ { 2 } ( t ) G ^ { 2 } \frac { \tau ( \tau - 1 ) ( 2 \tau - 1 ) } { 6 } + 2 \eta ( t ) ( \tau - 1 ) ] } \end{array}
+$$
+
+where we used the inequality in (38) and $\eta ( t ) \leq 1$ . Plugging (39) and (42) into (36) completes the proof of Lemma 2.
+
+# E Proof of Lemma 3
+
+We have
+
+$$
+\begin{array} { r l r } & { } & { \mathbb { E } \left[ \left. \widehat { \pmb { \theta } } ( t ) - \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right] = \mathbb { E } \left[ \left. \widehat { \pmb { \theta } } ( t ) \right. _ { 2 } ^ { 2 } \right] + \mathbb { E } \left[ \left. \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right] - 2 \mathbb { E } \left[ \langle \widehat { \pmb { \theta } } ( t ) , \pmb { \theta } ^ { * } \rangle \right] } \\ & { } & { \overset { ( ) } { = } \mathbb { E } \left[ \left. \widehat { \pmb { \theta } } ( t ) \right. _ { 2 } ^ { 2 } \right] + \mathbb { E } \left[ \left. \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right] - 2 \mathbb { E } \left[ \langle \pmb { \theta } ( t ) , \pmb { \theta } ^ { * } \rangle \right] , } \end{array}
+$$
+
+where (a) follows since
+
+$$
+\begin{array} { r } { \mathbb { E } \left[ \widehat { \pmb { \theta } } ( t ) \right] = \mathbb { E } \left[ \widehat { \pmb { \theta } } ( t - 1 ) \right] + \mathbb { E } \left[ \pmb { Q } \big ( \pmb { \theta } ( t ) - \widehat { \pmb { \theta } } ( t - 1 ) , q _ { 1 } \big ) \right] = \mathbb { E } \left[ \pmb { \theta } ( t ) \right] , } \end{array}
+$$
+
+where the last equality follows from (26a). In the following, we upper bound $\mathbb { E } \left[ \left. \widehat { \pmb { \theta } } ( t ) \right. _ { 2 } ^ { 2 } \right]$ . We have
+
+$$
+\begin{array} { r l r } { \mathbb { E } \left[ \left. \widehat { \theta } ( t ) \right. _ { 2 } ^ { 2 } \right] = \mathbb { E } \left[ \left. \widehat { \theta } ( t - 1 ) \right. _ { 2 } ^ { 2 } \right] + \mathbb { E } \left[ \left. Q \left( \theta ( t ) - \widehat { \theta } ( t - 1 ) , q _ { 1 } \right) \right. _ { 2 } ^ { 2 } \right] } & { } \\ & { } & { + 2 \mathbb { E } \left[ \left. \widehat { \theta } ( t - 1 ) , Q \left( \theta ( t ) - \widehat { \theta } ( t - 1 ) , q _ { 1 } \right) \right. \right] } \\ & { } & { \overset { ( ) } { \leq } \mathbb { E } \left[ \left. \widehat { \theta } ( t - 1 ) \right. _ { 2 } ^ { 2 } \right] + \mathbb { E } \left[ \left. \theta ( t ) - \widehat { \theta } ( t - 1 ) \right. _ { 2 } ^ { 2 } \right] + \frac { \varepsilon ( t ) d } { 4 q _ { 1 } ^ { 2 } } \mathbb { E } \left[ \left. \theta ( t ) - \widehat { \theta } ( t - 1 ) \right. _ { 2 } ^ { 2 } \right] } \\ & { } & { + 2 \mathbb { E } \left[ \left. \widehat { \theta } ( t - 1 ) , \theta ( t ) - \widehat { \theta } ( t - 1 ) \right. \right) } \\ & { } & { \overset { ( ) } { \leq } \mathbb { E } \left[ \left. \theta ( t ) \right. _ { 2 } ^ { 2 } \right] + \frac { \varepsilon d } { 4 q _ { 1 } ^ { 2 } ( t ) } \mathbb { E } \left[ \left. \theta ( t ) - \widehat { \theta } ( t - 1 ) \right. _ { 2 } ^ { 2 } \right] , \qquad ( 4 } \end{array}
+$$
+
+where (a) follows from (26) for some $0 \leq \varepsilon ( t ) \leq 1$ defined as
+
+$$
+\varepsilon ( t ) \triangleq \frac { \mathbb { E } \left[ \left( \operatorname* { m a x } \left\{ \Big | \theta ( t ) - \widehat { \theta } ( t - 1 ) \Big | \right\} - \operatorname* { m i n } \left\{ \Big | \theta ( t ) - \widehat { \theta } ( t - 1 ) \Big | \right\} \right) ^ { 2 } \right] } { \mathbb { E } \left[ \left\| \theta ( t ) - \widehat { \theta } ( t - 1 ) \right\| _ { 2 } ^ { 2 } \right] } ,
+$$
+
+noting that
+
+$$
+\pmb \theta ( t ) - \widehat \theta ( t - 1 ) = - \eta ( t - 1 ) \sum _ { m = 1 } ^ { M } \sum _ { i = 1 } ^ { \tau } \frac { B _ { m } } { B } \nabla F _ { m } \left( \pmb \theta _ { m } ^ { i } ( t - 1 ) , \xi _ { m } ^ { i } ( t - 1 ) \right) ,
+$$
+
+and in (b) we define $\varepsilon \triangleq \operatorname* { m a x } _ { t } \{ \varepsilon ( t ) \}$ . According to (47), from the convexity of $\left\| \cdot \right\| _ { 2 } ^ { 2 }$ , it follows that
+
+$$
+\begin{array} { r l } & { \mathbb { E } \left[ \left. \pmb { \theta } ( t ) - \widehat { \pmb { \theta } } ( t - 1 ) \right. _ { 2 } ^ { 2 } \right] \leq \eta ^ { 2 } ( t - 1 ) \displaystyle \sum _ { m = 1 } ^ { M } \displaystyle \sum _ { i = 1 } ^ { \tau } \displaystyle \sum _ { B } \mathbb { E } \left[ \left. \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t - 1 ) , \xi _ { m } ^ { i } ( t - 1 ) \right) \right. _ { 2 } ^ { 2 } \right] } \\ & { \qquad \overset { ( \ast ) } { \leq } \eta ^ { 2 } ( t - 1 ) \tau ^ { 2 } G ^ { 2 } , } \end{array}
+$$
+
+where (a) follows from Assumption 1. Accordingly, (45) reduces to
+
+$$
+\mathbb { E } \left[ \left. \widehat { \pmb { \theta } } ( t ) \right. _ { 2 } ^ { 2 } \right] \leq \mathbb { E } \left[ \left. \pmb { \theta } ( t ) \right. _ { 2 } ^ { 2 } \right] + \Big ( \frac { \eta ( t - 1 ) \tau G } { 2 q _ { 1 } } \Big ) ^ { 2 } \varepsilon d .
+$$
+
+Substituting the above inequality into (43) yields
+
+$$
+\begin{array} { r l } & { \mathbb { E } \left[ \left. \widehat { \pmb { \theta } } ( t ) - \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right] \leq \mathbb { E } \left[ \left. \pmb { \theta } ( t ) \right. _ { 2 } ^ { 2 } \right] + \mathbb { E } \left[ \left. \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right] - 2 \mathbb { E } \left[ \langle \pmb { \theta } ( t ) , \pmb { \theta } ^ { * } \rangle \right] + \left( \frac { \eta ( t - 1 ) \tau G } { 2 q _ { 1 } } \right) ^ { 2 } \varepsilon d } \\ & { \qquad = \mathbb { E } \left[ \left. \pmb { \theta } ( t ) - \pmb { \theta } ^ { * } \right. _ { 2 } ^ { 2 } \right] + \left( \frac { \eta ( t - 1 ) \tau G } { 2 q _ { 1 } } \right) ^ { 2 } \varepsilon d . } \end{array}
+$$
+
+# F Proof of Theorem 2
+
+According to the $L$ -smoothness of the loss functions $F _ { 1 } , \ldots , F _ { m }$ , we have
+
+$$
+F ( \pmb \theta ( t + 1 ) ) - F ( \pmb \theta ( t ) ) \leq \langle \pmb \theta ( t + 1 ) - \pmb \theta ( t ) , \nabla F ( \pmb \theta ( t ) ) \rangle + \frac { L } { 2 } \left\| \pmb \theta ( t + 1 ) - \pmb \theta ( t ) \right\| _ { 2 } ^ { 2 } .
+$$
+
+In the following we bound the average of the two terms on the RHS of the above inequality.
+
+Lemma 4. We have
+
+$$
+\begin{array} { r l } & { \mathbb { E } \big [ \langle \theta ( t + 1 ) - \theta ( t ) , \nabla F ( \theta ( t ) ) \rangle \big ] \leq \Big ( \frac { \eta ( t - 1 ) \tau G L } { 2 q _ { 1 } } \Big ) ^ { 2 } \Big ( \frac { \varepsilon d \eta ( t ) ( 2 \tau - 1 ) } { 2 } \Big ) } \\ & { \qquad + \eta ^ { 3 } ( t ) L ^ { 2 } G ^ { 2 } \frac { \tau ( \tau - 1 ) ( 2 \tau - 1 ) } { 6 } - \frac { \eta ( t ) \tau } { 2 } \mathbb { E } \big [ \| \nabla F ( \theta ( t ) ) \| _ { 2 } ^ { 2 } \big ] . } \end{array}
+$$
+
+Proof. See Appendix G.
+
+Lemma 5. We have
+
+$$
+\mathbb { E } \big [ \| \pmb { \theta } ( t + 1 ) - \pmb { \theta } ( t ) \| _ { 2 } ^ { 2 } \big ] \leq 2 \eta ^ { 2 } ( t ) \tau ^ { 2 } G ^ { 2 } + \Big ( \frac { \eta ( t - 1 ) \tau G } { 2 q _ { 1 } } \Big ) ^ { 2 } 2 \varepsilon d .
+$$
+
+Proof. See Appendix I.
+
+Substituting the results in Lemmas 4 and 5 into (51) yields
+
+$$
+\begin{array} { r l } & { \eta ( t ) \mathbb { E } \Big [ \big \| \nabla F ( \pmb { \theta } ( t ) ) \big \| _ { 2 } ^ { 2 } \Big ] \leq \frac { 2 } { \tau } \big ( \mathbb { E } \big [ F ( \pmb { \theta } ( t ) ) \big ] - \mathbb { E } \big [ F ( \pmb { \theta } ( t + 1 ) ) \big ] \big ) } \\ & { \qquad + \left( \frac { \eta ( t - 1 ) G } { 2 q _ { 1 } } \right) ^ { 2 } \big ( \eta ( t ) ( 2 \tau - 1 ) L + 2 \big ) \varepsilon d \tau L + 2 \eta ^ { 2 } ( t ) G ^ { 2 } \tau L + \eta ^ { 3 } ( t ) L ^ { 2 } G ^ { 2 } \frac { \big ( \tau - 1 \big ) \big ( 2 \tau - 1 \big ) } { 3 } . } \end{array}
+$$
+
+For any $T$ , by summing the above inequality over $t$ we have
+
+$$
+\begin{array} { l } { \displaystyle \sum _ { t = 0 } ^ { T - 1 } \eta ( t ) \mathbb { E } \Big [ \big \| \nabla F ( \pmb { \theta } ( t ) ) \big \| _ { 2 } ^ { 2 } \Big ] \leq \frac { 2 } { \tau } \big ( F ( \pmb { \theta } ( 0 ) ) - \mathbb { E } \big [ F ( \pmb { \theta } ( T ) ) \big ] \big ) } \\ { \displaystyle \qquad + \sum _ { t = 0 } ^ { T - 1 } \Big ( \frac { \eta ( t - 1 ) G } { 2 q _ { 1 } } \Big ) ^ { 2 } \big ( \eta ( t ) ( 2 \tau - 1 ) L + 2 \big ) \varepsilon d \tau L } \\ { \displaystyle \qquad + 2 G ^ { 2 } \tau L \sum _ { t = 0 } ^ { T - 1 } \eta ^ { 2 } ( t ) + L ^ { 2 } G ^ { 2 } \frac { \big ( \tau - 1 \big ) ( 2 \tau - 1 ) } { 3 } \sum _ { t = 0 } ^ { T - 1 } \eta ^ { 3 } ( t ) . } \end{array}
+$$
+
+We bound the first term on the RHS of the above inequality as follows:
+
+$$
+\begin{array} { r l r } {  { F ( \pmb { \theta } ( 0 ) ) - \mathbb { E } \big [ F ( \pmb { \theta } ( T ) ) \big ] \leq F ( \pmb { \theta } ( 0 ) ) - \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } F _ { m } ^ { * } } } \\ & { } & \\ & { } & { = F ( \pmb { \theta } ( 0 ) ) - F ^ { * } + F ^ { * } - \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } F _ { m } ^ { * } = F ( \pmb { \theta } ( 0 ) ) - F ^ { * } + \Gamma . } \end{array}
+$$
+
+Substituting the above results in (55) and dividing both sides of the inequality in (55) by $\textstyle \sum _ { t = 0 } ^ { T - 1 } \eta ( t )$ complete the proof of Theorem 2.
+
+# G Proof of Lemma 4
+
+We have
+
+$$
+\begin{array} { r l } & { \mathbb { E } \big [ \langle \theta ( t + 1 ) - \theta ( t ) , \nabla F ( \theta ( t ) ) \rangle \big ] = \mathbb { E } \big [ \langle \theta ( t + 1 ) - \hat { \theta } ( t ) - \theta ( t ) + \hat { \theta } ( t ) , \nabla F ( \theta ( t ) ) \rangle \big ] } \\ & { = \mathbb { E } \big [ \langle \theta ( t + 1 ) - \hat { \theta } ( t ) , \nabla F ( \theta ( t ) ) \rangle \big ] - \mathbb { E } \big [ \langle \theta ( t ) - \hat { \theta } ( t ) , \nabla F ( \theta ( t ) ) \rangle \big ] } \\ & { = \mathbb { E } \big [ \langle \theta ( t + 1 ) - \hat { \theta } ( t ) , \nabla F ( \theta ( t ) ) \rangle \big ] - \mathbb { E } \big [ \langle \theta ( t ) - \hat { \theta } ( t - 1 ) - Q \big ( \theta ( t ) - \hat { \theta } ( t - 1 ) , q _ { 1 } \big ) , \nabla F ( \theta ( t ) ) \rangle \big ] } \\ & { \overset { ( a ) } { = } \mathbb { E } \big [ \langle \theta ( t + 1 ) - \hat { \theta } ( t ) , \nabla F ( \theta ( t ) ) \rangle \big ] } \\ & { \overset { ( b ) } { = } \mathbb { E } \Big [ \langle - \eta ( t ) \displaystyle \sum _ { m = 1 } ^ { M } \sum _ { i = 1 } ^ { \tau } \frac { B _ { m } } { B } \nabla F _ { m } \big ( \theta _ { m } ^ { i } ( t ) \big ) , \nabla F ( \theta ( t ) ) \rangle \Big ] } \\ & { = - \eta ( t ) \mathbb { E } \Big [ \langle \nabla F \big ( \hat { \theta } ( t ) \big ) , \nabla F ( \theta ( t ) ) \rangle \Big ] - \eta ( t ) \displaystyle \sum _ { i = 2 } ^ { \tau } \mathbb { E } \Big [ \langle \sum _ { m = 1 } ^ { M } \frac { B _ { m } } { B } \nabla F _ { m } \big ( \theta _ { m } ^ { i } ( t ) \big ) , \nabla F ( \theta ( t ) ) \rangle \Big ] , \ ( 5 7 ) } \end{array}
+$$
+
+where (a) follows since $\mathbb { E } _ { \varphi } \big \lfloor Q ( { \pmb x } , q _ { 1 } ) \big \rfloor = { \pmb x }$ and the fact that $\pmb \theta ( t ) - \widehat \pmb \theta ( t - 1 )$ is independent of the stochastic quantization $Q \big ( \pmb \theta ( t ) - \hat { \pmb \theta } ( t - 1 ) , q _ { 1 } \big )$ , and $\mathbb { E } _ { \xi } \left[ \nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \right] =$ $\nabla F _ { m } \left( \pmb { \theta } _ { m } ^ { i } ( t ) \right) , \forall i , m$ , results (b). We bound the first term on the RHS of (57) as follows:
+
+$$
+\begin{array} { r l r } & { } & { - \eta ( t ) \mathbb { E } \Big [ \langle \nabla F \big ( \widehat { \theta } ( t ) \big ) , \nabla F ( { \theta } ( t ) ) \rangle \Big ] = \displaystyle \frac { \eta ( t ) } { 2 } \Big ( \mathbb { E } \Big [ \big \| \nabla F ( { \theta } ( t ) ) - \nabla F \big ( \widehat { \theta } ( t ) \big ) \big \| _ { 2 } ^ { 2 } \Big ] - \mathbb { E } \Big [ \big \| \nabla F ( { \theta } ( t ) ) \big \| _ { 2 } ^ { 2 } \Big ] } \\ & { } & { - \mathbb { E } \Big [ \big \| \nabla F \big ( \widehat { \theta } ( t ) \big ) \big \| _ { 2 } ^ { 2 } \Big ] \Big ) \leq \displaystyle \frac { \eta ( t ) L ^ { 2 } } { 2 } \mathbb { E } \Big [ \big \| \theta ( t ) - \widehat { \theta } ( t ) \big \| _ { 2 } ^ { 2 } \Big ] - \displaystyle \frac { \eta ( t ) } { 2 } \mathbb { E } \Big [ \big \| \nabla F ( { \theta } ( t ) ) \big \| _ { 2 } ^ { 2 } \Big ] . ~ ( 5 8 ) } \end{array}
+$$
+
+Lemma 6. We have
+
+$$
+\mathbb { E } \Big [ \big \lVert \pmb { \theta } ( t ) - \widehat { \pmb { \theta } } ( t ) \big \rVert _ { 2 } ^ { 2 } \Big ] \leq \Big ( \frac { \eta ( t - 1 ) \tau G } { 2 q _ { 1 } } \Big ) ^ { 2 } \varepsilon d .
+$$
+
+Proof. See Appendix H.
+
+Plugging (59) into (58) yields
+
+$$
+- \eta ( t ) \mathbb { E } \Big [ \langle \nabla F \big ( \widehat { \theta } ( t ) \big ) , \nabla F ( \theta ( t ) ) \rangle \Big ] \leq \Big ( \frac { \eta ( t - 1 ) \tau G L } { 2 q _ { 1 } } \Big ) ^ { 2 } \Big ( \frac { \varepsilon d \eta ( t ) } { 2 } \Big ) - \frac { \eta ( t ) } { 2 } \mathbb { E } \Big [ \big \| \nabla F ( \theta ( t ) ) \big \| _ { 2 } ^ { 2 } \Big ] .
+$$
+
+The second term on the RHS of (57) is bounded as follows:
+
+$$
+\begin{array} { r l } &  - \alpha \beta \underset { \geq } { \overset { n } { \sum } } \underset { \geq } { \overset { n } { \sum } } \underset { \geq } { \overset { n } { \sum } } \underset { \geq } { \overset { n } { \sum } } \underset { \geq } { \overset { n } { \sum } } \underset { \geq } { \overset { n } { \sum } } \underset { \geq } { \overset { n } { \sum } } \underset { \geq } { \overset { n } { \sum } } \underset { \geq } { \overset { n } { \sum } } \underset { \geq } { \overset { n } { \sum } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } { \overset { n } { \mathbb { F } } } \underset { \geq } \end{array}
+$$
+
+where (a) follows from the convexity of $\left\| \cdot \right\| _ { 2 } ^ { 2 }$ , and (b) follows from (38) and (59). Plugging (60) and (61) into (57) yields
+
+$$
+\begin{array} { r l } & { \mathbb { E } \big [ \langle \theta ( t + 1 ) - \theta ( t ) , \nabla F ( \theta ( t ) ) \rangle \big ] \leq \Big ( \frac { \eta ( t - 1 ) \tau G L } { 2 q _ { 1 } } \Big ) ^ { 2 } \Big ( \frac { \varepsilon d \eta ( t ) ( 2 \tau - 1 ) } { 2 } \Big ) } \\ & { \qquad + \eta ^ { 3 } ( t ) L ^ { 2 } G ^ { 2 } \frac { \tau ( \tau - 1 ) ( 2 \tau - 1 ) } { 6 } - \frac { \eta ( t ) \tau } { 2 } \mathbb { E } \big [ \big \| \nabla F ( \theta ( t ) ) \big \| _ { 2 } ^ { 2 } \big ] . } \end{array}
+$$
+
+# H Proof of Lemma 6
+
+We have
+
+$$
+\begin{array} { r l } { \mathbb { E } \Big [ \big \| \theta ( t ) - \widetilde { \theta } ( t ) \big \| _ { 2 } ^ { 2 } \Big ] = \mathbb { E } \Big [ \big \| \theta ( t ) - \widetilde { \theta } ( t - 1 ) - Q \big ( \theta ( t ) - \widetilde { \theta } ( t - 1 ) , g _ { 1 } \big ) \big \| _ { 2 } ^ { 2 } \Big ] } & { } \\ & { = \mathbb { E } \Big [ \big \| \theta ( t ) - \widetilde { \theta } ( t - 1 ) \big \| _ { 2 } ^ { 2 } \Big ] + \mathbb { E } \Big [ \big \| Q \big ( \theta ( t ) - \widetilde { \theta } ( t - 1 ) , g _ { 1 } \big ) \big \| _ { 2 } ^ { 2 } \Big ] } \\ & { \quad \quad - 2 \mathbb { E } \big [ \langle \theta ( t ) - \widetilde { \theta } ( t - 1 ) , Q \big ( \theta ( t - 1 ) , q _ { 1 } \big ) \rangle \big ] } \\ & { \stackrel { ( ) } { = } - \mathbb { E } \Big [ \big \| \theta ( t ) - \widetilde { \theta } ( t - 1 ) \big \| _ { 2 } ^ { 2 } \Big ] + \mathbb { E } \Big [ \big \| Q \big ( \theta ( t ) - \widetilde { \theta } ( t - 1 ) , q _ { 1 } \big ) \big \| _ { 2 } ^ { 2 } \Big ] } \\ & { \stackrel { ( ) } { \le } \frac { G ^ { 2 } } { 4 q _ { 1 } ^ { 2 } } \mathbb { E } \Big [ \big \| \theta ( t ) - \widetilde { \theta } ( t - 1 ) \big \| _ { 2 } ^ { 2 } \Big ] } \\ & { = \frac { G ^ { 2 } } { 4 q _ { 1 } ^ { 2 } } \mathbb { E } \Big [ \Big \| \eta ( t - 1 ) \sum _ { n = 1 } ^ { M } \sum _ { i = 1 } ^ { M } \frac { D } { B } \nabla F _ { n n } \big ( \theta _ { n n } ^ { i } ( t - 1 ) , \xi _ { m } ^ { i } ( t - 1 ) \big ) \Big \| _ { 2 } ^ { 2 } } \\ & { \stackrel { ( ) } { \le } \Big ( \frac { \eta } { 2 q _ { 1 } } { 2 q _ { 1 } } { \pi } \Big ) ^ { 2 } \mathcal { A } _ { d } , } \end{array}
+$$
+
+where (a) follows since $\pmb \theta ( t ) - \widehat \pmb \theta ( t - 1 )$ is independent of the stochastic quantization $Q \big ( \pmb \theta ( t ) -$ ${ \widehat { \pmb \theta } } ( t - 1 ) , q _ { 1 } )$ and $\mathbb { E } _ { \varphi } \left[ Q ( { \pmb x } , q _ { 1 } ) \right] = { \pmb x }$ , the second inequality in (1b) leads to (b), and (c) is the result of the convexity of $\left\| \cdot \right\| _ { 2 } ^ { 2 }$ and Assumption 1.
+
+# I Proof of Lemma 5
+
+We have
+
+$$
+\begin{array} { r } { \mathbb { E } \Big [ \big \| \pmb { \theta } ( t + 1 ) - \pmb { \theta } ( t ) \big \| _ { 2 } ^ { 2 } \Big ] \leq 2 \mathbb { E } \Big [ \big \| \pmb { \theta } ( t + 1 ) - \widehat { \pmb { \theta } } ( t ) \big \| _ { 2 } ^ { 2 } \Big ] + 2 \mathbb { E } \Big [ \big \| \pmb { \theta } ( t ) - \widehat { \pmb { \theta } } ( t ) \big \| _ { 2 } ^ { 2 } \Big ] . } \end{array}
+$$
+
+For the first term on the RHS of the above inequality, we have
+
+$$
+\begin{array} { r l } { 2 \mathbb { E } \Big [ \big \| \theta ( t + 1 ) - \widehat { \theta } ( t ) \big \| _ { 2 } ^ { 2 } \Big ] } & { = 2 \mathbb { E } \Big [ \Big \| \eta ( t ) \sum _ { m = 1 } ^ { M } \underset { i = 1 } { \overset { \tau } { \sum } } \frac { B _ { m } } { B } \nabla F _ { m } \left( \theta _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \Big \| _ { 2 } ^ { 2 } \Big ] } \\ & { \overset { ( \mathrm { a } ) } { \leq } 2 \eta ^ { 2 } ( t ) \tau \underset { m = 1 } { \overset { M } { \sum } } \underset { i = 1 } { \overset { \tau } { \sum } } \frac { B _ { m } } { B } \mathbb { E } \Big [ \big \| \nabla F _ { m } \left( \theta _ { m } ^ { i } ( t ) , \xi _ { m } ^ { i } ( t ) \right) \big \| _ { 2 } ^ { 2 } \Big ] } \\ & { \overset { ( \mathrm { b } ) } { \leq } 2 \eta ^ { 2 } ( t ) \tau ^ { 2 } G ^ { 2 } , } \end{array}
+$$
+
+where (a) and (b) follow from the convexity of $\left\| \cdot \right\| _ { 2 } ^ { 2 }$ and Assumption 1, respectively. Plugging (65) and (59) into (64) yields
+
+$$
+\mathbb { E } \Big [ \big \| \pmb { \theta } ( t + 1 ) - \pmb { \theta } ( t ) \big \| _ { 2 } ^ { 2 } \Big ] \leq 2 \eta ^ { 2 } ( t ) \tau ^ { 2 } G ^ { 2 } + \Big ( \frac { \eta ( t - 1 ) \tau G } { 2 q _ { 1 } } \Big ) ^ { 2 } 2 \varepsilon d .
+$$

parse/train/rkE8pVcle/rkE8pVcle.md

@@ -0,0 +1,402 @@
+# LEARNING THROUGH DIALOGUE INTERACTIONS BY ASKING QUESTIONS
+
+Jiwei Li, Alexander H. Miller, Sumit Chopra, Marc’Aurelio Ranzato, Jason Weston   
+Facebook AI Research,   
+New York, USA   
+{jiwel,ahm,spchopra,ranzato,jase}@fb.com
+
+# ABSTRACT
+
+A good dialogue agent should have the ability to interact with users by both responding to questions and by asking questions, and importantly to learn from both types of interaction. In this work, we explore this direction by designing a simulator and a set of synthetic tasks in the movie domain that allow such interactions between a learner and a teacher. We investigate how a learner can benefit from asking questions in both offline and online reinforcement learning settings, and demonstrate that the learner improves when asking questions. Finally, real experiments with Mechanical Turk validate the approach. Our work represents a first step in developing such end-to-end learned interactive dialogue agents.
+
+# 1 INTRODUCTION
+
+When a student is asked a question by a teacher, but is not confident about the answer, they may ask for clarification or hints. A good conversational agent (a learner/bot/student) should have this ability to interact with a dialogue partner (the teacher/user). However, recent efforts have mostly focused on learning through fixed answers provided in the training set, rather than through interactions. In that case, when a learner encounters a confusing situation such as an unknown surface form (phrase or structure), a semantically complicated sentence or an unknown word, the agent will either make a (usually poor) guess or will redirect the user to other resources (e.g., a search engine, as in Siri). Humans, in contrast, can adapt to many situations by asking questions.
+
+We identify three categories of mistakes a learner can make during dialogue1: (1) the learner has problems understanding the surface form of the text of the dialogue partner, e.g., the phrasing of a question; (2) the learner has a problem with reasoning, e.g. they fail to retrieve and connect the relevant knowledge to the question at hand; (3) the learner lacks the knowledge necessary to answer the question in the first place – that is, the knowledge sources the student has access to do not contain the needed information.
+
+All the situations above can be potentially addressed through interaction with the dialogue partner. Such interactions can be used to learn to perform better in future dialogues. If a human student has problems understanding a teacher’s question, they might ask the teacher to clarify the question. If the student doesn’t know where to start, they might ask the teacher to point out which known facts are most relevant. If the student doesn’t know the information needed at all, they might ask the teacher to tell them the knowledge they’re missing, writing it down for future use.
+
+In this work, we try to bridge the gap between how a human and an end-to-end machine learning dialogue agent deal with these situations: our student has to learn how to learn. We hence design a simulator and a set of synthetic tasks in the movie question answering domain that allow a bot to interact with a teacher to address the issues described above. Using this framework, we explore how a bot can benefit from interaction by asking questions in both offline supervised settings and online reinforcement learning settings, as well as how to choose when to ask questions in the latter setting. In both cases, we find that the learning system improves through interacting with users.
+
+Finally, we validate our approach on real data where the teachers are humans using Amazon Mechanical Turk, and observe similar results.
+
+# 2 RELATED WORK
+
+Learning language through interaction and feedback can be traced back to the 1950s, when Wittgenstein argued that the meaning of words is best understood from their use within given language games (Wittgenstein, 2010). The direction of interactive language learning through language games has been explored in the early seminal work of Winograd (Winograd, 1972), and in the recent SHRDLURN system (Wang et al., 2016). In a broader context, the usefulness of feedback and interactions has been validated in the setting of multiple language learning, such as second language learning (Bassiri, 2011) and learning by students (Higgins et al., 2002; Latham, 1997; Werts et al., 1995).
+
+In the context of dialogue, with the recent popularity of deep learning models, many neural dialogue systems have been proposed. These include the chit-chat type end-to-end dialogue systems (Vinyals & Le, 2015; Li et al., 2015; Sordoni et al., 2015), which directly generate a response given the previous history of user utterance. It also include a collection of goal-oriented dialogue systems (Wen et al., 2016; Su et al., 2016; Bordes & Weston, 2016), which complete a certain task such as booking a ticket or making a reservation at a restaurant. Another line of research focuses on supervised learning for question answering from dialogues (Dodge et al., 2015; Weston, 2016), using either a given database of knowledge (Bordes et al., 2015; Miller et al., 2016) or short stories (Weston et al., 2015). As far as we know, current dialogue systems mostly focus on learning through fixed supervised signals rather than interacting with users.
+
+Our work is closely related to the recent work of Weston (2016), which explores the problem of learning through conducting conversations, where supervision is given naturally in the response during the conversation. Their work introduced multiple learning schemes from dialogue utterances. In particular the authors discussed Imitation Learning, where the agent tries to learn by imitating the dialogue interactions between a teacher and an expert student; Reward-Based Imitation Learning, which only learns by imitating the dialogue interactions which have have correct answers; and Forward Prediction, which learns by predicting the teacher’s feedback to the student’s response. Despite the fact that Forward Prediction does not uses human-labeled rewards, the authors show that it yields promising results. However, their work did not fully explore the ability of an agent to learn via questioning and interaction. Our work can be viewed as a natural extension of theirs.
+
+# 3 THE TASKS
+
+In this section we describe the dialogue tasks we designed2. They are tailored for the three different situations described in Section 1 that motivate the bot to ask questions: (1) Question Clarification, in which the bot has problems understanding its dialogue partner’s text; (2) Knowledge Operation, in which the bot needs to ask for help to perform reasoning steps over an existing knowledge base; and (3) Knowledge Acquisition, in which the bot’s knowledge is incomplete and needs to be filled.
+
+For our experiments we adapt the WikiMovies dataset (Weston et al., 2015), which consists of roughly $1 0 0 \mathrm { k }$ questions over 75k entities based on questions with answers in the open movie dataset (OMDb). The training/dev/test sets respectively contain 181638 / 9702 / 9698 examples. The accuracy metric corresponds to the percentage of times the student gives correct answers to the teacher’s questions.
+
+Each dialogue takes place between a teacher and a bot. In this section we describe how we generate tasks using a simulator. Section 4.2 discusses how we test similar setups with real data using Mechanical Turk.
+
+The bot is first presented with facts from the OMDb KB. This allows us to control the exact knowledge the bot has access to. Then, we include several teacher-bot question-answer pairs unrelated to the question the bot needs to answer, which we call conversation histories3. In order to explore the benefits of asking clarification questions during a conversation, for each of the three scenarios, our simulator generated data for two different settings, namely, Question-Answering (denoted by QA), and Asking-Question (denoted by AQ). For both $Q A$ and $A Q$ , the bot needs to give an answer to the teacher’s original question at the end. The details of the simulator can be found in the appendix.
+
+# 3.1 QUESTION CLARIFICATION.
+
+In this setting, the bot does not understand the teacher’s question. We focus on a special situation where the bot does not understand the teacher because of typo/spelling mistakes, as shown in Figure 1. We intentionally misspell some words in the questions such as replacing the word “movie” with “movvie” or “star” with “sttar”.4 To make sure that the bot will have problems understanding the question, we guarantee that the bot has never encountered the misspellings before—the misspellingintroducing mechanisms in the training, dev and test sets are different, so the same word will be misspelled in different ways in different sets. We present two $A Q$ tasks: (i) Question Paraphrase where the student asks the teacher to use a paraphrase that does not contain spelling mistakes to clarify the question by asking “what do you mean?”; and (ii) Question Verification where the student asks the teacher whether the original typo-bearing question corresponds to another question without the spelling mistakes (e.g., “Do you mean which film did Tom Hanks appear in?”). The teacher will give feedback by giving a paraphrase of the original question without spelling mistakes (e.g., “I mean which film did Tom Hanks appear in”) in Question Paraphrase or positive/negative feedback in Question Verification. Next the student will give an answer and the teacher will give positive/negative feedback depending on whether the student’s answer is correct. Positive and negative feedback are variants of “No, that’s incorrect” or “Yes, that’s right”5. In these tasks, the bot has access to all relevant entries in the KB.
+
+# 3.2 KNOWLEDGE OPERATION
+
+The bot has access to all the relevant knowledge (facts) but lacks the ability to perform necessary reasoning operations over them; see Figure 2. We focus on a special case where the bot will try to understand what are the relevant facts. We explore two settings: Ask For Relevant Knowledge (Task 3) where the bot directly asks the teacher to point out the relevant KB fact and Knowledge Verification (Task 4) where the bot asks whether the teacher’s question is relevant to one particular KB fact. The teacher will point out the relevant KB fact in the Ask For Relevant Knowledge setting or give a positive or negative response in the Knowledge Verification setting. Then the bot will give an answer to the teacher’s original question and the teacher will give feedback on the answer.
+
+# 3.3 KNOWLEDGE ACQUISITION
+
+For the tasks in this subsection, the bot has an incomplete KB and there are entities important to the dialogue missing from it, see Figure 3. For example, given the question “Which movie did Tom Hanks star in?”, the missing part could either be the entity that the teacher is asking about (question entity for short, which is Tom Hanks in this example), the relation entity (starred actors), the answer to the question (Forrest Gump), or the combination of the three. In all cases, the bot has little chance of giving the correct answer due to the missing knowledge. It needs to ask the teacher the answer to acquire the missing knowledge. The teacher will give the answer and then move on to other questions (captured in the conversational history). They later will come back to reask the question. At this point, the bot needs to give an answer since the entity is not new any more.
+
+Though the correct answer has effectively been included in the earlier part of the dialogue as the answer to the bot’s question, as we will show later, many of the tasks are not as trivial as they look when the teacher reasks the question. This is because the bot’s model needs to memorize the missing entity and then construct the links between the missing entities and known ones. This is akin to the real world case where a student might make the same mistake again and again even though each time the teacher corrects them if their answer is wrong. We now detail each task in turn.
+
+![](images/bb347c16afa7fc669133cfc1383ea96a78cc81be4a96329a3e1d6fa423fdc3d3.jpg)  
+Figure 1: Sample dialogues for the two Question Clarification tasks (rows) using both the traditional $Q A$ setting (left column) and $A Q$ setting (right column). In each case the same example is given for simplicity. Black text prefixed by “kb:” denotes KB knowledge that the student has access to. Blue text is spoken by the teacher, denoted by T. $( + )$ denotes a reward of 1 (and 0 otherwise) that the teacher assigns to the bot. Red text denotes responses or questions posed by the bot, denoted by S. Brown denotes typos deliberately introduced by the authors. For the Question Verification setting, the student can either ask a correct (pertinent) question (as in this example) or an incorrect (irrelevant) one. The teacher will give positive or negative feedback based on the correctness of the student’s question. In our offline superised learning experiments, the probability of asking pertinent questions and correctly answering the original question from the teacher is set to 0.5. Finally, T/S denotes 5 pairs of questions and answers that are irrelevant to the rest of the conversation.
+
+Task 1: Question Paraphrase (AQ):   
+kb: Larry Crowne directed by Tom Hanks   
+kb: Forrest Gump starred actors Tom Hanks, Robin Wright, Gary Sinise   
+kb: Forrest Gump directed by Robert Zemeckis T/S : Conversation History.   
+T : Which movvie did Tom Hanks sttar in ? S : What do you mean ?   
+T : I mean which film did Tom Hanks appear in. T : Which movvie did Tom Hanks sttar in ? S : Forrest Gump   
+T : That’s correct. $( + )$
+
+![](images/1075a725d670bd1a1dafb1e56da35b31f24ecbe7344bf51254e61ca0005ecb92.jpg)  
+Figure 2: Sample dialogues for Knowledge Operation tasks.
+
+Missing Question Entity: The entity that the teacher is asking about is missing from the knowledge base. All KB facts containing the question entity will be hidden from the bot. In the example for Task 5 in Figure 3, since the teacher’s question contains the entity Tom Hanks, the KB facts that contain Tom Hanks are hidden from the bot.
+
+Task 5: Missing Question Entity (QA)   
+kb: Larry Crowne directed by Tom Hanks   
+kb: Forrest Gump starred actors Tom Hanks kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T/S : Conversation History.   
+T : Which movie did Tom Hanks star in ?   
+S : Forrest Gump   
+T : That’s correct. $( + )$ Task 5: Missing Question Entity (AQ)   
+kb: Larry Crowne directed by Tom Hanks   
+kb: Forrest Gump starred actors Tom Hanks, kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T : Which movie did Tom Hanks star in ?   
+S : I don’t know. What’s the answer?   
+T : The answer is Forrest Gump.   
+T/S : Conversation History.   
+T : Which movie did Tom Hanks star in ?   
+S : Forrest Gump   
+T : That’s correct. $( + )$ Task 6: Missing Answer Entity (AQ)   
+kb: Larry Crowne directed by Tom Hanks   
+kb: Forrest Gump starred actors Tom Hanks kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T : Which movie did Tom Hanks star in ?   
+S : I don’t know. What’s the answer?   
+T : The answer is Forrest Gump.   
+T/S : Conversation History.   
+T : Which movie did Tom Hanks star in ?   
+S : Forrest Gump   
+T : That’s correct. $( + )$ Task 7: Missing Relation Entity (AQ)   
+kb: Larry Crowne directed by Tom Hanks   
+kb: Forrest Gump starred actors Tom Hanks, kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T : Which movie did Tom Hanks star in ?   
+S : I don’t know. What’s the answer?   
+T : The answer is Forrest Gump.   
+T/S : Conversation History.   
+T : Which movie did Tom Hanks star in ?   
+S : Forrest Gump   
+T : That’s correct. $( + )$
+
+# Task 8: Missing Triple (AQ)
+
+kb: Larry Crowne directed by Tom Hanks   
+kb: Forrest Gump starred actors Tom Hanks kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T : Which movie did Tom Hanks star in ?   
+S : I don’t know. What’s the answer?   
+T : The answer is Forrest Gump.   
+T/S : Conversation History.   
+T : Which movie did Tom Hanks star in ?   
+S : Forrest Gump   
+T : That’s correct. (+) Task 9: Missing Everything (AQ)   
+kb: Larry Crowne directed by Tom Hanks   
+kb: Forrest Gump starred actors Tom Hanks, kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T : Which movie did Tom Hanks star in ?   
+S : I don’t know. What’s the answer?   
+T : The answer is Forrest Gump.   
+T/S : Conversation History.   
+T : Which movie did Tom Hanks star in ?   
+S : Forrest Gump   
+T : That’s correct. (+)
+
+Missing Answer Entity: The answer entity to the question is unknown to the bot. All KB facts that contain the answer entity will be hidden. Hence, in Task $6$ of Figure 3, all KB facts containing the answer entity Forrest Gump will be hidden from the bot.
+
+Missing Relation Entity: The relation type is unknown to the bot. In Task 7 of Figure 3, all KB facts that express the relation starred actors are hidden from the bot.
+
+Missing Triples: The triple that expresses the relation between the question entity and the answer entity is hidden from the bot. In Task 8 of Figure 3, the triple “Forrest Gump (question entity) starred actors Tom Hanks (answer entity)” will be hidden.
+
+Missing Everything: The question entity, the relation entity, the answer entity are all missing from the KB. All KB facts in Task 9 of Figure 3 will be removed since they either contain the relation entity (i.e., starred actors), the question entity (i.e., Forrest Gump) or the answer entity Tom Hanks.
+
+# 4 TRAIN/TEST REGIME
+
+We now discuss in detail the regimes we used to train and test our models, which are divided between evaluation within our simulator and using real data collected via Mechanical Turk.
+
+# 4.1 SIMULATOR
+
+Using our simulator, our objective was twofold. We first wanted to validate the usefulness of asking questions in all the settings described in Section 3. Second, we wanted to assess the ability of our student bot to learn when to ask questions. In order to accomplish these two objectives we explored training our models with our simulator using two methodologies, namely, Offline Supervised Learning and Online Reinforcement Learning.
+
+# 4.1.1 OFFLINE SUPERVISED LEARNING
+
+The motivation behind training our student models in an offline supervised setting was primarily to test the usefulness of the ability to ask questions. The dialogues are generated as described in the previous section, and the bot’s role is generated with a fixed policy. We chose a policy where answers to the teacher’s questions are correct answers $50 \%$ of the time, and incorrect otherwise, to add a degree of realism. Similarly, in tasks where questions can be irrelevant they are only asked correctly $50 \%$ of the time.6
+
+The offline setting explores different combinations of training and testing scenarios, which mimic different situations in the real world. The aim is to understand when and how observing interactions between two agents can help the bot improve its performance for different tasks. As a result we construct training and test sets in three ways across all tasks, resulting in 9 different scenarios per task, each of which correspond to a real world scenario.
+
+The three training sets we generated are referred to as TrainQA, TrainAQ, and TrainMix. TrainQA follows the QA setting discussed in the previous section: the bot never asks questions and only tries to immediately answer. TrainAQ follows the AQ setting: the student, before answering, first always asks a question in response to the teacher’s original question. TrainMix is a combination of the two where $5 0 \%$ of time the student asks a question and $5 0 \%$ does not.
+
+The three test sets we generated are referred to as TestQA, TestAQ, and TestModelAQ. TestQA and TestAQ are generated similarly to TrainQA and TrainAQ, but using a perfect fixed policy (rather than $50 \%$ correct) for evaluation purposes. In the TestModelAQ setting the model has to get the form of the question correct as well. In the Question Verification and Knowledge Verification tasks there are many possible ways of forming the question and some of them are correct – the model has to choose the right question to ask. E.g. it should ask “Does it have something to do with the fact that Larry Crowne directed by Tom Hanks?”rather than “Does it have something to do with the fact that Forrest Gump directed by Robert Zemeckis?” when the latter is irrelevant (the candidate list of questions is generated from the known knowledge base entries with respect to that question). The policy is trained using either the TrainAQ or TrainMix set, depending on the training scenario. The teacher will reply to the question, giving positive feedback if the student’s question is correct and no response and negative feedback otherwise. The student will then give the final answer. The difference between TestModelAQ and TestAQ only exists in the Question Verification and Knowledge Verification tasks; in other tasks there is only one way to ask the question and TestModelAQ and TestAQ are identical.
+
+To summarize, for every task listed in Section 3 we train one model for each of the three training sets (TrainQA, TrainAQ, TrainMix) and test each of these models on the three test sets (TestQA, TestAQ, and TestModelAQ), resulting in 9 combinations. For the purpose of notation the train/test combination is denoted by “TrainSetting+TestSetting”. For example, TrainA $Q +$ TestQA denotes a model which is trained using the TrainAQ dataset and tested on TestQA dataset. Each combination has a real world interpretation. For instance, $T r a i n A Q + T e s t Q A$ would refer to a scenario where a student can ask the teacher questions during learning but cannot to do so while taking an exam. Similarly, $T r a i n Q A + T e s t Q A$ describes a stoic teacher that never answers a student’s question at either learning or examination time. The setting $T r a i n Q A + T e s t A Q$ corresponds to the case where a lazy student never asks question at learning time but gets anxious during the examination and always asks a question.
+
+# 4.1.2 ONLINE REINFORCEMENT LEARNING (RL)
+
+We also explored scenarios where the student learns the ability to decide when to ask a question. In other words, the student learns how to learn.
+
+Although it is in the interest of the student to ask questions at every step of the conversation, since the response to its question will contain extra information, we don’t want our model to learn this behavior. Each time a human student asks a question, there’s a cost associated with that action. This cost is a reflection of the patience of the teacher, or more generally of the users interacting with the bot in the wild: users won’t find the bot engaging if it always asks clarification questions. The student should thus be judicious about asking questions and learn when and what to ask. For instance, if the student is confident about the answer, there is no need for it to ask. Or, if the teacher’s question is so hard that clarification is unlikely to help enough to get the answer right, then it should also refrain from asking.
