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+# WRPN: WIDE REDUCED-PRECISION NETWORKS
+
+Asit Mishra, Eriko Nurvitadhi, Jeffrey J Cook & Debbie Marr   
+Accelerator Architecture Lab   
+Intel Labs   
+{asit.k.mishra,eriko.nurvitadhi,jeffrey.j.cook,debbie.marr}@intel.com
+
+# ABSTRACT
+
+For computer vision applications, prior works have shown the efficacy of reducing numeric precision of model parameters (network weights) in deep neural networks. Activation maps, however, occupy a large memory footprint during both the training and inference step when using mini-batches of inputs. One way to reduce this large memory footprint is to reduce the precision of activations. However, past works have shown that reducing the precision of activations hurts model accuracy. We study schemes to train networks from scratch using reduced-precision activations without hurting accuracy. We reduce the precision of activation maps (along with model parameters) and increase the number of filter maps in a layer, and find that this scheme matches or surpasses the accuracy of the baseline full-precision network. As a result, one can significantly improve the execution efficiency (e.g. reduce dynamic memory footprint, memory bandwidth and computational energy) and speed up the training and inference process with appropriate hardware support. We call our scheme WRPN - wide reducedprecision networks. We report results and show that WRPN scheme is better than previously reported accuracies on ILSVRC-12 dataset while being computationally less expensive compared to previously reported reduced-precision networks.
+
+# 1 INTRODUCTION
+
+A promising approach to lower the compute and memory requirements of convolutional deeplearning workloads is through the use of low numeric precision algorithms. Operating in lower precision mode reduces computation as well as data movement and storage requirements. Due to such efficiency benefits, there are many existing works which propose low-precision deep neural networks (DNNs) (Zhou et al., 2017; Lin et al., 2015; Miyashita et al., 2016; Gupta et al., 2015b; Vanhoucke et al., 2011), even down to 2-bit ternary mode (Zhu et al., 2016; Li & Liu, 2016; Venkatesh et al., 2016) and 1-bit binary mode (Zhou et al., 2016; Courbariaux & Bengio, 2016; Rastegari et al., 2016; Courbariaux et al., 2015; Umuroglu et al., 2016). However, majority of existing works in low-precision DNNs sacrifice accuracy over the baseline full-precision networks. Further, most