+
+We now discuss how we model this problem under the Reinforcement Learning framework. The bot is presented with KB facts (some facts might be missing depending on the task) and a question. It needs to decide whether to ask a question or not at this point. The decision whether to ask is made by a binary policy $P _ { R L Q u e s t i o n }$ . If the student chooses to ask a question, it will be penalized by $c o s t _ { A Q }$ . We explored different values of $c o s t _ { A Q }$ ranging from $[ 0 , 2 ]$ , which we consider as modeling the patience of the teacher. The goal of this setting is to find the best policy for asking/notasking questions which would lead to the highest cumulative reward. The teacher will appropriately reply if the student asks a question. The student will eventually give an answer to the teacher’s initial question at the end using the policy $P _ { R L A n s w e r }$ , regardless of whether it had asked a question. The student will get a reward of $+ 1$ if its final answer is correct and $- 1$ otherwise. Note that the student can ask at most one question and that the type of question is always specified by the task under consideration. The final reward the student gets is the cumulative reward over the current dialogue episode. In particular the reward structure we propose is the following:
+
+Final Answer Correct Final Answer Incorrect
+
+<table><tr><td>Asking Question</td><td>Not asking Question</td></tr><tr><td>1-cost AQ</td><td>1</td></tr><tr><td>-1-cost AQ</td><td>-1</td></tr></table>
+
+For each of the tasks described in Section 3, we consider three different RL scenarios.
+
+Good-Student: The student will be presented with all relevant KB facts. There are no misspellings or unknown words in the teacher’s question. This represents a knowledgable student in the real world that knows as much as it needs to know (e.g., a large knowledge base, large vocabulary). This setting is identical across all missing entity tasks (5 - 9).
+
+Poor-Student: The KB facts or the questions presented to the student are flawed depending on each task. For example, for the Question Clarification tasks, the student does not understand the question due to spelling mistakes. For the Missing Question Entity task the entity that the teacher asks about is unknown by the student and all facts containing the entity will be hidden from the student. This setting is similar to a student that is underprepared for the tasks.
+
+Medium-Student: The combination of the previous two settings where for $5 0 \%$ of the questions, the student has access to the full KB and there are no new words or phrases or entities in the question, and $5 0 \%$ of the time the question and KB are taken from the Poor-Student setting.
+
+# 4.2 MECHANICAL TURK DATA
+
+Finally, to validate our approach beyond our simulator by using real language, we collected data via Amazon Mechanical Turk. Due to the cost of data collection, we focused on real language versions of Tasks 4 (Knowledge Verification) and 8 (Missing Triple), see Secs. 3.2 and 3.3 for the simulator versions. That is, we collect dialogues and use them in an offline supervised learning setup similar to Section 4.1.1. This setup allows easily reproducible experiments.
+
+For Mechanical Turk Task 4, the bot is asked a question by a human teacher, but before answering can ask the human if the question is related to one of the facts it knows about from its memory.
+
+![](images/672214b4a8794743c3342e4875ce41e848b13b459af37b01e3177bc323ed213e.jpg)  
+Reward:	1-CostAQ   
+Figure 4: An illustration of the poor-student setting for RL Task 1 (Question Paraphrase).
+
+It is then required to answer the original question, after some additional dialog turns relating to other question/answer pairs (called “conversational history”, as before). For Task 8, the bot is asked a question by a human but lacks the triple in its memory that would be needed to answer it. It is allowed to ask for the missing information, the human responds to the question in free-form language. The bot is then required to answer the original question, again after some “conversational history” has transpired.
+
+We collect around 10,000 episodes (dialogues) for training, 1000 for validation, and 2500 for testing for each of the two tasks. In each case, we give instructions to the Turkers that still follow the original form of the task, but make the tasks contain realistic language written by humans. The instructions given to the Turkers are given in the appendix.
+
+For both tasks, while the human turkers replace the simulator that the bot was previously conversing with, the bot’s dialogue actions (capabilities) are essentially unchanged from before. That is, when answering questions, now the bot is required to answer a human’s questions rather than templated questions from the simulator. When the bot is asking questions, the bot still asks in the same form as before, e.g. questions like “Does it have something to do with X” for Task 4 or “I don’t know. What’s the answer?” for Task 8. However, now its questions are answered by a human. In both cases (asking and answering) the human data is richer with potentially more complex language and lexical variability. Examples of the collected dialogues are given in Figure 5.
+
+![](images/2b392e8dccdaadfa7c2f1264720a5befd7b2eef240ac496cb5f97d66faf453c5.jpg)  
+Figure 5: Sample dialogues for Mechanical Turk versions of Tasks 4 and 8. Compared to the original tasks (see Figs 2 and 3) the teacher’s questions, and the teacher responses to the student’s questions, are written by humans and are more complex and contain more variety.
+
+# 5 MODELS
+
+For both offline supervised and online RL settings, we use the End-to-End Memory Network model (MemN2N) (Sukhbaatar et al., 2015) as a backbone. The model takes as input the last utterance of the dialogue history (the question from the teacher) as well as a set of memory contexts including short-term memories (the dialogue history between the bot and the teacher) and long-term memories (the knowledge base facts that the bot has access to), and outputs a label. We refer readers to the Appendix for more details about MemN2N.
+
+Offline Supervised Settings: The first learning strategy we adopt is the reward-based imitation strategy (denoted vanilla-MemN2N) described in (Weston, 2016), where at training time, the model maximizes the log likelihood probability of the correct answers the student gave (examples with incorrect final answers are discarded). Candidate answers are words that appear in the memories, which means the bot can only predict the entities that it has seen or known before.
+
+We also use a variation of MemN2N called “context MemN2N” (Cont-MemN2N for short) where we replace each word’s embedding with the average of its embedding (random for unseen words) and the embeddings of the other words that appear around it. We use both the preceeding and following words as context and the number of context words is a hyperparameter selected on the dev set.
+
+An issue with both vanilla-MemN2N and Cont-MemN2N is that the model only makes use of the bot’s answers as signals and ignores the teacher’s feedback. We thus propose to use a model that jointly predicts the bot’s answers and the teacher’s feedback (denoted as TrainQA $\left( + F P \right) )$ . The bot’s answers are predicted using a vanilla-MemN2N and the teacher’s feedback is predicted using the Forward Prediction (FP) model as described in (Weston, 2016). We refer the readers to the Appendix for the FP model details. At training time, the models learn to jointly predict the teacher’s feedback and the answers with positive reward. At test time, the model will only predict the bot’s answer.
+
+For the TestModelAQ setting described in Section 4, the model needs to decide the question to ask. Again, we use vanilla-MemN2N that takes as input the question and contexts, and outputs the question the bot will ask.
+
+Online RL Settings: A binary vanilla-MemN2N (denoted as $P _ { R L } ( Q u e s t i o n ) )$ is used to decide whether the bot should or should not ask a question, with the teacher replying if the bot does ask something. A second MemN2N is then used to decide the bot’s answer, denoted as $P _ { R L } ( A n s w e r )$ . $P _ { R L } ( A n s w e r )$ for $Q A$ and $A Q$ are two separate models, which means the bot will use different models for final-answer prediction depending on whether it chooses to ask a question or not.7
+
+We use the REINFORCE algorithm (Williams, 1992) to update $P _ { R L } ( Q u e s t i o n )$ and $P _ { R L } ( A n s w e r )$ . For each dialogue, the bot takes two sequential actions $( a _ { 1 } , a _ { 2 } )$ : to ask or not to ask a question (denoted as $a _ { 1 }$ ); and guessing the final answer (denoted as $a _ { 2 }$ ). Let $r ( a _ { 1 } , a _ { 2 } )$ denote the cumulative reward for the dialogue episode, computed using Table 1. The gradient to update the policy is given by:
+
+$$
+\begin{array} { r l } & { p ( a _ { 1 } , a _ { 2 } ) = P _ { R L } ( Q u e s t i o n ) ( a _ { 1 } ) \cdot P _ { R L } ( a n s w e r ) ( a _ { 2 } ) } \\ & { \nabla J ( \theta ) \approx \nabla \log p ( a _ { 1 } , a _ { 2 } ) [ r ( a _ { 1 } , a _ { 2 } ) - b ] } \end{array}
+$$
+
+where $b$ is the baseline value, which is estimated using another MemN2N model that takes as input the query $x$ and memory $C$ , and outputs a scalar $b$ denoting the estimation of the future reward. The baseline model is trained by minimizing the mean squared loss between the estimated reward $b$ and actual cumulative reward $r$ , $| | \boldsymbol { r } - \boldsymbol { b } | | ^ { 2 }$ . We refer the readers to (Ranzato et al., 2015; Zaremba & Sutskever, 2015) for more details. The baseline estimator model is independent from the policy models and the error is not backpropagated back to them.
+
+In practice, we find the following training strategy yields better results: first train only $P _ { R L } ( a n s w e r )$ , updating gradients only for the policy that predicts the final answer. After the bot’s final-answer policy is sufficiently learned, train both policies in parallel8. This has a real-world analogy where the bot first learns the basics of the task, and then learns to improve its performance via a question-asking policy tailored to the user’s patience (represented by $c o s t _ { A Q }$ ) and its own ability to asnwer questions.
+
+Table 2: Results for Cont-MemN2N on different tasks.   
+
+<table><tr><td></td><td colspan="5">Question Clarification</td><td colspan="5">Knowledge Operation</td></tr><tr><td></td><td colspan="2">Task 1:Q.Paraphrase TestAQ</td><td colspan="2">Task 2:Q.Verification</td><td colspan="2"></td><td colspan="2">Task 3:Ask For Relevant K.</td><td colspan="2">Task 4:K.Verification TestQA</td></tr><tr><td>Train\Test</td><td colspan="2">TestQA</td><td colspan="2">TestQA</td><td colspan="2">TestAQ TestQA</td><td colspan="2">TestAQ</td><td colspan="2">TestAQ</td></tr><tr><td>TrainQA (Context)</td><td>0.754</td><td>0.726</td><td>0.742</td><td>0.684</td><td></td><td>0.883 0.716</td><td>0.947</td><td>0.888</td><td>0.959</td><td></td></tr><tr><td>TrainAQ(Context)</td><td>0.640</td><td>0.889</td><td colspan="2">0.643</td><td colspan="2">0.807 0.789</td><td colspan="2">0.985</td><td>0.852 0.875</td><td>0.987</td></tr><tr><td>TrainMix (Context)</td><td>0.751</td><td>0.846</td><td colspan="2">0.740</td><td colspan="2">0.870</td><td colspan="2">0.985</td><td>0.985</td><td></td></tr><tr><td colspan="10">Knowledge Acquisition TestAQ TestAQ</td></tr><tr><td>TrainTest</td><td>TestQA</td><td>TestAQ</td><td colspan="2">TestQA</td><td colspan="2">TestQA</td><td colspan="2">TestQA</td><td>TestAQ</td><td>TestQA TestAQ</td></tr><tr><td></td><td>Task 5:Q.Entity</td><td></td><td colspan="2">Task 6:Answer Entity</td><td colspan="2">Task7:Relation Entity</td><td colspan="2">Task 8:Triple</td><td></td><td>Task 9:Everything</td></tr><tr><td>TrainQA (Context)</td><td>&lt;0.01</td><td>0.224</td><td colspan="2">&lt;0.01</td><td colspan="2">0.241</td><td colspan="2">0.339</td><td>0.251 &lt;0.01</td><td>0.058</td></tr><tr><td>TrainAQ(Context)</td><td>&lt;0.01</td><td>0.639</td><td colspan="2">&lt;0.01</td><td colspan="2">0.143</td><td colspan="2">0.154</td><td>0.884 &lt;0.01</td><td>0.908</td></tr><tr><td>TrainMix (Context)</td><td>&lt;0.01</td><td>0.632</td><td colspan="2">&lt;0.01</td><td colspan="2">0.216</td><td colspan="2">0.298</td><td>0.886 &lt;0.01</td><td>0.903</td></tr></table>
+
+# 6 EXPERIMENTS
+
+# 6.1 SIMULATOR
+
+Offline Results: Offline results are presented in Tables 2, 7 and 8 (the latter two are in the appendix). Table 7 presents results for the vanilla-MemN2N and Forward Prediction models. Table 2 presents results for Cont-MemN2N, which is better at handling unknown words. We repeat each experiment 10 times and report the best result. Finally, Table 8 presents results for the test scenario where the bot itself chooses when to ask questions. Observations can be summarized as as follows:
+
+- Asking questions helps at test time, which is intuitive since it provides additional evidence:
+
+• Train $1 Q + T e s t A Q$ (questions can be asked at both training and test time) performs the best across all the settings. Tra $i n Q A { + } T e s t A Q$ (questions can be asked at training time but not at test time) performs worse than $T r a i n Q A + T e s t Q A$ (questions can be asked at neither training nor test time) in tasks Question Clarification and Knowledge Operation due to the discrepancy between training and testing. $T r a i n Q A + T e s t A Q$ performs better than $T r a i n Q A + T e s t Q A$ on all Knowledge Acquisition tasks, the only exception being the Cont-MemN2N model on the Missing Triple setting. The explanation is that for most tasks in Knowledge Acquisition, the learner has no chance of giving the correct answer without asking questions. The benefit from asking is thus large enough to compensate for the negative effect introduced by data discrepancy between training and test time. TrainMix offers flexibility in bridging the gap between datasets generated using QA and AQ, very slightly underperforming TrainAQ+TestAQ, but gives competitive results on both TestQA and TestAQ in the Question Clarification and Knowledge Operations tasks. $I r a i n A Q + T e s t Q A$ (allowing questions at training time but forbid questions at test time) performs the worst, even worse than $T r a i n Q A + T e s t Q A$ . This has a real-world analogy where a student becomes dependent on the teacher answering their questions, later struggling to answer the test questions without help.   
+In the Missing Question Entity task (the student does not know about the question entity), the Missing Answer Entity task (the student does not know about the answer entity), and Missing Everything task, the bot achieves accuracy less than 0.01 if not asking questions at test time (i.e., TestQA). The performance of TestModelAQ, where the bot relies on its model to ask questions at test time (and thus can ask irrelevant questions) performs similarly to asking the correct question at test time (TestAQ) and better than not asking questions (TestQA).
+
+- Cont-MemN2N significantly outperforms vanilla-MemN2N. One explanation is that considering context provides significant evidence distinguishing correct answers from candidates in the dialogue history, especially in cases where the model encounters unfamiliar words.
+
+RL Results For the RL settings, we present results for Task 2 (Question Verification) and Task 6 (Missing Answer Entities) in Figure 6. Task 2 represents scenarios where different types of student have different abilities to correctly answer questions (e.g., a poor student can still sometimes give correct answers even when they do not fully understand the question). Task 6 represents tasks where a poor learner who lacks the knowledge necessary to answer the question can hardly give a correct answer. All types of students including the good student will theoretically benefit from asking questions (asking for the correct answer) in Task 6. We show the percentage of question-asking versus the cost of AQ on the test set and the accuracy of question-answering on the test set vs the cost of AQ. Our main findings were:
+
+![](images/ab2c6274536116f73ca7bb49b17c6f5b3d46d860d8ff9978b85bf21d6759c67c.jpg)  
+Figure 6: Results of online learning for Task 2 and Task 6
+
+• A good student does not need to ask questions in Task 2 (Question Verification), because they already understand the question. The student will raise questions asking for the correct answer when cost is low for Task 6 (Missing Answer Entities). A poor student always asks questions when the cost is low. As the cost increases, the frequency of question-asking declines. As the AQ cost increases gradually, good students will stop asking questions earlier than the medium and poor students. The explanation is intuitive: poor students benefit more from asking questions than good students, so they continue asking even with higher penalties.   
+• As the probability of question-asking declines, the accuracy for poor and medium students drops. Good students are more resilient to not asking questions.
+
+# 6.2 MECHANICAL TURK
+
+Results for the Mechanical Turk Tasks are given in Table 3. We again compare vanilla-MemN2N and Cont-MemN2N, using the same TrainAQ/TrainQA and TestAQ/TestQA combinations as before, for Tasks 4 and 8 as described in Section 4.2. We tune hyperparameters on the validation set and repeat each experiment 10 times and report the best result.
+
+While performance is lower than on the related Task 4 and Task 8 simulator tasks, we still arrive at the same trends and conclusions when real data from humans is used. The performance was expected to be lower because (i) real data has more lexical variety, complexity and noise; and (ii) the training set was smaller due to data collection costs (10k vs. 180k). We perform an analysis of the difference between simulated and real training data (or combining the two) in the appendix, which shows that using real data is indeed important and measurably superior to using simulated data.
+
+Table 3: Mechanical Turk Task Results. Asking Questions (AQ) outperforms only answering questions without asking (QA).   
+
+<table><tr><td></td><td colspan="4">vanilla-MemN2N</td><td colspan="4">Cont-MemN2N</td></tr><tr><td></td><td colspan="2">Task4:K.Verification</td><td colspan="2">Task 8:Triple</td><td colspan="2">Task 4:K.Verification</td><td colspan="2">Task 8:Triple</td></tr><tr><td>Train\Test</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td>TrainQA</td><td>0.331</td><td>0.313</td><td>0.133</td><td>0.162</td><td>0.712</td><td>0.703</td><td>0.308</td><td>0.234</td></tr><tr><td>TrainAQ</td><td>0.318</td><td>0.375</td><td>0.072</td><td>0.422</td><td>0.679</td><td>0.774</td><td>0.137</td><td>0.797</td></tr></table>
+
+More importantly, the same main conclusion is observed as before: TrainAQ $^ +$ TestAQ (questions can be asked at both training and test time) performs the best across all the settings. That is, we show that a bot asking questions to humans learns to outperform one that only answers them.
+
+# 7 CONCLUSIONS
+
+In this paper, we explored how an intelligent agent can benefit from interacting with users by asking questions. We developed tasks where interaction via asking questions is desired. We explore both online and offline settings that mimic different real world situations and show that in most cases, teaching a bot to interact with humans facilitates language understanding, and consequently leads to better question answering ability.
+
+# REFERENCES
+
+Mohammad Amin Bassiri. Interactional feedback and the impact of attitude and motivation on noticing l2 form. English Language and Literature Studies, 1(2):61, 2011.   
+Antoine Bordes and Jason Weston. Learning end-to-end goal-oriented dialog. arXiv preprint arXiv:1605.07683, 2016.   
+Antoine Bordes, Nicolas Usunier, Sumit Chopra, and Jason Weston. Large-scale simple question answering with memory networks. arXiv preprint arXiv:1506.02075, 2015.   
+Jesse Dodge, Andreea Gane, Xiang Zhang, Antoine Bordes, Sumit Chopra, Alexander Miller, Arthur Szlam, and Jason Weston. Evaluating prerequisite qualities for learning end-to-end dialog systems. arXiv preprint arXiv:1511.06931, 2015.   
+Richard Higgins, Peter Hartley, and Alan Skelton. The conscientious consumer: Reconsidering the role of assessment feedback in student learning. Studies in higher education, 27(1):53–64, 2002.   
+Andrew S Latham. Learning through feedback. Educational Leadership, 54(8):86–87, 1997.   
+Jiwei Li, Michel Galley, Chris Brockett, Jianfeng Gao, and Bill Dolan. A diversity-promoting objective function for neural conversation models. arXiv preprint arXiv:1510.03055, 2015.   
+Alexander Miller, Adam Fisch, Jesse Dodge, Amir-Hossein Karimi, Antoine Bordes, and Jason Weston. Key-value memory networks for directly reading documents. arXiv preprint arXiv:1606.03126, 2016.   
+Marc’Aurelio Ranzato, Sumit Chopra, Michael Auli, and Wojciech Zaremba. Sequence level training with recurrent neural networks. arXiv preprint arXiv:1511.06732, 2015.   
+Alessandro Sordoni, Michel Galley, Michael Auli, Chris Brockett, Yangfeng Ji, Margaret Mitchell, Jian-Yun Nie, Jianfeng Gao, and Bill Dolan. A neural network approach to context-sensitive generation of conversational responses. arXiv preprint arXiv:1506.06714, 2015.   
+Pei-Hao Su, Milica Gasic, Nikola Mrksic, Lina Rojas-Barahona, Stefan Ultes, David Vandyke, Tsung-Hsien Wen, and Steve Young. Continuously learning neural dialogue management. arXiv preprint arXiv:1606.02689, 2016.   
+Sainbayar Sukhbaatar, Jason Weston, Rob Fergus, et al. End-to-end memory networks. In Advances in neural information processing systems, pp. 2440–2448, 2015.
+
+Oriol Vinyals and Quoc Le. A neural conversational model. arXiv preprint arXiv:1506.05869, 2015.
+
+Sida I Wang, Percy Liang, and Christopher D Manning. Learning language games through interaction. arXiv preprint arXiv:1606.02447, 2016.
+
+Tsung-Hsien Wen, Milica Gasic, Nikola Mrksic, Lina M Rojas-Barahona, Pei-Hao Su, Stefan Ultes, David Vandyke, and Steve Young. A network-based end-to-end trainable task-oriented dialogue system. arXiv preprint arXiv:1604.04562, 2016.
+
+Margaret G Werts, Mark Wolery, Ariane Holcombe, and David L Gast. Instructive feedback: Review of parameters and effects. Journal of Behavioral Education, 5(1):55–75, 1995.
+
+Jason Weston. Dialog-based language learning. arXiv preprint arXiv:1604.06045, 2016.
+
+Jason Weston, Antoine Bordes, Sumit Chopra, Alexander M Rush, Bart van Merrienboer, Armand ¨ Joulin, and Tomas Mikolov. Towards ai-complete question answering: A set of prerequisite toy tasks. arXiv preprint arXiv:1502.05698, 2015.
+
+Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992.
+
+Terry Winograd. Understanding natural language. Cognitive psychology, 3(1):1–191, 1972.
+
+Ludwig Wittgenstein. Philosophical investigations. John Wiley & Sons, 2010.
+
+Wojciech Zaremba and Ilya Sutskever. Reinforcement learning neural turing machines. arXiv preprint arXiv:1505.00521, 362, 2015.
+
+# Appendix
+
+End-to-End Memory Networks The input to an end-to-end memory network model (MemN2N) is the last utterance of the dialogue history $x$ as well as a set of memories (context) $\scriptstyle { C = c _ { 1 } }$ , $c _ { 2 }$ , ..., $c _ { N } .$ ). Memory $C$ encodes both short-term memory, e..g, dialogue histories between the bot and the teacher and long-term memories, e.g., the knowledgebase facts that the bot has access to. Given the input $x$ and $C$ , the goal is to produce an output/label $a$ .
+
+In the first step, the query $x$ is transformed to a vector representation $u _ { 0 }$ by summing up its constituent word embeddings: $u _ { 0 } = A x$ . The input $\mathbf { X }$ is a bag-of-words vector and $A$ is the $d \times V$ word embedding matrix where $d$ denotes the vector dimensionality and $V$ denotes the vocabulary size. Each memory $c _ { i }$ is similarly transformed to vector $m _ { i }$ . The model will read information from the memory by linking input representation $q$ with memory vectors $m _ { i }$ using softmax weights:
+
+$$
+o _ { 1 } = \sum _ { i } p _ { i } ^ { 1 } m _ { i } \qquad p _ { i } ^ { 1 } = \mathsf { s o f t m a x } ( u _ { 0 } ^ { T } m _ { i } )
+$$
+
+The goal is to select memories relevant to the last utterance $x$ , i.e., the memories with large values of $p _ { i } ^ { 1 }$ . The queried memory vector $o _ { 1 }$ is the weighted sum of memory vectors. The queried memory vector $o _ { 1 }$ will be added on top of original input, $u _ { 1 } = o _ { 1 } + u _ { 0 } . \ u$ $u _ { 1 }$ is then used to query the memory vector. Such a process is repeated by querying the memory $_ \mathrm { N }$ times (so called “hops”). $N$ is set to three in all experiments in this paper.
+
+In the end, $u _ { N }$ is input to a softmax function for the final prediction:
+
+$$
+\boldsymbol { a } = \mathsf { s o f t m a x } ( u _ { N } ^ { T } y _ { 1 } , u _ { N } ^ { T } y _ { 2 } , . . . , u _ { N } ^ { T } y _ { L } )
+$$
+
+where $L$ denotes the number of candidate answers and $y$ denotes the representation of the answer. If the answer is a word, $y$ is the corresponding word embedding. If the answer is a sentence, $y$ is the embedding for the sentence achieved in the same way as we obtain embeddings for query $x$ and memory $c$ .
+
+Reward Based Imitation (RBI) and Forward Prediction (FP) RBI and $F P$ are two dialogue learning strategies proposed in (Weston, 2016) by harnessing different types of dialogue signals. RBI handles the case where the reward or the correctness of a bot’s answer is explicitly given (for example, $+ 1$ if the bot’s answer is correct and 0 otherwise). The model is directly trained to predict the correct answers (with label 1) at training time, which can be done using End-to-End Memory Networks (MemN2N) (Sukhbaatar et al., 2015) that map a dialogue input to a prediction.
+
+$F P$ handles the situation where a real-valued reward for a bot’s answer is not available, meaning that there is no $+ 1$ or 0 labels paired with a student’s utterance. However, the teacher will give a response to the bot’s answer, taking the form of a dialogue utterance. More formally, suppose that $x$ denotes the teacher’s question and $C { = } c _ { 1 }$ , $c _ { 2 }$ , ..., $c _ { N }$ denotes the dialogue history. In our $A Q$ settings, the bot will ask a question $a$ regarding the teacher’s question, denoted as $a \in \mathbb { A }$ , where A denotes the student’s question pool. The teacher will provide an utterance in response to the student question $a$ . In $F P$ , the model first maps the teacher’s initial question $x$ and dialogue history $C$ to vector representation $u$ using a memory network with multiple hops. Then the model will perform another hopof attention over all possible student’s questions in A, with an additional part that incorporates the information of which candidate (i.e., $a$ ) was actually selected in the dialogue:
+
+$$
+p _ { \hat { a } } = { \tt s o f t m a x } ( u ^ { T } y _ { \hat { a } } ) \quad o = \sum _ { \hat { a } \in \mathbb { A } } p _ { \hat { a } } ( y _ { \hat { a } } + \beta \cdot { \bf 1 } [ \hat { a } = a ] )
+$$
+
+where $y _ { \hat { a } }$ denotes the vector representation for the student’s question candidate $\hat { a }$ . $\beta$ is a ddimensional vector to signify the actual action $a$ that the student chooses. For tasks where the student only has one way to ask questions (e.g., “what do you mean”), there is no need to perform hops of attention over candidates since the cardinality of $\mathbb { A }$ is just 1. We thus directly assign a probability of 1 to the student’s question, making $o$ the sum of vector representation of $y _ { a }$ and $\beta$ .
+
+$o$ is then combined with $u$ to predict the teacher’s feedback $t$ using a softmax:
+
+$$
+\boldsymbol { u } _ { 1 } = o + \boldsymbol { u } \quad t = \mathrm { s o f t m a x } \big ( \boldsymbol { u } _ { 1 } ^ { T } x _ { r _ { 1 } } , \boldsymbol { u } _ { 1 } ^ { T } x _ { r _ { 2 } } , . . . , \boldsymbol { u } _ { 1 } ^ { T } x _ { r _ { N } } \big )
+$$
+
+where $\boldsymbol { x } _ { r _ { i } }$ denotes the embedding for the $i ^ { t h }$ response.
+
+Dialogue Simulator In this section we further detail the simulator and the datasets we generated in order to realize the various scenarios discussed in Section 3. We focused on the problem of movieQA where we adapted the WikiMovies dataset proposed in Weston et al. (2015). The dataset consists of roughly $1 0 0 \mathrm { k }$ questions with over $7 5 \mathrm { k }$ entities from the open movie dataset (OMDb).
+
+Each dialogue generated by the simulator takes place between a student and a teacher. The simulator samples a random question from the WikiMovies dataset and fetches the set of all KB facts relevant to the chosen question. This question is assumed to be the one the teacher asks its student, and is referred to as the “original” question. The student is first presented with the relevant KB facts followed by the original question. Providing the KB facts to the student allows us to control the exact knowledge the student is given access to while answering the questions. At this point, depending on the task at hand and the student’s ability to answer, the student might choose to directly answer it or ask a “followup” question. The nature of the followup question will depend on the scenario under consideration. If the student answers the question, it gets a response from the teacher about its correctness and the conversation ends. However if the student poses a followup question, the teacher gives an appropriate response, which should give additional information to the student to answer the original question. In order to make things more complicated, the simulator pads the conversation with several unrelated student-teacher question-answer pairs. These question-answer pairs can be viewed as distractions and are used to test the student’s ability to remember the additional knowledge provided by the teacher after it was queried. For each dialogue, the simulator incorporates 5 such pairs (10 sentences). We refer to these pairs as conversational histories.
+
+For the $Q A$ setting (see Section 3), the dialogues generated by the simulator are such that the student never asks a clarification question. Instead, it simply responds to the original question, even if it is wrong. For the dialogs in the $A Q$ setting, the student always asks a clarification question. The nature of the question asked is dependent on the scenario (whether it is Question Clarification, Knowledge Operation, or Knowledge Acquisition) under consideration. In order to simulate the case where the student sometimes choses to directly answer the original question and at other times choses to ask question, we created training datasets, which were a combination of $Q A$ and $A Q$ (called “Mixed”). For all these cases, the student needs to give an answer to the teacher’s original question at the end.
+
+# Instructions given to Turkers
+
+These are the instructions given for the textual feedback Mechanical Turk task (we also constructed a separate task to collect the questions to ask the bot with similar instructions, not described here):
+
+Task 4 (answers to bot’s questions):
+
+Title: Write brief responses to given dialogue exchanges (about $1 5 \mathrm { m i n }$ )
+
+Description: Write a brief response answering a provided question (25 questions per HIT).
+
+Directions:
+
+Each task consists of the following triplets:
+
+1) a question by the teacher   
+2) the correct answer(s) to the question (separated by “OR”), unknown to the student   
+3) a clarifying question asking for feedback from the teacher
+
+Consider the scenario where you are the teacher and have already asked the question, and received the reply from the student. Please compose a brief response replying to the student’s question. The correct answers are provided so that you know whether the student’s question was relevant or not.
+
+For example, given 1) question: “what is a color in the united states flag?”; 2) correct answer: “white OR blue OR red”; 3) student reply: “does this have to do with ‘US Flag has colors red,white,blue‘?”, your response could be something like “that’s right!”; for 3) reply: “does this have to do with ‘United States has population 320 million”, you might say “No, that fact is not relevant” or “Not really”.
+
+Please vary responses and try to minimize spelling mistakes. If the same responses are copied/pasted or similar responses are overused, we’ll reject the HIT. Avoid naming the student or addressing “the class” directly. We will consider bonuses for higher quality responses during review.
+
+Task 8: answers to bot’s questions:
+
+Title: Write brief responses to given dialogue exchanges (about 10 min)
+
+Description: Write a sentence describing the answer to a question (25 questions per HIT).
+
+Directions:
+
+Each task consists of the following triplets:
+
+1) a question by the teacher   
+2) the correct answer(s) to the question (separated by “OR”), unknown to the student   
+3) a question from the student asking the teacher for the answer
+
+Consider the scenario where you are the teacher and have already asked the question, and received the reply from the student. Please compose a brief response replying to the student’s question. The correct answers are provided so that you know which answers to provide.
+
+For example, given 1) question: “what is a color in the united states flag?”; 2) correct answer: “white OR blue OR red”; 3) student reply: “i dont know. what’s the answer ?”, your response could be something like “the color white is in the US flag” or “blue and red both appear in it”.
+
+Please vary responses and try to minimize spelling mistakes, and do not include the capitalized “OR” in your response. If the same responses are copied/pasted or similar responses are overused, we’ll reject the HIT. You don’t need to mention every correct answer in your response. Avoid naming the student or addressing “the class” directly. We will consider bonuses for higher quality responses during review.
+
+# Additional Mechanical Turk Experiments
+
+Here we provide additional experiments to supplement the ones described in Section 6.2. In the main paper, results were shown when training and testing on the collected Mechanical Turk data (around 10,000 episodes of training dialogues for training). As we collected the data in the same settings as Task 4 and 8 of our simulator, we could also consider supplementing training with simulated data as well, of which we have a larger amount (over 100,000 episodes). Note this is only for training, we will still test on the real (Mechanical Turk collected) data. Although the simulated data has less lexical variety as it is built from templates, the larger size might obtain improve results.
+
+Results are given Table 5 when training on the combination of real and simulator data, and testing on real data. This should be compared to training on only the real data (Table 4) and only on the simulator data (Table 6). The best results are obtained from the combination of simulator and real data. The best real data only results (selecting over algorithm and training strategy) on both tasks outperform the best results using simulator data, i.e. using Cont-MemN2N with the Train AQ / TestAQ setting) 0.774 and 0.797 is obtained vs. 0.714 and 0.788 for Tasks 4 and 8 respectively. This is despite there being far fewer examples of real data compared to simulator data. Overall we obtain two main conclusions from this additional experiment: (i) real data is indeed measurably superior to simulated data for training our models; (ii) in all cases (across different algorithms, tasks and data types – be they real data, simulated data or combinations) the bot asking questions (AQ) outperforms it only answering questions and not asking them (QA). The latter reinforces the main result of the paper.
+
+Table 4: Mechanical Turk Task Results, using real data for training and testing.   
+
+<table><tr><td></td><td colspan="4">vanilla-MemN2N</td><td colspan="3">Cont-MemN2N</td></tr><tr><td></td><td colspan="2">Task 4:K.Verification</td><td colspan="2">Task 8: Triple</td><td>Task 4:K.Verification</td><td colspan="2">Task 8:Triple</td></tr><tr><td>Train\Test</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td>TrainQA</td><td>0.331</td><td>0.313</td><td>0.133</td><td>0.162</td><td>0.712 0.703</td><td>0.308</td><td>0.234</td></tr><tr><td>TrainAQ</td><td>0.318</td><td>0.375</td><td>0.072</td><td>0.422</td><td>0.679 0.774</td><td>0.137</td><td>0.797</td></tr></table>
+
+<table><tr><td></td><td colspan="4">vanilla-MemN2N</td><td colspan="4">Cont-MemN2N</td></tr><tr><td></td><td colspan="2">Task 4:K.Verification</td><td colspan="2">Task 8: Triple</td><td colspan="2">Task 4:K.Verification</td><td colspan="2">Task 8: Triple</td></tr><tr><td>Train\Test</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td>TrainQA</td><td>0.356</td><td>0.311</td><td>0.128</td><td>0.174</td><td>0.733</td><td>0.717</td><td>0.368</td><td>0.352</td></tr><tr><td>TrainAQ</td><td>0.340</td><td>0.445</td><td>0.150</td><td>0.487</td><td>0.704</td><td>0.792</td><td>0.251</td><td>0.825</td></tr></table>
+
+Table 5: Results on Mechanical Turk Tasks using a combination of real and simulated data for training, testing on real data.
+
+Table 6: Results on Mechanical Turk Tasks using only simulated data for training, but testing on real data.   
+
+<table><tr><td></td><td colspan="4">vanilla-MemN2N</td><td colspan="3">Cont-MemN2N</td></tr><tr><td></td><td colspan="2">Task4:K.Verification</td><td colspan="2">Task 8: Triple</td><td>Task4:K.Verification</td><td colspan="2">Task 8:Triple</td></tr><tr><td>TrainTest</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td>TrainQA</td><td>0.340</td><td>0.311</td><td>0.120</td><td>0.165</td><td>0.665 0.648</td><td>0.349</td><td>0.342</td></tr><tr><td>TrainAQ</td><td>0.326</td><td>0.390</td><td>0.067</td><td>0.405</td><td>0.642 0.714</td><td>0.197</td><td>0.788</td></tr></table>
+
+Additional Offline Supervised Learning Experiments   
+
+<table><tr><td></td><td colspan="4">Question Clarification</td><td colspan="4">Knowledge Operation</td><td rowspan="3"></td></tr><tr><td></td><td colspan="2">Task1:Q.Paraphrase</td><td colspan="2">Task 2:Q.Verification</td><td colspan="2">Task3:AskForRelevantK.</td><td colspan="2">Task4:K.Verification</td></tr><tr><td>TrainTest</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td>TrainQA</td><td>0.338</td><td>0.284</td><td>0.340</td><td>0.271 0.373</td><td>0.462</td><td>0.344</td><td>0.482</td><td>0.322</td></tr><tr><td>TrainAQ</td><td>0.213</td><td>0.450</td><td>0.225</td><td></td><td>0.187 0.632 0.342</td><td></td><td>0.283</td><td>0.540</td></tr><tr><td>TrainAQ(+FP) TrainMix</td><td>0.288 0.326</td><td>0.464</td><td>0.146</td><td>0.320</td><td>0.631</td><td></td><td>0.311</td><td>0.524</td></tr><tr><td></td><td>0.373</td><td></td><td>0.329</td><td>0.326</td><td>0.442</td><td>0.558</td><td>0.476</td><td>0.491</td></tr><tr><td colspan="9">Knowledge Acquisition</td></tr><tr><td>TrainTest</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td></td><td>Task 5:Q.Entity</td><td></td><td>Task 6: Answer Entity</td><td></td><td>Task 7: Relation Entity</td><td>Task 8: Triple</td><td></td><td></td><td>Task 9:Everything</td></tr><tr><td>TrainQA (vanila)</td><td>&lt;0.01</td><td>0.223</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.109 0.129</td><td>0.201</td><td>0.259</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>TrainAQ (vanila)</td><td>&lt;0.01</td><td>0.660</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.082 0.156</td><td>0.124</td><td>0.664</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>TrainAQ(+FP)</td><td>&lt;0.01</td><td>0.742</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.085 0.188 0.152</td><td>0.064 0.180</td><td>0.702 0.572</td><td>&lt;0.01 &lt;0.01</td><td>&lt;0.01</td></tr><tr><td>Mix (vanila)</td><td>&lt;0.01</td><td>0.630</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.070</td><td></td><td></td><td></td><td>&lt;0.01</td></tr></table>
+
+Table 7: Results for offline settings using memory networks.   
+Table 8: Results for TestModelAQ settings.   
+
+<table><tr><td></td><td>Question Clarification</td><td>Knowledge Acquisition</td></tr><tr><td></td><td>Task 2:Q.Verification</td><td>Task4:K.Verification</td></tr><tr><td></td><td>TestModelAQ</td><td>TestModelAQ</td></tr><tr><td>TrainAQ</td><td>0.382</td><td>0.480</td></tr><tr><td>TrainAQ(+FP)</td><td>0.344</td><td>0.501</td></tr><tr><td>TrainMix</td><td>0.352</td><td>0.469</td></tr></table>

parse/train/rkE8pVcle/rkE8pVcle_content_list.json

@@ -0,0 +1,1864 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING THROUGH DIALOGUE INTERACTIONS BY ASKING QUESTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            758,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jiwei Li, Alexander H. Miller, Sumit Chopra, Marc’Aurelio Ranzato, Jason Weston   \nFacebook AI Research,   \nNew York, USA   \n{jiwel,ahm,spchopra,ranzato,jase}@fb.com ",
+        "bbox": [
+            183,
+            172,
+            764,
+            229
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            266,
+            544,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A good dialogue agent should have the ability to interact with users by both responding to questions and by asking questions, and importantly to learn from both types of interaction. In this work, we explore this direction by designing a simulator and a set of synthetic tasks in the movie domain that allow such interactions between a learner and a teacher. We investigate how a learner can benefit from asking questions in both offline and online reinforcement learning settings, and demonstrate that the learner improves when asking questions. Finally, real experiments with Mechanical Turk validate the approach. Our work represents a first step in developing such end-to-end learned interactive dialogue agents. ",
+        "bbox": [
+            233,
+            300,
+            764,
+            426
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            336,
+            476
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When a student is asked a question by a teacher, but is not confident about the answer, they may ask for clarification or hints. A good conversational agent (a learner/bot/student) should have this ability to interact with a dialogue partner (the teacher/user). However, recent efforts have mostly focused on learning through fixed answers provided in the training set, rather than through interactions. In that case, when a learner encounters a confusing situation such as an unknown surface form (phrase or structure), a semantically complicated sentence or an unknown word, the agent will either make a (usually poor) guess or will redirect the user to other resources (e.g., a search engine, as in Siri). Humans, in contrast, can adapt to many situations by asking questions. ",
+        "bbox": [
+            174,
+            494,
+            825,
+            606
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We identify three categories of mistakes a learner can make during dialogue1: (1) the learner has problems understanding the surface form of the text of the dialogue partner, e.g., the phrasing of a question; (2) the learner has a problem with reasoning, e.g. they fail to retrieve and connect the relevant knowledge to the question at hand; (3) the learner lacks the knowledge necessary to answer the question in the first place – that is, the knowledge sources the student has access to do not contain the needed information. ",
+        "bbox": [
+            174,
+            613,
+            825,
+            695
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "All the situations above can be potentially addressed through interaction with the dialogue partner. Such interactions can be used to learn to perform better in future dialogues. If a human student has problems understanding a teacher’s question, they might ask the teacher to clarify the question. If the student doesn’t know where to start, they might ask the teacher to point out which known facts are most relevant. If the student doesn’t know the information needed at all, they might ask the teacher to tell them the knowledge they’re missing, writing it down for future use. ",
+        "bbox": [
+            174,
+            704,
+            825,
+            787
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we try to bridge the gap between how a human and an end-to-end machine learning dialogue agent deal with these situations: our student has to learn how to learn. We hence design a simulator and a set of synthetic tasks in the movie question answering domain that allow a bot to interact with a teacher to address the issues described above. Using this framework, we explore how a bot can benefit from interaction by asking questions in both offline supervised settings and online reinforcement learning settings, as well as how to choose when to ask questions in the latter setting. In both cases, we find that the learning system improves through interacting with users. ",
+        "bbox": [
+            174,
+            794,
+            823,
+            891
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Finally, we validate our approach on real data where the teachers are humans using Amazon Mechanical Turk, and observe similar results. ",
+        "bbox": [
+            173,
+            103,
+            821,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            152,
+            344,
+            169
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Learning language through interaction and feedback can be traced back to the 1950s, when Wittgenstein argued that the meaning of words is best understood from their use within given language games (Wittgenstein, 2010). The direction of interactive language learning through language games has been explored in the early seminal work of Winograd (Winograd, 1972), and in the recent SHRDLURN system (Wang et al., 2016). In a broader context, the usefulness of feedback and interactions has been validated in the setting of multiple language learning, such as second language learning (Bassiri, 2011) and learning by students (Higgins et al., 2002; Latham, 1997; Werts et al., 1995). ",
+        "bbox": [
+            174,
+            185,
+            825,
+            284
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the context of dialogue, with the recent popularity of deep learning models, many neural dialogue systems have been proposed. These include the chit-chat type end-to-end dialogue systems (Vinyals & Le, 2015; Li et al., 2015; Sordoni et al., 2015), which directly generate a response given the previous history of user utterance. It also include a collection of goal-oriented dialogue systems (Wen et al., 2016; Su et al., 2016; Bordes & Weston, 2016), which complete a certain task such as booking a ticket or making a reservation at a restaurant. Another line of research focuses on supervised learning for question answering from dialogues (Dodge et al., 2015; Weston, 2016), using either a given database of knowledge (Bordes et al., 2015; Miller et al., 2016) or short stories (Weston et al., 2015). As far as we know, current dialogue systems mostly focus on learning through fixed supervised signals rather than interacting with users. ",
+        "bbox": [
+            174,
+            290,
+            825,
+            429
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our work is closely related to the recent work of Weston (2016), which explores the problem of learning through conducting conversations, where supervision is given naturally in the response during the conversation. Their work introduced multiple learning schemes from dialogue utterances. In particular the authors discussed Imitation Learning, where the agent tries to learn by imitating the dialogue interactions between a teacher and an expert student; Reward-Based Imitation Learning, which only learns by imitating the dialogue interactions which have have correct answers; and Forward Prediction, which learns by predicting the teacher’s feedback to the student’s response. Despite the fact that Forward Prediction does not uses human-labeled rewards, the authors show that it yields promising results. However, their work did not fully explore the ability of an agent to learn via questioning and interaction. Our work can be viewed as a natural extension of theirs. ",
+        "bbox": [
+            174,
+            435,
+            825,
+            574
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 THE TASKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            595,
+            302,
+            612
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section we describe the dialogue tasks we designed2. They are tailored for the three different situations described in Section 1 that motivate the bot to ask questions: (1) Question Clarification, in which the bot has problems understanding its dialogue partner’s text; (2) Knowledge Operation, in which the bot needs to ask for help to perform reasoning steps over an existing knowledge base; and (3) Knowledge Acquisition, in which the bot’s knowledge is incomplete and needs to be filled. ",
+        "bbox": [
+            174,
+            627,
+            825,
+            698
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "For our experiments we adapt the WikiMovies dataset (Weston et al., 2015), which consists of roughly $1 0 0 \\mathrm { k }$ questions over 75k entities based on questions with answers in the open movie dataset (OMDb). The training/dev/test sets respectively contain 181638 / 9702 / 9698 examples. The accuracy metric corresponds to the percentage of times the student gives correct answers to the teacher’s questions. ",
+        "bbox": [
+            174,
+            705,
+            825,
+            775
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Each dialogue takes place between a teacher and a bot. In this section we describe how we generate tasks using a simulator. Section 4.2 discusses how we test similar setups with real data using Mechanical Turk. ",
+        "bbox": [
+            174,
+            781,
+            823,
+            823
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The bot is first presented with facts from the OMDb KB. This allows us to control the exact knowledge the bot has access to. Then, we include several teacher-bot question-answer pairs unrelated to the question the bot needs to answer, which we call conversation histories3. In order to explore the benefits of asking clarification questions during a conversation, for each of the three scenarios, our simulator generated data for two different settings, namely, Question-Answering (denoted by QA), and Asking-Question (denoted by AQ). For both $Q A$ and $A Q$ , the bot needs to give an answer to the teacher’s original question at the end. The details of the simulator can be found in the appendix. ",
+        "bbox": [
+            176,
+            830,
+            823,
+            872
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 QUESTION CLARIFICATION. ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            178,
+            411,
+            193
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this setting, the bot does not understand the teacher’s question. We focus on a special situation where the bot does not understand the teacher because of typo/spelling mistakes, as shown in Figure 1. We intentionally misspell some words in the questions such as replacing the word “movie” with “movvie” or “star” with “sttar”.4 To make sure that the bot will have problems understanding the question, we guarantee that the bot has never encountered the misspellings before—the misspellingintroducing mechanisms in the training, dev and test sets are different, so the same word will be misspelled in different ways in different sets. We present two $A Q$ tasks: (i) Question Paraphrase where the student asks the teacher to use a paraphrase that does not contain spelling mistakes to clarify the question by asking “what do you mean?”; and (ii) Question Verification where the student asks the teacher whether the original typo-bearing question corresponds to another question without the spelling mistakes (e.g., “Do you mean which film did Tom Hanks appear in?”). The teacher will give feedback by giving a paraphrase of the original question without spelling mistakes (e.g., “I mean which film did Tom Hanks appear in”) in Question Paraphrase or positive/negative feedback in Question Verification. Next the student will give an answer and the teacher will give positive/negative feedback depending on whether the student’s answer is correct. Positive and negative feedback are variants of “No, that’s incorrect” or “Yes, that’s right”5. In these tasks, the bot has access to all relevant entries in the KB. ",
+        "bbox": [
+            174,
+            205,
+            825,
+            440
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 KNOWLEDGE OPERATION",
+        "text_level": 1,
+        "bbox": [
+            176,
+            459,
+            393,
+            473
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The bot has access to all the relevant knowledge (facts) but lacks the ability to perform necessary reasoning operations over them; see Figure 2. We focus on a special case where the bot will try to understand what are the relevant facts. We explore two settings: Ask For Relevant Knowledge (Task 3) where the bot directly asks the teacher to point out the relevant KB fact and Knowledge Verification (Task 4) where the bot asks whether the teacher’s question is relevant to one particular KB fact. The teacher will point out the relevant KB fact in the Ask For Relevant Knowledge setting or give a positive or negative response in the Knowledge Verification setting. Then the bot will give an answer to the teacher’s original question and the teacher will give feedback on the answer. ",
+        "bbox": [
+            174,
+            486,
+            825,
+            597
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.3 KNOWLEDGE ACQUISITION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            616,
+            405,
+            630
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For the tasks in this subsection, the bot has an incomplete KB and there are entities important to the dialogue missing from it, see Figure 3. For example, given the question “Which movie did Tom Hanks star in?”, the missing part could either be the entity that the teacher is asking about (question entity for short, which is Tom Hanks in this example), the relation entity (starred actors), the answer to the question (Forrest Gump), or the combination of the three. In all cases, the bot has little chance of giving the correct answer due to the missing knowledge. It needs to ask the teacher the answer to acquire the missing knowledge. The teacher will give the answer and then move on to other questions (captured in the conversational history). They later will come back to reask the question. At this point, the bot needs to give an answer since the entity is not new any more. ",
+        "bbox": [
+            173,
+            642,
+            825,
+            767
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Though the correct answer has effectively been included in the earlier part of the dialogue as the answer to the bot’s question, as we will show later, many of the tasks are not as trivial as they look when the teacher reasks the question. This is because the bot’s model needs to memorize the missing entity and then construct the links between the missing entities and known ones. This is akin to the real world case where a student might make the same mistake again and again even though each time the teacher corrects them if their answer is wrong. We now detail each task in turn. ",
+        "bbox": [
+            174,
+            775,
+            825,
+            858
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/bb347c16afa7fc669133cfc1383ea96a78cc81be4a96329a3e1d6fa423fdc3d3.jpg",
+        "image_caption": [
+            "Figure 1: Sample dialogues for the two Question Clarification tasks (rows) using both the traditional $Q A$ setting (left column) and $A Q$ setting (right column). In each case the same example is given for simplicity. Black text prefixed by “kb:” denotes KB knowledge that the student has access to. Blue text is spoken by the teacher, denoted by T. $( + )$ denotes a reward of 1 (and 0 otherwise) that the teacher assigns to the bot. Red text denotes responses or questions posed by the bot, denoted by S. Brown denotes typos deliberately introduced by the authors. For the Question Verification setting, the student can either ask a correct (pertinent) question (as in this example) or an incorrect (irrelevant) one. The teacher will give positive or negative feedback based on the correctness of the student’s question. In our offline superised learning experiments, the probability of asking pertinent questions and correctly answering the original question from the teacher is set to 0.5. Finally, T/S denotes 5 pairs of questions and answers that are irrelevant to the rest of the conversation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            103,
+            475,
+            421
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Task 1: Question Paraphrase (AQ):   \nkb: Larry Crowne directed by Tom Hanks   \nkb: Forrest Gump starred actors Tom Hanks, Robin Wright, Gary Sinise   \nkb: Forrest Gump directed by Robert Zemeckis T/S : Conversation History.   \nT : Which movvie did Tom Hanks sttar in ? S : What do you mean ?   \nT : I mean which film did Tom Hanks appear in. T : Which movvie did Tom Hanks sttar in ? S : Forrest Gump   \nT : That’s correct. $( + )$ ",
+        "bbox": [
+            500,
+            103,
+            802,
+            257
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/1075a725d670bd1a1dafb1e56da35b31f24ecbe7344bf51254e61ca0005ecb92.jpg",
+        "image_caption": [
+            "Figure 2: Sample dialogues for Knowledge Operation tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            616,
+            808,
+            804
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Missing Question Entity: The entity that the teacher is asking about is missing from the knowledge base. All KB facts containing the question entity will be hidden from the bot. In the example for Task 5 in Figure 3, since the teacher’s question contains the entity Tom Hanks, the KB facts that contain Tom Hanks are hidden from the bot. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            922
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Task 5: Missing Question Entity (QA)   \nkb: Larry Crowne directed by Tom Hanks   \nkb: Forrest Gump starred actors Tom Hanks kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T/S : Conversation History.   \nT : Which movie did Tom Hanks star in ?   \nS : Forrest Gump   \nT : That’s correct. $( + )$ Task 5: Missing Question Entity (AQ)   \nkb: Larry Crowne directed by Tom Hanks   \nkb: Forrest Gump starred actors Tom Hanks, kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T : Which movie did Tom Hanks star in ?   \nS : I don’t know. What’s the answer?   \nT : The answer is Forrest Gump.   \nT/S : Conversation History.   \nT : Which movie did Tom Hanks star in ?   \nS : Forrest Gump   \nT : That’s correct. $( + )$ Task 6: Missing Answer Entity (AQ)   \nkb: Larry Crowne directed by Tom Hanks   \nkb: Forrest Gump starred actors Tom Hanks kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T : Which movie did Tom Hanks star in ?   \nS : I don’t know. What’s the answer?   \nT : The answer is Forrest Gump.   \nT/S : Conversation History.   \nT : Which movie did Tom Hanks star in ?   \nS : Forrest Gump   \nT : That’s correct. $( + )$ Task 7: Missing Relation Entity (AQ)   \nkb: Larry Crowne directed by Tom Hanks   \nkb: Forrest Gump starred actors Tom Hanks, kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T : Which movie did Tom Hanks star in ?   \nS : I don’t know. What’s the answer?   \nT : The answer is Forrest Gump.   \nT/S : Conversation History.   \nT : Which movie did Tom Hanks star in ?   \nS : Forrest Gump   \nT : That’s correct. $( + )$ ",
+        "bbox": [
+            192,
+            164,
+            486,
+            279
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            514,
+            164,
+            803,
+            315
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            194,
+            330,
+            485,
+            481
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            514,
+            330,
+            803,
+            481
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Task 8: Missing Triple (AQ) ",
+        "text_level": 1,
+        "bbox": [
+            200,
+            501,
+            377,
+            513
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "kb: Larry Crowne directed by Tom Hanks   \nkb: Forrest Gump starred actors Tom Hanks kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T : Which movie did Tom Hanks star in ?   \nS : I don’t know. What’s the answer?   \nT : The answer is Forrest Gump.   \nT/S : Conversation History.   \nT : Which movie did Tom Hanks star in ?   \nS : Forrest Gump   \nT : That’s correct. (+) Task 9: Missing Everything (AQ)   \nkb: Larry Crowne directed by Tom Hanks   \nkb: Forrest Gump starred actors Tom Hanks, kb: Forrest Gump starred actors Sally Field kb: Forrest Gump directed by Robert Zemeckis T : Which movie did Tom Hanks star in ?   \nS : I don’t know. What’s the answer?   \nT : The answer is Forrest Gump.   \nT/S : Conversation History.   \nT : Which movie did Tom Hanks star in ?   \nS : Forrest Gump   \nT : That’s correct. (+) ",
+        "bbox": [
+            196,
+            513,
+            485,
+            652
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            514,
+            501,
+            803,
+            652
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Missing Answer Entity: The answer entity to the question is unknown to the bot. All KB facts that contain the answer entity will be hidden. Hence, in Task $6$ of Figure 3, all KB facts containing the answer entity Forrest Gump will be hidden from the bot. ",
+        "bbox": [
+            176,
+            747,
+            825,
+            789
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Missing Relation Entity: The relation type is unknown to the bot. In Task 7 of Figure 3, all KB facts that express the relation starred actors are hidden from the bot. ",
+        "bbox": [
+            174,
+            797,
+            821,
+            825
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Missing Triples: The triple that expresses the relation between the question entity and the answer entity is hidden from the bot. In Task 8 of Figure 3, the triple “Forrest Gump (question entity) starred actors Tom Hanks (answer entity)” will be hidden. ",
+        "bbox": [
+            174,
+            833,
+            825,
+            875
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Missing Everything: The question entity, the relation entity, the answer entity are all missing from the KB. All KB facts in Task 9 of Figure 3 will be removed since they either contain the relation entity (i.e., starred actors), the question entity (i.e., Forrest Gump) or the answer entity Tom Hanks. ",
+        "bbox": [
+            174,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 TRAIN/TEST REGIME ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            385,
+            118
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We now discuss in detail the regimes we used to train and test our models, which are divided between evaluation within our simulator and using real data collected via Mechanical Turk. ",
+        "bbox": [
+            173,
+            133,
+            823,
+            161
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 SIMULATOR ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            179,
+            299,
+            194
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Using our simulator, our objective was twofold. We first wanted to validate the usefulness of asking questions in all the settings described in Section 3. Second, we wanted to assess the ability of our student bot to learn when to ask questions. In order to accomplish these two objectives we explored training our models with our simulator using two methodologies, namely, Offline Supervised Learning and Online Reinforcement Learning. ",
+        "bbox": [
+            174,
+            205,
+            825,
+            276
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1.1 OFFLINE SUPERVISED LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            291,
+            460,
+            305
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The motivation behind training our student models in an offline supervised setting was primarily to test the usefulness of the ability to ask questions. The dialogues are generated as described in the previous section, and the bot’s role is generated with a fixed policy. We chose a policy where answers to the teacher’s questions are correct answers $50 \\%$ of the time, and incorrect otherwise, to add a degree of realism. Similarly, in tasks where questions can be irrelevant they are only asked correctly $50 \\%$ of the time.6 ",
+        "bbox": [
+            174,
+            315,
+            825,
+            398
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The offline setting explores different combinations of training and testing scenarios, which mimic different situations in the real world. The aim is to understand when and how observing interactions between two agents can help the bot improve its performance for different tasks. As a result we construct training and test sets in three ways across all tasks, resulting in 9 different scenarios per task, each of which correspond to a real world scenario. ",
+        "bbox": [
+            174,
+            406,
+            825,
+            476
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The three training sets we generated are referred to as TrainQA, TrainAQ, and TrainMix. TrainQA follows the QA setting discussed in the previous section: the bot never asks questions and only tries to immediately answer. TrainAQ follows the AQ setting: the student, before answering, first always asks a question in response to the teacher’s original question. TrainMix is a combination of the two where $5 0 \\%$ of time the student asks a question and $5 0 \\%$ does not. ",
+        "bbox": [
+            174,
+            483,
+            825,
+            553
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The three test sets we generated are referred to as TestQA, TestAQ, and TestModelAQ. TestQA and TestAQ are generated similarly to TrainQA and TrainAQ, but using a perfect fixed policy (rather than $50 \\%$ correct) for evaluation purposes. In the TestModelAQ setting the model has to get the form of the question correct as well. In the Question Verification and Knowledge Verification tasks there are many possible ways of forming the question and some of them are correct – the model has to choose the right question to ask. E.g. it should ask “Does it have something to do with the fact that Larry Crowne directed by Tom Hanks?”rather than “Does it have something to do with the fact that Forrest Gump directed by Robert Zemeckis?” when the latter is irrelevant (the candidate list of questions is generated from the known knowledge base entries with respect to that question). The policy is trained using either the TrainAQ or TrainMix set, depending on the training scenario. The teacher will reply to the question, giving positive feedback if the student’s question is correct and no response and negative feedback otherwise. The student will then give the final answer. The difference between TestModelAQ and TestAQ only exists in the Question Verification and Knowledge Verification tasks; in other tasks there is only one way to ask the question and TestModelAQ and TestAQ are identical. ",
+        "bbox": [
+            174,
+            560,
+            825,
+            753
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To summarize, for every task listed in Section 3 we train one model for each of the three training sets (TrainQA, TrainAQ, TrainMix) and test each of these models on the three test sets (TestQA, TestAQ, and TestModelAQ), resulting in 9 combinations. For the purpose of notation the train/test combination is denoted by “TrainSetting+TestSetting”. For example, TrainA $Q +$ TestQA denotes a model which is trained using the TrainAQ dataset and tested on TestQA dataset. Each combination has a real world interpretation. For instance, $T r a i n A Q + T e s t Q A$ would refer to a scenario where a student can ask the teacher questions during learning but cannot to do so while taking an exam. Similarly, $T r a i n Q A + T e s t Q A$ describes a stoic teacher that never answers a student’s question at either learning or examination time. The setting $T r a i n Q A + T e s t A Q$ corresponds to the case where a lazy student never asks question at learning time but gets anxious during the examination and always asks a question. ",
+        "bbox": [
+            174,
+            761,
+            825,
+            886
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1.2 ONLINE REINFORCEMENT LEARNING (RL) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            146,
+            529,
+            160
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We also explored scenarios where the student learns the ability to decide when to ask a question. In other words, the student learns how to learn. ",
+        "bbox": [
+            178,
+            170,
+            823,
+            199
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Although it is in the interest of the student to ask questions at every step of the conversation, since the response to its question will contain extra information, we don’t want our model to learn this behavior. Each time a human student asks a question, there’s a cost associated with that action. This cost is a reflection of the patience of the teacher, or more generally of the users interacting with the bot in the wild: users won’t find the bot engaging if it always asks clarification questions. The student should thus be judicious about asking questions and learn when and what to ask. For instance, if the student is confident about the answer, there is no need for it to ask. Or, if the teacher’s question is so hard that clarification is unlikely to help enough to get the answer right, then it should also refrain from asking. ",
+        "bbox": [
+            173,
+            205,
+            825,
+            330
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now discuss how we model this problem under the Reinforcement Learning framework. The bot is presented with KB facts (some facts might be missing depending on the task) and a question. It needs to decide whether to ask a question or not at this point. The decision whether to ask is made by a binary policy $P _ { R L Q u e s t i o n }$ . If the student chooses to ask a question, it will be penalized by $c o s t _ { A Q }$ . We explored different values of $c o s t _ { A Q }$ ranging from $[ 0 , 2 ]$ , which we consider as modeling the patience of the teacher. The goal of this setting is to find the best policy for asking/notasking questions which would lead to the highest cumulative reward. The teacher will appropriately reply if the student asks a question. The student will eventually give an answer to the teacher’s initial question at the end using the policy $P _ { R L A n s w e r }$ , regardless of whether it had asked a question. The student will get a reward of $+ 1$ if its final answer is correct and $- 1$ otherwise. Note that the student can ask at most one question and that the type of question is always specified by the task under consideration. The final reward the student gets is the cumulative reward over the current dialogue episode. In particular the reward structure we propose is the following: ",
+        "bbox": [
+            173,
+            338,
+            825,
+            518
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Final Answer Correct Final Answer Incorrect ",
+        "bbox": [
+            184,
+            542,
+            323,
+            568
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/a4ef10c631d3712dd371538ae31b74be63611c8b15cc2e533d2e519389c285c8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Asking Question</td><td>Not asking Question</td></tr><tr><td>1-cost AQ</td><td>1</td></tr><tr><td>-1-cost AQ</td><td>-1</td></tr></table>",
+        "bbox": [
+            338,
+            529,
+            581,
+            569
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For each of the tasks described in Section 3, we consider three different RL scenarios. ",
+        "bbox": [
+            174,
+            607,
+            735,
+            621
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Good-Student: The student will be presented with all relevant KB facts. There are no misspellings or unknown words in the teacher’s question. This represents a knowledgable student in the real world that knows as much as it needs to know (e.g., a large knowledge base, large vocabulary). This setting is identical across all missing entity tasks (5 - 9). ",
+        "bbox": [
+            174,
+            622,
+            825,
+            676
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Poor-Student: The KB facts or the questions presented to the student are flawed depending on each task. For example, for the Question Clarification tasks, the student does not understand the question due to spelling mistakes. For the Missing Question Entity task the entity that the teacher asks about is unknown by the student and all facts containing the entity will be hidden from the student. This setting is similar to a student that is underprepared for the tasks. ",
+        "bbox": [
+            174,
+            676,
+            823,
+            746
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Medium-Student: The combination of the previous two settings where for $5 0 \\%$ of the questions, the student has access to the full KB and there are no new words or phrases or entities in the question, and $5 0 \\%$ of the time the question and KB are taken from the Poor-Student setting. ",
+        "bbox": [
+            174,
+            746,
+            825,
+            787
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 MECHANICAL TURK DATA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            792,
+            400,
+            808
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, to validate our approach beyond our simulator by using real language, we collected data via Amazon Mechanical Turk. Due to the cost of data collection, we focused on real language versions of Tasks 4 (Knowledge Verification) and 8 (Missing Triple), see Secs. 3.2 and 3.3 for the simulator versions. That is, we collect dialogues and use them in an offline supervised learning setup similar to Section 4.1.1. This setup allows easily reproducible experiments. ",
+        "bbox": [
+            174,
+            819,
+            823,
+            888
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For Mechanical Turk Task 4, the bot is asked a question by a human teacher, but before answering can ask the human if the question is related to one of the facts it knows about from its memory. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/672214b4a8794743c3342e4875ce41e848b13b459af37b01e3177bc323ed213e.jpg",
+        "image_caption": [
+            "Reward:\t1-CostAQ ",
+            "Figure 4: An illustration of the poor-student setting for RL Task 1 (Question Paraphrase). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            326,
+            127,
+            678,
+            247
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "It is then required to answer the original question, after some additional dialog turns relating to other question/answer pairs (called “conversational history”, as before). For Task 8, the bot is asked a question by a human but lacks the triple in its memory that would be needed to answer it. It is allowed to ask for the missing information, the human responds to the question in free-form language. The bot is then required to answer the original question, again after some “conversational history” has transpired. ",
+        "bbox": [
+            174,
+            311,
+            825,
+            395
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We collect around 10,000 episodes (dialogues) for training, 1000 for validation, and 2500 for testing for each of the two tasks. In each case, we give instructions to the Turkers that still follow the original form of the task, but make the tasks contain realistic language written by humans. The instructions given to the Turkers are given in the appendix. ",
+        "bbox": [
+            174,
+            402,
+            825,
+            458
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For both tasks, while the human turkers replace the simulator that the bot was previously conversing with, the bot’s dialogue actions (capabilities) are essentially unchanged from before. That is, when answering questions, now the bot is required to answer a human’s questions rather than templated questions from the simulator. When the bot is asking questions, the bot still asks in the same form as before, e.g. questions like “Does it have something to do with X” for Task 4 or “I don’t know. What’s the answer?” for Task 8. However, now its questions are answered by a human. In both cases (asking and answering) the human data is richer with potentially more complex language and lexical variability. Examples of the collected dialogues are given in Figure 5. ",
+        "bbox": [
+            173,
+            465,
+            825,
+            577
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/2b392e8dccdaadfa7c2f1264720a5befd7b2eef240ac496cb5f97d66faf453c5.jpg",
+        "image_caption": [
+            "Figure 5: Sample dialogues for Mechanical Turk versions of Tasks 4 and 8. Compared to the original tasks (see Figs 2 and 3) the teacher’s questions, and the teacher responses to the student’s questions, are written by humans and are more complex and contain more variety. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            225,
+            593,
+            772,
+            747
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 MODELS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            833,
+            279,
+            849
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For both offline supervised and online RL settings, we use the End-to-End Memory Network model (MemN2N) (Sukhbaatar et al., 2015) as a backbone. The model takes as input the last utterance of the dialogue history (the question from the teacher) as well as a set of memory contexts including short-term memories (the dialogue history between the bot and the teacher) and long-term memories (the knowledge base facts that the bot has access to), and outputs a label. We refer readers to the Appendix for more details about MemN2N. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Offline Supervised Settings: The first learning strategy we adopt is the reward-based imitation strategy (denoted vanilla-MemN2N) described in (Weston, 2016), where at training time, the model maximizes the log likelihood probability of the correct answers the student gave (examples with incorrect final answers are discarded). Candidate answers are words that appear in the memories, which means the bot can only predict the entities that it has seen or known before. ",
+        "bbox": [
+            174,
+            138,
+            823,
+            208
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We also use a variation of MemN2N called “context MemN2N” (Cont-MemN2N for short) where we replace each word’s embedding with the average of its embedding (random for unseen words) and the embeddings of the other words that appear around it. We use both the preceeding and following words as context and the number of context words is a hyperparameter selected on the dev set. ",
+        "bbox": [
+            174,
+            215,
+            825,
+            272
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "An issue with both vanilla-MemN2N and Cont-MemN2N is that the model only makes use of the bot’s answers as signals and ignores the teacher’s feedback. We thus propose to use a model that jointly predicts the bot’s answers and the teacher’s feedback (denoted as TrainQA $\\left( + F P \\right) )$ . The bot’s answers are predicted using a vanilla-MemN2N and the teacher’s feedback is predicted using the Forward Prediction (FP) model as described in (Weston, 2016). We refer the readers to the Appendix for the FP model details. At training time, the models learn to jointly predict the teacher’s feedback and the answers with positive reward. At test time, the model will only predict the bot’s answer. ",
+        "bbox": [
+            174,
+            277,
+            825,
+            376
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For the TestModelAQ setting described in Section 4, the model needs to decide the question to ask. Again, we use vanilla-MemN2N that takes as input the question and contexts, and outputs the question the bot will ask. ",
+        "bbox": [
+            174,
+            382,
+            825,
+            424
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Online RL Settings: A binary vanilla-MemN2N (denoted as $P _ { R L } ( Q u e s t i o n ) )$ is used to decide whether the bot should or should not ask a question, with the teacher replying if the bot does ask something. A second MemN2N is then used to decide the bot’s answer, denoted as $P _ { R L } ( A n s w e r )$ . $P _ { R L } ( A n s w e r )$ for $Q A$ and $A Q$ are two separate models, which means the bot will use different models for final-answer prediction depending on whether it chooses to ask a question or not.7 ",
+        "bbox": [
+            174,
+            431,
+            825,
+            502
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We use the REINFORCE algorithm (Williams, 1992) to update $P _ { R L } ( Q u e s t i o n )$ and $P _ { R L } ( A n s w e r )$ . For each dialogue, the bot takes two sequential actions $( a _ { 1 } , a _ { 2 } )$ : to ask or not to ask a question (denoted as $a _ { 1 }$ ); and guessing the final answer (denoted as $a _ { 2 }$ ). Let $r ( a _ { 1 } , a _ { 2 } )$ denote the cumulative reward for the dialogue episode, computed using Table 1. The gradient to update the policy is given by: ",
+        "bbox": [
+            174,
+            508,
+            825,
+            579
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e949cd5536d2f8396707ab808237b0343de330c8833910b7d56fc635b8670e2.jpg",
+        "text": "$$\n\\begin{array} { r l } & { p ( a _ { 1 } , a _ { 2 } ) = P _ { R L } ( Q u e s t i o n ) ( a _ { 1 } ) \\cdot P _ { R L } ( a n s w e r ) ( a _ { 2 } ) } \\\\ & { \\nabla J ( \\theta ) \\approx \\nabla \\log p ( a _ { 1 } , a _ { 2 } ) [ r ( a _ { 1 } , a _ { 2 } ) - b ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            588,
+            681,
+            628
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "where $b$ is the baseline value, which is estimated using another MemN2N model that takes as input the query $x$ and memory $C$ , and outputs a scalar $b$ denoting the estimation of the future reward. The baseline model is trained by minimizing the mean squared loss between the estimated reward $b$ and actual cumulative reward $r$ , $| | \\boldsymbol { r } - \\boldsymbol { b } | | ^ { 2 }$ . We refer the readers to (Ranzato et al., 2015; Zaremba & Sutskever, 2015) for more details. The baseline estimator model is independent from the policy models and the error is not backpropagated back to them. ",
+        "bbox": [
+            173,
+            637,
+            825,
+            722
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In practice, we find the following training strategy yields better results: first train only $P _ { R L } ( a n s w e r )$ , updating gradients only for the policy that predicts the final answer. After the bot’s final-answer policy is sufficiently learned, train both policies in parallel8. This has a real-world analogy where the bot first learns the basics of the task, and then learns to improve its performance via a question-asking policy tailored to the user’s patience (represented by $c o s t _ { A Q }$ ) and its own ability to asnwer questions. ",
+        "bbox": [
+            174,
+            728,
+            825,
+            813
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/4a52db1948daea5d350ec98d22eae0198576537371f6233ecb8b6d4a1ad69f5b.jpg",
+        "table_caption": [
+            "Table 2: Results for Cont-MemN2N on different tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"5\">Question Clarification</td><td colspan=\"5\">Knowledge Operation</td></tr><tr><td></td><td colspan=\"2\">Task 1:Q.Paraphrase TestAQ</td><td colspan=\"2\">Task 2:Q.Verification</td><td colspan=\"2\"></td><td colspan=\"2\">Task 3:Ask For Relevant K.</td><td colspan=\"2\">Task 4:K.Verification TestQA</td></tr><tr><td>Train\\Test</td><td colspan=\"2\">TestQA</td><td colspan=\"2\">TestQA</td><td colspan=\"2\">TestAQ TestQA</td><td colspan=\"2\">TestAQ</td><td colspan=\"2\">TestAQ</td></tr><tr><td>TrainQA (Context)</td><td>0.754</td><td>0.726</td><td>0.742</td><td>0.684</td><td></td><td>0.883 0.716</td><td>0.947</td><td>0.888</td><td>0.959</td><td></td></tr><tr><td>TrainAQ(Context)</td><td>0.640</td><td>0.889</td><td colspan=\"2\">0.643</td><td colspan=\"2\">0.807 0.789</td><td colspan=\"2\">0.985</td><td>0.852 0.875</td><td>0.987</td></tr><tr><td>TrainMix (Context)</td><td>0.751</td><td>0.846</td><td colspan=\"2\">0.740</td><td colspan=\"2\">0.870</td><td colspan=\"2\">0.985</td><td>0.985</td><td></td></tr><tr><td colspan=\"10\">Knowledge Acquisition TestAQ TestAQ</td></tr><tr><td>TrainTest</td><td>TestQA</td><td>TestAQ</td><td colspan=\"2\">TestQA</td><td colspan=\"2\">TestQA</td><td colspan=\"2\">TestQA</td><td>TestAQ</td><td>TestQA TestAQ</td></tr><tr><td></td><td>Task 5:Q.Entity</td><td></td><td colspan=\"2\">Task 6:Answer Entity</td><td colspan=\"2\">Task7:Relation Entity</td><td colspan=\"2\">Task 8:Triple</td><td></td><td>Task 9:Everything</td></tr><tr><td>TrainQA (Context)</td><td>&lt;0.01</td><td>0.224</td><td colspan=\"2\">&lt;0.01</td><td colspan=\"2\">0.241</td><td colspan=\"2\">0.339</td><td>0.251 &lt;0.01</td><td>0.058</td></tr><tr><td>TrainAQ(Context)</td><td>&lt;0.01</td><td>0.639</td><td colspan=\"2\">&lt;0.01</td><td colspan=\"2\">0.143</td><td colspan=\"2\">0.154</td><td>0.884 &lt;0.01</td><td>0.908</td></tr><tr><td>TrainMix (Context)</td><td>&lt;0.01</td><td>0.632</td><td colspan=\"2\">&lt;0.01</td><td colspan=\"2\">0.216</td><td colspan=\"2\">0.298</td><td>0.886 &lt;0.01</td><td>0.903</td></tr></table>",
+        "bbox": [
+            173,
+            101,
+            867,
+            231
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            276,
+            326,
+            292
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.1 SIMULATOR ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            309,
+            299,
+            323
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Offline Results: Offline results are presented in Tables 2, 7 and 8 (the latter two are in the appendix). Table 7 presents results for the vanilla-MemN2N and Forward Prediction models. Table 2 presents results for Cont-MemN2N, which is better at handling unknown words. We repeat each experiment 10 times and report the best result. Finally, Table 8 presents results for the test scenario where the bot itself chooses when to ask questions. Observations can be summarized as as follows: ",
+        "bbox": [
+            174,
+            335,
+            825,
+            405
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "- Asking questions helps at test time, which is intuitive since it provides additional evidence: ",
+        "bbox": [
+            174,
+            411,
+            779,
+            426
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "• Train $1 Q + T e s t A Q$ (questions can be asked at both training and test time) performs the best across all the settings. Tra $i n Q A { + } T e s t A Q$ (questions can be asked at training time but not at test time) performs worse than $T r a i n Q A + T e s t Q A$ (questions can be asked at neither training nor test time) in tasks Question Clarification and Knowledge Operation due to the discrepancy between training and testing. $T r a i n Q A + T e s t A Q$ performs better than $T r a i n Q A + T e s t Q A$ on all Knowledge Acquisition tasks, the only exception being the Cont-MemN2N model on the Missing Triple setting. The explanation is that for most tasks in Knowledge Acquisition, the learner has no chance of giving the correct answer without asking questions. The benefit from asking is thus large enough to compensate for the negative effect introduced by data discrepancy between training and test time. TrainMix offers flexibility in bridging the gap between datasets generated using QA and AQ, very slightly underperforming TrainAQ+TestAQ, but gives competitive results on both TestQA and TestAQ in the Question Clarification and Knowledge Operations tasks. $I r a i n A Q + T e s t Q A$ (allowing questions at training time but forbid questions at test time) performs the worst, even worse than $T r a i n Q A + T e s t Q A$ . This has a real-world analogy where a student becomes dependent on the teacher answering their questions, later struggling to answer the test questions without help.   \nIn the Missing Question Entity task (the student does not know about the question entity), the Missing Answer Entity task (the student does not know about the answer entity), and Missing Everything task, the bot achieves accuracy less than 0.01 if not asking questions at test time (i.e., TestQA). The performance of TestModelAQ, where the bot relies on its model to ask questions at test time (and thus can ask irrelevant questions) performs similarly to asking the correct question at test time (TestAQ) and better than not asking questions (TestQA). ",
+        "bbox": [
+            214,
+            438,
+            825,
+            834
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "- Cont-MemN2N significantly outperforms vanilla-MemN2N. One explanation is that considering context provides significant evidence distinguishing correct answers from candidates in the dialogue history, especially in cases where the model encounters unfamiliar words. ",
+        "bbox": [
+            176,
+            847,
+            825,
+            888
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "RL Results For the RL settings, we present results for Task 2 (Question Verification) and Task 6 (Missing Answer Entities) in Figure 6. Task 2 represents scenarios where different types of student have different abilities to correctly answer questions (e.g., a poor student can still sometimes give correct answers even when they do not fully understand the question). Task 6 represents tasks where a poor learner who lacks the knowledge necessary to answer the question can hardly give a correct answer. All types of students including the good student will theoretically benefit from asking questions (asking for the correct answer) in Task 6. We show the percentage of question-asking versus the cost of AQ on the test set and the accuracy of question-answering on the test set vs the cost of AQ. Our main findings were: ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab2c6274536116f73ca7bb49b17c6f5b3d46d860d8ff9978b85bf21d6759c67c.jpg",
+        "image_caption": [
+            "Figure 6: Results of online learning for Task 2 and Task 6 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            261,
+            102,
+            736,
+            410
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            472,
+            825,
+            569
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "• A good student does not need to ask questions in Task 2 (Question Verification), because they already understand the question. The student will raise questions asking for the correct answer when cost is low for Task 6 (Missing Answer Entities). A poor student always asks questions when the cost is low. As the cost increases, the frequency of question-asking declines. As the AQ cost increases gradually, good students will stop asking questions earlier than the medium and poor students. The explanation is intuitive: poor students benefit more from asking questions than good students, so they continue asking even with higher penalties.   \n• As the probability of question-asking declines, the accuracy for poor and medium students drops. Good students are more resilient to not asking questions. ",
+        "bbox": [
+            215,
+            580,
+            825,
+            734
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "6.2 MECHANICAL TURK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            751,
+            356,
+            766
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Results for the Mechanical Turk Tasks are given in Table 3. We again compare vanilla-MemN2N and Cont-MemN2N, using the same TrainAQ/TrainQA and TestAQ/TestQA combinations as before, for Tasks 4 and 8 as described in Section 4.2. We tune hyperparameters on the validation set and repeat each experiment 10 times and report the best result. ",
+        "bbox": [
+            174,
+            776,
+            825,
+            833
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "While performance is lower than on the related Task 4 and Task 8 simulator tasks, we still arrive at the same trends and conclusions when real data from humans is used. The performance was expected to be lower because (i) real data has more lexical variety, complexity and noise; and (ii) the training set was smaller due to data collection costs (10k vs. 180k). We perform an analysis of the difference between simulated and real training data (or combining the two) in the appendix, which shows that using real data is indeed important and measurably superior to using simulated data. ",
+        "bbox": [
+            173,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/b90e4f2e67ce03c8c73584fa2b7d2c52a28761b2088fcfdac77434acedce67bc.jpg",
+        "table_caption": [
+            "Table 3: Mechanical Turk Task Results. Asking Questions (AQ) outperforms only answering questions without asking (QA). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">vanilla-MemN2N</td><td colspan=\"4\">Cont-MemN2N</td></tr><tr><td></td><td colspan=\"2\">Task4:K.Verification</td><td colspan=\"2\">Task 8:Triple</td><td colspan=\"2\">Task 4:K.Verification</td><td colspan=\"2\">Task 8:Triple</td></tr><tr><td>Train\\Test</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td>TrainQA</td><td>0.331</td><td>0.313</td><td>0.133</td><td>0.162</td><td>0.712</td><td>0.703</td><td>0.308</td><td>0.234</td></tr><tr><td>TrainAQ</td><td>0.318</td><td>0.375</td><td>0.072</td><td>0.422</td><td>0.679</td><td>0.774</td><td>0.137</td><td>0.797</td></tr></table>",
+        "bbox": [
+            228,
+            101,
+            769,
+            156
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "More importantly, the same main conclusion is observed as before: TrainAQ $^ +$ TestAQ (questions can be asked at both training and test time) performs the best across all the settings. That is, we show that a bot asking questions to humans learns to outperform one that only answers them. ",
+        "bbox": [
+            174,
+            219,
+            825,
+            262
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            282,
+            328,
+            299
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we explored how an intelligent agent can benefit from interacting with users by asking questions. We developed tasks where interaction via asking questions is desired. We explore both online and offline settings that mimic different real world situations and show that in most cases, teaching a bot to interact with humans facilitates language understanding, and consequently leads to better question answering ability. ",
+        "bbox": [
+            174,
+            314,
+            825,
+            385
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            406,
+            285,
+            421
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Mohammad Amin Bassiri. Interactional feedback and the impact of attitude and motivation on noticing l2 form. English Language and Literature Studies, 1(2):61, 2011.   \nAntoine Bordes and Jason Weston. Learning end-to-end goal-oriented dialog. arXiv preprint arXiv:1605.07683, 2016.   \nAntoine Bordes, Nicolas Usunier, Sumit Chopra, and Jason Weston. Large-scale simple question answering with memory networks. arXiv preprint arXiv:1506.02075, 2015.   \nJesse Dodge, Andreea Gane, Xiang Zhang, Antoine Bordes, Sumit Chopra, Alexander Miller, Arthur Szlam, and Jason Weston. Evaluating prerequisite qualities for learning end-to-end dialog systems. arXiv preprint arXiv:1511.06931, 2015.   \nRichard Higgins, Peter Hartley, and Alan Skelton. The conscientious consumer: Reconsidering the role of assessment feedback in student learning. Studies in higher education, 27(1):53–64, 2002.   \nAndrew S Latham. Learning through feedback. Educational Leadership, 54(8):86–87, 1997.   \nJiwei Li, Michel Galley, Chris Brockett, Jianfeng Gao, and Bill Dolan. A diversity-promoting objective function for neural conversation models. arXiv preprint arXiv:1510.03055, 2015.   \nAlexander Miller, Adam Fisch, Jesse Dodge, Amir-Hossein Karimi, Antoine Bordes, and Jason Weston. Key-value memory networks for directly reading documents. arXiv preprint arXiv:1606.03126, 2016.   \nMarc’Aurelio Ranzato, Sumit Chopra, Michael Auli, and Wojciech Zaremba. Sequence level training with recurrent neural networks. arXiv preprint arXiv:1511.06732, 2015.   \nAlessandro Sordoni, Michel Galley, Michael Auli, Chris Brockett, Yangfeng Ji, Margaret Mitchell, Jian-Yun Nie, Jianfeng Gao, and Bill Dolan. A neural network approach to context-sensitive generation of conversational responses. arXiv preprint arXiv:1506.06714, 2015.   \nPei-Hao Su, Milica Gasic, Nikola Mrksic, Lina Rojas-Barahona, Stefan Ultes, David Vandyke, Tsung-Hsien Wen, and Steve Young. Continuously learning neural dialogue management. arXiv preprint arXiv:1606.02689, 2016.   \nSainbayar Sukhbaatar, Jason Weston, Rob Fergus, et al. End-to-end memory networks. In Advances in neural information processing systems, pp. 2440–2448, 2015. ",
+        "bbox": [
+            171,
+            429,
+            826,
+            929
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Oriol Vinyals and Quoc Le. A neural conversational model. arXiv preprint arXiv:1506.05869, 2015. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            119
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Sida I Wang, Percy Liang, and Christopher D Manning. Learning language games through interaction. arXiv preprint arXiv:1606.02447, 2016. ",
+        "bbox": [
+            171,
+            126,
+            821,
+            156
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Tsung-Hsien Wen, Milica Gasic, Nikola Mrksic, Lina M Rojas-Barahona, Pei-Hao Su, Stefan Ultes, David Vandyke, and Steve Young. A network-based end-to-end trainable task-oriented dialogue system. arXiv preprint arXiv:1604.04562, 2016. ",
+        "bbox": [
+            176,
+            162,
+            823,
+            207
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Margaret G Werts, Mark Wolery, Ariane Holcombe, and David L Gast. Instructive feedback: Review of parameters and effects. Journal of Behavioral Education, 5(1):55–75, 1995. ",
+        "bbox": [
+            171,
+            214,
+            823,
+            244
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jason Weston. Dialog-based language learning. arXiv preprint arXiv:1604.06045, 2016. ",
+        "bbox": [
+            169,
+            251,
+            753,
+            267
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Jason Weston, Antoine Bordes, Sumit Chopra, Alexander M Rush, Bart van Merrienboer, Armand ¨ Joulin, and Tomas Mikolov. Towards ai-complete question answering: A set of prerequisite toy tasks. arXiv preprint arXiv:1502.05698, 2015. ",
+        "bbox": [
+            174,
+            275,
+            825,
+            318
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992. ",
+        "bbox": [
+            171,
+            325,
+            823,
+            354
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Terry Winograd. Understanding natural language. Cognitive psychology, 3(1):1–191, 1972. ",
+        "bbox": [
+            173,
+            363,
+            772,
+            378
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Ludwig Wittgenstein. Philosophical investigations. John Wiley & Sons, 2010. ",
+        "bbox": [
+            173,
+            386,
+            691,
+            401
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Wojciech Zaremba and Ilya Sutskever. Reinforcement learning neural turing machines. arXiv preprint arXiv:1505.00521, 362, 2015. ",
+        "bbox": [
+            176,
+            410,
+            825,
+            439
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Appendix ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            444,
+            258,
+            460
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "End-to-End Memory Networks The input to an end-to-end memory network model (MemN2N) is the last utterance of the dialogue history $x$ as well as a set of memories (context) $\\scriptstyle { C = c _ { 1 } }$ , $c _ { 2 }$ , ..., $c _ { N } .$ ). Memory $C$ encodes both short-term memory, e..g, dialogue histories between the bot and the teacher and long-term memories, e.g., the knowledgebase facts that the bot has access to. Given the input $x$ and $C$ , the goal is to produce an output/label $a$ . ",
+        "bbox": [
+            173,
+            467,
+            825,
+            537
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the first step, the query $x$ is transformed to a vector representation $u _ { 0 }$ by summing up its constituent word embeddings: $u _ { 0 } = A x$ . The input $\\mathbf { X }$ is a bag-of-words vector and $A$ is the $d \\times V$ word embedding matrix where $d$ denotes the vector dimensionality and $V$ denotes the vocabulary size. Each memory $c _ { i }$ is similarly transformed to vector $m _ { i }$ . The model will read information from the memory by linking input representation $q$ with memory vectors $m _ { i }$ using softmax weights: ",
+        "bbox": [
+            173,
+            542,
+            825,
+            614
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/97e5db2b6ce01ea2a7117028f7b755d43eb78df90ef605ce4b7c62d7a4548e49.jpg",
+        "text": "$$\no _ { 1 } = \\sum _ { i } p _ { i } ^ { 1 } m _ { i } \\qquad p _ { i } ^ { 1 } = \\mathsf { s o f t m a x } ( u _ { 0 } ^ { T } m _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            618,
+            648,
+            651
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The goal is to select memories relevant to the last utterance $x$ , i.e., the memories with large values of $p _ { i } ^ { 1 }$ . The queried memory vector $o _ { 1 }$ is the weighted sum of memory vectors. The queried memory vector $o _ { 1 }$ will be added on top of original input, $u _ { 1 } = o _ { 1 } + u _ { 0 } . \\ u$ $u _ { 1 }$ is then used to query the memory vector. Such a process is repeated by querying the memory $_ \\mathrm { N }$ times (so called “hops”). $N$ is set to three in all experiments in this paper. ",
+        "bbox": [
+            173,
+            656,
+            825,
+            728
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the end, $u _ { N }$ is input to a softmax function for the final prediction: ",
+        "bbox": [
+            173,
+            733,
+            622,
+            748
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/20949eec3c29e6c7be2463c804f57540c65ecb5920300ebc4acaba2c479e1fc2.jpg",
+        "text": "$$\n\\boldsymbol { a } = \\mathsf { s o f t m a x } ( u _ { N } ^ { T } y _ { 1 } , u _ { N } ^ { T } y _ { 2 } , . . . , u _ { N } ^ { T } y _ { L } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            753,
+            632,
+            772
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $L$ denotes the number of candidate answers and $y$ denotes the representation of the answer. If the answer is a word, $y$ is the corresponding word embedding. If the answer is a sentence, $y$ is the embedding for the sentence achieved in the same way as we obtain embeddings for query $x$ and memory $c$ . ",
+        "bbox": [
+            173,
+            776,
+            825,
+            833
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Reward Based Imitation (RBI) and Forward Prediction (FP) RBI and $F P$ are two dialogue learning strategies proposed in (Weston, 2016) by harnessing different types of dialogue signals. RBI handles the case where the reward or the correctness of a bot’s answer is explicitly given (for example, $+ 1$ if the bot’s answer is correct and 0 otherwise). The model is directly trained to predict the correct answers (with label 1) at training time, which can be done using End-to-End Memory Networks (MemN2N) (Sukhbaatar et al., 2015) that map a dialogue input to a prediction. ",
+        "bbox": [
+            173,
+            839,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "$F P$ handles the situation where a real-valued reward for a bot’s answer is not available, meaning that there is no $+ 1$ or 0 labels paired with a student’s utterance. However, the teacher will give a response to the bot’s answer, taking the form of a dialogue utterance. More formally, suppose that $x$ denotes the teacher’s question and $C { = } c _ { 1 }$ , $c _ { 2 }$ , ..., $c _ { N }$ denotes the dialogue history. In our $A Q$ settings, the bot will ask a question $a$ regarding the teacher’s question, denoted as $a \\in \\mathbb { A }$ , where A denotes the student’s question pool. The teacher will provide an utterance in response to the student question $a$ . In $F P$ , the model first maps the teacher’s initial question $x$ and dialogue history $C$ to vector representation $u$ using a memory network with multiple hops. Then the model will perform another hopof attention over all possible student’s questions in A, with an additional part that incorporates the information of which candidate (i.e., $a$ ) was actually selected in the dialogue: ",
+        "bbox": [
+            173,
+            103,
+            825,
+            243
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6065697e012f9b13d1db2dec5d9e133bf0eb8c338d6cedeecc0a65b636190d29.jpg",
+        "text": "$$\np _ { \\hat { a } } = { \\tt s o f t m a x } ( u ^ { T } y _ { \\hat { a } } ) \\quad o = \\sum _ { \\hat { a } \\in \\mathbb { A } } p _ { \\hat { a } } ( y _ { \\hat { a } } + \\beta \\cdot { \\bf 1 } [ \\hat { a } = a ] )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            247,
+            687,
+            282
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $y _ { \\hat { a } }$ denotes the vector representation for the student’s question candidate $\\hat { a }$ . $\\beta$ is a ddimensional vector to signify the actual action $a$ that the student chooses. For tasks where the student only has one way to ask questions (e.g., “what do you mean”), there is no need to perform hops of attention over candidates since the cardinality of $\\mathbb { A }$ is just 1. We thus directly assign a probability of 1 to the student’s question, making $o$ the sum of vector representation of $y _ { a }$ and $\\beta$ . ",
+        "bbox": [
+            174,
+            286,
+            825,
+            357
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$o$ is then combined with $u$ to predict the teacher’s feedback $t$ using a softmax: ",
+        "bbox": [
+            174,
+            363,
+            683,
+            378
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0917ce3a10c24de000ca6e6d6998a2645a9021d7c19c6b235fa6b5e0810d4431.jpg",
+        "text": "$$\n\\boldsymbol { u } _ { 1 } = o + \\boldsymbol { u } \\quad t = \\mathrm { s o f t m a x } \\big ( \\boldsymbol { u } _ { 1 } ^ { T } x _ { r _ { 1 } } , \\boldsymbol { u } _ { 1 } ^ { T } x _ { r _ { 2 } } , . . . , \\boldsymbol { u } _ { 1 } ^ { T } x _ { r _ { N } } \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            383,
+            683,
+            402
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { x } _ { r _ { i } }$ denotes the embedding for the $i ^ { t h }$ response. ",
+        "bbox": [
+            174,
+            406,
+            531,
+            421
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Dialogue Simulator In this section we further detail the simulator and the datasets we generated in order to realize the various scenarios discussed in Section 3. We focused on the problem of movieQA where we adapted the WikiMovies dataset proposed in Weston et al. (2015). The dataset consists of roughly $1 0 0 \\mathrm { k }$ questions with over $7 5 \\mathrm { k }$ entities from the open movie dataset (OMDb). ",
+        "bbox": [
+            174,
+            428,
+            825,
+            484
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Each dialogue generated by the simulator takes place between a student and a teacher. The simulator samples a random question from the WikiMovies dataset and fetches the set of all KB facts relevant to the chosen question. This question is assumed to be the one the teacher asks its student, and is referred to as the “original” question. The student is first presented with the relevant KB facts followed by the original question. Providing the KB facts to the student allows us to control the exact knowledge the student is given access to while answering the questions. At this point, depending on the task at hand and the student’s ability to answer, the student might choose to directly answer it or ask a “followup” question. The nature of the followup question will depend on the scenario under consideration. If the student answers the question, it gets a response from the teacher about its correctness and the conversation ends. However if the student poses a followup question, the teacher gives an appropriate response, which should give additional information to the student to answer the original question. In order to make things more complicated, the simulator pads the conversation with several unrelated student-teacher question-answer pairs. These question-answer pairs can be viewed as distractions and are used to test the student’s ability to remember the additional knowledge provided by the teacher after it was queried. For each dialogue, the simulator incorporates 5 such pairs (10 sentences). We refer to these pairs as conversational histories. ",
+        "bbox": [
+            174,
+            491,
+            825,
+            714
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For the $Q A$ setting (see Section 3), the dialogues generated by the simulator are such that the student never asks a clarification question. Instead, it simply responds to the original question, even if it is wrong. For the dialogs in the $A Q$ setting, the student always asks a clarification question. The nature of the question asked is dependent on the scenario (whether it is Question Clarification, Knowledge Operation, or Knowledge Acquisition) under consideration. In order to simulate the case where the student sometimes choses to directly answer the original question and at other times choses to ask question, we created training datasets, which were a combination of $Q A$ and $A Q$ (called “Mixed”). For all these cases, the student needs to give an answer to the teacher’s original question at the end. ",
+        "bbox": [
+            174,
+            719,
+            825,
+            832
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Instructions given to Turkers ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            838,
+            379,
+            853
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "These are the instructions given for the textual feedback Mechanical Turk task (we also constructed a separate task to collect the questions to ask the bot with similar instructions, not described here): ",
+        "bbox": [
+            174,
+            859,
+            825,
+            888
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Task 4 (answers to bot’s questions): ",
+        "bbox": [
+            176,
+            910,
+            408,
+            922
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Title: Write brief responses to given dialogue exchanges (about $1 5 \\mathrm { m i n }$ ) ",
+        "bbox": [
+            174,
+            103,
+            645,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Description: Write a brief response answering a provided question (25 questions per HIT). ",
+        "bbox": [
+            169,
+            125,
+            764,
+            140
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Directions: ",
+        "bbox": [
+            174,
+            147,
+            248,
+            160
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Each task consists of the following triplets: ",
+        "bbox": [
+            176,
+            167,
+            455,
+            181
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "1) a question by the teacher   \n2) the correct answer(s) to the question (separated by “OR”), unknown to the student   \n3) a clarifying question asking for feedback from the teacher ",
+        "bbox": [
+            173,
+            183,
+            736,
+            223
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Consider the scenario where you are the teacher and have already asked the question, and received the reply from the student. Please compose a brief response replying to the student’s question. The correct answers are provided so that you know whether the student’s question was relevant or not. ",
+        "bbox": [
+            176,
+            224,
+            823,
+            265
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For example, given 1) question: “what is a color in the united states flag?”; 2) correct answer: “white OR blue OR red”; 3) student reply: “does this have to do with ‘US Flag has colors red,white,blue‘?”, your response could be something like “that’s right!”; for 3) reply: “does this have to do with ‘United States has population 320 million”, you might say “No, that fact is not relevant” or “Not really”. ",
+        "bbox": [
+            174,
+            265,
+            825,
+            334
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Please vary responses and try to minimize spelling mistakes. If the same responses are copied/pasted or similar responses are overused, we’ll reject the HIT. Avoid naming the student or addressing “the class” directly. We will consider bonuses for higher quality responses during review. ",
+        "bbox": [
+            176,
+            335,
+            823,
+            376
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Task 8: answers to bot’s questions: ",
+        "bbox": [
+            176,
+            397,
+            403,
+            411
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Title: Write brief responses to given dialogue exchanges (about 10 min) ",
+        "bbox": [
+            174,
+            417,
+            650,
+            433
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Description: Write a sentence describing the answer to a question (25 questions per HIT). ",
+        "bbox": [
+            173,
+            439,
+            756,
+            454
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Directions: ",
+        "bbox": [
+            174,
+            462,
+            248,
+            474
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Each task consists of the following triplets: ",
+        "bbox": [
+            176,
+            483,
+            457,
+            496
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "1) a question by the teacher   \n2) the correct answer(s) to the question (separated by “OR”), unknown to the student   \n3) a question from the student asking the teacher for the answer ",
+        "bbox": [
+            174,
+            498,
+            730,
+            537
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Consider the scenario where you are the teacher and have already asked the question, and received the reply from the student. Please compose a brief response replying to the student’s question. The correct answers are provided so that you know which answers to provide. ",
+        "bbox": [
+            176,
+            539,
+            821,
+            579
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For example, given 1) question: “what is a color in the united states flag?”; 2) correct answer: “white OR blue OR red”; 3) student reply: “i dont know. what’s the answer ?”, your response could be something like “the color white is in the US flag” or “blue and red both appear in it”. ",
+        "bbox": [
+            174,
+            580,
+            823,
+            621
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Please vary responses and try to minimize spelling mistakes, and do not include the capitalized “OR” in your response. If the same responses are copied/pasted or similar responses are overused, we’ll reject the HIT. You don’t need to mention every correct answer in your response. Avoid naming the student or addressing “the class” directly. We will consider bonuses for higher quality responses during review. ",
+        "bbox": [
+            174,
+            622,
+            825,
+            690
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Additional Mechanical Turk Experiments ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            710,
+            465,
+            726
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Here we provide additional experiments to supplement the ones described in Section 6.2. In the main paper, results were shown when training and testing on the collected Mechanical Turk data (around 10,000 episodes of training dialogues for training). As we collected the data in the same settings as Task 4 and 8 of our simulator, we could also consider supplementing training with simulated data as well, of which we have a larger amount (over 100,000 episodes). Note this is only for training, we will still test on the real (Mechanical Turk collected) data. Although the simulated data has less lexical variety as it is built from templates, the larger size might obtain improve results. ",
+        "bbox": [
+            174,
+            733,
+            825,
+            830
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Results are given Table 5 when training on the combination of real and simulator data, and testing on real data. This should be compared to training on only the real data (Table 4) and only on the simulator data (Table 6). The best results are obtained from the combination of simulator and real data. The best real data only results (selecting over algorithm and training strategy) on both tasks outperform the best results using simulator data, i.e. using Cont-MemN2N with the Train AQ / TestAQ setting) 0.774 and 0.797 is obtained vs. 0.714 and 0.788 for Tasks 4 and 8 respectively. This is despite there being far fewer examples of real data compared to simulator data. Overall we obtain two main conclusions from this additional experiment: (i) real data is indeed measurably superior to simulated data for training our models; (ii) in all cases (across different algorithms, tasks and data types – be they real data, simulated data or combinations) the bot asking questions (AQ) outperforms it only answering questions and not asking them (QA). The latter reinforces the main result of the paper. ",
+        "bbox": [
+            174,
+            837,
+            825,
+            920
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            188
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/ec0d80bc6065ddb32404cf771a72618304cf44734997d3953935af71df59a391.jpg",
+        "table_caption": [
+            "Table 4: Mechanical Turk Task Results, using real data for training and testing. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">vanilla-MemN2N</td><td colspan=\"3\">Cont-MemN2N</td></tr><tr><td></td><td colspan=\"2\">Task 4:K.Verification</td><td colspan=\"2\">Task 8: Triple</td><td>Task 4:K.Verification</td><td colspan=\"2\">Task 8:Triple</td></tr><tr><td>Train\\Test</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td>TrainQA</td><td>0.331</td><td>0.313</td><td>0.133</td><td>0.162</td><td>0.712 0.703</td><td>0.308</td><td>0.234</td></tr><tr><td>TrainAQ</td><td>0.318</td><td>0.375</td><td>0.072</td><td>0.422</td><td>0.679 0.774</td><td>0.137</td><td>0.797</td></tr></table>",
+        "bbox": [
+            228,
+            200,
+            769,
+            256
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/255524efbaa2ce427e6165bbbdce5d1bd70939c0546d0ccfa50942e28be14d51.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">vanilla-MemN2N</td><td colspan=\"4\">Cont-MemN2N</td></tr><tr><td></td><td colspan=\"2\">Task 4:K.Verification</td><td colspan=\"2\">Task 8: Triple</td><td colspan=\"2\">Task 4:K.Verification</td><td colspan=\"2\">Task 8: Triple</td></tr><tr><td>Train\\Test</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td>TrainQA</td><td>0.356</td><td>0.311</td><td>0.128</td><td>0.174</td><td>0.733</td><td>0.717</td><td>0.368</td><td>0.352</td></tr><tr><td>TrainAQ</td><td>0.340</td><td>0.445</td><td>0.150</td><td>0.487</td><td>0.704</td><td>0.792</td><td>0.251</td><td>0.825</td></tr></table>",
+        "bbox": [
+            230,
+            314,
+            767,
+            369
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Results on Mechanical Turk Tasks using a combination of real and simulated data for training, testing on real data. ",
+        "bbox": [
+            171,
+            378,
+            823,
+            407
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/0dfddf65d6359210e45514ac7c486e0eef151abbc306537b7baba07cd242771a.jpg",
+        "table_caption": [
+            "Table 6: Results on Mechanical Turk Tasks using only simulated data for training, but testing on real data. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">vanilla-MemN2N</td><td colspan=\"3\">Cont-MemN2N</td></tr><tr><td></td><td colspan=\"2\">Task4:K.Verification</td><td colspan=\"2\">Task 8: Triple</td><td>Task4:K.Verification</td><td colspan=\"2\">Task 8:Triple</td></tr><tr><td>TrainTest</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td>TrainQA</td><td>0.340</td><td>0.311</td><td>0.120</td><td>0.165</td><td>0.665 0.648</td><td>0.349</td><td>0.342</td></tr><tr><td>TrainAQ</td><td>0.326</td><td>0.390</td><td>0.067</td><td>0.405</td><td>0.642 0.714</td><td>0.197</td><td>0.788</td></tr></table>",
+        "bbox": [
+            228,
+            441,
+            769,
+            497
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/adfeaa84ed6e1b85fc492f510755460d962aea9e0d2cf774922f0d4f9feaf03d.jpg",
+        "table_caption": [
+            "Additional Offline Supervised Learning Experiments "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">Question Clarification</td><td colspan=\"4\">Knowledge Operation</td><td rowspan=\"3\"></td></tr><tr><td></td><td colspan=\"2\">Task1:Q.Paraphrase</td><td colspan=\"2\">Task 2:Q.Verification</td><td colspan=\"2\">Task3:AskForRelevantK.</td><td colspan=\"2\">Task4:K.Verification</td></tr><tr><td>TrainTest</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td>TrainQA</td><td>0.338</td><td>0.284</td><td>0.340</td><td>0.271 0.373</td><td>0.462</td><td>0.344</td><td>0.482</td><td>0.322</td></tr><tr><td>TrainAQ</td><td>0.213</td><td>0.450</td><td>0.225</td><td></td><td>0.187 0.632 0.342</td><td></td><td>0.283</td><td>0.540</td></tr><tr><td>TrainAQ(+FP) TrainMix</td><td>0.288 0.326</td><td>0.464</td><td>0.146</td><td>0.320</td><td>0.631</td><td></td><td>0.311</td><td>0.524</td></tr><tr><td></td><td>0.373</td><td></td><td>0.329</td><td>0.326</td><td>0.442</td><td>0.558</td><td>0.476</td><td>0.491</td></tr><tr><td colspan=\"9\">Knowledge Acquisition</td></tr><tr><td>TrainTest</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td><td>TestQA</td><td>TestAQ</td></tr><tr><td></td><td>Task 5:Q.Entity</td><td></td><td>Task 6: Answer Entity</td><td></td><td>Task 7: Relation Entity</td><td>Task 8: Triple</td><td></td><td></td><td>Task 9:Everything</td></tr><tr><td>TrainQA (vanila)</td><td>&lt;0.01</td><td>0.223</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.109 0.129</td><td>0.201</td><td>0.259</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>TrainAQ (vanila)</td><td>&lt;0.01</td><td>0.660</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.082 0.156</td><td>0.124</td><td>0.664</td><td>&lt;0.01</td><td>&lt;0.01</td></tr><tr><td>TrainAQ(+FP)</td><td>&lt;0.01</td><td>0.742</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.085 0.188 0.152</td><td>0.064 0.180</td><td>0.702 0.572</td><td>&lt;0.01 &lt;0.01</td><td>&lt;0.01</td></tr><tr><td>Mix (vanila)</td><td>&lt;0.01</td><td>0.630</td><td>&lt;0.01</td><td>&lt;0.01</td><td>0.070</td><td></td><td></td><td></td><td>&lt;0.01</td></tr></table>",
+        "bbox": [
+            173,
+            573,
+            856,
+            723
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/f266e7b60c8663b46615dd2e827738360363dc5c126bd4a6d0edd3a379ffd9ce.jpg",
+        "table_caption": [
+            "Table 7: Results for offline settings using memory networks. ",
+            "Table 8: Results for TestModelAQ settings. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Question Clarification</td><td>Knowledge Acquisition</td></tr><tr><td></td><td>Task 2:Q.Verification</td><td>Task4:K.Verification</td></tr><tr><td></td><td>TestModelAQ</td><td>TestModelAQ</td></tr><tr><td>TrainAQ</td><td>0.382</td><td>0.480</td></tr><tr><td>TrainAQ(+FP)</td><td>0.344</td><td>0.501</td></tr><tr><td>TrainMix</td><td>0.352</td><td>0.469</td></tr></table>",
+        "bbox": [
+            326,
+            767,
+            668,
+            834
+        ],
+        "page_idx": 15
+    }
+]

parse/train/rylb3eBtwr/rylb3eBtwr.md

@@ -0,0 +1,672 @@
+# ROBUST SUBSPACE RECOVERY LAYER FOR UNSUPERVISED ANOMALY DETECTION
+
+Chieh-Hsin Lai∗, Dongmian Zou∗& Gilad Lerman
+
+School of Mathematics   
+University of Minnesota   
+Minneapolis, MN 55455   
+{laixx313, dzou, lerman}@umn.edu
+
+# ABSTRACT
+
+We propose a neural network for unsupervised anomaly detection with a novel robust subspace recovery layer (RSR layer). This layer seeks to extract the underlying subspace from a latent representation of the given data and removes outliers that lie away from this subspace. It is used within an autoencoder. The encoder maps the data into a latent space, from which the RSR layer extracts the subspace. The decoder then smoothly maps back the underlying subspace to a “manifold” close to the original inliers. Inliers and outliers are distinguished according to the distances between the original and mapped positions (small for inliers and large for outliers). Extensive numerical experiments with both image and document datasets demonstrate state-of-the-art precision and recall.
+
+# 1 INTRODUCTION
+
+Finding and utilizing patterns in data is a common task for modern machine learning systems. However, there is often some anomalous information that does not follow a common pattern and has to be recognized. For this purpose, anomaly detection aims to identify data points that “do not conform to expected behavior” (Chandola et al., 2009). We refer to such points as either anomalous or outliers. In many applications, there is no ground truth available to distinguish anomalous from normal points, and they need to be detected in an unsupervised fashion. For example, one may need to remove anomalous images from a set of images obtained by a search engine without any prior knowledge about how a normal image should look (Xia et al., 2015). Similarly, one may need to distinguish unusual news items from a large collection of news documents without any information whether a news item is usual or not (Kannan et al., 2017). In these examples, the only assumptions are that normal data points appear more often than anomalous ones and have a simple underlying structure which is unknown to the user.
+
+Some early methods for anomaly detection relied on Principal Component Analysis (PCA) (Shyu et al., 2003). Here one assumes that the underlying unknown structure of the normal samples is linear. However, PCA is sensitive to outliers and will often not succeed in recovering the linear structure or identifying the outliers (Lerman & Maunu, 2018; Vaswani & Narayanamurthy, 2018). More recent ideas of Robust PCA (RPCA) (Wright et al., 2009; Vaswani & Narayanamurthy, 2018) have been considered for some specific problems of anomaly detection or removal (Zhou & Paffenroth, 2017; Paffenroth et al., 2018). RPCA assumes sparse corruption, that is, few elements of the data matrix are corrupted. This assumption is natural for some special problems in computer vision, in particular, background subtraction (De La Torre & Black, 2003; Wright et al., 2009; Vaswani & Narayanamurthy, 2018). However, a natural setting of anomaly detection with hidden linear structure may assume instead that a large portion of the data points are fully corrupted. The mathematical framework that addresses this setting is referred to as robust subspace recovery (RSR) (Lerman & Maunu, 2018).
+
+While Robust PCA and RSR try to extract linear structure or identify outliers lying away from such structure, the underlying geometric structure of many real datasets is nonlinear. Therefore, one needs to extract crucial features of the nonlinear structure of the data while being robust to outliers. In order to achieve this goal, we propose to use an autoencoder (composed of an encoder and a decoder) with an RSR layer. We refer to it as RSRAE (RSR autoencoder). It aims to robustly and nonlinearly reduce the dimension of the data in the following way. The encoder maps the data into a high-dimensional space. The RSR layer linearly maps the embedded points into a low-dimensional subspace that aims to learn the hidden linear structure of the embedded normal points. The decoder maps the points from this subspace to the original space. It aims to map the normal points near their original locations, and the anomalous points far from their original locations.
+
+Ideally, the encoder maps the normal data to a linear space and any anomalies lie away from this subspace. In this ideal scenario, anomalies can be removed by an RSR method directly applied to the data embedded by the encoder. Since the linear model for the normal data embedded by the encoder is only approximate, we do not directly apply RSR to the embedded data. Instead, we minimize a sum of the reconstruction error of the autoencoder and the RSR error for the data embedded by the encoder. We advocate for an alternating procedure, so that the parameters of the autoencoder and the RSR layer are optimized in turn.
+
+# 1.1 STRUCTURE OF THE REST OF THE PAPER
+
+Section 2 reviews works that are directly related to the proposed RSRAE and highlights the original contributions of this paper. Section 3 explains the proposed RSRAE, and in particular, its RSR layer and total energy function. Section 4 includes extensive experimental evidence demonstrating effectiveness of RSRAE with both image and document data. Section 5 discusses theory for the relationship of the RSR penalty with the WGAN penalty. Section 6 summarizes this work and mentions future directions.
+
+# 2 RELATED WORKS AND CONTRIBUTION
+
+We review related works in Section 2.1 and highlight our contribution in Section 2.2.
+
+# 2.1 RELATED WORKS
+
+Several recent works have used autoencoders for anomaly detection. Xia et al. (2015) proposed the earliest work on anomaly detection via an autoencoder, while utilizing large reconstruction error of outliers. They apply an iterative and cyclic scheme, where in each iteration, they determine the inliers and use them for updating the parameters of the autoencoder. Aytekin et al. (2018) apply $\ell _ { 2 }$ normalization for the latent code of the autoencoder and also consider the case of multiple modes for the normal samples. Instead of using the reconstruction error, they apply $k$ -means clustering for the latent code, and identify outliers as points whose latent representations are far from all the cluster centers. Zong et al. (2018) also use an autoencoder with clustered latent code, but they fit a Gaussian Mixture Model using an additional neural network. Restricted Boltzmann Machines (RBMs) are similar to autoencoders. Zhai et al. (2016) define “energy functions” for RBMs that are similar to the reconstruction losses for autoencoders. They identify anomalous samples according to large energy values. Chalapathy et al. (2017) propose using ideas of RPCA within an autoencoder, where they alternatively optimize the parameters of the autoencoder and a sparse residual matrix.
+
+The above works are designed for datasets with a small fraction of outliers. However, when this fraction increases, outliers are often not distinguished by high reconstruction errors or low similarity scores. In order to identify them, additional assumptions on the structure of the normal data need to be incorporated. For example, Zhou & Paffenroth (2017) decompose the input data into two parts: low-rank and sparse (or column-sparse). The low-rank part is fed into an autoencoder and the sparse part is imposed as a penalty term with the $\ell _ { 1 }$ -norm (or $\ell _ { 2 , 1 }$ -norm for column-sparsity).
+
+In this work, we use a term analogous to the $\ell _ { 2 , 1 }$ -norm, which can be interpreted as the sum of absolute deviations from a latent subspace. However, we do not decompose the data a priori, but minimize an energy combining this term and the reconstruction error. Minimization of the former term is known as least absolute deviations in RSR (Lerman & Maunu, 2018). It was first suggested for RSR and related problems in Watson (2001); Ding et al. (2006); Zhang et al. (2009). The robustness to outliers of this energy, or of relaxed versions of it, was studied in McCoy & Tropp (2011); Xu et al. (2012); Lerman & Zhang (2014); Zhang & Lerman (2014); Lerman et al. (2015); Lerman & Maunu (2017); Maunu et al. (2017). In particular, Maunu et al. (2017) established its well-behaved landscape under special, though natural, deterministic conditions. Under similar conditions, they guaranteed fast subspace recovery by a simple algorithm that aims to minimize this energy.
+
+Another directly related idea for extracting useful latent features is an addition of a linear selfexpressive layer to an autoencoder (Ji et al., 2017). It is used in the different setting of unsupervised subspace clustering. By imposing the self-expressiveness, the autoencoder is robust to an increasing number of clusters. Although self-expressiveness also improves robustness to noise and outliers, Ji et al. (2017) aims at clustering and thus its goal is different than ours. Furthermore, their selfexpressive energy does not explicitly consider robustness, while ours does. Lezama et al. (2018) consider a somewhat parallel idea of imposing a loss function to increase the robustness of representation. However, their goal is to increase the margin between classes and their method only applies to a supervised setting in anomaly detection, where the normal data is multi-modal.
+
+# 2.2 CONTRIBUTION OF THIS WORK
+
+This work introduces an RSR layer within an autoencoder. It incorporates a special regularizer that enforces an outliers-robust linear structure in the embedding obtained by the encoder. We clarify that the method does not alternate between application of the autoencoder and the RSR layer, but fully integrates these two components. Our experiments demonstrate that a simple incorporation of a “robust loss” within a regular autoencoder does not work well for anomaly detection. We try to explain this and also the improvement obtained by incorporating an additional RSR layer.
+
+Our proposed architecture is simple to implement. Furthermore, the RSR layer is not limited to a specific design of RSRAE but can be put into any well-designed autoencoder structure. The epoch time of the proposed algorithm is comparable to those of other common autoencoders. Furthermore, our experiments show that RSRAE competitively performs in unsupervised anomaly detection tasks.
+
+RSRAE addresses the unsupervised setting, but is not designed to be highly competitive in the semisupervised or supervised settings, where one has access to training data from the normal class or from both classes, respectively. In these settings, RSRAE functions like a regular autoencoder without taking an advantage of its RSR layer, unless the training data for the normal class is corrupted with outliers.
+
+The use of RSR is not restricted to autoencoders. We establish some preliminary analysis for RSR within a generative adversarial network (GAN) (Goodfellow et al., 2014; Arjovsky et al., 2017) in Section 5. More precisely, we show that a linear WGAN intrinsically incorporates RSR in some special settings, although it is unclear how to impose an RSR layer.
+
+# 3 RSR LAYER FOR OUTLIER REMOVAL
+
+We assume input data $\{ \mathbf { x } ^ { ( t ) } \} _ { t = 1 } ^ { N }$ in $\mathbb { R } ^ { M }$ , and denote by $\mathbf { X }$ its corresponding data matrix, whose $t$ -th column is $\mathbf { x } ^ { ( t ) }$ . The encoder of RSRAE, $\mathcal { E } ^ { \mathcal { O } }$ , is a neural network that maps each data point, $\mathbf { x } ^ { ( t ) }$ , to its latent code $\mathbf { z } ^ { ( t ) } = \boldsymbol { \mathcal { E } } \big ( \mathbf { x } ^ { ( t ) } \big ) \in \mathbb { R } ^ { D }$ . The RSR layer is a linear transformation $\mathbf { A } \in \mathbb { R } ^ { \bar { d } \times D }$ that reduces the dimension to $d$ . That is, $\widetilde { \mathbf { z } } ^ { ( t ) } = \mathbf { A } \mathbf { z } ^ { ( t ) } \in \mathbb { R } ^ { d }$ . The decoder $\mathcal { D }$ is a neural network that maps $\tilde { \mathbf { z } } ^ { ( t ) }$ to $\tilde { \mathbf { x } } ^ { ( t ) }$ in the original ambient space $\mathbb { R } ^ { M }$ .
+
+We can write the forward maps in a compact form using the corresponding data matrices as follows:
+
+$$
+\begin{array} { r } { \mathbf { Z } = \boldsymbol { \mathcal { E } } ( \mathbf { X } ) , \tilde { \mathbf { Z } } = \mathbf { A } \mathbf { Z } , \tilde { \mathbf { X } } = \boldsymbol { \mathcal { D } } ( \tilde { \mathbf { Z } } ) . } \end{array}
+$$
+
+Ideally, we would like to optimize RSRAE so it only maintains the underlying structure of the normal data. We assume that the original normal data lies on a $d$ -dimensional “manifold” in $\mathbb { R } ^ { D }$ and thus the RSR layer embeds its latent code into $\mathbb { R } ^ { d }$ . In this ideal optimization setting, the similarity between the input and the output of RSRAE is large whenever the input is normal and small whenever the input is anomalous. Therefore, by thresholding a similarity measure, one may distinguish between normal and anomalous data points.
+
+In practice, the matrix A and the parameters of $\mathcal { E } ^ { \sigma }$ and $\mathcal { D }$ are obtained by minimizing a loss function, which is a sum of two parts: the reconstruction loss from the autoencoder and the loss from the RSR layer. For $p > 0$ , an $\ell _ { 2 , p }$ reconstruction loss for the autoencoder is
+
+$$
+L _ { \mathrm { A E } } ^ { p } ( \mathcal { E } , \mathbf { A } , \mathcal { D } ) = \sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \tilde { \mathbf { x } } ^ { ( t ) } \right\| _ { 2 } ^ { p } .
+$$
+
+In order to motivate our choice of RSR loss, we review a common formulation for the original RSR problem. In this problem one needs to recover a linear subspace, or equivalently an orthogonal projection $\mathbf { P }$ onto this subspace. Assume a dataset $\{ \mathbf { y } ^ { ( t ) } \} _ { t = 1 } ^ { N }$ and let $\mathbf { I }$ denote the identity matrix in the ambient space of the dataset. The goal is to find an orthogonal projector of dimension whose subspace robustly approximates this dataset. The least $q$ -th power deviations formulation for $q > 0$ , or least absolute deviations when $q = 1$ (Lerman & Maunu, 2018), seeks $\mathbf { P }$ that minimizes
+
+$$
+\hat { L } ( \mathbf { P } ) = \sum _ { t = 1 } ^ { N } \left\| \left( \mathbf { I } - \mathbf { P } \right) \mathbf { y } ^ { ( t ) } \right\| _ { 2 } ^ { q } .
+$$
+
+The solution of this problem is robust to some outliers when $q \ \leq \ 1$ (Lerman & Zhang, 2014; Lerman $\&$ Maunu, 2017); furthermore, $q < 1$ can result in a wealth of local minima and thus $q = 1$ is preferable (Lerman & Zhang, 2014; Lerman & Maunu, 2017).
+
+A similar loss function to (3) for RSRAE is
+
+$$
+\begin{array} { r l } & { L _ { \mathrm { R S R } } ^ { q } ( \mathbf { A } ) \ = \ \lambda _ { 1 } L _ { \mathrm { R S R } _ { 1 } } ( \mathbf { A } ) + \lambda _ { 2 } L _ { \mathrm { R S R } _ { 2 } } ( \mathbf { A } ) } \\ & { \mathrel { \mathop : } = \ \lambda _ { 1 } \displaystyle \sum _ { t = 1 } ^ { N } \left\| \mathbf { z } ^ { ( t ) } - \mathbf { A } ^ { \mathrm { T } } \underbrace { \mathbf { A } \mathbf { z } ^ { ( t ) } } _ { \tilde { \mathbf { z } } ^ { ( t ) } } \right\| _ { 2 } ^ { q } + \lambda _ { 2 } \left\| \mathbf { A } \mathbf { A } ^ { \mathrm { T } } - \mathbf { I } _ { d } \right\| _ { \mathrm { F } } ^ { 2 } \ , } \end{array}
+$$
+
+where ${ \bf A } ^ { \mathrm { T } }$ denotes the transpose of A, $\mathbf { I } _ { d }$ denotes the $d \times d$ identity matrix and $\lVert \cdot \rVert _ { \mathrm { F } }$ denotes the Frobenius norm. Here $\lambda _ { 1 } , \lambda _ { 2 } > 0$ are predetermined hyperparameters, though we later show that one may solve the underlying problem without using them. We note that the first term in the weighted sum of (4) is close to (3) as long as $\mathbf { A } ^ { \mathrm { T } } \mathbf { A }$ is close to an orthogonal projector. To enforce this requirement we introduced the second term in the weighted sum of (4). In Appendix C we discuss further properties of the RSR energy and its minimization.
+
+To emphasize the effect of outlier removal, we take $p = 1$ in (2) and $q = 1$ in (4). That is, we use the $l _ { 2 , 1 }$ norm, or the formulation of least absolute deviations, for both reconstruction and RSR. The loss function of RSRAE is the sum of the two loss terms in (2) and (4), that is,
+
+$$
+L _ { \mathrm { R S R A E } } ( \boldsymbol { \mathcal { E } } , \mathbf { A } , \mathcal { D } ) = L _ { \mathrm { A E } } ^ { 1 } ( \boldsymbol { \mathcal { E } } , \mathbf { A } , \mathcal { D } ) + L _ { \mathrm { R S R } } ^ { 1 } ( \mathbf { A } ) .
+$$
+
+We remark that the sole minimization of $L _ { \mathrm { A E } } ^ { 1 }$ , without $L _ { \mathrm { R S R } } ^ { 1 }$ , is not effective for anomaly detection.   
+We numerically demonstrate this in Section 4.3 and also try to explain it in Section 5.1.
+
+Our proposed algorithm for optimizing (5), which we refer to as the RSRAE algorithm, uses alternating minimization. It iteratively backpropagates the three terms $L _ { \mathrm { A E } } ^ { 1 }$ , $\boldsymbol { L } _ { \mathrm { R S R _ { 1 } } }$ , $\boldsymbol { L } _ { \mathrm { { R S R } _ { 2 } } }$ and accordingly updates the parameters of the RSR autoencoder. For clarity, we describe this basic procedure in Algorithm 1 of Appendix A. It is independent of the values of the parameters $\lambda _ { 1 }$ and $\lambda _ { 2 }$ . Note that the additional gradient step with respect to the RSR loss just updates the parameters in A. Therefore it does not significantly increase the epoch time of a standard autoencoder for anomaly detection. Another possible method, which we refer to as RSRAE+, is direct minimization of $L _ { \mathrm { R S R A E } }$ with predetermined $\lambda _ { 1 }$ and $\lambda _ { 2 }$ via auto-differentiation (see Algorithm 2 of Appendix A). Section 4.3 and Appendix I.2 demonstrate that in general, RSRAE performs better than RSRAE+, though it is possible that similar performance can be achieved by carefully tuning the parameters $\lambda _ { 1 }$ and $\lambda _ { 2 }$ when implementing RSRAE+.
+
+We remark that a standard autoencoder is obtained by minimizing only $L _ { \mathrm { A E } } ^ { 2 }$ , without the RSR loss. One might hope that minimizing $L _ { \mathrm { A E } } ^ { 1 }$ may introduce the needed robustness. However, Section 4.3 AE and Appendix I.2 demonstrate that results obtained by minimizing $L _ { \mathrm { A E } } ^ { 1 }$ or $L _ { \mathrm { A E } } ^ { 2 }$ are comparable, and are worse than those of RSRAE and RSRAE $^ +$ .
+
+# 4 EXPERIMENTAL RESULTS
+
+We test our method 1on five datasets: Caltech 101 (Fei-Fei et al., 2007), Fashion-MNIST (Xiao et al., 2017), Tiny Imagenet (a small subset of Imagenet (Russakovsky et al., 2015)), Reuters-21578 (Lewis, 1997) and 20 Newsgroups (Lang, 1995).
+
+Caltech 101 contains 9,146 RGB images labeled according to 101 distinct object categories. We take the 11 categories that contain at least 100 images and randomly choose 100 images per category. We preprocess all 1100 images to have size $3 2 \times 3 2 \times 3$ and pixel values normalized between $- 1$ and 1. In each experiment, the inliers are the 100 images from a certain category and we sample $c$ $\times ~ 1 0 0$ outliers from the rest of 1000 images of other categories, where $c \in \{ 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 \}$ .
+
+Fashion-MNIST contains $2 8 \times 2 8$ grayscale images of clothing and accessories, which are categorized into 10 classes. We use the test set which contains 10,000 images and normalize pixel values to lie in $[ - 1 , 1 ]$ . In each experiment, we fix a class and the inliers are the test images in this class. We randomly sample $c \times 1 { , } 0 0 0$ outliers from the rest of classes (here and below $c$ is as above). Since there are around 1000 test images in each class, the outlier ratio is approximately $c$ .
+
+Tiny Imagenet contains 200 classes of RGB images from a distinct subset of Imagenet. We select 10 classes with 500 training images per class. We preprocess the images to have size $3 2 \times 3 2 \times 3$ and pixel values in $[ - 1 , 1 ]$ . We further represent the images by deep features obtained by a ResNet (He et al., 2016) with dimension 256 (Appendix I.1 provides results for the raw images). In each experiment, 500 inliers are from a fixed class and $c \times 5 0 0$ outliers are from the rest of classes.
+
+Reuters-21578 contains 90 text categories with multi-labels. We consider the five largest classes with single labels and randomly sample from them 360 documents per class. The documents are preprocessed into vectors of size 26,147 by sequentially applying the TFIDF transformer and Hashing vectorizer (Rajaraman & Ullman, 2011). In each experiment, the inliers are the documents of a fixed class and $c \times 3 6 0$ outliers are randomly sampled from the other classes.
+
+20 Newsgroups contains newsgroup documents with 20 different labels. We sample 360 documents per class and preprocess them as above into vectors of size 10,000. In each experiment, the inliers are the documents from a fixed class and $c \times 3 6 0$ outliers are sampled from the other classes.
+
+# 4.1 BENCHMARKS AND SETTING
+
+We compare RSRAE with the following benchmarks: Local Outlier Factor (LOF) (Breunig et al., 2000), One-Class SVM (OCSVM) (Scholkopf et al., 2000; Amer et al., 2013), Isolation Forest (IF) ¨ (Liu et al., 2012), Deep Structured Energy Based Models (DSEBMs) (Zhai et al., 2016), Geometric Transformations (GT) (Golan & El-Yaniv, 2018), and Deep Autoencoding Gaussian Mixture Model (DAGMM) (Zong et al., 2018). Of those benchmarks, LOF, OCSVM and IF are traditional, while powerful methods, for unsupervised anomaly detection and do not involve neural networks. DSEBMs, DAGMM and GT are more recent and all involve neural networks. DSEBMs is built for unsupervised anomaly detection. DAGMM and GT are designed for semi-supervised anomaly detection, but allow corruption. We use them to learn a model for the inliers and assign anomaly scores using the combined set of both inliers and outliers. GT only applies to image data. We briefly describe these methods in Appendix E.
+
+We implemented DSEBMs, DAGMM and GT using the codes2 from Golan & El-Yaniv (2018) with minimal modification so that they adapt to the data described above and the available GPUs in our machine. The LOF, OCSVM and IF methods are adapted from the scikit-learn packages.
+
+We describe the structure of the RSRAE as follows. For the image datasets without deep features, the encoder consists of three convolutional layers: $5 \times 5$ kernels with 32 output channels, strides 2; $5 \times 5$ kernels with 64 output channels, strides 2; and $3 \times 3$ kernels with 128 output channels, strides 2. The output of the encoder is flattened and the RSR layer transforms it into a 10-dimensional vector. That is, we fix $d = 1 0$ in all experiments. The decoder consists of a dense layer that maps the output of the RSR layer into a vector of the same shape as the output of the encoder, and three deconvolutional layers: $3 \times 3$ kernels with 64 output channels, strides 2; $5 \times 5$ kernels with 32 output channels, strides 2; $5 \times 5$ kernels with 1 (grayscale) or 3 (RGB) output channels, strides 2. For the preprocessed document datasets or the deep features of Tiny Imagenet, the encoder is a fully connected network with size (32, 64, 128), the RSR layer linearly maps the output of the encoder to dimension 10, and the decoder is a fully connected network with size (128, 64, 32, $D$ ) where $D$ is the dimension of the input. Batch normalization is applied to each layer of the encoders and the decoders. The output of the RSR layer is $\ell _ { 2 }$ -normalized before applying the decoder. For DSEBMs and DAGMM we use the same number of layers and the same dimensions in each layer for the autoencoder as in RSRAE. For each experiment, the RSRAE model is optimized with Adam using a learning rate of 0.00025 and 200 epochs. The batch size is 128 for each gradient step. The setting of training is consistent for all the neural network based methods.
+
+The two main hyperparameters of RSRAE are the intrinsic dimension $d$ and learning rate. Their values were fixed above. Appendix G demonstrates stability to changes in these values.
+
+All experiments were executed on a Linux machine with 64GB RAM and four GTX1080Ti GPUs. For all experiments with neural networks, we used TensorFlow and Keras. We report runtimes in Appendix H.
+
+# 4.2 RESULTS
+
+We summarize the precision and recall of our experiments by the AUC (area under curve) and AP (average precision) scores. For completeness, we include the definitions of these common scores in Appendix E. We compute them by considering the outliers as “positive”. We remark that we did not record the precision-recall-F1 scores, as in Xia et al. (2015); Zong et al. (2018), since in practice it requires knowledge of the outlier ratio.
+
+Figs. 1 and 2 present the AUC and AP scores of RSRAE and the methods described in Section 4.1 for the datasets described above, where GT is only applied to image data without deep features. For each constant $c$ (the outlier ratio) and each method, we average the AUC and AP scores over 5 runs with different random initializations and also compute the standard deviations. For brevity of presentation, we report the averaged scores among all classes and designate the averaged standard deviations by bars.
+
+The results indicates that RSRAE clearly outperforms other methods in most cases, especially when $c$ is large. Indeed, the RSR layer was designed to handle large outlier ratios. For Fashion MNIST and Tiny Imagenet with deep features, IF performs similarly to RSRAE, but IF performs poorly on the document datasets. OCSVM is the closest to RSRAE for the document datasets but it is generally not so competitive for the image datasets.
+
+# 4.3 COMPARISON WITH VARIATIONS OF RSRAE
+
+We use one image dataset (Caltech 101) and one document dataset (Reuters-21578) and compare between RSRAE and three variations of it. The first one is RSRAE $^ +$ (see Section 3) with $\lambda _ { 1 } =$ $\lambda _ { 2 } ~ = ~ 0 . 1$ in (4) (these parameters were optimized on 20 Newsgroup, though results with other choices of parameters are later demonstrated in Section G.3). The next two are simpler autoencoders without RSR layers: AE-1 minimizes $L _ { \mathrm { A E } } ^ { 1 }$ , the $\ell _ { 2 , 1 }$ reconstruction loss; and AE minimizes $L _ { \mathrm { A E } } ^ { 2 }$ , the $\ell _ { 2 , 2 }$ reconstruction loss (it is a regular autoencoder for anomaly detection). We maintain the same architecture as that of RSRAE, including the matrix A, but use different loss functions.
+
+Fig. 3 reports the AUC and AP scores. We see that for the two datasets ${ \mathrm { R S R A E } } +$ with the prespecified $\lambda _ { 1 }$ and $\lambda _ { 2 }$ does not perform as well as RSRAE, but its performance is still better than AE and AE-1. This is expected since we chose $\lambda _ { 1 }$ and $\lambda _ { 2 }$ after few trials with a different dataset, whereas RSRAE is independent of these parameters. The performance of AE and AE-1 is clearly worse, and they are also not as good as some methods compared with in Section 4.2. At last, AE is generally comparable with AE-1. Similar results are noticed for the other datasets in Appendix I.2.
+
+# 5 RELATED THEORY FOR THE RSR PENALTY
+
+We explain here why we find it natural to incorporate RSR within a neural network. In Section 5.1 we first review the mathematical idea of an autoencoder and discuss the robustness of a linear autoencoder with an $\ell _ { 2 , 1 }$ loss (i.e., RSR loss). We then explain why a general autoencoder with an $\ell _ { 2 , 1 }$ loss is not expected to be robust to outliers and why an RSR layer can improve its robustness. Section 5.2 is a first step of extending this view to a generative network. It establishes some robustness of WGAN with a linear generator, but the extension of an RSR layer to WGAN is left as an open problem.
+
+# 5.1 ROBUSTNESS AND RELATED PROPERTIES OF AUTOENCODERS
+
+Mathematically, an autoencoder for a dataset $\{ \mathbf { x } ^ { ( t ) } \} _ { t = 1 } ^ { N } \subset \mathbb { R } _ { . } ^ { D }$ and a latent dimension $d < D$ is composed of an encoder $\mathcal { E } : \mathbb { R } ^ { D }  \mathbb { R } ^ { d }$ and a decoder $\mathbf { \dot { \mathcal { D } } } : \mathbb { R } ^ { d }  \mathbb { R } ^ { D }$ that minimize the following energy function with $p = 2$ :
+
+$$
+\sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathcal { D } \circ \mathcal { E } \big ( \mathbf { x } ^ { ( t ) } \big ) \right\| _ { 2 } ^ { p } ,
+$$
+
+where $\circ$ denotes function decomposition. It is a natural nonlinear generalization of PCA (Goodfellow et al., 2016). Indeed, in the case of a linear autoencoder, $\mathcal { E }$ and $\mathcal { D }$ are linear maps represented by matrices $\mathbf { E } \in \mathbb { R } ^ { d \times D }$ and $\mathbf { D } \in \mathbb { R } ^ { D \times d }$ , respectively, that need to minimize (among such matrices) the following loss function with $p = 2$
+
+$$
+\sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { D E x } ^ { ( t ) } \right\| _ { 2 } ^ { p } .
+$$
+
+We explain in Appendix D.1 that if $( \mathbf { D } ^ { \star } , \mathbf { E } ^ { \star } )$ is a minimizer of (7) with $p = 2$ (among $\mathbf { E } \in \mathbb { R } ^ { d \times D }$ and $\mathbf { D } \in \mathbb { R } ^ { D \times d } ,$ , then $\mathbf { D } ^ { \star } \mathbf { E } ^ { \star }$ is the orthoprojector on the $d$ -dimensional PCA subspace. This means, that the latent code $\{ \mathbf { E } ^ { \star } \mathbf { x } ^ { ( t ) } \} _ { t = 1 } ^ { N }$ parametrizes the PCA subspace and an additional application of $\mathbf { D } ^ { \star }$ to $\{ \mathbf { E } ^ { \star } \mathbf { x } ^ { ( t ) } \} _ { t = 1 } ^ { N }$ results in the projections of the data points $\{ \mathbf { x } ^ { ( t ) } \} _ { t = 1 } ^ { N }$ onto the PCA subspace. The recovery error for data points on this subspace is zero (as $\mathbf { D } ^ { \star } \mathbf { E } ^ { \star }$ is the identity on this subspace), and in general, this error is the Euclidean distance to the PCA subspace, $\big \| \mathbf { x } ^ { ( t ) } - \mathbf { D } ^ { \star } \mathbf { E } ^ { \star } \mathbf { x } ^ { ( t ) } \big \| _ { 2 }$ .
+
+Intuitively, the idea of a general autoencoder is the same. It aims to fit a nice structure, such as a manifold, to the data, where ideally $\mathcal { D } \circ \mathcal { E } ^ { \circ }$ is a projection onto this nice structure. This idea can only be made rigorous for data approximated by simple geometric structure, e.g., by a graph of a sufficiently smooth function.
+
+![](images/463fdcd915cfe260f68e7cd287643cbc19d35c3b63131a6b9b96504f638d9e72.jpg)  
+Figure 1: AUC and AP scores for RSRAE using Caltech 101 and Fashion MNIST.
+
+![](images/b83b9b1970901fe795e095f05e4b31d2694e98639254178f54eb3c1d316db156.jpg)  
+Figure 2: AUC and AP scores for RSRAE using Tiny Imagenet with deep features, Reuters-21578 and 20 Newsgroups.
+
+In order to extend these methods to anomaly detection, one needs to incorporate robust strategies, so that the methods can still recover the underlying structure of the inliers, and consequently assign lower recovery errors for the inliers and higher recovery errors for the outliers. For example, in the linear case, one may assume a set of inliers lying on and around a subspace and an arbitrary set of outliers (with some restriction on their fraction). PCA, and equivalently, the linear autoencoder that minimizes (7) with $p = 2$ , is not robust to general outliers. Thus it is not expected to distinguish well between inliers and outliers in this setting. As explained in Appendix D.1, minimizing (7) with $p = 1$ gives rise to the least absolute deviations subspace. This subspace can be robust to outliers under some conditions, but these conditions are restrictive (see examples in Lerman & Zhang (2014)). In order to deal with more adversarial outliers, it is advised to first normalize the data to the sphere (after appropriate centering) and then estimate the least absolute deviations subspace. This procedure was theoretically justified for a general setting of adversarial outliers in Maunu & Lerman (2019).
+
+As in the linear case, an autoencoder that uses the loss function in (6) with $p = 1$ may not be robust to adversarial outliers. Unlike the linear case, there are no simple normalizations for this case. Indeed, the normalization to the sphere can completely distort the structure of an underlying manifold and it is also hard to center in this case. Furthermore, there are some obstacles of establishing robustness for the nonlinear case even under special assumptions.
+
+Our basic idea for a robust autoencoder is to search for a latent low-dimensional code for the inliers within a larger embedding space. The additional RSR loss focuses on parametrizing the lowdimensional subspace of the encoded inliers, while being robust to outliers. Following the above discussion, we enhance such robustness by applying a normalization similar to the one discussed above, but adapted better to the structure of the network (see Section 4.1). The emphasis of the RSR layer is on appropriately encoding the inliers, where the encoding of the outliers does not matter. It is okay for the encoded outliers to lie within the subspace of the encoded inliers, as this will result in large recovery errors for the outliers. However, in general, most encoded outliers lie away from this subspace, and this is why such a mechanism is needed (otherwise, a regular autoencoder may obtain a good embedding).
+
+![](images/53365a7399a0d2f62c30ad510e2566fa04d3452c38d7013a7d34cd323b438aa9.jpg)  
+Figure 3: AUC and AP scores for RSRAE and alternative formulations using Caltech 101 and Reuters-21578.
+
+# 5.2 RELATIONSHIP OF THE RSR LOSS WITH LINEARLY GENERATED WGAN
+
+An open problem is whether RSR can be used within other neural network structures for unsupervised learning, such as variational autoencoders (VAEs) (Kingma & Welling, 2013) and generative adversarial networks (GANs) (Goodfellow et al., 2014). The latter two models are used in anomaly detection with a score function similar to the reconstruction error (An & Cho, 2015; Vasilev et al., 2018; Zenati et al., 2018; Kliger & Fleishman, 2018).
+
+While we do not solve this problem, we establish a natural relationship between RSR and Wasserstein-GAN (WGAN) (Arjovsky et al., 2017; Gulrajani et al., 2017) with a linear generator, which is analogous to the example of a linear autoencoder mentioned above.
+
+Let $W _ { p }$ denote the $p$ -Wasserstein distance in $\mathbb { R } ^ { D }$ ( $p \geq 1 \AA ,$ ). That is, for two probability distributions $\mu , \nu$ on $\mathbb { R } ^ { D }$ ,
+
+$$
+W _ { p } ( \mu , \nu ) = \left( \operatorname* { i n f } _ { \pi \in \Pi ( \mu , \nu ) } \mathbb { E } _ { ( \mathbf { x } , \mathbf { y } ) \sim \pi } \| \mathbf { x } - \mathbf { y } \| _ { 2 } ^ { p } \right) ^ { 1 / p } ,
+$$
+
+where $\Pi ( \mu , \nu )$ is the set of joint distributions with $\mu , \nu$ as marginals. We formulate the following proposition (while prove it later in Appendix D.2) and then interpret it.
+
+Proposition 5.1. Let $p \geq 1$ and $\mu$ be a Gaussian distribution on $\mathbb { R } ^ { D }$ with mean $\mathbf { m } _ { X } \in \mathbb { R } ^ { D }$ and full-rank covariance matrix $\pmb { \Sigma } _ { X } \in \mathbb { R } ^ { D \times D }$ (that is, $\mu$ is $\mathcal { N } ( \mathbf { m } _ { X } , \pmb { \Sigma } _ { X } ) )$ ). Then
+
+$$
+\begin{array} { r l } { \underset { \nu \mathrm { i s } } { \mathrm { m i n } } } & { W _ { p } ( \mu , \nu ) } \\ { \mathrm { s . t . } \quad } & { \mathbf { m } _ { Y } \in \mathbb { R } ^ { D } } \\ & { \mathrm { r a n k } ( \Sigma _ { \mathrm { Y } } ) = \mathrm { d } } \end{array}
+$$
+
+is achieved when $\mathbf { m } _ { Y } = \mathbf { m } _ { X }$ and $\Sigma _ { Y } = \mathbf { P } _ { \mathcal { L } } \pmb { \Sigma } _ { X } \mathbf { P } _ { \mathcal { L } }$ , where for $X \sim \mu$
+
+$$
+\begin{array} { r } { \mathcal { L } = \underset { \mathrm { d i m } \mathcal { L } = \mathrm { d } } { \mathrm { a r g m i n } } \ : \mathbb { E } \left\| X - { \bf P } \mathcal { L } X \right\| _ { 2 } ^ { p } . } \end{array}
+$$
+
+The setting of this proposition implicitly assumes a linear generator of WGAN. Indeed, the linear mapping, which can be represented by a $d \times D$ matrix, maps a distribution in $\mathcal { N } ( \mathbf { m } _ { X } , \pmb { \Sigma } _ { X } )$ into a distribution in $\mathcal { N } ( \mathbf { m } _ { Y } , \bar { \pmb { \Sigma } } _ { Y } )$ and reduces the rank of the covariance matrix from $D$ to $d$ . The proposition states that in this setting the underlying minimization is closely related to minimizing the loss function (3). Note that here $p \geq 1$ , however, if one further corrupts the sample, then $p = 1$ is the suitable choice (Lerman $\&$ Maunu, 2018). This choice is also more appropriate for WGAN, since there is no $p$ -WGAN for $p \neq 1$ .
+
+Nevertheless, training a WGAN is not exactly the same as minimizing the $W _ { 1 }$ distance (Gulrajani et al., 2017), since it is difficult to impose the Lipschitz constraint for a neural network. Furthermore, in practice, the WGAN generator, which is a neural network, is nonlinear, and thus its output is typically non-Gaussian. The robustness of WGAN with a linear autoencoder, which we established here, does not extend to a general WGAN (this is similar to our earlier observation that the robustness of a linear autoencoder with an RSR loss does not generalize to a nonlinear autoencoder). We believe that a similar structure like the RSR layer has to be imposed for enhancing the robustness of WGAN, and possibly also other generative networks, but we leave its effective implementation as an open problem.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+We constructed a simple but effective RSR layer within the autoencoder structure for anomaly detection. It is easy to use and adapt. We have demonstrated competitive results for image and document data and believe that it can be useful in many other applications.
+
+There are several directions for further exploration of the RSR loss in unsupervised deep learning models for anomaly detection. First, we are interested in theoretical guarantees for RSRAE. A more direct subproblem is understanding the geometric structure of the “manifold” learned by RSRAE. Second, it is possible that there are better geometric methods to robustly embed the manifold of inliers. For example, one may consider a multiscale incorporation of RSR layers, which we expand on in Appendix D.3. Third, one may try to incorporate an RSR layer in other neural networks for anomaly detection that use nonlinear dimension reduction. We hope that some of these methods may be easier to directly analyze than our proposed method. For example, we are curious about successful incorporation of robust metrics for GANs or WGANs. In particular, we wonder about extensions of the theory proposed here for WGAN when considering a more general setting.
+
+# ACKNOWLEDGMENTS
+
+This research has been supported by NSF award DMS18-30418. Part of this work was pursued when Dongmian Zou was a postdoctoral associate at the Institute for Mathematics and its Applications at the University of Minnesota. We thank Teng Zhang for his help with proving Proposition 5.1 (we discussed a related but different proposition with similar ideas of proofs). We thank Madeline Handschy for commenting on an earlier version of this paper.
+
+# REFERENCES
+
+Mennatallah Amer, Markus Goldstein, and Slim Abdennadher. Enhancing one-class support vector machines for unsupervised anomaly detection. In Proceedings of the ACM SIGKDD Workshop on Outlier Detection and Description, pp. 8–15. ACM, 2013.
+
+Jinwon An and Sungzoon Cho. Variational autoencoder based anomaly detection using reconstruction probability. Special Lecture on IE, 2(1), 2015.
+
+Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein generative adversarial networks. ´ In Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pp. 214–223, International Convention Centre, Sydney, Australia, 06–11 Aug 2017. PMLR. URL http://proceedings.mlr.press/v70/arjovsky17a.html.
+
+Caglar Aytekin, Xingyang Ni, Francesco Cricri, and Emre Aksu. Clustering and unsupervised anomaly detection with $l _ { 2 }$ normalized deep auto-encoder representations. In 2018 International
+
+Joint Conference on Neural Networks (IJCNN), pp. 1–6, July 2018. doi: 10.1109/IJCNN.2018.   
+8489068.
+
+Markus M Breunig, Hans-Peter Kriegel, Raymond T Ng, and Jorg Sander. LOF: identifying density- ¨ based local outliers. In ACM sigmod record, volume 29, 2, pp. 93–104. ACM, 2000.
+
+Raghavendra Chalapathy, Aditya Krishna Menon, and Sanjay Chawla. Robust, deep and inductive anomaly detection. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 36–51. Springer, 2017.
+
+Varun Chandola, ArindaFm Banerjee, and Vipin Kumar. Anomaly detection: A survey. ACM computing surveys (CSUR), 41(3):15, 2009.
+
+Jesse Davis and Mark Goadrich. The relationship between precision-recall and roc curves. In Proceedings of the 23rd International Conference on Machine Learning, ICML $^ { , 0 6 }$ , pp. 233– 240, New York, NY, USA, 2006. ACM. ISBN 1-59593-383-2. doi: 10.1145/1143844.1143874. URL http://doi.acm.org/10.1145/1143844.1143874.
+
+Fernando De La Torre and Michael J Black. A framework for robust subspace learning. International Journal of Computer Vision, 54(1-3):117–142, 2003.
+
+Chris Ding, Ding Zhou, Xiaofeng He, and Hongyuan Zha. R1-PCA: rotational invariant $l _ { 1 }$ -norm principal component analysis for robust subspace factorization. In Proceedings of the 23rd international conference on Machine learning, pp. 281–288. ACM, 2006.
+
+Li Fei-Fei, Rob Fergus, and Pietro Perona. Learning generative visual models from few training examples: An incremental bayesian approach tested on 101 object categories. Computer vision and Image understanding, 106(1):59–70, 2007.
+
+Izhak Golan and Ran El-Yaniv. Deep anomaly detection using geometric transformations. In Advances in Neural Information Processing Systems, pp. 9781–9791, 2018.
+
+Markus Goldstein and Seiichi Uchida. A comparative evaluation of unsupervised anomaly detection algorithms for multivariate data. PloS one, 11(4):e0152173, 2016.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014.
+
+Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep learning. MIT press, 2016.
+
+Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville. Improved training of Wasserstein GANs. In Advances in Neural Information Processing Systems, pp. 5767–5777, 2017.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In European conference on computer vision, pp. 630–645. Springer, 2016.
+
+Pan Ji, Tong Zhang, Hongdong Li, Mathieu Salzmann, and Ian Reid. Deep subspace clustering networks. In Advances in Neural Information Processing Systems, pp. 24–33, 2017.
+
+Peter W Jones. Rectifiable sets and the traveling salesman problem. Invent Math, 102(1):1–15, 1990.
+
+Ramakrishnan Kannan, Hyenkyun Woo, Charu C. Aggarwal, and Haesun Park. Outlier detection for text data. In Proceedings of the 2017 SIAM International Conference on Data Mining, pp. 489–497, 2017. URL https://epubs.siam.org/doi/abs/10.1137/1.9781611974973.55.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In 3rd International Conference for Learning Representations. arXiv:1412.6980, 2014.
+
+Diederik P Kingma and Max Welling. Auto-encoding variational Bayes. In 2nd International Conference for Learning Representations. arXiv:1312.6114, 2013.
+
+Mark Kliger and Shachar Fleishman. Novelty detection with GAN, 2018. URL https://openreview. net/forum?id=Hy7EPh10W.
+
+Ken Lang. Newsweeder: Learning to filter netnews. In Proceedings of the Twelfth International Conference on Machine Learning, pp. 331–339, 1995.
+
+Gilad Lerman and Tyler Maunu. Fast, robust and non-convex subspace recovery. Information and Inference: A Journal of the IMA, 7(2):277–336, 2017.
+
+Gilad Lerman and Tyler Maunu. An overview of robust subspace recovery. Proceedings of the IEEE, 106(8):1380–1410, 2018.
+
+Gilad Lerman and Teng Zhang. $l _ { p }$ -recovery of the most significant subspace among multiple subspaces with outliers. Constructive Approximation, 40(3):329–385, 2014.
+
+Gilad Lerman, Michael B McCoy, Joel A Tropp, and Teng Zhang. Robust computation of linear models by convex relaxation. Foundations of Computational Mathematics, 15(2):363–410, 2015.
+
+David Lewis. Reuters-21578 text categorization test collection. Distribution 1.0, AT&T LabsResearch, 1997.
+
+Jose Lezama, Qiang Qiu, Pablo Mus ´ e, and Guillermo Sapiro. OL ´ E: Orthogonal low-rank ´ embedding-a plug and play geometric loss for deep learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 8109–8118, 2018.
+
+Fei Tony Liu, Kai Ming Ting, and Zhi-Hua Zhou. Isolation-based anomaly detection. ACM Transactions on Knowledge Discovery from Data (TKDD), 6(1):3, 2012.
+
+Tyler Maunu and Gilad Lerman. Robust subspace recovery with adversarial outliers. CoRR, abs/1904.03275, 2019. URL http://arxiv.org/abs/1904.03275.
+
+Tyler Maunu, Teng Zhang, and Gilad Lerman. A well-tempered landscape for non-convex robust subspace recovery. arXiv preprint arXiv:1706.03896, 2017.
+
+Michael McCoy and Joel A Tropp. Two proposals for robust PCA using semidefinite programming. Electronic Journal of Statistics, 5:1123–1160, 2011.
+
+Randy Paffenroth, Kathleen Kay, and Les Servi. Robust PCA for anomaly detection in cyber networks. arXiv preprint arXiv:1801.01571, 2018.
+
+Fabian Pedregosa, Gael Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier ¨ Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, Jake VanderPlas, Alexandre Passos, David Cournapeau, Matthieu Brucher, Matthieu Perrot, and Edouard Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research, 12: 2825–2830, 2011.
+
+Anand Rajaraman and Jeffrey David Ullman. Mining of massive datasets. Cambridge University Press, 2011.
+
+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International journal of computer vision, 115(3):211–252, 2015.
+
+Bernhard Scholkopf, Robert C Williamson, Alex J Smola, John Shawe-Taylor, and John C Platt. ¨ Support vector method for novelty detection. In Advances in neural information processing systems, pp. 582–588, 2000.
+
+Mei-Ling Shyu, Shu-Ching Chen, Kanoksri Sarinnapakorn, and LiWu Chang. A novel anomaly detection scheme based on principal component classifier. In Proc. ICDM Foundation and New Direction of Data Mining workshop, 2003, pp. 172–179, 2003.
+
+Aleksei Vasilev, Vladimir Golkov, Ilona Lipp, Eleonora Sgarlata, Valentina Tomassini, Derek K Jones, and Daniel Cremers. q-space novelty detection with variational autoencoders. arXiv preprint arXiv:1806.02997, 2018.
+
+Namrata Vaswani and Praneeth Narayanamurthy. Static and dynamic robust PCA and matrix completion: A review. Proceedings of the IEEE, 106(8):1359–1379, 2018.   
+G. Alistair Watson. Some Problems in Orthogonal Distance and Non-Orthogonal Distance Regression. Defense Technical Information Center, 2001. URL http://books.google.com/books?id= WKKWGwAACAAJ.   
+John Wright, Arvind Ganesh, Shankar Rao, Yigang Peng, and Yi Ma. Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization. In Advances in neural information processing systems, pp. 2080–2088, 2009.   
+Yan Xia, Xudong Cao, Fang Wen, Gang Hua, and Jian Sun. Learning discriminative reconstructions for unsupervised outlier removal. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1511–1519, 2015.   
+Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017.   
+Huan Xu, Constantine Caramanis, and Sujay Sanghavi. Robust PCA via outlier pursuit. IEEE Trans. Information Theory, 58(5):3047–3064, 2012. doi: 10.1109/TIT.2011.2173156.   
+Houssam Zenati, Chuan Sheng Foo, Bruno Lecouat, Gaurav Manek, and Vijay Ramaseshan Chandrasekhar. Efficient GAN-based anomaly detection, 2018. URL https://openreview.net/forum? id=BkXADmJDM.   
+Shuangfei Zhai, Yu Cheng, Weining Lu, and Zhongfei Zhang. Deep structured energy based models for anomaly detection. In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, pp. 1100–1109, 2016.   
+Teng Zhang and Gilad Lerman. A novel M-estimator for robust PCA. Journal of Machine Learning Research, 15(1):749–808, 2014.   
+Teng Zhang, Arthur Szlam, and Gilad Lerman. Median K-flats for hybrid linear modeling with many outliers. In Computer Vision Workshops (ICCV Workshops), 2009 IEEE 12th International Conference on, pp. 234–241. IEEE, 2009.   
+Chong Zhou and Randy C Paffenroth. Anomaly detection with robust deep autoencoders. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 665–674. ACM, 2017.   
+Bo Zong, Qi Song, Martin Renqiang Min, Wei Cheng, Cristian Lumezanu, Daeki Cho, and Haifeng Chen. Deep autoencoding gaussian mixture model for unsupervised anomaly detection. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id= BJJLHbb0-.
+
+# A DETAILS OF RSRAE AND RSRAE $^ +$
+
+The implementations of both RSRAE and ${ \mathrm { R S R A E } } +$ are simple. For completeness we provide here their details in algorithm boxes. The codes will be later posted in a supplementary webpage. Algorithm 1 describes RSRAE, which minimizes (5) by alternating minimization. It denotes the vectors of parameters of the encoder and decoder by $\pmb \theta$ and $\varphi$ , respectively.
+
+# Algorithm 1 RSRAE
+
+Input: Data $\{ \mathbf { x } ^ { ( t ) } \} _ { t = 1 } ^ { N }$ ; thresholds $\epsilon _ { \mathrm { A E } }$ , $\epsilon _ { \mathrm { R S R _ { 1 } } }$ , $\mathrm { \epsilon _ { R S R _ { 2 } } }$ , $\epsilon _ { \mathrm { T } }$ ; architecture and initial parameters of ${ \mathcal { E } } ^ { \mathcal { C } } , { \mathcal { D } }$ , A (including number of columns of $\mathbf { A }$ ); number of epochs batches; learning rate for backpropagation; similarity measure
+
+Output: Labels of data points as normal or anomalous   
+1: for each epoch do   
+2: Divide input data into batches   
+3: for each batch do   
+4: if $L _ { \mathrm { A E } } ^ { 1 } ( \pmb { \theta } , \mathbf { A } , \pmb { \varphi } ) > \epsilon _ { \mathrm { A E } }$ then   
+5: Backpropagate $L _ { \mathrm { A E } } ^ { 1 } ( \theta , \mathbf { A } , \varphi )$ w.r.t. $\theta , \mathbf { A } , \varphi \&$ update $\theta , \mathbf { A } , \varphi$   
+6: end if   
+7: if $L _ { \mathrm { R S R } _ { 1 } } ^ { 1 } ( \mathbf { A } ) > \epsilon _ { \mathrm { R S R } _ { 1 } }$ then   
+8: Backpropagate $L _ { \mathrm { R S R } _ { 1 } } ^ { 1 } ( \mathbf { A } )$ w.r.t. A $\&$ update A   
+9: end if   
+10: if $L _ { \mathrm { R S R } _ { 2 } } ^ { 1 } ( \mathbf { A } ) > \epsilon _ { \mathrm { R S R } _ { 2 } }$ then   
+11: Backpropagate $L _ { \mathrm { R S R _ { 2 } } } ^ { 1 } ( \mathbf { A } )$ w.r.t. A $\&$ update A   
+12: end if   
+13: end for   
+14: end for   
+15: for $t = 1 , \ldots , N$ do   
+16: Calculate similarity between $\mathbf { x } ^ { ( t ) }$ and $\tilde { \mathbf { x } } ^ { ( t ) }$   
+17: if similarity $\geq \epsilon _ { \mathrm { T } }$ then   
+18: $\mathbf { x } ^ { ( t ) }$ is normal   
+19: else   
+20: $\mathbf { x } ^ { ( t ) }$ is anomalous   
+21: end if   
+22: end for   
+23: return Normality labels for $t = 1 , \ldots , N$
+
+We clarify some guidelines for choosing default parameters, which we follow in all reported experiments. We set $\epsilon _ { \mathrm { A E } }$ , $\epsilon _ { \mathrm { R S R _ { 1 } } }$ and $\mathrm { \epsilon _ { R S R _ { 2 } } }$ to be zero. In general, we use networks with dense layers but for image data we use convolutional layers. We prefer using tanh as the activation function due to its smoothness. However, for a dataset that does not lie in the unit cube, we use either a ReLU function if all of its coordinates are positive, or a leaky ReLU function otherwise. The network parameters and the elements of A are initialized to be i.i.d. standard normal. In all numerical experiments, we set the number of columns of A to be 10, that is, $d = 1 0$ . The learning rate is chosen so that there is a sufficient improvement of the loss values after each epoch. Instead of fixing $\epsilon _ { \mathrm { T } }$ , we report the AUC and AP scores for different values of $\epsilon _ { \mathrm { T } }$ .
+
+Algorithm 2 describes RSRAE $^ +$ , which minimizes (5) with fixed $\lambda _ { 1 }$ and $\lambda _ { 2 }$ by auto-differentiation.
+
+# Algorithm 2 RSRAE+
+
+Input: Data ing num $\{ \mathbf { x } ^ { ( t ) } \} _ { t = 1 } ^ { N }$ ; threshmns of ds AE, ); para $\epsilon _ { \mathrm { T } }$ ; architecture and initial paraers of the the energy function f ; $\mathcal { E } ^ { \mathcal { O } }$ , $\mathcal { D }$ , A (includ-er of epochs $\mathbf { A }$ $\lambda _ { 1 } , \lambda _ { 2 }$ $\&$ batches; learning rate for backpropagation; similarity measure
+
+Output: Labels of data points as normal or anomalous   
+1: for each epoch do   
+2: Divide input data into batches   
+3: for each batch do   
+4: if $L _ { \mathrm { A E } } ^ { 1 } ( \pmb { \theta } , \mathbf { A } , \pmb { \varphi } ) > \epsilon _ { \mathrm { A E } }$ then   
+5: Backpropagate $L _ { \mathrm { A E } } ^ { 1 } ( \theta , \mathbf { A } , \varphi ) + \lambda _ { 1 } L _ { \mathrm { R S R } _ { 1 } } ^ { 1 } ( \mathbf { A } ) + \lambda _ { 2 } L _ { \mathrm { R S R } _ { 2 } } ^ { 1 } ( \mathbf { A } )$ w.r.t. $\theta , \mathbf { A } , \varphi$ & update   
+θ, A, ϕ   
+6: end if   
+7: end for   
+8: end for   
+9: for $t = 1 , \ldots , N$ do   
+10: Calculate similarity between $\mathbf { x } ^ { ( t ) }$ and $\tilde { \mathbf { x } } ^ { ( t ) }$   
+11: if similarity $\geq \epsilon _ { \mathrm { T } }$ then   
+12: $\mathbf { x } ^ { ( t ) }$ is normal   
+13: else   
+14: $\mathbf { x } ^ { ( t ) }$ is anomalous   
+15: end if   
+16: end for   
+17: return Normality labels for $t = 1 , \ldots , N$
+
+# B DEMONSTRATION OF RSRAE FOR ARTIFICIAL DATA
+
+For illustrating the performance of RSRAE, in comparison with a regular autoencoder, we consider a simple artificial geometric example. We assume corrupted data whose normal part is embedded in a “Swiss roll manifold”3, which is a two-dimensional manifold in $\mathbb { R } ^ { 3 }$ . More precisely, the normal part is obtained by mapping 1,000 points uniformly sampled from the rectangle $[ 3 \pi / \dot { 2 } , 9 \pi / 2 ] \times [ 0 , \dot { 2 } 1 ]$ into $\mathbb { R } ^ { 3 }$ by the function
+
+$$
+( s , t ) \mapsto ( t \cos ( t ) , s , t \sin ( t ) ) .
+$$
+
+The anomalous part is obtained by i.i.d. sampling of 500 points from an isotropic Gaussian distribution in $\mathbb { R } ^ { 3 }$ with zero mean and standard deviation 2 in any direction. Fig. 4a illustrates such a sample, where the inliers are in black and the outliers are in blue. We remark that Fig 5a is identical.
+
+We construct the RSRAE with the following structure. The encoder is composed of fully-connected layers of sizes (32, 64, 128). The decoder is composed of fully connected layers of sizes (128, 64, 32, 3). Each fully connected layer is activated by the leaky ReLU function with $\alpha = 0 . 2$ . The intrinsic dimension for the RSR layer, that, is the number of columns of $\mathbf { A }$ , is $d = 2$ .
+
+For comparison, we construct the regular autoencoder AE (see Section 4.3). Recall that both of them have the same architecture (including the linear map A), but AE minimizes the $\ell _ { 2 }$ loss function in (6) (with $p = 2 \AA$ ) without an additional RSR loss. We optimize both models with 10,000 epochs and a batch gradient descent using Adam (Kingma & Ba, 2014) with a learning rate of 0.01.
+
+The reconstructed data $( \tilde { \mathbf { X } } )$ using RSRAE and AE are plotted in Figs. 4d and 5d, respectively. We further demonstrate the output obtained by the encoder and the RSR layer. The output of the encoder, $\mathbf { Z } = { \mathcal { E } } ( \mathbf { X } )$ , lies in $\mathbb { R } ^ { 1 2 8 }$ . For visualization purposes we project it onto a $\mathbb { R } ^ { 3 }$ as follows. We first find two vectors that span the image of A and we add to it the “principal direction” of $\mathbf { Z }$ orthogonal to the span of A. We project $\mathbf { Z }$ onto the span of these 3 vectors. Figs. 4b and 5b show these projections for RSRAE and AE, respectively. Figs. 4c and 5c demonstrate the respective mappings of $\mathbf { Z }$ by $\mathbf { A }$ during the RSR layer.
+
+Figs. 4d and 5d imply that the set of reconstructed normal points in RSRAE seem to lie on the original manifold, whereas the reconstructed normal points by AE seem to only lie near, but often not on the Swiss roll manifold. More importantly, the anomalous points reconstructed by RSRAE seem to be sufficiently far from the set of original anomalous points, unlike the reconstructed points by AE. Therefore, RSRAE can better distinguish anomalies using the distance between the original and reconstructed points, where small values are obtained for normal points and large ones for anomalous ones. Fig. 6 demonstrates this claim. They plot the histograms of the distance between the original and reconstructed points when applying RSRAE and AE, where distances for normal and anomalous points are distinguished by color. Clearly, RSRAE distinguishes normal and anomalous data better than AE.
+
+![](images/c5aba650bc85302a8766e5e78f1b8e4cd269a288c422e1ed0a989109d57fa4de.jpg)  
+Figure 4: Demonstration of the output of the encoder, RSR layer and decoder of RSRAE on a corrupted Swiss roll dataset.
+
+![](images/1078bb35f49a0ceec613f441286aab974f3b108803c5a24648ea101cc1bde37a.jpg)  
+Figure 5: Demonstration of the output of the encoder, mapping by A, and decoder of AE on a corrupted Swiss roll dataset.
+
+![](images/f84ecd5c4b210c1c8c19492c4aee183cf32ef13c503358057886e2e19690fd48.jpg)  
+Figure 6: Demonstration of the reconstruction error distribution for RSRAE and AE.
+
+# C FURTHER DISCUSSION OF THE RSR TERM
+
+The RSR energy in (4) includes two different terms. The proposition below indicates that the second term of (4) is zero when plugging into it the solution of the minimization of the first term of (4) with the additional requirement that A has full rank. That is, in theory, one may only minimize the first term of (4) over the set of matrices $\mathbf { A } \in \mathbb { R } ^ { d \times D }$ with full rank. We then discuss computational issues of this different minimization.
+
+Proposition C.1. Assume that $\{ \mathbf { z } ^ { ( t ) } \} _ { t = 1 } ^ { N } \subset \mathbb { R } ^ { D }$ spans $\mathbb { R } ^ { D }$ , $d \leqslant D$ and let
+
+$$
+\mathbf { A } ^ { \star } = \underset { \mathbf { A } \in \mathbb { R } ^ { d \times D } } { \operatorname { a r g m i n } } \sum _ { t = 1 } ^ { N } \left\| \mathbf { z } ^ { ( t ) } - \mathbf { A } ^ { \mathrm { T } } \mathbf { A } \mathbf { z } ^ { ( t ) } \right\| _ { 2 } .
+$$
+
+Then $\mathbf { A } ^ { \star } \mathbf { A } ^ { \star \mathrm { T } } = \mathbf { I } _ { d }$
+
+Proof. Let $\mathbf { A } ^ { \star }$ be an optimizer of (12) and $\mathbf { P } ^ { \star }$ denote the orthogonal projection onto the range of $\mathbf { A } ^ { \star \mathrm { T } } \mathbf { A } ^ { \star }$ . Note that $\mathbf { P } ^ { \star }$ can be written as $\tilde { \mathbf { A } } ^ { \mathrm { T } } \tilde { \mathbf { A } }$ , where $\tilde { \mathbf { A } }$ is a $d \times D$ matrix composed of an orthonormal basis of the range of $\mathbf { P } ^ { \star }$ . Therefore, being an optimum of (12), $\mathbf { A } ^ { \star }$ satisfies
+
+$$
+\left\| \mathbf { z } ^ { ( t ) } - \mathbf { P } ^ { \star } \mathbf { z } ^ { ( t ) } \right\| _ { 2 } \geq \left\| \mathbf { z } ^ { ( t ) } - \mathbf { A } ^ { \star \mathrm { T } } \mathbf { A } ^ { \star } \mathbf { z } ^ { ( t ) } \right\| _ { 2 } , \quad t = 1 , \cdots , N .
+$$
+
+On the other hand, the definition of orthogonal projection implies that
+
+$$
+\left\| \mathbf { z } ^ { ( t ) } - \mathbf { P } ^ { \star } \mathbf { z } ^ { ( t ) } \right\| _ { 2 } \leq \left\| \mathbf { z } ^ { ( t ) } - \mathbf { A } ^ { \star \mathrm { T } } \mathbf { A } ^ { \star } \mathbf { z } ^ { ( t ) } \right\| _ { 2 } , \quad t = 1 , \cdots , N .
+$$
+
+That is, equality is obtained in (13) and (14). This equality and the fact that $\mathbf { P } ^ { \star }$ is a projection on the range of $\mathbf { A } ^ { \star \mathrm { T } } \mathbf { A } ^ { \star }$ imply that
+
+$$
+{ \bf P } ^ { \star } { \bf z } ^ { ( t ) } = { \bf A } ^ { \star \mathrm { T } } { \bf A } ^ { \star } { \bf z } ^ { ( t ) } , \quad t = 1 , \cdots , N .
+$$
+
+Since $\{ \mathbf { z } ^ { ( t ) } \} _ { t = 1 } ^ { N }$ spans $\mathbb { R } ^ { D }$ , (15) results in
+
+$$
+\mathbf { P ^ { \star } } = \mathbf { A ^ { \star \mathrm { T } } A ^ { \star } } ,
+$$
+
+which further implies that
+
+$$
+\mathbf { A } ^ { \star } \mathbf { A } ^ { \star \mathrm { T } } \mathbf { A } ^ { \star } = \mathbf { A } ^ { \star } \mathbf { P } ^ { \star } = \mathbf { A } ^ { \star } \mathbf { \Lambda } .
+$$
+
+Combining this observation $( \mathbf { A } ^ { \star } \mathbf { A } ^ { \star \mathrm { T } } \mathbf { A } ^ { \star } \mathbf { \Lambda } = \mathbf { \Lambda } \mathbf { A } ^ { \star } \mathbf { \Lambda } )$ ) with the constraint that $\mathbf { A } ^ { \star }$ has a full rank, we conclude that $\mathbf { A } ^ { \star } \mathbf { A } ^ { \star \mathrm { T } } = \mathbf { I } _ { d }$ .
+
+The minimization in (12) is nonconvex and intractable. Nevertheless, Lerman & Maunu (2017) propose a heuristic to solve it with some weak guarantees and Maunu et al. (2017) propose an algorithm with guarantees under some conditions. However, such a minimization is even more difficult when applied to the combined energy in (5), instead of (4). Therefore, we find it necessary to include the second term in (4) that imposes the nearness of $\mathbf { A ^ { \mathrm { T } } A }$ to an orthogonal projection (equivalently, of $\mathbf { A A } ^ { \mathrm { T } }$ to the identity).
+
+# D MORE ON RELATED THEORY FOR THE RSR PENALTY
+
+In Section D.1 we characterize the solution of (7) via a subspace problem. Special case solutions to this problem include both the PCA subspace and the least absolute deviations subspace. In Section D.2 we prove Proposition 5.1. In Section D.3 we review some pure mathematical work that we find relevant to this discussion.
+
+# D.1 PROPERTY OF LINEAR AUTOENCODERS
+
+The following proposition expresses the solution of (7) in terms of another minimization problem.   
+After proving it, we clarify that the other minimization problem is related to both PCA and RSR.
+
+Proposition D.1. Let $p \geq 1$ , $d < D$ , and $\{ \mathbf { x } ^ { ( t ) } \} _ { t = 1 } ^ { N } \subset \mathbb { R } ^ { D }$ be a dataset with rank at least d. If $( \mathbf { D } ^ { \hat { \star } } , \mathbf { E } ^ { \star } ) \in \mathbb { R } ^ { D \times d } \times \hat { \mathbb { R } } ^ { d \times D }$ is a minimizer of (7), then
+
+$$
+\mathbf { D } ^ { \star } \mathbf { E } ^ { \star } = \mathbf { P } ^ { \star } \ ,
+$$
+
+where $\mathbf { P } ^ { \star } \in \mathbb { R } ^ { D \times D }$ is a minimizer of
+
+$$
+\sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { P x } ^ { ( t ) } \right\| _ { 2 } ^ { p } ,
+$$
+
+among all orthoprojectors $\mathbf { P }$ (that is, $\mathbf { P } = \mathbf { P } ^ { T }$ and $\mathbf { P } ^ { 2 } = \mathbf { P } .$ ) of rank $d$
+
+Proof. Let $\mathbf { P } ^ { \diamond }$ be a minimizer of (19) and $( \mathbf { D } ^ { \star } , \mathbf { E } ^ { \star } )$ be a minimizer of (7). Since $\mathbf { P } ^ { \diamond }$ is an orthoprojector of rank $d$ it can be written as $\mathbf { P } ^ { \circ } = \mathbf { U } ^ { \circ } \mathbf { U } ^ { \circ \mathrm { T } }$ , where $\mathbf { U } ^ { \diamond } \in \mathbb { R } ^ { D \times d }$ , and thus
+
+$$
+\sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { D } ^ { \star } \mathbf { E } ^ { \star } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } ^ { p } \leq \sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { U } ^ { \diamond } \mathbf { U } ^ { \diamond \mathrm { T } } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } ^ { p } = \sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { P } ^ { \diamond } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } ^ { p } .
+$$
+
+Let $\mathcal { L }$ denote the column space of $\mathbf { D } ^ { \star } \mathbf { E } ^ { \star }$ . Then by the property of orthoprojection
+
+$$
+\left\| { \bf x } ^ { ( t ) } - { \bf D } ^ { \star } { \bf E } ^ { \star } { \bf x } ^ { ( t ) } \right\| _ { 2 } \geq \left\| { \bf x } ^ { ( t ) } - { \bf P } _ { \mathcal { L } } { \bf x } ^ { ( t ) } \right\| _ { 2 } \mathrm { f o r } 1 \leq t \leq N
+$$
+
+and consequently
+
+$$
+\sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { D } ^ { \star } \mathbf { E } ^ { \star } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } ^ { p } \geq \sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { P } _ { \mathcal { L } } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } ^ { p } \geq \sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { P } ^ { \diamond } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } ^ { p } .
+$$
+
+The combination of (20) and (22) yields the following two equalities
+
+$$
+\sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { P } _ { \mathcal { L } } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } ^ { p } = \sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { P } ^ { \diamond } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } ^ { p } ,
+$$
+
+$$
+\sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { D } ^ { \star } \mathbf { E } ^ { \star } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } ^ { p } = \sum _ { t = 1 } ^ { N } \left\| \mathbf { x } ^ { ( t ) } - \mathbf { P } \mathcal { L } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } ^ { p } .
+$$
+
+We note that (23) implies that $\mathbf { P } _ { \mathcal { L } }$ is a minimizer of (19) (among all rank $d$ orthoprojectors). We further note that (21) and (24) yield that for all $1 \leq t \leq N$
+
+$$
+\left\| \mathbf { x } ^ { ( t ) } - \mathbf { D } ^ { \star } \mathbf { E } ^ { \star } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } = \left\| \mathbf { x } ^ { ( t ) } - \mathbf { P } _ { \mathcal { L } } \mathbf { x } ^ { ( t ) } \right\| _ { 2 } .
+$$
+
+Since $\mathbf { D } ^ { \star } \mathbf { E } ^ { \star } \mathbf { x } ^ { ( t ) } \in \mathcal { L }$ and $\mathbf { P } _ { \mathcal { L } }$ is an orthoprojector we conclude from (25) that
+
+$$
+\mathbf { D } ^ { \star } \mathbf { E } ^ { \star } \mathbf { x } ^ { ( t ) } = \mathbf { P } _ { \mathcal { L } } \mathbf { x } ^ { ( t ) } \mathrm { ~ f o r ~ } 1 \leq t \leq N .
+$$
+
+We note that the definition of $( \mathbf { D } ^ { \star } , \mathbf { E } ^ { \star } )$ implies that $\mathcal { L }$ (which is the column space of $\mathbf { D } ^ { \star } \mathbf { E } ^ { \star } )$ i s contained in the span of $\{ \mathbf { x } ^ { ( t ) } \} _ { t = 1 } ^ { N }$ . We also recall that the dimension of the span of $\{ \mathbf { x } ^ { ( t ) } \} _ { t = 1 } ^ { N }$ is at least the dimension of $\mathcal { L }$ , that is, . Combining the latter facts with (26) we obtain that $\mathbf { D ^ { \star } } \mathbf { E ^ { \star } } = \mathbf { P } _ { \mathcal { L } }$ . This and the fact that $\mathbf { P } _ { \mathcal { L } }$ is a minimizer of (19) (which was derived from (23)) concludes (18).
+
+Note that when $p = 2$ , the energy function in (19) corresponds to PCA. More precisely, a minimizer $\mathbf { P } ^ { \star }$ of (19) (among rank $d$ orthoprojectors) is an orthoprojector on a $d$ -dimensional PCA subspace, equivalently, a subspace spanned by top $d$ eigenvectors of the sample covariance (we assume for simplicity linear, and not affine, autoencoder, so the PCA subspace is linear and thus when $p = 2$ the data is centered at the origin). This minimizer is unique if and only if the $d$ -th eigenvalue of the sample covariance is larger than the $( d + 1 )$ -st eigenvalue. These elementary facts are reviewed in Section II-A of Lerman & Maunu (2018).
+
+When $p = 1$ , the minimizer $\mathbf { P } ^ { \star }$ of (19) (among rank $d$ orthoprojectors) is an orthoprojector on the $d$ -dimensional least absolute deviations subspace. This subspace is reviewed in Section II-D of Lerman & Maunu (2018) as a common approach for RSR. The minimizer is often not unique, where sufficient and necessary conditions for local minima of (19) are studied in Lerman & Zhang (2014).
+
+# D.2 PROOF OF PROPOSITION 5.1
+
+Proof. We denote the subspace $\mathcal { L }$ in the left hand side of (10) by ${ \mathcal { L } } ^ { \star }$ in order to distinguish it from the generic notation $\mathcal { L }$ for subspaces. Consider the random variable $X \sim \mu$ , Where $\mu$ is $\mathcal { N } ( { \bf m } _ { X } , \bar { \Sigma } _ { X } )$ . Fix $\pi \in \Pi ( \mu , \nu )$ . We note that
+
+$$
+\begin{array} { r l } & { { \mathbb { E } } _ { ( X , Y ) \sim \pi } \left\| X - Y \right\| _ { 2 } ^ { p } } \\ { = } & { \displaystyle \int _ { { \mathbb R } ^ { D } } \displaystyle \int _ { { \mathbb R } ^ { D } } \| { \bf x } - { \bf y } \| _ { 2 } ^ { p } \pi ( { \bf x } , { \bf y } ) \mathrm { d } { \bf x } \ \mathrm { d } { \bf y } } \\ & { \displaystyle \geq \ \operatorname* { m i n } _ { \mathrm { d i m } \mathcal { L } = \mathrm { d } } \displaystyle \int _ { { \mathbb R } ^ { D } } \mathrm { d i s t } ( { \bf x } , \mathcal { L } ) ^ { \mathrm { p } } \int _ { { \mathbb R } ^ { D } } \pi ( { \bf x } , { \bf y } ) \mathrm { d } { \bf y } \ \mathrm { d } { \bf x } } \\ { = } & { \displaystyle \operatorname* { m i n } _ { \mathrm { d i m } \mathcal { L } = \mathrm { d } } \displaystyle \int _ { { \mathbb R } ^ { D } } \mathrm { d i s t } ( { \bf x } , \mathcal { L } ) ^ { \mathrm { p } } \mu ( { \bf x } ) \ \mathrm { d } { \bf x } } \\ { = } & { \displaystyle \operatorname* { m i n } _ { \mathrm { d i m } \mathcal { L } = \mathrm { d } } { \mathbb { E } } \| X - { \bf P } _ { \mathcal { L } } X \| _ { 2 } ^ { p } \ . } \end{array}
+$$
+
+The inequality in (27) holds since $X$ is fixed and $Y$ satisfies $( X , Y ) \sim \pi$ , so the distribution of $Y$ is $\mathcal { N } ( \mathbf { m } _ { Y } , \pmb { \Sigma } _ { Y } )$ . Therefore, almost surely, $Y$ takes values in the $d$ -dimensional affine subspace $\{ \mathbf { y } \in \mathbb { R } ^ { D } : \mathbf { y } - \mathbf { m } _ { Y } \in \mathrm { r a n g e } ( \pmb { \Sigma } _ { \mathrm { Y } } ) \}$ . Furthermore, we note that equality in (27) is achieved when $Y = \mathbf { P } _ { \mathcal { L } ^ { \star } } X$ .
+
+We conclude the proof by showing that
+
+$$
+\mathbf { m } _ { X } \in \mathcal { L } ^ { \star } .
+$$
+
+Indeed, (28) implies that the orthogonal projection of $X \sim \mathcal { N } ( \mathbf { m } _ { X } , \pmb { \Sigma } _ { X } )$ onto ${ \mathcal { L } } ^ { \star }$ results in a random variable with distribution $\nu$ which is $\mathcal { N } ( \mathbf { m } _ { X } , \mathbf { P } _ { \mathcal { L } ^ { \star } } \pmb { \Sigma } _ { X } \mathbf { P } _ { \mathcal { L } ^ { \star } } )$ . By the above observation about the optimality of $Y = \mathbf { P } _ { \mathcal { L } ^ { \star } } X$ , the density of this distribution is the optimal solution of (9).
+
+To prove (28), we assume without loss of generality that $\mathbf { m } _ { X } = \mathbf { 0 }$ . Denote the orthogonal projection of the origin onto the affine subspace ${ \mathcal { L } } ^ { \star }$ by $\mathbf { m } \mathcal { L } ^ { \star }$ and let $\mathcal { L } _ { 0 } = \mathcal { L } ^ { \star } - \mathbf { m } _ { \mathcal { L } ^ { \star } }$ . We need to show that ${ \mathcal L } ^ { \star } = { \mathcal L } _ { 0 }$ , or equivalently, $\mathbf { m } _ { \mathcal { L } ^ { \star } } = \mathbf { 0 }$ . We note $\mathcal { L } _ { 0 }$ is a linear subspace, $\mathbf { m } \mathcal { L } ^ { \star }$ is orthogonal to $\mathcal { L } _ { 0 }$ and thus there exists a rotation matrix $\mathbf { O }$ such that
+
+$$
+\begin{array} { r } {  { \mathbf { O } } \mathcal { L } _ { 0 } = \{ ( 0 , \cdots , 0 , z _ { D - d + 1 } , \cdots , z _ { D } ) : z _ { D - d + 1 } , \cdots z _ { D } \in \mathbb { R } \} , } \end{array}
+$$
+
+and
+
+$$
+\mathbf { O m } _ { \mathcal { L } ^ { \star } } = ( m _ { 1 } , \cdots , m _ { D - d } , 0 , \cdots , 0 ) .
+$$
+
+For any $\mathbf { x } \in \mathbb { R } ^ { D }$ we note that $\mu ( \mathbf { x } ) = \mu ( - \mathbf { x } )$ since $\mu$ is Gaussian. Using this observation, other basic observations and the notation $\mathbf { O x } = ( x _ { 1 } ^ { \prime } , \cdot \cdot \cdot , x _ { D } ^ { \prime } )$ we obtain that
+
+$$
+\begin{array} { r l } & { \mathrm { d i s t } ( \mathbf { x } , \mathcal { L } ^ { \star } ) ^ { \mathrm { p } } \boldsymbol { \mu } ( \mathbf { x } ) + \mathrm { d i s t } ( - \mathbf { x } , \mathcal { L } ^ { \star } ) ^ { \mathrm { p } } \boldsymbol { \mu } ( - \mathbf { x } ) } \\ { = } & { ( \mathrm { d i s t } ( \mathbf { x } , \mathcal { L } ^ { \star } ) ^ { \mathrm { p } } + \mathrm { d i s t } ( - \mathbf { x } , \mathcal { L } ^ { \star } ) ^ { \mathrm { p } } ) \boldsymbol { \mu } ( \mathbf { x } ) } \\ { = } & { ( \mathrm { d i s t } ( \mathbf { O } \mathbf { x } , \mathbf { O } \mathcal { L } ^ { \star } ) ^ { \mathrm { p } } + \mathrm { d i s t } ( - \mathbf { O } \mathbf { x } , \mathbf { O } \mathcal { L } ^ { \star } ) ^ { \mathrm { p } } ) \boldsymbol { \mu } ( \mathbf { x } ) } \\ { = } & { \left( \left( \displaystyle \sum _ { i = 1 } ^ { D - d } ( x _ { i } ^ { \prime } - m _ { i } ) ^ { 2 } \right) ^ { p / 2 } + \left( \displaystyle \sum _ { i = 1 } ^ { D - d } ( - x _ { i } ^ { \prime } - m _ { i } ) ^ { 2 } \right) ^ { p / 2 } \right) \boldsymbol { \mu } ( \mathbf { x } ) } \end{array}
+$$
+
+$$
+\begin{array} { r l } { = } & { \left( \left( \displaystyle \sum _ { i = 1 } ^ { D - d } ( x _ { i } ^ { \prime } - m _ { i } ) ^ { 2 } \right) ^ { p / 2 } + \left( \displaystyle \sum _ { i = 1 } ^ { D - d } ( x _ { i } ^ { \prime } + m _ { i } ) ^ { 2 } \right) ^ { p / 2 } \right) \mu ( \mathbf { x } ) } \\ & { \geq 2 \left( \displaystyle \sum _ { i = 1 } ^ { D - d } x _ { i } ^ { \prime } \right) ^ { p / 2 } \mu ( \mathbf { x } ) } \\ & { = 2 \mathrm { d i s t } ( 0 \mathbf { x } , 0 . \mathcal { L } _ { 0 } ) ^ { \mathrm { P } } \mu ( \mathbf { x } ) } \\ & { = \mathrm { ~ } 2 \mathrm { d i s t } ( \mathbf { x } , \mathcal { L } _ { 0 } ) ^ { \mathrm { P } } \mu ( \mathbf { x } ) } \\ & { = \mathrm { ~ } ( \mathrm { d i s t } ( \mathbf { x } , \mathcal { L } _ { 0 } ) ^ { \mathrm { P } } + \mathrm { d i s t } ( - \mathbf { x } , \mathcal { L } _ { 0 } ) ^ { \mathrm { P } } ) \mu ( \mathbf { x } ) } \\ & { = \mathrm { ~ } \mathrm { d i s t } ( \mathbf { x } , \mathcal { L } _ { 0 } ) ^ { \mathrm { P } } \mu ( \mathbf { x } ) + \mathrm { d i s t } ( - \mathbf { x } , \mathcal { L } _ { 0 } ) ^ { \mathrm { P } } \mu ( - \mathbf { x } ) . } \end{array}
+$$
+
+The inequality in (31) follows from the fact that for $p \geq 1$ , the function $\| \cdot \| _ { 2 } ^ { p }$ is convex as it is a composition of the convex function $\| \cdot \| _ { 2 } : \mathbb { R } ^ { d } \to \mathbb { R } _ { + }$ and the increasing convex function $( \cdot ) ^ { p } :$ $\mathbb { R } _ { + } \to \mathbb { R } _ { + }$ . Equality is achieved in (31) if $m _ { i } = 0$ for $i = 1 , \cdots , D - d$ , that is, ${ \mathcal L } ^ { \star } = { \mathcal L } _ { 0 }$ .
+
+Integrating the left and right hand sides of (31) over $\mathbb { R } ^ { D }$ results in
+
+$$
+\int _ { { \mathbb R } ^ { D } } \mathrm { d i s t } ( { \bf x } , \mathscr { L } ^ { \star } ) ^ { \mathrm { p } } \mu ( { \bf x } ) \mathrm { d } { \bf x } \geq \int _ { { \mathbb R } ^ { \mathrm { D } } } \mathrm { d i s t } ( { \bf x } , \mathscr { L } _ { 0 } ) ^ { \mathrm { p } } \mu ( { \bf x } ) \mathrm { d } { \bf x } .
+$$
+
+Since ${ \mathcal { L } } ^ { \star }$ is a minimizer among all affine subspaces of rank $d$ of $\begin{array} { r } { \int _ { \mathbb { R } ^ { D } } \mathrm { d i s t } ( \mathbf { x } , \mathcal { L } ) ^ { \mathrm { p } } \mu ( \mathbf { x } ) \ \mathrm { d } \mathbf { x } \ = } \end{array}$ $\mathbb { E } \| \mathrm { X } - \mathbf { P } _ { \mathcal { L } } \mathrm { X } \| _ { 2 } ^ { \mathrm { p } }$ , equality is obtained in (32). Consequently, equality is obtained, almost everywhere, in (31). Therefore, ${ \mathcal L } ^ { \star } = { \mathcal L } _ { 0 }$ and the claim is proved.
+
+# D.3 RELEVANT MATHEMATICAL THEORY
+
+We note that a complex network can represent a large class of functions. Consequently, for a sufficiently complex network, minimizing the loss function in (6) results in minimum value zero. In this case the minimizing “manifold” contains the original data, including the outliers. On the other hand, the RSR loss term imposes fitting a subspace that robustly fits only part of the data and thus cannot result in minimum value zero. Nevertheless, imposing a subspace constraint might be too restrictive, even in the latent space. A seminal work by Jones (1990) studies optimal types of curves that contain general sets. This work relates the construction and optimal properties of these curves with multiscale approximation of the underlying set by lines. It was generalized to higher dimensions in (?) and to a setting relevant to outliers in (?). These works suggest loss functions that incorporate several linear RSR layers from different scales. Nevertheless, their pure setting does not directly apply to our setting. We have also noticed various technical difficulties when trying to directly implement these ideas to our setting.
+
+# E BRIEF DESCRIPTION OF THE BASELINES AND METRICS
+
+We first clarify the methods used as baselines in Section 4.
+
+Local Outlier Factor (LOF) measures the local deviation of a given data point with respect to its neighbors. If the LOF of a data point is too large then the point is determined to be an outlier.
+
+One-Class SVM (OCSVM) learns a margin for a class of data. Since outliers contribute less than the normal class, it also applies to the unsupervised setting (Goldstein & Uchida, 2016). It is usually applied with a non-linear kernel.
+
+Isolation Forest (IF) determines outliers by looking at the number of splittings needed for isolating a sample. It constructs random decision trees. A short path length for separating a data point implies a higher probability that the point is an outlier.
+
+Geometric Transformations (GT) applies a variety of geometric transforms to input images and consequently creates a self-labeled dataset, where the labels are the types of transformations. Its anomaly detection is based on Dirichlet Normality score according to the softmax output from a classification network for the labels.
+
+Deep Structured Energy-Based Models (DSEBMs) outputs an energy function which is the negative log probability that a sample follows the data distribution. The energy based model is connected to an autoencoder to avoid the need of complex sampling methods.
+
+Deep Autoencoding Gaussian Mixture Model (DAGMM) is also a deep autoencoder model. It optimizes an end-to-end structure that contains both an autoencoder and an estimator for Gaussian Mixture Model. The anomaly detection is done after modeling the density function of the Gaussian Mixture Model.
+
+Next, we review the definitions of the two metrics that we used: the AUC and AP scores (Davis & Goadrich, 2006). In computing these metrics we identify the outliers as “positive”.
+
+AUC (area-under-curve) is the area under the Receiver Operating Characteristic (ROC) curve. Recall that the True Positive Rate (TPR), or Recall, is the number of samples correctly labeled as positive divided by the total number of actual positive samples. The False Positive Rate (FPR), on the other hand, is the number of negative samples incorrectly labeled as positive divided by the total number of actual negative samples. The ROC curve is a graph of TPR as a function of FPR. It is drawn by recording values of FPR and TPR for different choices of $\epsilon _ { \mathrm { T } }$ in Algorithm 1.
+
+AP (average-precision) is the area under the Precision-Recall Curve. While Recall is the TPR, Precision is the number of samples correctly labeled as positive divided by the total number of predicted positives. The Precision-Recall curve is the graph of Precision as a function of Recall. It is drawn by recording values of Precision and Recall for different choices of $\epsilon _ { \mathrm { T } }$ in Algorithm 1.
+
+Both AUC and AP can be computed using the corresponding functions in the scikit-learn package (Pedregosa et al., 2011).
+
+# F COMPARISON WITH RSR AND RCAE
+
+We demonstrate basic properties of our framework by comparing it to two different frameworks. The first framework is direct RSR, which tries to model the inliers by a low-dimensional subspace, as opposed to the nonlinear model discussed in here. Based on careful comparison of RSR methods in Lerman & Maunu (2018), we use the Fast Median Subspace (FMS) algorithm (Lerman & Maunu, 2017) and its normalized version, the Spherical FMS (SFMS). The other framework can be viewed a nonlinear version of RPCA, instead of RSR. It assumes sparse elementwise corruption of the data matrix, instead of corruption of whole data points, or equivalently, of some columns of the data matrix. For this purpose we use the Robust Convolutional Autoencoder (RCAE) algorithm of Chalapathy et al. (2017), who advocate it as “extension of robust PCA to allow for a nonlinear manifold that explains most of the data”. We adopt the same network structures as in Section 4.1.
+
+Fig. 7 reports comparisons of RSRAE, FMS, SFMS and RCAE on the datasets used in Section 4.2. We first note that both FMS and SFMS are not effective for the datasets we have been using. That is, the inliers in these datasets are not well-approximated by a linear model. It is also interesting to notice that without normalization to the sphere, FMS can be much worse than SFMS. That is, SFMS is often way more robust to outliers than FMS. This observation and the fact that there are no obvious normalization procedures a general autoencoder (see Section 5) clarifies why the mere use of the $L _ { \mathrm { A E } } ^ { 1 }$ loss for an autoencoder is not expected to be robust enough to outliers.
+
+Comparing with RSRAE, we note that RCAE is not a competitive method for these datasets. This is not surprising since the model of RCAE, which assumes sparse elementwise corruption, does not fit well to the problem of anomaly detection, but to other problems, such as background detection.
+
+![](images/36d4087a4804b6faed8cde3c6ef95e0635a5e007f473b3e39d466e40a2c71169.jpg)  
+Figure 7: AUC and AP scores for RSRAE, FMS, SFMS and RCAE. From top to bottom are the results using Caltech 101, Fashion MNIST, Tiny Imagenet with deep features, Reuters-21578 and 20 Newsgroups.
+
+# G SENSITIVITY TO HYPERPARAMETERS
+
+We examine the sensitivity of some of the reported results to changes in the hyperparameters. Section G.1 tests the sensitivity of RSRAE to changes in the intrinsic dimension $d$ . Section G.2 tests the sensitivity of RSRAE to changes in the learning rate. Section G.3 tests the sensitivity of RSRAE $^ +$ to changes in $\lambda _ { 1 }$ and $\lambda _ { 2 }$ .
+
+# G.1 SENSITIVITY TO THE INTRINSIC DIMENSION
+
+In the experiments reported in Section 4 we fixed $d = 1 0$ . Here we check the sensitivity of the reported results to changes in $d$ . We use the same datasets of Section 4.2 with an outlier ratio of $c = 0 . 5$ and test the following values of $d$ $\cdot 1 , 2 , 5 , 8 , 1 0 , 1 2 , 1 5 , 2 0 , 3 0 , 4 0 , 5 0$ . Fig. 8 reports the AUC and AP scores for these choice of $d$ and for these datasets with $c = 0 . 5$ . We note that, in general, our results are not sensitive to choices of $d \leq 3 0$ .
+
+We believe that the structure of these datasets is complex, and is not represented by a smooth manifold of a fixed dimension. Therefore, low-dimensional encoding of the inliers is beneficial with various choices of low dimensions.
+
+When $d$ gets closer to $D$ the performance deteriorates. Such a decrease in accuracy is noticeable for Reuters-21578 and 20 Newsgroups, where for both datasets $D = 1 2 8$ . For the image data sets (without deep features) $D = 1 1 5 2$ and thus only relatively small values of $d$ were tested. As an example of large $d$ for an image dataset, we consider the case of $d = D = 1 1 5 2$ in Caltech101 with $c = 0 . 5$ . In this case, $\mathrm { A U C } = 0 . 6 1 9$ and $\mathbf { A P } = 0 . 5 1 2$ , which are very low scores.
+
+![](images/e235994d3403e16e585f343b6deac678b38c1e4467424f172234fa1212adcaab.jpg)  
+We conclude that in our experiments (with $c = 0 . 5$ ), RSRAE was stable in $d$ around our choice of $d = 1 0$ .   
+Figure 8: AUC and AP scores for different choices of $d$ . The datasets are the same as those in Section 4.2, where the outlier ratio is $c = 0 . 5$ .
+
+In the experiments reported in Section 4 we fixed the learning rate for RSRAE to be 0.00025. Here we check the sensitivity of the reported results to changes in the learning rate. We use the same datasets of Section 4.2 with an outlier ratio of $c = 0 . 5$ and test the following values of the learning rate: 0.0001, 0.00025, 0.0005, 0.001, 0.0025, 0.005, 0.01, 0.025, 0.05, 0.1. Fig. 9 reports the AUC and AP scores for these values and for these datasets (with $c = 0 . 5$ ). We note that the performance is stable for learning rates not exceeding 0.01.
+
+![](images/237121dae508826e93d73b88ee594f19b06d4859644dd3e084bf1d4461be0baa.jpg)  
+Figure 9: AUC and AP scores for various learning rates. The datasets are the same as those in Section 4.2, where the outlier ratio is $c = 0 . 5$ .
+
+# G.3 SENSITIVITY OF RSRAE $^ +$ TO $\lambda _ { 1 }$ AND $\lambda _ { 2 }$
+
+We study the sensitivity of RSRAE $^ +$ to different choices of $\lambda _ { 1 }$ and $\lambda _ { 2 }$ . We recall that RSRAE does not require these parameters. It is still interesting to check such sensitivity and find out whether careful tuning of these parameters in ${ \mathrm { R S R A E } } +$ can yield better scores than those of RSRAE. We use the same datasets of Section 4.2 with an outlier ratio of $c = 0 . 5$ and simultaneously test the following values of either $\lambda _ { 1 }$ or $\lambda _ { 2 }$ $: 0 . 0 1 , 0 . 0 2 , 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 5 , 1 . 0 , 2 . 0 .$ . Figs. 10 and 11 report the AUC and AP scores for these values and datasets (with $c = 0 . 5$ ). For each subfigure, the above values of $\lambda _ { 1 }$ and $\lambda _ { 2 }$ are recorded on the $x$ and $y$ axes, respectively. The darker colors of the heat map correspond to larger scores. For comparison, the corresponding AUC or AP score of RSRAE is indicated in the title of each subfigure.
+
+We note that ${ \mathrm { R S R A E } } +$ is more sensitive to $\lambda _ { 1 }$ than $\lambda _ { 2 }$ . Furthermore, as $\lambda _ { 1 }$ increases the scores are often more stable to changes in $\lambda _ { 1 }$ . That is, the magnitudes of the derivatives of the scores with respect to $\lambda _ { 1 }$ seem to generally decrease with $\lambda _ { 1 }$ . In Section 4.3 we used $\lambda _ { 1 } = \lambda _ { 2 } = 0 . 1$ as this choice seemed optimal for the independent set of 20 Newsgroup. We note though that optimal hyperparameters depend on the dataset and it is thus not a good idea to optimize them using different datasets. They also depend on the choice of $c$ , but for brevity we only test them with $c = 0 . 5$ .
+
+At last we note that the AUC and AP scores of RSRAE are comparable to the fine-tuned ones of ${ \mathrm { R S R A E } } +$ (where $c = 0 . 5$ ). We thus advocate using the alternating minimization of RSRAE, which is independent of $\lambda _ { 1 }$ and $\lambda _ { 2 }$ .
+
+![](images/45aceed4332994262c3d9c9117d5262e5d00b24a477214b544b8b0c2c50fad5d.jpg)  
+Figure 10: AUC and AP scores for RSRAE $^ +$ with various choices of $\lambda _ { 1 }$ and $\lambda _ { 2 }$ for Caltech 101, Fashion MNIST and Tiny Imagenet with deep features, where $c = 0 . 5$ .
+
+![](images/7baa066c5edf09732b4cc6b2fa0d931183e3705195e69842d4155e676bcd8fde.jpg)  
+Figure 11: AUC and AP scores for RSRAE $^ +$ with various choices of $\lambda _ { 1 }$ and $\lambda _ { 2 }$ using Reuters-21578 and 20 Newsgroup, where $c = 0 . 5$ .
+
+# H RUNTIME COMPARISON
+
+Table 1 records runtimes for all the methods and datasets in Section 4.2 with the choice of $c = 0 . 5$ . More precisely, a runtime is the the time needed to complete a single experiment, where 200 epoches were used for the neural networks. The table averages each runtime over the different classes.
+
+Note that LOF, OCSVM and IF are faster than the rest of methods since they do not require training neural networks. We also note that the runtime of RSRAE is competitive in comparison to the other tested methods, that is, DSEBMs, DAGMM, and GT. The neural network structures of these four methods are the same, and thus the difference in runtime is mainly due to different pre and post processing.
+
+Table 1: Runtime comparison: runtimes (in seconds) are reported for all methods and datasets in Section 4.2, where the outlier ratio is $c = 0 . 5$ . Since GT was only applied to the image datasets without deep features, its runtime is not available (N/A) for the last three datasets.   
+
+<table><tr><td rowspan=1 colspan=1>DatasetsBenchmarks</td><td rowspan=1 colspan=1>Caltech 101</td><td rowspan=1 colspan=1>Fashion MNIST</td><td rowspan=1 colspan=1>Tiny Imagenet</td><td rowspan=1 colspan=1>Reuters-21578</td><td rowspan=1 colspan=1>20 Newsgroups</td></tr><tr><td rowspan=1 colspan=1>LOF</td><td rowspan=1 colspan=1>0.233</td><td rowspan=1 colspan=1>7.163</td><td rowspan=1 colspan=1>0.707</td><td rowspan=1 colspan=1>25.342</td><td rowspan=1 colspan=1>10.516</td></tr><tr><td rowspan=1 colspan=1>OCSVM</td><td rowspan=1 colspan=1>0.120</td><td rowspan=1 colspan=1>3.151</td><td rowspan=1 colspan=1>0.473</td><td rowspan=1 colspan=1>8.726</td><td rowspan=1 colspan=1>4.169</td></tr><tr><td rowspan=1 colspan=1>IF</td><td rowspan=1 colspan=1>0.339</td><td rowspan=1 colspan=1>1.485</td><td rowspan=1 colspan=1>0.511</td><td rowspan=1 colspan=1>20.481</td><td rowspan=1 colspan=1>6.751</td></tr><tr><td rowspan=1 colspan=1>GT</td><td rowspan=1 colspan=1>21.681</td><td rowspan=1 colspan=1>87.729</td><td rowspan=1 colspan=1>N/A</td><td rowspan=1 colspan=1>N/A</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td rowspan=1 colspan=1>DSEBMs</td><td rowspan=1 colspan=1>14.293</td><td rowspan=1 colspan=1>46.933</td><td rowspan=1 colspan=1>25.194</td><td rowspan=1 colspan=1>41.083</td><td rowspan=1 colspan=1>33.852</td></tr><tr><td rowspan=1 colspan=1>DAGMM</td><td rowspan=1 colspan=1>21.066</td><td rowspan=1 colspan=1>71.632</td><td rowspan=1 colspan=1>41.211</td><td rowspan=1 colspan=1>83.551</td><td rowspan=1 colspan=1>60.720</td></tr><tr><td rowspan=1 colspan=1>RSRAE</td><td rowspan=1 colspan=1>6.305</td><td rowspan=1 colspan=1>33.853</td><td rowspan=1 colspan=1>10.940</td><td rowspan=1 colspan=1>32.061</td><td rowspan=1 colspan=1>18.869</td></tr></table>
+
+# I ADDITIONAL RESULTS
+
+We include some supplementary numerical results. In Section I.1 we show the results for Tiny Imagenet without deep features. In Section I.2 we extend the results reported in section 4.3 for the other datasets.
+
+# I.1 TINY IMAGENET WITHOUT DEEP FEATURES
+
+Fig. 12 presents the results for Tiny Imagenet without deep features. We see that RSRAE performs the best, but in general all the methods do not perform well. Indeed, the performance is significantly worse to that with deep features.
+
+![](images/c680e8846d9a811bd92c7f692a7e0b9abb185d132912ba3ca07d2a6a5cc5239c.jpg)  
+Figure 12: AUC and AP scores for the Tiny Imagenet without using the deep features.
+
+# I.2 ADDITIONAL COMPARISON WITH VARIATIONS OF RSRAE
+
+Figs. 13 and 14 extend the comparisons in Section 4.3 for additional datasets. The conclusion is the same. In general, RSRAE performs better by a large margin than AE and AE-1. On the other hand, ${ \mathrm { R S R A E } } +$ is often in between RSRAE and AE/AE-1. However, for 20 Newsgroups, RSRAE $^ +$ performs similarly to RSRAE, and possibly slightly better, than RSRAE. It seems that in this case our choice of $\lambda _ { 1 }$ and $\lambda _ { 2 }$ is good.
+
+![](images/46274ea60a09c264ba35054d1cddaed544a382826e35014a3d7871266e56aa2c.jpg)  
+Figure 13: AUC and AP scores for RSRAE and alternative formulations using Fashion MNIST and deep features of Tiny Imagenet, where $c = 0 . 5$ .
+
+![](images/5d39c122fe898e67cec077abef3c0b0777f6b6d4de4610eb09cc571c92fb1033.jpg)  
+Figure 14: AUC and AP scores for RSRAE and alternative formulations using Tiny Imagenet (images) and 20 Newsgroup, where $c = 0 . 5$ .

parse/train/s-NI4H4e3Rf/s-NI4H4e3Rf.md

@@ -0,0 +1,205 @@
+# H-NeRF: Neural Radiance Fields for Rendering and Temporal Reconstruction of Humans in Motion
+
+Hongyi Xu Google Research hongyixu@google.com
+
+Thiemo Alldieck Google Research alldieck@google.com
+
+Cristian Sminchisescu Google Research sminchisescu@google.com
+
+# Abstract
+
+We present neural radiance fields for rendering and temporal (4D) reconstruction of humans in motion (H-NeRF), as captured by a sparse set of cameras or even from a monocular video. Our approach combines ideas from neural scene representation, novel-view synthesis, and implicit statistical geometric human representations, coupled using novel loss functions. Instead of learning a radiance field with a uniform occupancy prior, we constrain it by a structured implicit human body model, represented using signed distance functions. This allows us to robustly fuse information from sparse views and generalize well beyond the poses or views observed in training. Moreover, we apply geometric constraints to co-learn the structure of the observed subject – including both body and clothing – and to regularize the radiance field to geometrically plausible solutions. Extensive experiments on multiple datasets demonstrate the robustness and the accuracy of our approach, its generalization capabilities significantly outside a small training set of poses and views, and statistical extrapolation beyond the observed shape.
+
+# 1 Introduction
+
+Enabling free-viewpoint video of a human in motion, based on a sparse set of views, is extremely challenging, but has many applications. Our work is motivated by a breadth of transformative 3D use cases, including immersive visualization of photographs, virtual clothing and try-on, fitness, as well as AR and VR for improved communication or collaboration. However, so far static scenes and rigid objects have been the primary subject of research. In pursuing realistic novel-view synthesis two schools of thought have been established: 1) 3D reconstruction methods aim to recover the geometry of the observed scene as accurately as possible, with novel views generated using classical rendering pipelines [30, 42]. 2) Image-based rendering techniques [7, 10, 40] and very recently neural radiance fields [29] primarily aim at image production quality without explicitly constructing an accurate 3D geometric model. While these techniques explicitly or implicitly reconstruct the scene geometry sometimes, this is however not guaranteed to accurately resemble the true scene geometry. We argue that novel-view rendering and reconstruction are two sides of the same coin, and that reliable viewpoint generalization, especially given relatively few input views, would require good quality for both. To this end, we propose a unified model in order to support both robust reconstruction and photo-realistic rendering. Dynamic scenes, especially those capturing a human in motion, add considerable complexity to the problem: while static scenes can be observed from many views by a camera moving through the scene, any configuration of a dynamic scene is typically observed only from sparse views. Moreover, the scene geometry and its appearance may change considerably over time. To cope with few views, some methods integrate scene knowledge over time by warping observations into a common reference frame [35, 37]. At test time, the information is warped back to the desired state and rendered from a novel view. Extrapolating to unseen motion, however, remains challenging. For scenes capturing people, this means that only poses seen during training can be rendered at test time. For some applications, however, rendering the subject over a broad range of motions, or in novel poses, is desirable. To make generalisation over poses and views possible, we rely on additional problem domain knowledge in the form of a human body model, imGHUM [4]. imGHUM is an implicit signed distance function (SDF) conditioned on generative shape and pose codes learned from a large corpus of dynamic human scans. In this work, imGHUM is used as the common reference frame for a neural radiance field and as a structured prior for robust reconstruction. Additionally, since imGHUM can represent a broad distribution of statistically valid human poses and shapes, we can render the reconstructed subject in novel poses and even with modified body shapes. In summary, our system supports photo-realistic free-view point temporal rendering of a human subject given only sparse camera observations. By conditioning on an implicit human body model, we can render considerably different viewpoints, body poses, and body shapes compared to those observed in training. Our carefully designed losses ensure not only good image quality but also plausible temporal reconstructions that can be used for the free-viewpoint visualisation of human performance capture.
+
+![](images/6faeec44f9caf451971a47878320c6e86db2e12fcdeb0855d577e5e019081c0d.jpg)  
+Figure 1: Overview of H-NeRF. Given a set of images of a human performer collected from a sparse set of calibrated cameras, we train a geometry-aware neural radiance field by co-learning a deformable signed distance field. First, we estimate the body shape $\beta$ and track the articulated pose $\theta , \mathbf { T }$ using the implicit statistical human body model imGHUM (orange). imGHUM encodes all 3D spatial points $\mathbf { x }$ across frames with a 4D point descriptor $( \mathbf { s } , d )$ referencing a canonical frame. Using the foreground mask M, we co-train a residual SDF and a NeRF network in that canonical space, in order to integrate all image observations into a consistent implicit 3D geometry $\Delta d$ and its view-depended $( \mathbf { v } )$ radiance representation $( \mathbf { c } , \sigma )$ . The trainable residual SDF and NeRF networks (in blue) are conditioned on the body pose $\pmb { \theta }$ and the root transformation $\mathbf { T }$ to model pose-dependent geometric and appearance variations. Our framework supports both accurate 3D geometric reconstruction and free-viewpoint rendering, and generalizes well to novel views, shapes, and poses.
+
+# 2 Related Work
+
+Human performance capture is the process of reconstructing the dynamic (4D) geometry of a human subject as observed in one or multiple synchronized views. Pioneering methods deform a rigged template mesh against silhouettes observed in multiple views [6, 50], estimated skeleton key-points [12], or image correspondences [9]. Later systems relying on RGB-D video streams [5] or monocular capture [3, 2, 17, 52, 16, 15] have also been presented. All these methods rely either on a pre-scanned template mesh or on a body model that is deformed to explain the image evidence.
+
+Neural representation for view synthesis. With the advent of neural networks, researchers have begun to explore alternative solutions to represent a scene in order to support novel view synthesis and photo-realistic rendering; we refer the reader to [48] for a survey. Different scene representation have been explored. Some methods use voxel grids to embed the scene [23, 45]. For rendering, the voxel grid is probed by shooting a ray and by linearly interpolating voxel values. These samples are then transformed into color values using a neural network. Others have proposed neural textures [49, 43] that can be rendered based on view-dependent effects or texture synthesis networks [22] to realistically render meshes. In a similar spirit, other methods render point clouds, where each point carries local appearance information [26, 1]. Riegler and Koltun [40] reproject features from nearby views and rely on a SfM reconstruction scaffold for novel view synthesis. With the recent advent of implicit function networks [25, 34, 8, 27], such 3D representations have been explored for rendering and view synthesis with great success. In contrast to discrete voxel representations, textures, or point clouds, these methods represent the scene as a continuous function and thus are not bound to a specific image or volume resolution. In the pioneering work of Sitzmann et al. an implicit function produces features displayed using a neural renderer [46]. Follow up methods focus on 3D geometric reconstruction [44], and use 2D supervision [31, 53].
+
+Neural Radiance Fields (NeRFs) are a recent approach to represent scenes for novel view synthesis. Mildenhall et al. [29] introduce Neural Radiance Fields resented as fully connected neural networks, where the input is a spatial query point and a viewing direction. The output is a volume density and the emitted radiance at the query location in the direction of the viewer. By ray-tracing using this simple representation, one can generate photo-realistic images from novel views. Despite the excellent quality of results, one drawback is the slow rendering time. To this end, researchers have presented faster versions that e.g. transform the radiance field into more efficient sparse grids [18] or remove the dependence on viewing-direction during rendering by estimating a spherical harmonic representation of the radiance function [54]. Others improve fidelity or rendering time by tackling ambiguities in the original formulation [56], spatial decomposition into multiple NeRFs [39], or combining NeRF with sparse voxel fields [21]. Initial work on adapting NeRF to dynamic scenes has been presented as well. Park et al. [35] produce “Nerfies” (NeRF-Selfies) from videos where subjects carefully move a camera around their head. The scene information is fused by warping query points into a canonical reference frame. Similarly, Pumarola et al. [37] produce dynamic NeRFs from synthetic animation data. Related to our approach, some methods integrate human body models to fuse information over time. A-NeRF [47] uses a skeleton to rigidly transform NeRF features to refine estimated 3D poses. A similar approach is followed in NARF [32] for view synthesis. Most related, Neural Body [36] attaches learnable features to the vertices of a SMPL body model [24]. These features are processed using a sparse 3D convolutional network, where the output forms a neural radiance field. In contrast to our approach, the resolution is bounded by the spatial resolution of the 3D convolutional network and no geometric supervision is used. We highlight differences in $\ S 5$ .
+
+# 3 Background
+
+Given a collection of images capturing a dynamic scene of a human in motion, observed from a sparse set of calibrated camera views (in the limit a monocular camera, as we will show), we aim to learn both the detailed temporal geometry of the human in motion, and to render the sequence from novel camera views and for different human poses. To this end, our work unifies two main methodologies: 1) implicit 3D human representations, and 2) volumetric radiance fields. In this section, we provide the relevant background on both representations, which we co-learn in a joint framework.
+
+Neural Radiance Fields. A neural radiance field (NeRF) [29] represents a 3D scene as a continuous function of color volume densities. The model consists of a neural network function $F _ { \omega }$ that maps a 3D spatial point $\mathbf { x } \in \mathbf { R } ^ { 3 }$ and a viewing direction $\mathbf { v } \in \mathbf { R } ^ { 3 }$ to a volume density $\sigma \in \mathbf { R } ^ { + }$ and a radiance $\mathbf { c } ( \mathbf { x } , \mathbf { \bar { v } } ) \in \mathbf { R } ^ { 3 }$ emitted towards the viewer. In practice, NeRF encodes the inputs $\mathbf { x }$ and $\mathbf { v }$ using a sinusoidal positional encoding $\gamma : \mathbf { R } ^ { 3 }  \mathbf { R } ^ { 3 + \bar { 6 } m }$ that projects a coordinate vector into a high-dimensional space using a set of sine and cosine functions of $m$ increasing frequencies. Given a ray $\mathbf { r } = \mathbf { o } + s \mathbf { v }$ with $N$ samples $\{ { \bf x } \}$ originating from a camera location o, NeRF integrates radiance values along the ray by means of alpha blending. The pixel/ray color is approximated with numerical quadrature [33],
+
+$$
+\mathbf { C } ( \mathbf { r } ) = \sum _ { i = 1 } ^ { N } \alpha ( \mathbf { x } _ { i } ) \prod _ { j < i } ( 1 - \alpha ( \mathbf { x } _ { j } ) ) \mathbf { c } ( \mathbf { x } _ { i } , \mathbf { v } ) ,
+$$
+
+$$
+\alpha ( \mathbf { x } _ { i } ) = 1 - \exp \bigl ( - \sigma ( \mathbf { x } _ { i } ) \delta _ { i } \bigr ) ,
+$$
+
+where $\alpha ( \mathbf { x } _ { i } )$ is the transparency by accumulating transmittance along the ray, and $\delta _ { i } = | \mathbf { x } _ { i + 1 } - \mathbf { x } _ { i } |$ is the distance between adjacent samples. The NeRF function $F _ { w }$ is fully differentiable and its network parameters $w$ can be optimized using an image reconstruction loss [29]. An approximate 3D scene geometry $\mathbf { S } _ { F } = \{ \mathbf { x } | \sigma ( \mathbf { x } ) = \sigma _ { h } \}$ can be extracted from the trained opacity field via Marching Cubes [20] at a density threshold $\sigma _ { h }$ .
+
+Implicit Generative Human Models. Implicit human body surfaces are typically represented as the decision boundary of either binary occupancy classifiers [11, 41, 28] or signed distance functions [14, 4]. Specifically, our work builds upon the SOTA statistical implicit human model imGHUM [4] $\bar { H } _ { \omega } : ( \mathbf { T } ^ { - 1 } \mathbf { x } , \bar { } \mathbf { \beta } , \pmb { \theta } )  ( d , \mathbf { s } )$ that maps a 3D spatial point $\mathbf { x }$ , unposed with root joint transformation $\dot { \mathbf { T } } \in \mathbf { R } ^ { 4 \times 3 }$ , to its signed distance value $d \in \mathbf { R }$ with respect to a body surface parameterized with body shape $\boldsymbol { \beta } \in { \bf R } ^ { 1 6 }$ and articulated pose ${ \pmb \theta } \in { \bf R } ^ { 1 1 8 }$ . In addition to $d$ , imGHUM returns implicit continuous semantics $\textbf { s } \in \ \mathbf { R } ^ { 3 }$ of the query point which corresponds to the 3D coordinate of the nearest surface point defined on a canonical surface. We refer to the original paper [4] for details. Essentially, imGHUM builds a human-centric 4D semantic descriptor $( d , \mathbf { s } )$ for all spatial points in the neighborhood of the surface. imGHUM is trained with a large collection of human scans of diverse body shapes and poses, sharing the same generative shape and pose latent code with the mesh-based statistical human model GHUM [51]. The implicit articulated 3D human body $\mathbf { S } _ { H } ( \beta , \pmb \theta , \mathbf { T } ) = \{ \mathbf { x } | d ( \mathbf { x } ) = 0 \}$ is defined by the the zero-isosurface of the signed distance field.
+
+# 4 Method
+
+We present the details of our main contribution, H-NeRF, a novel neural network (fig. 1) that exploits the power of volumetric radiance fields to learn complex human structure and appearance, by relying on statistical implicit human pose and shape signed distance functions for accurate geometric reconstruction. Further, H-NeRF relies on imGHUM, an implicit model of articulated human pose and shape, as a rich geometric prior, to integrate scene information over time, and to represent human articulation. Co-learning both a radiance field and a signed distance function of scene geometry consistently in a unified framework, enables accurate 3D geometric reconstruction and volume rendering of a dynamic human in motion, through 1) consistent integration of image observations over time, and 2) good generalization capability for novel viewpoints, human poses, and for statistically extrapolated shapes, given only very few training poses and camera views.
+
+Given a video of a human in motion, observed by a sparse set of calibrated cameras, our objective is to generate free-viewpoint video of the observed person and to reconstruct the underlying 4D geometry. The set of input images, with a resolution of $w \times h$ pixels, is denoted as $\{ \mathbf { I } _ { t } ^ { c } \mathbf { \bar { \Psi } } \in \mathbf { \bar { ~ R } } ^ { w \times \mathbf { \breve { h } } \times 3 } | c \mathbf { \Psi } = \mathbf { \bar { ~ \Psi } }$ $1 , \dots , N _ { c } , t = 1 , \dots , N _ { t } \}$ , where $c$ is the camera index, $N _ { c }$ is the number of cameras, $t$ is the frame index, and $N _ { t }$ is the number of frames. For each image, we apply [13] to obtain the binary foreground human mask ${ \bf M } _ { t } ^ { c } \in { \bf R } ^ { w \times h }$ . In addition, we obtain a temporally consistent imGHUM latent shape and pose codes, $\beta$ and $\left( \boldsymbol { \theta } _ { t } , \mathbf { T } _ { t } \right)$ , respectively, at each frame index $t$ , by optimizing an imGHUMequivalent parametric replica under multi-view keypoint and body segmentation losses [55]. We refer to our Sup. Mat. for details on the imGHUM fitting process.
+
+In the sequel we first explain our adaptations to NeRF for the static case. imGHUM is used as a prior in order to bias the reconstruction of the observed scene towards a more accurate human geometry. We continue by explaining the additional methodological innovation needed in the dynamic case. Hereby, imGHUM provides spatial-temporal correspondences and is used as the common reference frame to fuse information across different views and time instances.
+
+# 4.1 Static Semantic Human NeRF
+
+We first formulate H-NeRF for a human capture at a single moment in time $N _ { t } ~ = ~ 1$ ). From multi-view image observations, we co-learn a radiance field $F _ { \omega } : \mathbf { x } , \mathbf { v }  ( \mathbf { c } , \sigma )$ for free-viewpoint rendering, and a signed distance function $\hat { H } _ { \omega } : \mathbf { x }  \hat { d }$ for 3D geometric reconstruction. We use $\hat { H } _ { \omega }$ for the dressed subject in order to distinguish it from the body’s imGHUM SDF $H _ { \omega }$ .
+
+Like NeRF, we formulate an $L _ { 1 }$ image reconstruction loss to optimize $F _ { w }$ as
+
+$$
+\mathcal { L } _ { \mathrm { r e c } } = \sum _ { \mathbf { r } \in \mathcal { R } } \| \bar { \mathbf { C } } ( \mathbf { r } ) - \mathbf { C } ( \mathbf { r } ) \| _ { 1 } ,
+$$
+
+where $\mathcal { R }$ denotes the batched set of all pixels/rays, and $\bar { \mathbf { C } } ( \mathbf { r } ) , \mathbf { C } ( \mathbf { r } )$ are observed and rendered pixel colors, cf. (1), respectively. However, under sparse training camera views, the volumetric radiance field is not well regularized, leading to poor generalization to novel viewpoints, cf. fig. 2. Specifically, we observe that the model encounters difficulties in correctly representing the scene and in separating the person from the background. The model fails to learn a semantically meaningful opacity field, i.e. $\alpha = 0$ in free space, and 1 if occupied.
+
+Coarse Scene Structuring. Using imGHUM fits for all training images, we can spatially locate the person in 3D. We define a 3D bounding box $\mathbf { B } \in [ \underline { { \mathbf { S } } } _ { H } - \epsilon , \overline { { \mathbf { S } _ { H } } } + \epsilon ]$ around the detected person, where $\underline { { \mathbf { S } } } _ { H } , \overline { { \mathbf { S } _ { H } } }$ are the minimal and maximal coordinates of the human body surface S and $\epsilon$ is a spatial margin reserved for geometry not modeled by imGHUM. All radiance points associated to the rendering of the person should reside inside the bounding box, leading to a 3D segmentation loss
+
+$$
+\begin{array} { r l r } & { } & { \tilde { \mathbf { C } } ( \mathbf { r } ) = \displaystyle \sum _ { i = 1 } ^ { N } b ( \mathbf { x } _ { i } ) \alpha ( \mathbf { x } _ { i } ) \prod _ { j < i } ( 1 - b ( \mathbf { x } _ { j } ) \alpha ( \mathbf { x } _ { j } ) ) \mathbf { c } ( \mathbf { x } _ { i } , \mathbf { v } ) , } \\ & { } & { \mathcal { L } _ { \mathrm { m a s k } } = \displaystyle \sum _ { \mathbf { r } \in \mathcal { R } } \left\| \mathbf { M } ( \mathbf { r } ) \big ( \bar { \mathbf { C } } ( \mathbf { r } ) - \tilde { \mathbf { C } } ( \mathbf { r } ) \big ) \right\| _ { 1 } , } \end{array}
+$$
+
+where $b ( \mathbf { x } _ { i } )$ is 0 if outside of $\mathbf { B }$ and 1 otherwise, and $\mathbf { M } ( \mathbf { r } )$ the image mask. We rely on the mask, so the loss is only applied to image observations from the subject, and not the remaining scene geometry.
+
+Unifying SDF with NeRF. After structuring the scene coarsely, we now couple the estimated radiance field with an implicit signed distance-based 3D reconstruction, in order to further regularize the opacity distribution. A signed distance function associated to the detected person in the image naturally comes with a 3D classifier for all spatial points, where $\hat { d } ( \mathbf { x } _ { i } ) > 0$ means $\mathbf { x } _ { i }$ is in free space, whereas $\mathbf { x } _ { i }$ lies within the subject when $\hat { d } ( \mathbf { x } _ { i } ) < = 0$ . We rely on this insight in order to co-learn a SDF of the performer and constrain the radiance field. To this end, we introduce a pseudo alpha value $\dot { \alpha } ( \mathbf x _ { i } ) = \phi ( \gamma \hat { d } ( \mathbf x _ { i } ) )$ where $\phi$ is a Sigmoid activation function and $\gamma$ controls the sharpness of the boundary. To refine the NeRF opacity semantics, especially for the volume in the neighborhood of the human surface $\mathbf { B }$ , we formulate two losses
+
+$$
+\begin{array} { l } { { \displaystyle { \hat { \bf C } } ( { \bf r } ) = \sum _ { i = 1 } ^ { N } \hat { \alpha } ( { \bf x } _ { i } ) \prod _ { j < i } ( 1 - \hat { \alpha } ( { \bf x } _ { j } ) ) { \bf c } ( { \bf x } _ { i } , { \bf v } ) } , \quad \hat { \alpha } ( { \bf x } _ { i } ) = b ( { \bf x } _ { i } ) \dot { \alpha } ( { \bf x } _ { i } ) + ( 1 - b ( { \bf x } _ { i } ) ) \alpha ( { \bf x } _ { i } ) } ,  \\ { { \displaystyle { \mathcal { L } } _ { \mathrm { b l e n d } } = \sum _ { { \bf r } \in { \mathcal R } } \left( \| { \bf M } ( { \bf r } ) \big ( { \bar { \bf C } } ( { \bf r } ) - \hat { \bf C } ( { \bf r } ) \big ) \| _ { 1 } + \eta \| \big ( 1 - { \bf M } ( { \bf r } ) \big ) \big ( { \bar { \bf C } } ( { \bf r } ) - \hat { \bf C } ( { \bf r } ) \big ) \| _ { 1 } \right) } , \quad { \mathrm { ~ i ~ f ~ } } \quad { \mathrm { ~ i ~ f ~ } } \quad i = { \bf r } \quad { \bf r } \quad - \Pi _ { i } ^ { \prime } , } \\ { { \displaystyle { \mathcal { L } } _ { \mathrm { g e o m } } ( { \bf r } ) = \sum _ { i = 1 } ^ { N } b ( { \bf x } _ { i } ) \mathrm { B C E } \big ( \phi \big ( \lambda ( \sigma _ { h } - \sigma ( { \bf x } _ { i } ) ) \big ) , \dot { \alpha } ( { \bf x } _ { i } ) \big ) } , } \end{array}
+$$
+
+where $\hat { \mathbf { C } } ( \mathbf { r } )$ is the rendered pixel color with blended alpha values $\hat { \alpha }$ replacing NeRF alpha values $\alpha$ with SDF-based pseudo alpha $\dot { \alpha }$ for all ray points inside the bounding box $\mathbf { B }$ . The first term in ${ \mathcal { L } } _ { \mathrm { b l e n d } }$ requires the rendered pixel color for all the intersecting rays within the human mask to come from the surface, whereas the second term assumes that all background color is formed from ray samples outside of $\mathbf { B }$ . We set $\eta = 1$ when no other geometry than the person is inside $\mathbf { B }$ and tune $\eta$ down if the assumption is violated (e.g. person standing on a floor). The term $\mathcal { L } _ { \mathrm { g e o m } }$ uses the binary cross entropy loss to couple the NeRF surface boundary with the zero-isosurface of the signed distance function describing the subject. While the coupling terms given by (7) and (8) act as strong priors for the opacity distribution, during test time, we still rely on volumetric radiance rendering with learned NeRF alpha values $\alpha$ to support transparency effects and complex geometry for structures like hair.
+
+Image-based SDF learning. Learning the signed distance field $\hat { H } _ { \omega }$ from scratch using a sparse set of training images is still challenging. The model often fails to reconstruct reasonable human geometry, even when using coupling losses (7) and (8). We therefore leverage imGHUM as an inner layer for the target human reconstruction and combine it with a light-weight residual SDF network $\Delta H _ { \omega } : { \bf x } \to \Delta d$ . The residual SDF models surface details, including hair and clothing, that are not represented by imGHUM. The final signed distance for $\mathbf { x } _ { i }$ becomes $\hat { d } ( \mathbf { x } _ { i } ) = d ( \mathbf { x } _ { i } | \beta , \pmb { \theta } , \mathbf { T } ) + \Delta d ( \mathbf { x } _ { i } )$ . Given the training images, we learn the personalized residual SDF using our coupling losses, given by (7) and (8), and additionally apply geometric regularization
+
+$$
+\mathcal { L } _ { \mathrm { s e g } } = \sum _ { \mathbf { r } \in \mathcal { R } } \operatorname { B C E } ( \mathbf { M } ( \mathbf { r } ) , \hat { d } _ { \mathrm { m i n } } ( \mathbf { r } ) ) ,
+$$
+
+$$
+\mathcal { L } _ { \mathrm { e i k } } ( \mathbf { r } ) = \sum _ { i = 1 } ^ { N } b ( \mathbf { x } _ { i } ) ( \| \nabla _ { \mathbf { x } _ { i } } \hat { d } ( \mathbf { x } _ { i } ) \| _ { 2 } - 1 ) ^ { 2 } , \quad \mathcal { L } _ { \mathrm { r e g } } = \| \psi ( \Delta d ) \| _ { 1 } ,
+$$
+
+where $\hat { d } _ { \mathrm { m i n } } ( \mathbf { r } )$ denotes the minimal signed distance for all sampled ray points and $\psi$ is the ReLU activation function. The term $\mathcal { L } _ { \mathrm { s e g } }$ ensures that if a pixel is inside the human segmentation mask, there should be at least one intersection between the ray and the 3D human surface and therefore $\hat { d } _ { \mathrm { m i n } } ( \mathbf { r } )$ should be non-positive. Otherwise, all ray samples should have positive signed distances. Using $\mathcal { L } _ { \mathrm { e i k } }$ , we enforce the composite SDF $\hat { d }$ to be approximately a signed distance function, i.e. incorporating Eikonal regularization [14]. The term $\mathcal { L } _ { \mathrm { r e g } }$ regularizes the residual distance $\Delta d$ to be non-positive. This is because imGHUM should reside within the geometry and personalized geometric details given by $\Delta d$ should be modeled on top of the skin surface.
+
+# 4.2 Dynamic Semantic Human NeRF
+
+We now extend the framework to model dynamic human motion. To implicitly model scenes where the human moves, we learn a consistent, continuous function $G _ { \omega } : \bar { ( \mathbf { c } , \alpha , \hat { d } | \mathbf { z } ) } \to ( \mathbf { c } ^ { \prime } , \alpha ^ { \prime } , \hat { d } ^ { \prime } | \mathbf { z } ^ { \prime } )$ for both the geometry and the appearance variations across frames. For generalization to novel poses or shapes, $G _ { \omega }$ should be conditioned on a semantically meaningful latent code $\mathbf { z }$ that can be interpolated and should ideally have good extrapolation properties. To integrate scene information over time in a single radiance field, we follow the approaches in [35, 37] and warp observations into a canonical reference frame. Given our constraints to $G _ { \omega }$ , imGHUM, conditioned on its semantic shape and pose latent code $( \beta , \pmb \theta , \mathbf { T } )$ , naturally provides spatial correspondences across frames. Given two spatial points $\mathbf { x }$ and $\mathbf { x } ^ { \prime }$ in the space of two human instances $( \beta , \pmb \theta , \mathbf { T } )$ and $( \beta ^ { \prime } , \pmb \theta ^ { \prime } , \mathbf { T } ^ { \prime } )$ respectively, we decide these are in correspondence if they have the same semantics and signed distances $( \bar { \bf s } , d | \beta , \pmb \theta , { \bf T } ) = ( { \bf s } ^ { \prime } , d ^ { \prime } | \beta ^ { \prime } , \pmb \theta ^ { \prime } , { \bf T } ^ { \prime } )$ . Essentially, imGHUM assigns a 4D point descriptor $( \mathbf { s } , d ) \in \mathbf { R } ^ { 4 }$ to any spatial point and deforms the volume continuously with respect to the parameterized articulated human body surface. Specifically, we apply $H _ { \omega } : ( \bar { \mathbf { T } } ^ { - 1 } \mathbf { x } , \beta , \pmb { \dot { \theta } } )  ( d , \mathbf { s } )$ to map a spatial point $\mathbf { x }$ to a canonical point descriptor $( \mathbf { s } , d )$ . We then modify the input to the NeRF function with the point descriptor as $F _ { w } : ( \mathbf { s } , d ) \to ( \mathbf { c } , \sigma )$ , and similarly for the residual SDF function as $\Delta H _ { \omega } : ( \mathbf { s } , d )  \Delta d$ . imGHUM is pre-trained using a large corpus of 3D human scans, thus we keep it fixed in our network and directly use this prior to integrate structured geometric and appearance information across training frames. Given that the background scene is invariant to the human motion, we only apply the imGHUM warping function to spatial points inside the bounding box B, thus largely improving computation time and lowering memory usage.
+
+Pose-Dependent Geometry and Appearance. Training a single canonical NeRF and a residual SDF from a sequence integrates image observations into a consistent implicit geometric and volumetric representation of the radiance. However, we observe that fine-grained geometric and appearance variations caused by human motion are missing in the canonical NeRF and the residual SDF. To model such variations in geometry (clothing deformation) and appearance (lighting, self-shadows), we condition the NeRF function $F _ { \omega }$ – the volume density $\sigma$ (geometry) and color value c (appearance) – as well as the residual SDF $\Delta H _ { \omega }$ (geometry), on the body pose code $\pmb \theta$ . In addition, we also notice that the relative position and rotation of the person with respect to the scene largely affects appearance (due to illumination effects), but should not affect the geometry. Based on this observation, we additionally condition the NeRF color on the person’s root transformation as $\mathbf { c } ( \mathbf { x } | \mathbf { \boldsymbol { \theta } } , \mathbf { T } )$ .
+
+H-NeRF Articulation. Differently from prior work [35, 37], our H-NeRF modules are conditioned on semantic human latent codes $( \beta , \pmb \theta , \mathbf { T } )$ , leading to an SDF and radiance field that can be interpolated. Moreover, due to imGHUM’s capability to accurately model geometric volume deformation for diverse human poses and shapes, H-NeRF enjoys strong generalisation to novel poses and even body shapes. Concretely, during inference, one can simply assign a different set of $( \beta , \pmb \theta )$ to transfer the learned geometry and appearance to an unseen pose and shape configuration. For better generalization, we add Gaussian noise to the input code $( \pmb \theta , \mathbf T )$ , preventing network overfitting to pose-dependent deformation and appearance effects experienced during training. When only a sparse set of camera views is available, We apply the same technique to NeRF, conditioned on viewing direction $\mathbf { v }$ .
+
+Fine-tuning imGHUM Pose and Shape. We observe that imGHUM fitting can be a source of error when learning $F _ { \omega }$ and $\Delta H _ { \omega }$ from dynamic image sequences. Instead of keeping parameters fixed, we further improve the fit during training by fine tuning a time-consistent shape correction $\Delta \beta$ and per-frame pose correction $\Delta \theta ( t )$ with
+
+$$
+\begin{array} { r l } & { \displaystyle \mathrm { \sum _ { f i t } } = \sum _ { \mathbf { r } \in \mathcal { R } } \mathrm { B C E } ( \mathbf { M } ( \mathbf { r } ) , d _ { \operatorname* { m i n } } ( \mathbf { r } | \boldsymbol { \beta } + \Delta \boldsymbol { \beta } , \pmb { \theta } + \Delta \pmb { \theta } ( t ) ) ) , \quad \mathcal { L } _ { \operatorname* { i n c } } = \| \Delta \boldsymbol { \beta } \| _ { 2 } + \frac { 1 } { N _ { t } } \sum _ { t = 1 } ^ { N _ { t } } \| \Delta \pmb { \theta } \| _ { 2 } , } \end{array}
+$$
+
+where ${ \mathcal { L } } _ { \mathrm { f i t } }$ aligns the imGHUM fit with the human foreground segmentation and ${ \mathcal { L } } _ { \mathrm { i n c } }$ regularizes the incremental corrections.
+
+# 5 Experiments
+
+We quantitatively and qualitatively evaluate H-NeRF through ablations and by comparing with other methods. Please refer to our Sup. Mat. for additional results and ablation experiments. We first detail our model architecture, as well as the datasets and evaluation metrics used.
+
+Architecture and Training. We use the same network architecture for all of our experiments, where the trainable modules $F _ { \omega }$ and $\Delta H _ { \omega }$ consist of eight 256-dimensional and six 128-dimensional MLPs respectively, with a skip connection to the middle layer and with Swish activation [38]. As in the original NeRF [29], we use 256 coarse and fine-level ray samples, for each of which we use 8- and 1-dimensional positional encoding for $F _ { \omega }$ and $\Delta H _ { \omega }$ , respectively. We train the network using the Adam optimizer with a learning rate of 0.001 exponentially decayed by a factor 0.1 until the maximum number of iterations is reached ( $1 0 k$ iteration of $4 k$ ray batch size). We apply Gaussian noise with $\sigma = 0 . 1$ to the NeRF conditioning code $( { \pmb \theta } , \mathbf T , \mathbf v )$ .
+
+![](images/62edcca9d6dc33c339a157fd14b80255d3de684df111c2098d20d712becfd354.jpg)  
+Figure 2: Left: H-NeRF (red, $\mathbf { X }$ -axis: #cameras) outperforms the original NeRF (blue) and IDR (green) in both novel view synthesis and 3D reconstruction of a static scene (16 test cameras). NeRF and IDR fail under sparse camera views (solid line: RenderPeople scan; dashed line: GHS3D scan). Right: we qualitatively show novel views and reconstructed geometry for NeRF, IDR, and H-NeRF (from left to right) trained using four cameras.
+
+Table 1: Quantitative evaluation of dynamic sequences for various datasets. When two metrics are reported they correspond to evaluation of training poses and novel poses for test cameras, respectively. PeopleSnapshot and Human3.6M are evaluated on novel poses under training views (where groundtruth images exist). Geometric metrics are only reported when ground-truth is available.   
+
+<table><tr><td>Model</td><td>Dataset</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>Ch ×10-3 ↓</td><td>NC↑</td><td>IoU↑</td></tr><tr><td rowspan="3">NeuralBody [36]</td><td>RenderPeople</td><td>27.33/23.52</td><td>0.888/0.827</td><td>0.117/0.247</td><td>0.536/0.63</td><td>0.908/0.892</td><td>0.864/0.824</td></tr><tr><td>GHS3D</td><td>24.7</td><td>0.829</td><td>0.236</td><td>0.79</td><td>0.887</td><td>0.81</td></tr><tr><td>PeopleSnapshot</td><td>24.62</td><td>0.849</td><td>0.160</td><td>1</td><td>一</td><td>1</td></tr><tr><td rowspan="4">H-NeRF (ours)</td><td>Human3.6M</td><td>24.86</td><td>0.82</td><td>0.189</td><td>1</td><td>一</td><td>1</td></tr><tr><td>RenderPeople</td><td>28.78/24.31</td><td>0.913/0.856</td><td>0.125/0.246</td><td>0.217/0.274</td><td>0.950/0.939</td><td>0.917/0.9</td></tr><tr><td>GHS3D</td><td>24.92</td><td>0.852</td><td>0.232</td><td>0.218</td><td>0.932</td><td>0.89</td></tr><tr><td>PeopleSnapshot Human3.6M</td><td>26.33 25.01</td><td>0.868 0.83</td><td>0.159 0.17</td><td>一</td><td>一</td><td>1</td></tr></table>
+
+Dataset and Metrics. We provide qualitative and quantitative evaluation on four different datasets: RenderPeople (8 sequences), GHS3D [51] (14 sequences), PeopleSnapshot [3] (7 sequences) and Human3.6M [19] (5 sequences). The first two are synthetic, rendering animated (RenderPeople) or 4D human scans (GHS3D) from four orthogonal cameras, where ground-truth geometry paired with images is available. We evaluate both the image and the 3D geometry reconstruction quality from two novel cameras, and qualitatively demonstrate generalization to shapes and poses not in the training set. The remaining datasets are real captured videos (PeopleSnapshot: monocular, Human3.6M: four cameras) without paired 3D geometry, where we only evaluate the rendered images. For image metrics, we adopt peak signal-to-noise ratio (PSNR $\uparrow$ ), structural similarity index (SSIM $\uparrow$ ) and learned perceptual image patch similarity (LPIPS $\downarrow$ ) [57]. For evaluation, we render the NeRF field inside the human bounding box $\mathbf { B }$ and compare to the ground-truth segmented image within a region of interest around the person. To evaluate geometric reconstruction quality, we report bi-directional Chamfer (Ch) $L _ { 2 }$ distance, Normal Consistency (NC) and Volumetric Intersection over Union (IoU), evaluated on the mesh produced by applying Marching Cubes on the SDF, with a resolution of $2 5 6 ^ { 3 }$ .
+
+Static Human Reconstruction under Sparse Viewpoints. H-NeRF learns a geometrically regularized volumetric radiance field constrained by an imGHUM-based SDF, which significantly improves robustness under sparse training camera views. In fig. 2, we show reconstruction of a static RenderPeople and GHS3D scan, respectively, using the original NeRF [29], the state-of-the-art multi-view reconstruction approach IDR [53], and H-NeRF, for increasing number of cameras. Both novel view image quality and geometric accuracy improve for all methods as the number of training views increases. However, in contrast to competitors, H-NeRF produces good quality output even under sparse training views (2-8), demonstrating the generalization capability of a co-training approach.
+
+Dynamic Human Reconstruction and Rendering. In the following, we evaluate H-NeRF’s capabilities to reconstruct and render dynamic scenes (fig. 5). We compare H-NeRF against NeuralBody [36] (tab. 1), the current state-of-the-art for novel view synthesis of humans in motion. For fair comparisons, we have trained NeuralBody with the same data as H-NeRF and use GHUM meshes (instead of SMPL). GHUM meshes with the same pose and shape representation as in our model are used for NeuralBody’s structured latent codes. We further compare with Nerfies [35]. However, Nerfies has difficulties with the high range of motion in our test sequences and fails for most of them. We consider such results less meaningful and we report them only in the Sup. Mat. for completeness.
+
+![](images/6434043d8a0c1cc7eb871d5b1fa2bbb7a9947eec176810cd6e3176f5e1ab3fdd.jpg)  
+Figure 3: Frame Number Ablation. H-NeRF (red, x-axis: #training video frames) outperforms Neural Body (blue), evaluated on a RenderPeople (dashed) and a GHS3D (solid) sequence.
+
+![](images/1b4631173a50f0445731ff0e95a4e084d98ff478f3ae720174fc73307228385e.jpg)  
+Figure 4: Qualitative comparisons with NeuralBody on four different datasets. Left: GHS3D (top) and RenderPeople (bottom) ground-truth image, our result, NeuralBody, ground-truth geometry (front and back), our geometry, NeuralBody’s geometry. Right: PeopleSnapshot (top) and Human3.6M (bottom) ground-truth image, our result, NeuralBody, our geometry, NeuralBody’s geometry. Pay attention to the sharp renderings and the complete and detailed geometry produced by our method. Digital zoom-in recommended.
+
+H-NeRF overperforms NeuralBody across datasets and metrics, especially for geometric reconstruction. One possible explanation for NeuralBody’s lower performance on view synthesis, is that NeuralBody is not conditioned on the root transformation. Thus, global lighting effects are handled less well. Furthermore, pose depended effects that go beyond body articulation are only indirectly modeled through relative distances of input mesh vertices. Finally, NeuralBody conditions on the frame index, which is set to zero for novel poses. On the PeopleSnapshot dataset where no view-dependent and only subtle pose-dependent effects are present, these limitations are not entirely apparent. In contrast, H-NeRF pays special attention to modeling both detailed geometry and appearance, and its explicit conditioning on pose and the root transformation captures pose-dependent geometric and appearance effects. This strategy pays off, especially in more complex scenarios.
+
+Frame Number Ablation. We have illustrated H-NeRF’s rendering and reconstruction capabilities for static and dynamic sequences. Next, we evaluate the necessary number of different poses or time-frames H-NeRF needs during training for good pose generalisation. We again compare against NeuralBody. To this end, we have trained both methods with an increasing number of images, uniformly distributed over the full sequence. Results are shown in fig. 3. As expected, both methods perform better when trained with more data. However, H-NeRF performs overall better and more importantly, the quality degrades less in the sparse training regime, especially for geometric reconstruction. Our results support H-NeRF’s robustness to sparse training data and its capability to reconstruct accurate geometry. In practical terms, H-NeRF can be robustly trained with as little as 10 temporal frames per camera (in a four camera set-up), resulting in a performance capture effort of only a few minutes.
+
+Qualitative Results for Pose Generalisation and Shape Extrapolation. Finally, we illustrate HNeRF’s rendering and reconstruction accuracy qualitatively (fig. 4). Consistently with the reported metrics, our synthesized novel poses appear sharper, more detailed, and contain less noise than the current state-of-the-art. The estimated geometry is complete, smooth, and contains much of the detail present in the original scan, e.g. the clothing folds on the back of the person on the left, the first row.
+
+![](images/e469af6acf42ce9ddfbe2dea143a520eca9a85c6d19405478198be73e6954db7.jpg)  
+Figure 5: Qualitative results of H-NeRF for novel image synthesis. We show ground truth images and scans (left, if available) side-by-side with our results (right). Red icons correspond to test, green icons to training configurations. Symbols correspond to camera, pose, and shape, respectively. From top to bottom: RenderPeople, GHS3D, PeopleSnapshot and Human3.6M.
+
+In contrast, geometries produced by NeuralBody are noisier and sometimes incomplete. To further demonstrate the versatility of our approach, in fig. 5 we show examples of synthesized images and reconstructed geometry from novel viewpoints, in novel poses, and for modified body shape.
+
+Limitations and Broader Impact. In the current setup, we assume full-body views of a single person in every-day clothing. Inconsistency with these assumptions, e.g. by placing another person in the scene and occluding the performer, would break the method although models of partial views, or representation estimates of multiple people could be used for generality. Furthermore, our method exhibits some sensitivity to the estimated body shapes and poses, as well as the quality of image segmentation. Although the process of learning to render could absorb certain inaccuracies, as pose, shape or segmentation are increasingly degraded, the method would fail eventually.
+
+Our method paves the way towards immersive AR and VR applications. In contrast, our approach is not targeted, or particularly useful for applications like visual surveillance or person identification as a particular set-up and a cooperating subject is needed. We do not build an audio-visual model that would be necessary for deep fakes.
+
+# 6 Conclusions
+
+We have presented novel neural radiance field models (H-NeRF) for the photo-realistic rendering and the temporal reconstruction of humans in motion. Our objective is to extend the scope of the previous state of the art, focused on static scenes observed by a large number of cameras, by building dynamic models, which based on a sparse set of views, can generalize well to novel camera views, human body poses, and extrapolate over human body shapes. Our key contribution is in developing a new model with associated multiple losses, in order to specialize and constrain a generic NeRF formulation by using a compatible implicit statistical 3D human pose and shape model, represented using signed distance functions. Our implicit geometric formulation captures not just the statistical regularities of the human body, but also hair and clothing represented as an implicit residual network. Our model is trained end-to-end based on several novel losses, and achieves good results for both 3D reconstruction and photorealistic rendering. Training the body model in an end-to-end rendering framework carries the promise to learn complex implicit skinning functions based on images only, a performance previously possible only in the realm of human capture using complex and expensive 3D body scanners, in the laboratory. We illustrate the favorable capabilities of H-NeRF through extensive experimentation using several datasets and against other state of the art techniques.
+
+# References
+
+arXiv:1906.08240. [2] Thiemo Alldieck, Marcus Magnor, Bharat Lal Bhatnagar, Christian Theobalt, and Gerard Pons-Moll. Learning to reconstruct people in clothing from a single RGB camera. In IEEE Conf. Comput. Vis. Pattern Recog., pages 1175–1186. IEEE, 2019. [3] Thiemo Alldieck, Marcus Magnor, Weipeng Xu, Christian Theobalt, and Gerard Pons-Moll. Video based reconstruction of 3D people models. In IEEE Conf. Comput. Vis. Pattern Recog., pages 8387–8397. IEEE, 2018.   
+[4] Thiemo Alldieck, Hongyi Xu, and Cristian Sminchisescu. imGHUM: Implicit generative models of 3D human shape and articulated pose. In Int. Conf. Comput. Vis., 2021. [5] Federica Bogo, Michael J. Black, Matthew Loper, and Javier Romero. Detailed full-body reconstructions of moving people from monocular RGB-D sequences. In IEEE International Conference on Computer Vision, pages 2300–2308. IEEE, 2015.   
+[6] Joel Carranza, Christian Theobalt, Marcus A Magnor, and Hans-Peter Seidel. Free-viewpoint video of human actors. ACM Trans. Graph., 22(3):569–577, 2003. [7] Shenchang Eric Chen and Lance Williams. View interpolation for image synthesis. In Proceedings of the 20th annual conference on Computer graphics and interactive techniques, pages 279–288, 1993. [8] Zhiqin Chen and Hao Zhang. Learning implicit fields for generative shape modeling. In IEEE Conf. Comput. Vis. Pattern Recog., pages 5939–5948, 2019. [9] Edilson De Aguiar, Carsten Stoll, Christian Theobalt, Naveed Ahmed, Hans-Peter Seidel, and Sebastian Thrun. Performance capture from sparse multi-view video. In ACM Trans. Graph., pages 1–10. 2008.   
+[10] Paul E Debevec, Camillo J Taylor, and Jitendra Malik. Modeling and rendering architecture from photographs: A hybrid geometry-and image-based approach. In Proceedings of the 23rd annual conference on Computer graphics and interactive techniques, pages 11–20, 1996.   
+[11] Boyang Deng, JP Lewis, Timothy Jeruzalski, Gerard Pons-Moll, Geoffrey Hinton, Mohammad Norouzi, and Andrea Tagliasacchi. Neural articulated shape approximation. In Eur. Conf. Comput. Vis. Springer, August 2020.   
+[12] Juergen Gall, Carsten Stoll, Edilson De Aguiar, Christian Theobalt, Bodo Rosenhahn, and Hans-Peter Seidel. Motion capture using joint skeleton tracking and surface estimation. In IEEE Conf. Comput. Vis. Pattern Recog., pages 1746–1753. IEEE, 2009.   
+[13] Ke Gong, Xiaodan Liang, Yicheng Li, Yimin Chen, Ming Yang, and Liang Lin. Instance-level human parsing via part grouping network. In Proceedings of the European Conference on Computer Vision (ECCV), pages 770–785, 2018.   
+[14] Amos Gropp, Lior Yariv, Niv Haim, Matan Atzmon, and Yaron Lipman. Implicit geometric regularization for learning shapes. In Int. Conf. on Mach. Learn., pages 3569–3579. 2020.   
+[15] Marc Habermann, Lingjie Liu, Weipeng Xu, Michael Zollhoefer, Gerard Pons-Moll, and Christian Theobalt. Real-time deep dynamic characters. ACM Trans. Graph., 40(4), aug 2021.   
+[16] Marc Habermann, Weipeng Xu, Michael Zollhoefer, Gerard Pons-Moll, and Christian Theobalt. Deepcap: Monocular human performance capture using weak supervision. In IEEE Conf. Comput. Vis. Pattern Recog. IEEE, jun 2020.   
+[17] Marc Habermann, Weipeng Xu, Michael Zollhöfer, Gerard Pons-Moll, and Christian Theobalt. Livecap: Real-time human performance capture from monocular video. ACM Trans. Graph., 38(2):14:1–14:17, 2019.   
+[18] Peter Hedman, Pratul P Srinivasan, Ben Mildenhall, Jonathan T Barron, and Paul Debevec. Baking neural radiance fields for real-time view synthesis. In Int. Conf. Comput. Vis., 2021.   
+[19] Catalin Ionescu, Dragos Papava, Vlad Olaru, and Cristian Sminchisescu. Human3.6M: Large scale datasets and predictive methods for 3d human sensing in natural environments. IEEE Trans. Pattern Anal. Mach. Intell., 2014.   
+[20] Thomas Lewiner, Hélio Lopes, Antônio Wilson Vieira, and Geovan Tavares. Efficient implementation of marching cubes’ cases with topological guarantees. Journal of Graphics Tools, 8(2):1–15, 2003.   
+[21] Lingjie Liu, Jiatao Gu, Kyaw Zaw Lin, Tat-Seng Chua, and Christian Theobalt. Neural sparse voxel fields. NeurIPS, 2020.   
+[22] Stephen Lombardi, Jason Saragih, Tomas Simon, and Yaser Sheikh. Deep appearance models for face rendering. ACM Trans. Graph., 37(4):1–13, 2018.   
+[23] Stephen Lombardi, Tomas Simon, Jason Saragih, Gabriel Schwartz, Andreas Lehrmann, and Yaser Sheikh. Neural volumes: Learning dynamic renderable volumes from images. ACM Trans. Graph., 38(4):65:1– 65:14, July 2019.   
+[24] Matthew Loper, Naureen Mahmood, Javier Romero, Gerard Pons-Moll, and Michael J Black. SMPL: A skinned multi-person linear model. ACM Trans. Graph., 34(6):248:1–248:16, 2015.   
+[25] Lars M. Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger. Occupancy networks: Learning 3d reconstruction in function space. In IEEE Conf. Comput. Vis. Pattern Recog., pages 4460–4470, 2019.   
+[26] Moustafa Meshry, Dan B Goldman, Sameh Khamis, Hugues Hoppe, Rohit Pandey, Noah Snavely, and Ricardo Martin-Brualla. Neural rerendering in the wild. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6878–6887, 2019. Deep level sets: Implicit surface representations for 3D shape inference. arXiv preprint arXiv:1901.06802, 2019.   
+[28] Marko Mihajlovic, Yan Zhang, Michael J. Black, and Siyu Tang. LEAP: Learning articulated occupancy of people. In Proceedings IEEE/CVF Conf. on Computer Vision and Pattern Recognition (CVPR), June 2021.   
+[29] Ben Mildenhall, Pratul P. Srinivasan, Matthew Tancik, Jonathan T. Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. In Eur. Conf. Comput. Vis., 2020.   
+[30] Roger Mohr, Long Quan, and Françoise Veillon. Relative 3d reconstruction using multiple uncalibrated images. The International Journal of Robotics Research, 14(6):619–632, 1995.   
+[31] Michael Niemeyer, Lars Mescheder, Michael Oechsle, and Andreas Geiger. Differentiable volumetric rendering: Learning implicit 3d representations without 3d supervision. In IEEE Conf. Comput. Vis. Pattern Recog., June 2020.   
+[32] Atsuhiro Noguchi, Xiao Sun, Stephen Lin, and Tatsuya Harada. Neural articulated radiance field. In Int. Conf. Comput. Vis., 2021.   
+[33] Michael Oechsle, Songyou Peng, and Andreas Geiger. Unisurf: Unifying neural implicit surfaces and radiance fields for multi-view reconstruction. In Int. Conf. Comput. Vis., 2021.   
+[34] Jeong Joon Park, Peter Florence, Julian Straub, Richard A. Newcombe, and Steven Lovegrove. Deepsdf: Learning continuous signed distance functions for shape representation. In IEEE Conf. Comput. Vis. Pattern Recog., pages 165–174, 2019.   
+[35] Keunhong Park, Utkarsh Sinha, Jonathan T. Barron, Sofien Bouaziz, Dan B Goldman, Steven M Seitz, and Ricardo Martin-Brualla. Nerfies: Deformable neural radiance fields. In Int. Conf. Comput. Vis., 2021.   
+[36] Sida Peng, Yuanqing Zhang, Yinghao Xu, Qianqian Wang, Qing Shuai, Hujun Bao, and Xiaowei Zhou. Neural body: Implicit neural representations with structured latent codes for novel view synthesis of dynamic humans. In IEEE Conf. Comput. Vis. Pattern Recog., 2021.   
+[37] Albert Pumarola, Enric Corona, Gerard Pons-Moll, and Francesc Moreno-Noguer. D-nerf: Neural radiance fields for dynamic scenes. In IEEE Conf. Comput. Vis. Pattern Recog. IEEE, jun 2021.   
+[38] Prajit Ramachandran, Barret Zoph, and Quoc V Le. Searching for activation functions. arXiv preprint arXiv:1710.05941, 2017.   
+[39] Daniel Rebain, Wei Jiang, Soroosh Yazdani, Ke Li, Kwang Moo Yi, and Andrea Tagliasacchi. Derf: Decomposed radiance fields. In IEEE Conf. Comput. Vis. Pattern Recog., 2021.   
+[40] Gernot Riegler and Vladlen Koltun. Free view synthesis. In European Conference on Computer Vision, pages 623–640. Springer, 2020.   
+[41] Shunsuke Saito, Zeng Huang, Ryota Natsume, Shigeo Morishima, Angjoo Kanazawa, and Hao Li. Pifu: Pixel-aligned implicit function for high-resolution clothed human digitization. In Int. Conf. Comput. Vis., pages 2304–2314, 2019.   
+[42] Johannes L Schonberger and Jan-Michael Frahm. Structure-from-motion revisited. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4104–4113, 2016.   
+[43] Aliaksandra Shysheya, Egor Zakharov, Kara-Ali Aliev, Renat Bashirov, Egor Burkov, Karim Iskakov, Aleksei Ivakhnenko, Yury Malkov, Igor Pasechnik, Dmitry Ulyanov, et al. Textured neural avatars. In IEEE Conf. Comput. Vis. Pattern Recog., pages 2387–2397, 2019.   
+[44] Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Implicit neural representations with periodic activation functions. Adv. Neural Inform. Process. Syst., 33, 2020.   
+[45] Vincent Sitzmann, Justus Thies, Felix Heide, Matthias Nießner, Gordon Wetzstein, and Michael Zollhöfer. Deepvoxels: Learning persistent 3d feature embeddings. In Proc. Computer Vision and Pattern Recognition (CVPR), IEEE, 2019.   
+[46] Vincent Sitzmann, Michael Zollhöfer, and Gordon Wetzstein. Scene representation networks: Continuous 3d-structure-aware neural scene representations. In Adv. Neural Inform. Process. Syst., 2019.   
+[47] Shih-Yang Su, Frank Yu, Michael Zollhoefer, and Helge Rhodin. A-nerf: Surface-free human 3d pose refinement via neural rendering. In Adv. Neural Inform. Process. Syst., 2021.   
+[48] Ayush Tewari, Ohad Fried, Justus Thies, Vincent Sitzmann, Stephen Lombardi, Kalyan Sunkavalli, Ricardo Martin-Brualla, Tomas Simon, Jason Saragih, Matthias Nießner, et al. State of the art on neural rendering. In Comput. Graph. Forum, volume 39, pages 701–727. Wiley Online Library, 2020.   
+[49] Justus Thies, Michael Zollhöfer, and Matthias Nießner. Deferred neural rendering: Image synthesis using neural textures. ACM Transactions on Graphics (TOG), 38(4):1–12, 2019.   
+[50] Daniel Vlasic, Ilya Baran, Wojciech Matusik, and Jovan Popovic. Articulated mesh animation from ´ multi-view silhouettes. In ACM Trans. Graph., pages 1–9. 2008.   
+[51] Hongyi Xu, Eduard Gabriel Bazavan, Andrei Zanfir, William T Freeman, Rahul Sukthankar, and Cristian Sminchisescu. GHUM & GHUML: Generative 3d human shape and articulated pose models. In IEEE Conf. Comput. Vis. Pattern Recog., pages 6184–6193, 2021.   
+[52] Weipeng Xu, Avishek Chatterjee, Michael Zollhoefer, Helge Rhodin, Dushyant Mehta, Hans-Peter Seidel, and Christian Theobalt. Monoperfcap: Human performance capture from monocular video. ACM Trans. Graph., 2018.   
+[53] Lior Yariv, Yoni Kasten, Dror Moran, Meirav Galun, Matan Atzmon, Basri Ronen, and Yaron Lipman. Multiview neural surface reconstruction by disentangling geometry and appearance. Adv. Neural Inform. Process. Syst., 33, 2020.   
+[54] Alex Yu, Ruilong Li, Matthew Tancik, Hao Li, Ren Ng, and Angjoo Kanazawa. Plenoctrees for real-time rendering of neural radiance fields. In Int. Conf. Comput. Vis., 2021.   
+[55] Mihai Zanfir, Elisabeta Oneata, Alin-Ionut Popa, Andrei Zanfir, and Cristian Sminchisescu. Human synthesis and scene compositing. In AAAI, pages 12749–12756, 2020.   
+[56] Kai Zhang, Gernot Riegler, Noah Snavely, and Vladlen Koltun. Nerf $^ { + + }$ : Analyzing and improving neural radiance fields. arXiv preprint arXiv:2010.07492, 2020.   
+[57] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In CVPR, 2018.

parse/train/s-NI4H4e3Rf/s-NI4H4e3Rf_content_list.json

@@ -0,0 +1,868 @@
+[
+    {
+        "type": "text",
+        "text": "H-NeRF: Neural Radiance Fields for Rendering and Temporal Reconstruction of Humans in Motion ",
+        "text_level": 1,
+        "bbox": [
+            184,
+            122,
+            813,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hongyi Xu Google Research hongyixu@google.com ",
+        "bbox": [
+            196,
+            226,
+            366,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Thiemo Alldieck Google Research alldieck@google.com ",
+        "bbox": [
+            397,
+            226,
+            566,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Cristian Sminchisescu Google Research sminchisescu@google.com ",
+        "bbox": [
+            596,
+            226,
+            800,
+            268
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            290,
+            535,
+            305
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present neural radiance fields for rendering and temporal (4D) reconstruction of humans in motion (H-NeRF), as captured by a sparse set of cameras or even from a monocular video. Our approach combines ideas from neural scene representation, novel-view synthesis, and implicit statistical geometric human representations, coupled using novel loss functions. Instead of learning a radiance field with a uniform occupancy prior, we constrain it by a structured implicit human body model, represented using signed distance functions. This allows us to robustly fuse information from sparse views and generalize well beyond the poses or views observed in training. Moreover, we apply geometric constraints to co-learn the structure of the observed subject – including both body and clothing – and to regularize the radiance field to geometrically plausible solutions. Extensive experiments on multiple datasets demonstrate the robustness and the accuracy of our approach, its generalization capabilities significantly outside a small training set of poses and views, and statistical extrapolation beyond the observed shape. ",
+        "bbox": [
+            233,
+            309,
+            766,
+            502
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            522,
+            312,
+            539
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Enabling free-viewpoint video of a human in motion, based on a sparse set of views, is extremely challenging, but has many applications. Our work is motivated by a breadth of transformative 3D use cases, including immersive visualization of photographs, virtual clothing and try-on, fitness, as well as AR and VR for improved communication or collaboration. However, so far static scenes and rigid objects have been the primary subject of research. In pursuing realistic novel-view synthesis two schools of thought have been established: 1) 3D reconstruction methods aim to recover the geometry of the observed scene as accurately as possible, with novel views generated using classical rendering pipelines [30, 42]. 2) Image-based rendering techniques [7, 10, 40] and very recently neural radiance fields [29] primarily aim at image production quality without explicitly constructing an accurate 3D geometric model. While these techniques explicitly or implicitly reconstruct the scene geometry sometimes, this is however not guaranteed to accurately resemble the true scene geometry. We argue that novel-view rendering and reconstruction are two sides of the same coin, and that reliable viewpoint generalization, especially given relatively few input views, would require good quality for both. To this end, we propose a unified model in order to support both robust reconstruction and photo-realistic rendering. Dynamic scenes, especially those capturing a human in motion, add considerable complexity to the problem: while static scenes can be observed from many views by a camera moving through the scene, any configuration of a dynamic scene is typically observed only from sparse views. Moreover, the scene geometry and its appearance may change considerably over time. To cope with few views, some methods integrate scene knowledge over time by warping observations into a common reference frame [35, 37]. At test time, the information is warped back to the desired state and rendered from a novel view. Extrapolating to unseen motion, however, remains challenging. For scenes capturing people, this means that only poses seen during training can be rendered at test time. For some applications, however, rendering the subject over a broad range of motions, or in novel poses, is desirable. To make generalisation over poses and views possible, we rely on additional problem domain knowledge in the form of a human body model, imGHUM [4]. imGHUM is an implicit signed distance function (SDF) conditioned on generative shape and pose codes learned from a large corpus of dynamic human scans. In this work, imGHUM is used as the common reference frame for a neural radiance field and as a structured prior for robust reconstruction. Additionally, since imGHUM can represent a broad distribution of statistically valid human poses and shapes, we can render the reconstructed subject in novel poses and even with modified body shapes. In summary, our system supports photo-realistic free-view point temporal rendering of a human subject given only sparse camera observations. By conditioning on an implicit human body model, we can render considerably different viewpoints, body poses, and body shapes compared to those observed in training. Our carefully designed losses ensure not only good image quality but also plausible temporal reconstructions that can be used for the free-viewpoint visualisation of human performance capture. ",
+        "bbox": [
+            174,
+            545,
+            825,
+            902
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/6faeec44f9caf451971a47878320c6e86db2e12fcdeb0855d577e5e019081c0d.jpg",
+        "image_caption": [
+            "Figure 1: Overview of H-NeRF. Given a set of images of a human performer collected from a sparse set of calibrated cameras, we train a geometry-aware neural radiance field by co-learning a deformable signed distance field. First, we estimate the body shape $\\beta$ and track the articulated pose $\\theta , \\mathbf { T }$ using the implicit statistical human body model imGHUM (orange). imGHUM encodes all 3D spatial points $\\mathbf { x }$ across frames with a 4D point descriptor $( \\mathbf { s } , d )$ referencing a canonical frame. Using the foreground mask M, we co-train a residual SDF and a NeRF network in that canonical space, in order to integrate all image observations into a consistent implicit 3D geometry $\\Delta d$ and its view-depended $( \\mathbf { v } )$ radiance representation $( \\mathbf { c } , \\sigma )$ . The trainable residual SDF and NeRF networks (in blue) are conditioned on the body pose $\\pmb { \\theta }$ and the root transformation $\\mathbf { T }$ to model pose-dependent geometric and appearance variations. Our framework supports both accurate 3D geometric reconstruction and free-viewpoint rendering, and generalizes well to novel views, shapes, and poses. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            88,
+            785,
+            210
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            385,
+            825,
+            523
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            542,
+            321,
+            559
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Human performance capture is the process of reconstructing the dynamic (4D) geometry of a human subject as observed in one or multiple synchronized views. Pioneering methods deform a rigged template mesh against silhouettes observed in multiple views [6, 50], estimated skeleton key-points [12], or image correspondences [9]. Later systems relying on RGB-D video streams [5] or monocular capture [3, 2, 17, 52, 16, 15] have also been presented. All these methods rely either on a pre-scanned template mesh or on a body model that is deformed to explain the image evidence. ",
+        "bbox": [
+            174,
+            566,
+            825,
+            650
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Neural representation for view synthesis. With the advent of neural networks, researchers have begun to explore alternative solutions to represent a scene in order to support novel view synthesis and photo-realistic rendering; we refer the reader to [48] for a survey. Different scene representation have been explored. Some methods use voxel grids to embed the scene [23, 45]. For rendering, the voxel grid is probed by shooting a ray and by linearly interpolating voxel values. These samples are then transformed into color values using a neural network. Others have proposed neural textures [49, 43] that can be rendered based on view-dependent effects or texture synthesis networks [22] to realistically render meshes. In a similar spirit, other methods render point clouds, where each point carries local appearance information [26, 1]. Riegler and Koltun [40] reproject features from nearby views and rely on a SfM reconstruction scaffold for novel view synthesis. With the recent advent of implicit function networks [25, 34, 8, 27], such 3D representations have been explored for rendering and view synthesis with great success. In contrast to discrete voxel representations, textures, or point clouds, these methods represent the scene as a continuous function and thus are not bound to a specific image or volume resolution. In the pioneering work of Sitzmann et al. an implicit function produces features displayed using a neural renderer [46]. Follow up methods focus on 3D geometric reconstruction [44], and use 2D supervision [31, 53]. ",
+        "bbox": [
+            174,
+            656,
+            825,
+            876
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Neural Radiance Fields (NeRFs) are a recent approach to represent scenes for novel view synthesis. Mildenhall et al. [29] introduce Neural Radiance Fields resented as fully connected neural networks, where the input is a spatial query point and a viewing direction. The output is a volume density and the emitted radiance at the query location in the direction of the viewer. By ray-tracing using this simple representation, one can generate photo-realistic images from novel views. Despite the excellent quality of results, one drawback is the slow rendering time. To this end, researchers have presented faster versions that e.g. transform the radiance field into more efficient sparse grids [18] or remove the dependence on viewing-direction during rendering by estimating a spherical harmonic representation of the radiance function [54]. Others improve fidelity or rendering time by tackling ambiguities in the original formulation [56], spatial decomposition into multiple NeRFs [39], or combining NeRF with sparse voxel fields [21]. Initial work on adapting NeRF to dynamic scenes has been presented as well. Park et al. [35] produce “Nerfies” (NeRF-Selfies) from videos where subjects carefully move a camera around their head. The scene information is fused by warping query points into a canonical reference frame. Similarly, Pumarola et al. [37] produce dynamic NeRFs from synthetic animation data. Related to our approach, some methods integrate human body models to fuse information over time. A-NeRF [47] uses a skeleton to rigidly transform NeRF features to refine estimated 3D poses. A similar approach is followed in NARF [32] for view synthesis. Most related, Neural Body [36] attaches learnable features to the vertices of a SMPL body model [24]. These features are processed using a sparse 3D convolutional network, where the output forms a neural radiance field. In contrast to our approach, the resolution is bounded by the spatial resolution of the 3D convolutional network and no geometric supervision is used. We highlight differences in $\\ S 5$ . ",
+        "bbox": [
+            174,
+            883,
+            823,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            825,
+            353
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Background ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            378,
+            308,
+            396
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a collection of images capturing a dynamic scene of a human in motion, observed from a sparse set of calibrated camera views (in the limit a monocular camera, as we will show), we aim to learn both the detailed temporal geometry of the human in motion, and to render the sequence from novel camera views and for different human poses. To this end, our work unifies two main methodologies: 1) implicit 3D human representations, and 2) volumetric radiance fields. In this section, we provide the relevant background on both representations, which we co-learn in a joint framework. ",
+        "bbox": [
+            174,
+            410,
+            825,
+            494
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Neural Radiance Fields. A neural radiance field (NeRF) [29] represents a 3D scene as a continuous function of color volume densities. The model consists of a neural network function $F _ { \\omega }$ that maps a 3D spatial point $\\mathbf { x } \\in \\mathbf { R } ^ { 3 }$ and a viewing direction $\\mathbf { v } \\in \\mathbf { R } ^ { 3 }$ to a volume density $\\sigma \\in \\mathbf { R } ^ { + }$ and a radiance $\\mathbf { c } ( \\mathbf { x } , \\mathbf { \\bar { v } } ) \\in \\mathbf { R } ^ { 3 }$ emitted towards the viewer. In practice, NeRF encodes the inputs $\\mathbf { x }$ and $\\mathbf { v }$ using a sinusoidal positional encoding $\\gamma : \\mathbf { R } ^ { 3 }  \\mathbf { R } ^ { 3 + \\bar { 6 } m }$ that projects a coordinate vector into a high-dimensional space using a set of sine and cosine functions of $m$ increasing frequencies. Given a ray $\\mathbf { r } = \\mathbf { o } + s \\mathbf { v }$ with $N$ samples $\\{ { \\bf x } \\}$ originating from a camera location o, NeRF integrates radiance values along the ray by means of alpha blending. The pixel/ray color is approximated with numerical quadrature [33], ",
+        "bbox": [
+            173,
+            500,
+            825,
+            626
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6c79645fc8ba756236f334c29dacd41ac02689d0ff4950ed7a2936e5222c6201.jpg",
+        "text": "$$\n\\mathbf { C } ( \\mathbf { r } ) = \\sum _ { i = 1 } ^ { N } \\alpha ( \\mathbf { x } _ { i } ) \\prod _ { j < i } ( 1 - \\alpha ( \\mathbf { x } _ { j } ) ) \\mathbf { c } ( \\mathbf { x } _ { i } , \\mathbf { v } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            616,
+            642,
+            661
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c5501423d8f2f962aab87e44c02d40ad5d7d5c07546e94405988e3260bfc7696.jpg",
+        "text": "$$\n\\alpha ( \\mathbf { x } _ { i } ) = 1 - \\exp \\bigl ( - \\sigma ( \\mathbf { x } _ { i } ) \\delta _ { i } \\bigr ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            662,
+            596,
+            681
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha ( \\mathbf { x } _ { i } )$ is the transparency by accumulating transmittance along the ray, and $\\delta _ { i } = | \\mathbf { x } _ { i + 1 } - \\mathbf { x } _ { i } |$ is the distance between adjacent samples. The NeRF function $F _ { w }$ is fully differentiable and its network parameters $w$ can be optimized using an image reconstruction loss [29]. An approximate 3D scene geometry $\\mathbf { S } _ { F } = \\{ \\mathbf { x } | \\sigma ( \\mathbf { x } ) = \\sigma _ { h } \\}$ can be extracted from the trained opacity field via Marching Cubes [20] at a density threshold $\\sigma _ { h }$ . ",
+        "bbox": [
+            174,
+            683,
+            823,
+            753
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Implicit Generative Human Models. Implicit human body surfaces are typically represented as the decision boundary of either binary occupancy classifiers [11, 41, 28] or signed distance functions [14, 4]. Specifically, our work builds upon the SOTA statistical implicit human model imGHUM [4] $\\bar { H } _ { \\omega } : ( \\mathbf { T } ^ { - 1 } \\mathbf { x } , \\bar { } \\mathbf { \\beta } , \\pmb { \\theta } )  ( d , \\mathbf { s } )$ that maps a 3D spatial point $\\mathbf { x }$ , unposed with root joint transformation $\\dot { \\mathbf { T } } \\in \\mathbf { R } ^ { 4 \\times 3 }$ , to its signed distance value $d \\in \\mathbf { R }$ with respect to a body surface parameterized with body shape $\\boldsymbol { \\beta } \\in { \\bf R } ^ { 1 6 }$ and articulated pose ${ \\pmb \\theta } \\in { \\bf R } ^ { 1 1 8 }$ . In addition to $d$ , imGHUM returns implicit continuous semantics $\\textbf { s } \\in \\ \\mathbf { R } ^ { 3 }$ of the query point which corresponds to the 3D coordinate of the nearest surface point defined on a canonical surface. We refer to the original paper [4] for details. Essentially, imGHUM builds a human-centric 4D semantic descriptor $( d , \\mathbf { s } )$ for all spatial points in the neighborhood of the surface. imGHUM is trained with a large collection of human scans of diverse body shapes and poses, sharing the same generative shape and pose latent code with the mesh-based statistical human model GHUM [51]. The implicit articulated 3D human body $\\mathbf { S } _ { H } ( \\beta , \\pmb \\theta , \\mathbf { T } ) = \\{ \\mathbf { x } | d ( \\mathbf { x } ) = 0 \\}$ is defined by the the zero-isosurface of the signed distance field. ",
+        "bbox": [
+            174,
+            758,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 Method ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            138,
+            271,
+            155
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We present the details of our main contribution, H-NeRF, a novel neural network (fig. 1) that exploits the power of volumetric radiance fields to learn complex human structure and appearance, by relying on statistical implicit human pose and shape signed distance functions for accurate geometric reconstruction. Further, H-NeRF relies on imGHUM, an implicit model of articulated human pose and shape, as a rich geometric prior, to integrate scene information over time, and to represent human articulation. Co-learning both a radiance field and a signed distance function of scene geometry consistently in a unified framework, enables accurate 3D geometric reconstruction and volume rendering of a dynamic human in motion, through 1) consistent integration of image observations over time, and 2) good generalization capability for novel viewpoints, human poses, and for statistically extrapolated shapes, given only very few training poses and camera views. ",
+        "bbox": [
+            174,
+            162,
+            825,
+            301
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given a video of a human in motion, observed by a sparse set of calibrated cameras, our objective is to generate free-viewpoint video of the observed person and to reconstruct the underlying 4D geometry. The set of input images, with a resolution of $w \\times h$ pixels, is denoted as $\\{ \\mathbf { I } _ { t } ^ { c } \\mathbf { \\bar { \\Psi } } \\in \\mathbf { \\bar { ~ R } } ^ { w \\times \\mathbf { \\breve { h } } \\times 3 } | c \\mathbf { \\Psi } = \\mathbf { \\bar { ~ \\Psi } }$ $1 , \\dots , N _ { c } , t = 1 , \\dots , N _ { t } \\}$ , where $c$ is the camera index, $N _ { c }$ is the number of cameras, $t$ is the frame index, and $N _ { t }$ is the number of frames. For each image, we apply [13] to obtain the binary foreground human mask ${ \\bf M } _ { t } ^ { c } \\in { \\bf R } ^ { w \\times h }$ . In addition, we obtain a temporally consistent imGHUM latent shape and pose codes, $\\beta$ and $\\left( \\boldsymbol { \\theta } _ { t } , \\mathbf { T } _ { t } \\right)$ , respectively, at each frame index $t$ , by optimizing an imGHUMequivalent parametric replica under multi-view keypoint and body segmentation losses [55]. We refer to our Sup. Mat. for details on the imGHUM fitting process. ",
+        "bbox": [
+            173,
+            308,
+            825,
+            433
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the sequel we first explain our adaptations to NeRF for the static case. imGHUM is used as a prior in order to bias the reconstruction of the observed scene towards a more accurate human geometry. We continue by explaining the additional methodological innovation needed in the dynamic case. Hereby, imGHUM provides spatial-temporal correspondences and is used as the common reference frame to fuse information across different views and time instances. ",
+        "bbox": [
+            174,
+            438,
+            825,
+            508
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 Static Semantic Human NeRF ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            523,
+            421,
+            539
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We first formulate H-NeRF for a human capture at a single moment in time $N _ { t } ~ = ~ 1$ ). From multi-view image observations, we co-learn a radiance field $F _ { \\omega } : \\mathbf { x } , \\mathbf { v }  ( \\mathbf { c } , \\sigma )$ for free-viewpoint rendering, and a signed distance function $\\hat { H } _ { \\omega } : \\mathbf { x }  \\hat { d }$ for 3D geometric reconstruction. We use $\\hat { H } _ { \\omega }$ for the dressed subject in order to distinguish it from the body’s imGHUM SDF $H _ { \\omega }$ . ",
+        "bbox": [
+            173,
+            545,
+            825,
+            604
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Like NeRF, we formulate an $L _ { 1 }$ image reconstruction loss to optimize $F _ { w }$ as ",
+        "bbox": [
+            173,
+            611,
+            676,
+            626
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d0eb6a3bdec595a333048ee0cb3fce356fec22043b4fc930f1f156acc33cd77b.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { r e c } } = \\sum _ { \\mathbf { r } \\in \\mathcal { R } } \\| \\bar { \\mathbf { C } } ( \\mathbf { r } ) - \\mathbf { C } ( \\mathbf { r } ) \\| _ { 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            627,
+            596,
+            661
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { R }$ denotes the batched set of all pixels/rays, and $\\bar { \\mathbf { C } } ( \\mathbf { r } ) , \\mathbf { C } ( \\mathbf { r } )$ are observed and rendered pixel colors, cf. (1), respectively. However, under sparse training camera views, the volumetric radiance field is not well regularized, leading to poor generalization to novel viewpoints, cf. fig. 2. Specifically, we observe that the model encounters difficulties in correctly representing the scene and in separating the person from the background. The model fails to learn a semantically meaningful opacity field, i.e. $\\alpha = 0$ in free space, and 1 if occupied. ",
+        "bbox": [
+            173,
+            662,
+            825,
+            747
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Coarse Scene Structuring. Using imGHUM fits for all training images, we can spatially locate the person in 3D. We define a 3D bounding box $\\mathbf { B } \\in [ \\underline { { \\mathbf { S } } } _ { H } - \\epsilon , \\overline { { \\mathbf { S } _ { H } } } + \\epsilon ]$ around the detected person, where $\\underline { { \\mathbf { S } } } _ { H } , \\overline { { \\mathbf { S } _ { H } } }$ are the minimal and maximal coordinates of the human body surface S and $\\epsilon$ is a spatial margin reserved for geometry not modeled by imGHUM. All radiance points associated to the rendering of the person should reside inside the bounding box, leading to a 3D segmentation loss ",
+        "bbox": [
+            176,
+            752,
+            825,
+            828
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2d59cec32cda1fa6df37ed1bdf916a2ab1043bb7253bd0217a0754c5de13ad5c.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { \\tilde { \\mathbf { C } } ( \\mathbf { r } ) = \\displaystyle \\sum _ { i = 1 } ^ { N } b ( \\mathbf { x } _ { i } ) \\alpha ( \\mathbf { x } _ { i } ) \\prod _ { j < i } ( 1 - b ( \\mathbf { x } _ { j } ) \\alpha ( \\mathbf { x } _ { j } ) ) \\mathbf { c } ( \\mathbf { x } _ { i } , \\mathbf { v } ) , } \\\\ & { } & { \\mathcal { L } _ { \\mathrm { m a s k } } = \\displaystyle \\sum _ { \\mathbf { r } \\in \\mathcal { R } } \\left\\| \\mathbf { M } ( \\mathbf { r } ) \\big ( \\bar { \\mathbf { C } } ( \\mathbf { r } ) - \\tilde { \\mathbf { C } } ( \\mathbf { r } ) \\big ) \\right\\| _ { 1 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            829,
+            676,
+            910
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $b ( \\mathbf { x } _ { i } )$ is 0 if outside of $\\mathbf { B }$ and 1 otherwise, and $\\mathbf { M } ( \\mathbf { r } )$ the image mask. We rely on the mask, so the loss is only applied to image observations from the subject, and not the remaining scene geometry. ",
+        "bbox": [
+            171,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Unifying SDF with NeRF. After structuring the scene coarsely, we now couple the estimated radiance field with an implicit signed distance-based 3D reconstruction, in order to further regularize the opacity distribution. A signed distance function associated to the detected person in the image naturally comes with a 3D classifier for all spatial points, where $\\hat { d } ( \\mathbf { x } _ { i } ) > 0$ means $\\mathbf { x } _ { i }$ is in free space, whereas $\\mathbf { x } _ { i }$ lies within the subject when $\\hat { d } ( \\mathbf { x } _ { i } ) < = 0$ . We rely on this insight in order to co-learn a SDF of the performer and constrain the radiance field. To this end, we introduce a pseudo alpha value $\\dot { \\alpha } ( \\mathbf x _ { i } ) = \\phi ( \\gamma \\hat { d } ( \\mathbf x _ { i } ) )$ where $\\phi$ is a Sigmoid activation function and $\\gamma$ controls the sharpness of the boundary. To refine the NeRF opacity semantics, especially for the volume in the neighborhood of the human surface $\\mathbf { B }$ , we formulate two losses ",
+        "bbox": [
+            173,
+            125,
+            825,
+            258
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bea1ba03ba53b89e97009300f8d3cffac243ac85bf4c6c7b2072ee7630f62d2f.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle { \\hat { \\bf C } } ( { \\bf r } ) = \\sum _ { i = 1 } ^ { N } \\hat { \\alpha } ( { \\bf x } _ { i } ) \\prod _ { j < i } ( 1 - \\hat { \\alpha } ( { \\bf x } _ { j } ) ) { \\bf c } ( { \\bf x } _ { i } , { \\bf v } ) } , \\quad \\hat { \\alpha } ( { \\bf x } _ { i } ) = b ( { \\bf x } _ { i } ) \\dot { \\alpha } ( { \\bf x } _ { i } ) + ( 1 - b ( { \\bf x } _ { i } ) ) \\alpha ( { \\bf x } _ { i } ) } ,  \\\\ { { \\displaystyle { \\mathcal { L } } _ { \\mathrm { b l e n d } } = \\sum _ { { \\bf r } \\in { \\mathcal R } } \\left( \\| { \\bf M } ( { \\bf r } ) \\big ( { \\bar { \\bf C } } ( { \\bf r } ) - \\hat { \\bf C } ( { \\bf r } ) \\big ) \\| _ { 1 } + \\eta \\| \\big ( 1 - { \\bf M } ( { \\bf r } ) \\big ) \\big ( { \\bar { \\bf C } } ( { \\bf r } ) - \\hat { \\bf C } ( { \\bf r } ) \\big ) \\| _ { 1 } \\right) } , \\quad { \\mathrm { ~ i ~ f ~ } } \\quad { \\mathrm { ~ i ~ f ~ } } \\quad i = { \\bf r } \\quad { \\bf r } \\quad - \\Pi _ { i } ^ { \\prime } , } \\\\ { { \\displaystyle { \\mathcal { L } } _ { \\mathrm { g e o m } } ( { \\bf r } ) = \\sum _ { i = 1 } ^ { N } b ( { \\bf x } _ { i } ) \\mathrm { B C E } \\big ( \\phi \\big ( \\lambda ( \\sigma _ { h } - \\sigma ( { \\bf x } _ { i } ) ) \\big ) , \\dot { \\alpha } ( { \\bf x } _ { i } ) \\big ) } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            258,
+            767,
+            383
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { \\mathbf { C } } ( \\mathbf { r } )$ is the rendered pixel color with blended alpha values $\\hat { \\alpha }$ replacing NeRF alpha values $\\alpha$ with SDF-based pseudo alpha $\\dot { \\alpha }$ for all ray points inside the bounding box $\\mathbf { B }$ . The first term in ${ \\mathcal { L } } _ { \\mathrm { b l e n d } }$ requires the rendered pixel color for all the intersecting rays within the human mask to come from the surface, whereas the second term assumes that all background color is formed from ray samples outside of $\\mathbf { B }$ . We set $\\eta = 1$ when no other geometry than the person is inside $\\mathbf { B }$ and tune $\\eta$ down if the assumption is violated (e.g. person standing on a floor). The term $\\mathcal { L } _ { \\mathrm { g e o m } }$ uses the binary cross entropy loss to couple the NeRF surface boundary with the zero-isosurface of the signed distance function describing the subject. While the coupling terms given by (7) and (8) act as strong priors for the opacity distribution, during test time, we still rely on volumetric radiance rendering with learned NeRF alpha values $\\alpha$ to support transparency effects and complex geometry for structures like hair. ",
+        "bbox": [
+            173,
+            386,
+            825,
+            526
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Image-based SDF learning. Learning the signed distance field $\\hat { H } _ { \\omega }$ from scratch using a sparse set of training images is still challenging. The model often fails to reconstruct reasonable human geometry, even when using coupling losses (7) and (8). We therefore leverage imGHUM as an inner layer for the target human reconstruction and combine it with a light-weight residual SDF network $\\Delta H _ { \\omega } : { \\bf x } \\to \\Delta d$ . The residual SDF models surface details, including hair and clothing, that are not represented by imGHUM. The final signed distance for $\\mathbf { x } _ { i }$ becomes $\\hat { d } ( \\mathbf { x } _ { i } ) = d ( \\mathbf { x } _ { i } | \\beta , \\pmb { \\theta } , \\mathbf { T } ) + \\Delta d ( \\mathbf { x } _ { i } )$ . Given the training images, we learn the personalized residual SDF using our coupling losses, given by (7) and (8), and additionally apply geometric regularization ",
+        "bbox": [
+            173,
+            532,
+            825,
+            647
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f48b2dd78bf2b52fc25a3145a9f2feed699c28d1b931705e6dce83c62115db51.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { s e g } } = \\sum _ { \\mathbf { r } \\in \\mathcal { R } } \\operatorname { B C E } ( \\mathbf { M } ( \\mathbf { r } ) , \\hat { d } _ { \\mathrm { m i n } } ( \\mathbf { r } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            388,
+            648,
+            607,
+            683
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/60b36ebfaaee4eb9f5d58fec08d1a133467ba95b7b7d30e58379822c5f0ddd95.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { e i k } } ( \\mathbf { r } ) = \\sum _ { i = 1 } ^ { N } b ( \\mathbf { x } _ { i } ) ( \\| \\nabla _ { \\mathbf { x } _ { i } } \\hat { d } ( \\mathbf { x } _ { i } ) \\| _ { 2 } - 1 ) ^ { 2 } , \\quad \\mathcal { L } _ { \\mathrm { r e g } } = \\| \\psi ( \\Delta d ) \\| _ { 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            685,
+            705,
+            727
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\hat { d } _ { \\mathrm { m i n } } ( \\mathbf { r } )$ denotes the minimal signed distance for all sampled ray points and $\\psi$ is the ReLU activation function. The term $\\mathcal { L } _ { \\mathrm { s e g } }$ ensures that if a pixel is inside the human segmentation mask, there should be at least one intersection between the ray and the 3D human surface and therefore $\\hat { d } _ { \\mathrm { m i n } } ( \\mathbf { r } )$ should be non-positive. Otherwise, all ray samples should have positive signed distances. Using $\\mathcal { L } _ { \\mathrm { e i k } }$ , we enforce the composite SDF $\\hat { d }$ to be approximately a signed distance function, i.e. incorporating Eikonal regularization [14]. The term $\\mathcal { L } _ { \\mathrm { r e g } }$ regularizes the residual distance $\\Delta d$ to be non-positive. This is because imGHUM should reside within the geometry and personalized geometric details given by $\\Delta d$ should be modeled on top of the skin surface. ",
+        "bbox": [
+            173,
+            731,
+            826,
+            849
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Dynamic Semantic Human NeRF ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            444,
+            106
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We now extend the framework to model dynamic human motion. To implicitly model scenes where the human moves, we learn a consistent, continuous function $G _ { \\omega } : \\bar { ( \\mathbf { c } , \\alpha , \\hat { d } | \\mathbf { z } ) } \\to ( \\mathbf { c } ^ { \\prime } , \\alpha ^ { \\prime } , \\hat { d } ^ { \\prime } | \\mathbf { z } ^ { \\prime } )$ for both the geometry and the appearance variations across frames. For generalization to novel poses or shapes, $G _ { \\omega }$ should be conditioned on a semantically meaningful latent code $\\mathbf { z }$ that can be interpolated and should ideally have good extrapolation properties. To integrate scene information over time in a single radiance field, we follow the approaches in [35, 37] and warp observations into a canonical reference frame. Given our constraints to $G _ { \\omega }$ , imGHUM, conditioned on its semantic shape and pose latent code $( \\beta , \\pmb \\theta , \\mathbf { T } )$ , naturally provides spatial correspondences across frames. Given two spatial points $\\mathbf { x }$ and $\\mathbf { x } ^ { \\prime }$ in the space of two human instances $( \\beta , \\pmb \\theta , \\mathbf { T } )$ and $( \\beta ^ { \\prime } , \\pmb \\theta ^ { \\prime } , \\mathbf { T } ^ { \\prime } )$ respectively, we decide these are in correspondence if they have the same semantics and signed distances $( \\bar { \\bf s } , d | \\beta , \\pmb \\theta , { \\bf T } ) = ( { \\bf s } ^ { \\prime } , d ^ { \\prime } | \\beta ^ { \\prime } , \\pmb \\theta ^ { \\prime } , { \\bf T } ^ { \\prime } )$ . Essentially, imGHUM assigns a 4D point descriptor $( \\mathbf { s } , d ) \\in \\mathbf { R } ^ { 4 }$ to any spatial point and deforms the volume continuously with respect to the parameterized articulated human body surface. Specifically, we apply $H _ { \\omega } : ( \\bar { \\mathbf { T } } ^ { - 1 } \\mathbf { x } , \\beta , \\pmb { \\dot { \\theta } } )  ( d , \\mathbf { s } )$ to map a spatial point $\\mathbf { x }$ to a canonical point descriptor $( \\mathbf { s } , d )$ . We then modify the input to the NeRF function with the point descriptor as $F _ { w } : ( \\mathbf { s } , d ) \\to ( \\mathbf { c } , \\sigma )$ , and similarly for the residual SDF function as $\\Delta H _ { \\omega } : ( \\mathbf { s } , d )  \\Delta d$ . imGHUM is pre-trained using a large corpus of 3D human scans, thus we keep it fixed in our network and directly use this prior to integrate structured geometric and appearance information across training frames. Given that the background scene is invariant to the human motion, we only apply the imGHUM warping function to spatial points inside the bounding box B, thus largely improving computation time and lowering memory usage. ",
+        "bbox": [
+            173,
+            111,
+            825,
+            392
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Pose-Dependent Geometry and Appearance. Training a single canonical NeRF and a residual SDF from a sequence integrates image observations into a consistent implicit geometric and volumetric representation of the radiance. However, we observe that fine-grained geometric and appearance variations caused by human motion are missing in the canonical NeRF and the residual SDF. To model such variations in geometry (clothing deformation) and appearance (lighting, self-shadows), we condition the NeRF function $F _ { \\omega }$ – the volume density $\\sigma$ (geometry) and color value c (appearance) – as well as the residual SDF $\\Delta H _ { \\omega }$ (geometry), on the body pose code $\\pmb \\theta$ . In addition, we also notice that the relative position and rotation of the person with respect to the scene largely affects appearance (due to illumination effects), but should not affect the geometry. Based on this observation, we additionally condition the NeRF color on the person’s root transformation as $\\mathbf { c } ( \\mathbf { x } | \\mathbf { \\boldsymbol { \\theta } } , \\mathbf { T } )$ . ",
+        "bbox": [
+            173,
+            397,
+            825,
+            536
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "H-NeRF Articulation. Differently from prior work [35, 37], our H-NeRF modules are conditioned on semantic human latent codes $( \\beta , \\pmb \\theta , \\mathbf { T } )$ , leading to an SDF and radiance field that can be interpolated. Moreover, due to imGHUM’s capability to accurately model geometric volume deformation for diverse human poses and shapes, H-NeRF enjoys strong generalisation to novel poses and even body shapes. Concretely, during inference, one can simply assign a different set of $( \\beta , \\pmb \\theta )$ to transfer the learned geometry and appearance to an unseen pose and shape configuration. For better generalization, we add Gaussian noise to the input code $( \\pmb \\theta , \\mathbf T )$ , preventing network overfitting to pose-dependent deformation and appearance effects experienced during training. When only a sparse set of camera views is available, We apply the same technique to NeRF, conditioned on viewing direction $\\mathbf { v }$ . ",
+        "bbox": [
+            173,
+            541,
+            826,
+            667
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Fine-tuning imGHUM Pose and Shape. We observe that imGHUM fitting can be a source of error when learning $F _ { \\omega }$ and $\\Delta H _ { \\omega }$ from dynamic image sequences. Instead of keeping parameters fixed, we further improve the fit during training by fine tuning a time-consistent shape correction $\\Delta \\beta$ and per-frame pose correction $\\Delta \\theta ( t )$ with ",
+        "bbox": [
+            173,
+            672,
+            825,
+            728
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c9b10ac1623ea2b0d74f22322004de7c4649fc8312912c4ff8a54e2d6721b09e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\mathrm { \\sum _ { f i t } } = \\sum _ { \\mathbf { r } \\in \\mathcal { R } } \\mathrm { B C E } ( \\mathbf { M } ( \\mathbf { r } ) , d _ { \\operatorname* { m i n } } ( \\mathbf { r } | \\boldsymbol { \\beta } + \\Delta \\boldsymbol { \\beta } , \\pmb { \\theta } + \\Delta \\pmb { \\theta } ( t ) ) ) , \\quad \\mathcal { L } _ { \\operatorname* { i n c } } = \\| \\Delta \\boldsymbol { \\beta } \\| _ { 2 } + \\frac { 1 } { N _ { t } } \\sum _ { t = 1 } ^ { N _ { t } } \\| \\Delta \\pmb { \\theta } \\| _ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            207,
+            724,
+            761,
+            763
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\mathcal { L } } _ { \\mathrm { f i t } }$ aligns the imGHUM fit with the human foreground segmentation and ${ \\mathcal { L } } _ { \\mathrm { i n c } }$ regularizes the incremental corrections. ",
+        "bbox": [
+            174,
+            765,
+            823,
+            792
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            811,
+            313,
+            829
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We quantitatively and qualitatively evaluate H-NeRF through ablations and by comparing with other methods. Please refer to our Sup. Mat. for additional results and ablation experiments. We first detail our model architecture, as well as the datasets and evaluation metrics used. ",
+        "bbox": [
+            174,
+            835,
+            825,
+            877
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Architecture and Training. We use the same network architecture for all of our experiments, where the trainable modules $F _ { \\omega }$ and $\\Delta H _ { \\omega }$ consist of eight 256-dimensional and six 128-dimensional MLPs respectively, with a skip connection to the middle layer and with Swish activation [38]. As in the original NeRF [29], we use 256 coarse and fine-level ray samples, for each of which we use 8- and 1-dimensional positional encoding for $F _ { \\omega }$ and $\\Delta H _ { \\omega }$ , respectively. We train the network using the Adam optimizer with a learning rate of 0.001 exponentially decayed by a factor 0.1 until the maximum number of iterations is reached ( $1 0 k$ iteration of $4 k$ ray batch size). We apply Gaussian noise with $\\sigma = 0 . 1$ to the NeRF conditioning code $( { \\pmb \\theta } , \\mathbf T , \\mathbf v )$ . ",
+        "bbox": [
+            171,
+            883,
+            826,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/62edcca9d6dc33c339a157fd14b80255d3de684df111c2098d20d712becfd354.jpg",
+        "image_caption": [
+            "Figure 2: Left: H-NeRF (red, $\\mathbf { X }$ -axis: #cameras) outperforms the original NeRF (blue) and IDR (green) in both novel view synthesis and 3D reconstruction of a static scene (16 test cameras). NeRF and IDR fail under sparse camera views (solid line: RenderPeople scan; dashed line: GHS3D scan). Right: we qualitatively show novel views and reconstructed geometry for NeRF, IDR, and H-NeRF (from left to right) trained using four cameras. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            85,
+            825,
+            214
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/10efee4795e131a735f4a76c0b0d71e4dcf01e81554100aeb7d88b72d14ffe4b.jpg",
+        "table_caption": [
+            "Table 1: Quantitative evaluation of dynamic sequences for various datasets. When two metrics are reported they correspond to evaluation of training poses and novel poses for test cameras, respectively. PeopleSnapshot and Human3.6M are evaluated on novel poses under training views (where groundtruth images exist). Geometric metrics are only reported when ground-truth is available. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Dataset</td><td>PSNR↑</td><td>SSIM↑</td><td>LPIPS↓</td><td>Ch ×10-3 ↓</td><td>NC↑</td><td>IoU↑</td></tr><tr><td rowspan=\"3\">NeuralBody [36]</td><td>RenderPeople</td><td>27.33/23.52</td><td>0.888/0.827</td><td>0.117/0.247</td><td>0.536/0.63</td><td>0.908/0.892</td><td>0.864/0.824</td></tr><tr><td>GHS3D</td><td>24.7</td><td>0.829</td><td>0.236</td><td>0.79</td><td>0.887</td><td>0.81</td></tr><tr><td>PeopleSnapshot</td><td>24.62</td><td>0.849</td><td>0.160</td><td>1</td><td>一</td><td>1</td></tr><tr><td rowspan=\"4\">H-NeRF (ours)</td><td>Human3.6M</td><td>24.86</td><td>0.82</td><td>0.189</td><td>1</td><td>一</td><td>1</td></tr><tr><td>RenderPeople</td><td>28.78/24.31</td><td>0.913/0.856</td><td>0.125/0.246</td><td>0.217/0.274</td><td>0.950/0.939</td><td>0.917/0.9</td></tr><tr><td>GHS3D</td><td>24.92</td><td>0.852</td><td>0.232</td><td>0.218</td><td>0.932</td><td>0.89</td></tr><tr><td>PeopleSnapshot Human3.6M</td><td>26.33 25.01</td><td>0.868 0.83</td><td>0.159 0.17</td><td>一</td><td>一</td><td>1</td></tr></table>",
+        "bbox": [
+            174,
+            296,
+            823,
+            396
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            477,
+            825,
+            560
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Dataset and Metrics. We provide qualitative and quantitative evaluation on four different datasets: RenderPeople (8 sequences), GHS3D [51] (14 sequences), PeopleSnapshot [3] (7 sequences) and Human3.6M [19] (5 sequences). The first two are synthetic, rendering animated (RenderPeople) or 4D human scans (GHS3D) from four orthogonal cameras, where ground-truth geometry paired with images is available. We evaluate both the image and the 3D geometry reconstruction quality from two novel cameras, and qualitatively demonstrate generalization to shapes and poses not in the training set. The remaining datasets are real captured videos (PeopleSnapshot: monocular, Human3.6M: four cameras) without paired 3D geometry, where we only evaluate the rendered images. For image metrics, we adopt peak signal-to-noise ratio (PSNR $\\uparrow$ ), structural similarity index (SSIM $\\uparrow$ ) and learned perceptual image patch similarity (LPIPS $\\downarrow$ ) [57]. For evaluation, we render the NeRF field inside the human bounding box $\\mathbf { B }$ and compare to the ground-truth segmented image within a region of interest around the person. To evaluate geometric reconstruction quality, we report bi-directional Chamfer (Ch) $L _ { 2 }$ distance, Normal Consistency (NC) and Volumetric Intersection over Union (IoU), evaluated on the mesh produced by applying Marching Cubes on the SDF, with a resolution of $2 5 6 ^ { 3 }$ . ",
+        "bbox": [
+            173,
+            565,
+            826,
+            760
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Static Human Reconstruction under Sparse Viewpoints. H-NeRF learns a geometrically regularized volumetric radiance field constrained by an imGHUM-based SDF, which significantly improves robustness under sparse training camera views. In fig. 2, we show reconstruction of a static RenderPeople and GHS3D scan, respectively, using the original NeRF [29], the state-of-the-art multi-view reconstruction approach IDR [53], and H-NeRF, for increasing number of cameras. Both novel view image quality and geometric accuracy improve for all methods as the number of training views increases. However, in contrast to competitors, H-NeRF produces good quality output even under sparse training views (2-8), demonstrating the generalization capability of a co-training approach. ",
+        "bbox": [
+            174,
+            766,
+            825,
+            877
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Dynamic Human Reconstruction and Rendering. In the following, we evaluate H-NeRF’s capabilities to reconstruct and render dynamic scenes (fig. 5). We compare H-NeRF against NeuralBody [36] (tab. 1), the current state-of-the-art for novel view synthesis of humans in motion. For fair comparisons, we have trained NeuralBody with the same data as H-NeRF and use GHUM meshes (instead of SMPL). GHUM meshes with the same pose and shape representation as in our model are used for NeuralBody’s structured latent codes. We further compare with Nerfies [35]. However, Nerfies has difficulties with the high range of motion in our test sequences and fails for most of them. We consider such results less meaningful and we report them only in the Sup. Mat. for completeness. ",
+        "bbox": [
+            174,
+            883,
+            821,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/6434043d8a0c1cc7eb871d5b1fa2bbb7a9947eec176810cd6e3176f5e1ab3fdd.jpg",
+        "image_caption": [
+            "Figure 3: Frame Number Ablation. H-NeRF (red, x-axis: #training video frames) outperforms Neural Body (blue), evaluated on a RenderPeople (dashed) and a GHS3D (solid) sequence. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            87,
+            825,
+            166
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/1b4631173a50f0445731ff0e95a4e084d98ff478f3ae720174fc73307228385e.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative comparisons with NeuralBody on four different datasets. Left: GHS3D (top) and RenderPeople (bottom) ground-truth image, our result, NeuralBody, ground-truth geometry (front and back), our geometry, NeuralBody’s geometry. Right: PeopleSnapshot (top) and Human3.6M (bottom) ground-truth image, our result, NeuralBody, our geometry, NeuralBody’s geometry. Pay attention to the sharp renderings and the complete and detailed geometry produced by our method. Digital zoom-in recommended. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            209,
+            816,
+            343
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            449,
+            825,
+            532
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "H-NeRF overperforms NeuralBody across datasets and metrics, especially for geometric reconstruction. One possible explanation for NeuralBody’s lower performance on view synthesis, is that NeuralBody is not conditioned on the root transformation. Thus, global lighting effects are handled less well. Furthermore, pose depended effects that go beyond body articulation are only indirectly modeled through relative distances of input mesh vertices. Finally, NeuralBody conditions on the frame index, which is set to zero for novel poses. On the PeopleSnapshot dataset where no view-dependent and only subtle pose-dependent effects are present, these limitations are not entirely apparent. In contrast, H-NeRF pays special attention to modeling both detailed geometry and appearance, and its explicit conditioning on pose and the root transformation captures pose-dependent geometric and appearance effects. This strategy pays off, especially in more complex scenarios. ",
+        "bbox": [
+            174,
+            539,
+            825,
+            678
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Frame Number Ablation. We have illustrated H-NeRF’s rendering and reconstruction capabilities for static and dynamic sequences. Next, we evaluate the necessary number of different poses or time-frames H-NeRF needs during training for good pose generalisation. We again compare against NeuralBody. To this end, we have trained both methods with an increasing number of images, uniformly distributed over the full sequence. Results are shown in fig. 3. As expected, both methods perform better when trained with more data. However, H-NeRF performs overall better and more importantly, the quality degrades less in the sparse training regime, especially for geometric reconstruction. Our results support H-NeRF’s robustness to sparse training data and its capability to reconstruct accurate geometry. In practical terms, H-NeRF can be robustly trained with as little as 10 temporal frames per camera (in a four camera set-up), resulting in a performance capture effort of only a few minutes. ",
+        "bbox": [
+            174,
+            683,
+            825,
+            835
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Qualitative Results for Pose Generalisation and Shape Extrapolation. Finally, we illustrate HNeRF’s rendering and reconstruction accuracy qualitatively (fig. 4). Consistently with the reported metrics, our synthesized novel poses appear sharper, more detailed, and contain less noise than the current state-of-the-art. The estimated geometry is complete, smooth, and contains much of the detail present in the original scan, e.g. the clothing folds on the back of the person on the left, the first row. ",
+        "bbox": [
+            174,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e469af6acf42ce9ddfbe2dea143a520eca9a85c6d19405478198be73e6954db7.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative results of H-NeRF for novel image synthesis. We show ground truth images and scans (left, if available) side-by-side with our results (right). Red icons correspond to test, green icons to training configurations. Symbols correspond to camera, pose, and shape, respectively. From top to bottom: RenderPeople, GHS3D, PeopleSnapshot and Human3.6M. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            83,
+            820,
+            371
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In contrast, geometries produced by NeuralBody are noisier and sometimes incomplete. To further demonstrate the versatility of our approach, in fig. 5 we show examples of synthesized images and reconstructed geometry from novel viewpoints, in novel poses, and for modified body shape. ",
+        "bbox": [
+            174,
+            448,
+            825,
+            489
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Limitations and Broader Impact. In the current setup, we assume full-body views of a single person in every-day clothing. Inconsistency with these assumptions, e.g. by placing another person in the scene and occluding the performer, would break the method although models of partial views, or representation estimates of multiple people could be used for generality. Furthermore, our method exhibits some sensitivity to the estimated body shapes and poses, as well as the quality of image segmentation. Although the process of learning to render could absorb certain inaccuracies, as pose, shape or segmentation are increasingly degraded, the method would fail eventually. ",
+        "bbox": [
+            174,
+            496,
+            825,
+            594
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our method paves the way towards immersive AR and VR applications. In contrast, our approach is not targeted, or particularly useful for applications like visual surveillance or person identification as a particular set-up and a cooperating subject is needed. We do not build an audio-visual model that would be necessary for deep fakes. ",
+        "bbox": [
+            174,
+            599,
+            825,
+            655
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            678,
+            307,
+            694
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented novel neural radiance field models (H-NeRF) for the photo-realistic rendering and the temporal reconstruction of humans in motion. Our objective is to extend the scope of the previous state of the art, focused on static scenes observed by a large number of cameras, by building dynamic models, which based on a sparse set of views, can generalize well to novel camera views, human body poses, and extrapolate over human body shapes. Our key contribution is in developing a new model with associated multiple losses, in order to specialize and constrain a generic NeRF formulation by using a compatible implicit statistical 3D human pose and shape model, represented using signed distance functions. Our implicit geometric formulation captures not just the statistical regularities of the human body, but also hair and clothing represented as an implicit residual network. Our model is trained end-to-end based on several novel losses, and achieves good results for both 3D reconstruction and photorealistic rendering. Training the body model in an end-to-end rendering framework carries the promise to learn complex implicit skinning functions based on images only, a performance previously possible only in the realm of human capture using complex and expensive 3D body scanners, in the laboratory. We illustrate the favorable capabilities of H-NeRF through extensive experimentation using several datasets and against other state of the art techniques. ",
+        "bbox": [
+            173,
+            704,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            266,
+            106
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "arXiv:1906.08240. [2] Thiemo Alldieck, Marcus Magnor, Bharat Lal Bhatnagar, Christian Theobalt, and Gerard Pons-Moll. Learning to reconstruct people in clothing from a single RGB camera. In IEEE Conf. Comput. Vis. Pattern Recog., pages 1175–1186. IEEE, 2019. [3] Thiemo Alldieck, Marcus Magnor, Weipeng Xu, Christian Theobalt, and Gerard Pons-Moll. Video based reconstruction of 3D people models. In IEEE Conf. Comput. Vis. Pattern Recog., pages 8387–8397. IEEE, 2018.   \n[4] Thiemo Alldieck, Hongyi Xu, and Cristian Sminchisescu. imGHUM: Implicit generative models of 3D human shape and articulated pose. In Int. Conf. Comput. Vis., 2021. [5] Federica Bogo, Michael J. Black, Matthew Loper, and Javier Romero. Detailed full-body reconstructions of moving people from monocular RGB-D sequences. In IEEE International Conference on Computer Vision, pages 2300–2308. IEEE, 2015.   \n[6] Joel Carranza, Christian Theobalt, Marcus A Magnor, and Hans-Peter Seidel. Free-viewpoint video of human actors. ACM Trans. Graph., 22(3):569–577, 2003. [7] Shenchang Eric Chen and Lance Williams. View interpolation for image synthesis. In Proceedings of the 20th annual conference on Computer graphics and interactive techniques, pages 279–288, 1993. [8] Zhiqin Chen and Hao Zhang. Learning implicit fields for generative shape modeling. In IEEE Conf. Comput. Vis. Pattern Recog., pages 5939–5948, 2019. [9] Edilson De Aguiar, Carsten Stoll, Christian Theobalt, Naveed Ahmed, Hans-Peter Seidel, and Sebastian Thrun. Performance capture from sparse multi-view video. In ACM Trans. Graph., pages 1–10. 2008.   \n[10] Paul E Debevec, Camillo J Taylor, and Jitendra Malik. Modeling and rendering architecture from photographs: A hybrid geometry-and image-based approach. In Proceedings of the 23rd annual conference on Computer graphics and interactive techniques, pages 11–20, 1996.   \n[11] Boyang Deng, JP Lewis, Timothy Jeruzalski, Gerard Pons-Moll, Geoffrey Hinton, Mohammad Norouzi, and Andrea Tagliasacchi. Neural articulated shape approximation. In Eur. Conf. Comput. Vis. Springer, August 2020.   \n[12] Juergen Gall, Carsten Stoll, Edilson De Aguiar, Christian Theobalt, Bodo Rosenhahn, and Hans-Peter Seidel. Motion capture using joint skeleton tracking and surface estimation. In IEEE Conf. Comput. Vis. Pattern Recog., pages 1746–1753. IEEE, 2009.   \n[13] Ke Gong, Xiaodan Liang, Yicheng Li, Yimin Chen, Ming Yang, and Liang Lin. Instance-level human parsing via part grouping network. In Proceedings of the European Conference on Computer Vision (ECCV), pages 770–785, 2018.   \n[14] Amos Gropp, Lior Yariv, Niv Haim, Matan Atzmon, and Yaron Lipman. Implicit geometric regularization for learning shapes. In Int. Conf. on Mach. Learn., pages 3569–3579. 2020.   \n[15] Marc Habermann, Lingjie Liu, Weipeng Xu, Michael Zollhoefer, Gerard Pons-Moll, and Christian Theobalt. Real-time deep dynamic characters. ACM Trans. Graph., 40(4), aug 2021.   \n[16] Marc Habermann, Weipeng Xu, Michael Zollhoefer, Gerard Pons-Moll, and Christian Theobalt. Deepcap: Monocular human performance capture using weak supervision. In IEEE Conf. Comput. Vis. Pattern Recog. IEEE, jun 2020.   \n[17] Marc Habermann, Weipeng Xu, Michael Zollhöfer, Gerard Pons-Moll, and Christian Theobalt. Livecap: Real-time human performance capture from monocular video. ACM Trans. Graph., 38(2):14:1–14:17, 2019.   \n[18] Peter Hedman, Pratul P Srinivasan, Ben Mildenhall, Jonathan T Barron, and Paul Debevec. Baking neural radiance fields for real-time view synthesis. In Int. Conf. Comput. Vis., 2021.   \n[19] Catalin Ionescu, Dragos Papava, Vlad Olaru, and Cristian Sminchisescu. Human3.6M: Large scale datasets and predictive methods for 3d human sensing in natural environments. IEEE Trans. Pattern Anal. Mach. Intell., 2014.   \n[20] Thomas Lewiner, Hélio Lopes, Antônio Wilson Vieira, and Geovan Tavares. Efficient implementation of marching cubes’ cases with topological guarantees. Journal of Graphics Tools, 8(2):1–15, 2003.   \n[21] Lingjie Liu, Jiatao Gu, Kyaw Zaw Lin, Tat-Seng Chua, and Christian Theobalt. Neural sparse voxel fields. NeurIPS, 2020.   \n[22] Stephen Lombardi, Jason Saragih, Tomas Simon, and Yaser Sheikh. Deep appearance models for face rendering. ACM Trans. Graph., 37(4):1–13, 2018.   \n[23] Stephen Lombardi, Tomas Simon, Jason Saragih, Gabriel Schwartz, Andreas Lehrmann, and Yaser Sheikh. Neural volumes: Learning dynamic renderable volumes from images. ACM Trans. Graph., 38(4):65:1– 65:14, July 2019.   \n[24] Matthew Loper, Naureen Mahmood, Javier Romero, Gerard Pons-Moll, and Michael J Black. SMPL: A skinned multi-person linear model. ACM Trans. Graph., 34(6):248:1–248:16, 2015.   \n[25] Lars M. Mescheder, Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger. Occupancy networks: Learning 3d reconstruction in function space. In IEEE Conf. Comput. Vis. Pattern Recog., pages 4460–4470, 2019.   \n[26] Moustafa Meshry, Dan B Goldman, Sameh Khamis, Hugues Hoppe, Rohit Pandey, Noah Snavely, and Ricardo Martin-Brualla. Neural rerendering in the wild. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6878–6887, 2019. Deep level sets: Implicit surface representations for 3D shape inference. arXiv preprint arXiv:1901.06802, 2019.   \n[28] Marko Mihajlovic, Yan Zhang, Michael J. Black, and Siyu Tang. LEAP: Learning articulated occupancy of people. In Proceedings IEEE/CVF Conf. on Computer Vision and Pattern Recognition (CVPR), June 2021.   \n[29] Ben Mildenhall, Pratul P. Srinivasan, Matthew Tancik, Jonathan T. Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. In Eur. Conf. Comput. Vis., 2020.   \n[30] Roger Mohr, Long Quan, and Françoise Veillon. Relative 3d reconstruction using multiple uncalibrated images. The International Journal of Robotics Research, 14(6):619–632, 1995.   \n[31] Michael Niemeyer, Lars Mescheder, Michael Oechsle, and Andreas Geiger. Differentiable volumetric rendering: Learning implicit 3d representations without 3d supervision. In IEEE Conf. Comput. Vis. Pattern Recog., June 2020.   \n[32] Atsuhiro Noguchi, Xiao Sun, Stephen Lin, and Tatsuya Harada. Neural articulated radiance field. In Int. Conf. Comput. Vis., 2021.   \n[33] Michael Oechsle, Songyou Peng, and Andreas Geiger. Unisurf: Unifying neural implicit surfaces and radiance fields for multi-view reconstruction. In Int. Conf. Comput. Vis., 2021.   \n[34] Jeong Joon Park, Peter Florence, Julian Straub, Richard A. Newcombe, and Steven Lovegrove. Deepsdf: Learning continuous signed distance functions for shape representation. In IEEE Conf. Comput. Vis. Pattern Recog., pages 165–174, 2019.   \n[35] Keunhong Park, Utkarsh Sinha, Jonathan T. Barron, Sofien Bouaziz, Dan B Goldman, Steven M Seitz, and Ricardo Martin-Brualla. Nerfies: Deformable neural radiance fields. In Int. Conf. Comput. Vis., 2021.   \n[36] Sida Peng, Yuanqing Zhang, Yinghao Xu, Qianqian Wang, Qing Shuai, Hujun Bao, and Xiaowei Zhou. Neural body: Implicit neural representations with structured latent codes for novel view synthesis of dynamic humans. In IEEE Conf. Comput. Vis. Pattern Recog., 2021.   \n[37] Albert Pumarola, Enric Corona, Gerard Pons-Moll, and Francesc Moreno-Noguer. D-nerf: Neural radiance fields for dynamic scenes. In IEEE Conf. Comput. Vis. Pattern Recog. IEEE, jun 2021.   \n[38] Prajit Ramachandran, Barret Zoph, and Quoc V Le. Searching for activation functions. arXiv preprint arXiv:1710.05941, 2017.   \n[39] Daniel Rebain, Wei Jiang, Soroosh Yazdani, Ke Li, Kwang Moo Yi, and Andrea Tagliasacchi. Derf: Decomposed radiance fields. In IEEE Conf. Comput. Vis. Pattern Recog., 2021.   \n[40] Gernot Riegler and Vladlen Koltun. Free view synthesis. In European Conference on Computer Vision, pages 623–640. Springer, 2020.   \n[41] Shunsuke Saito, Zeng Huang, Ryota Natsume, Shigeo Morishima, Angjoo Kanazawa, and Hao Li. Pifu: Pixel-aligned implicit function for high-resolution clothed human digitization. In Int. Conf. Comput. Vis., pages 2304–2314, 2019.   \n[42] Johannes L Schonberger and Jan-Michael Frahm. Structure-from-motion revisited. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4104–4113, 2016.   \n[43] Aliaksandra Shysheya, Egor Zakharov, Kara-Ali Aliev, Renat Bashirov, Egor Burkov, Karim Iskakov, Aleksei Ivakhnenko, Yury Malkov, Igor Pasechnik, Dmitry Ulyanov, et al. Textured neural avatars. In IEEE Conf. Comput. Vis. Pattern Recog., pages 2387–2397, 2019.   \n[44] Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Implicit neural representations with periodic activation functions. Adv. Neural Inform. Process. Syst., 33, 2020.   \n[45] Vincent Sitzmann, Justus Thies, Felix Heide, Matthias Nießner, Gordon Wetzstein, and Michael Zollhöfer. Deepvoxels: Learning persistent 3d feature embeddings. In Proc. Computer Vision and Pattern Recognition (CVPR), IEEE, 2019.   \n[46] Vincent Sitzmann, Michael Zollhöfer, and Gordon Wetzstein. Scene representation networks: Continuous 3d-structure-aware neural scene representations. In Adv. Neural Inform. Process. Syst., 2019.   \n[47] Shih-Yang Su, Frank Yu, Michael Zollhoefer, and Helge Rhodin. A-nerf: Surface-free human 3d pose refinement via neural rendering. In Adv. Neural Inform. Process. Syst., 2021.   \n[48] Ayush Tewari, Ohad Fried, Justus Thies, Vincent Sitzmann, Stephen Lombardi, Kalyan Sunkavalli, Ricardo Martin-Brualla, Tomas Simon, Jason Saragih, Matthias Nießner, et al. State of the art on neural rendering. In Comput. Graph. Forum, volume 39, pages 701–727. Wiley Online Library, 2020.   \n[49] Justus Thies, Michael Zollhöfer, and Matthias Nießner. Deferred neural rendering: Image synthesis using neural textures. ACM Transactions on Graphics (TOG), 38(4):1–12, 2019.   \n[50] Daniel Vlasic, Ilya Baran, Wojciech Matusik, and Jovan Popovic. Articulated mesh animation from ´ multi-view silhouettes. In ACM Trans. Graph., pages 1–9. 2008.   \n[51] Hongyi Xu, Eduard Gabriel Bazavan, Andrei Zanfir, William T Freeman, Rahul Sukthankar, and Cristian Sminchisescu. GHUM & GHUML: Generative 3d human shape and articulated pose models. In IEEE Conf. Comput. Vis. Pattern Recog., pages 6184–6193, 2021.   \n[52] Weipeng Xu, Avishek Chatterjee, Michael Zollhoefer, Helge Rhodin, Dushyant Mehta, Hans-Peter Seidel, and Christian Theobalt. Monoperfcap: Human performance capture from monocular video. ACM Trans. Graph., 2018.   \n[53] Lior Yariv, Yoni Kasten, Dror Moran, Meirav Galun, Matan Atzmon, Basri Ronen, and Yaron Lipman. Multiview neural surface reconstruction by disentangling geometry and appearance. Adv. Neural Inform. Process. Syst., 33, 2020.   \n[54] Alex Yu, Ruilong Li, Matthew Tancik, Hao Li, Ren Ng, and Angjoo Kanazawa. Plenoctrees for real-time rendering of neural radiance fields. In Int. Conf. Comput. Vis., 2021.   \n[55] Mihai Zanfir, Elisabeta Oneata, Alin-Ionut Popa, Andrei Zanfir, and Cristian Sminchisescu. Human synthesis and scene compositing. In AAAI, pages 12749–12756, 2020.   \n[56] Kai Zhang, Gernot Riegler, Noah Snavely, and Vladlen Koltun. Nerf $^ { + + }$ : Analyzing and improving neural radiance fields. arXiv preprint arXiv:2010.07492, 2020.   \n[57] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. In CVPR, 2018. ",
+        "bbox": [
+            171,
+            138,
+            828,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            106,
+            828,
+            888
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            92,
+            826,
+            190
+        ],
+        "page_idx": 11
+    }
+]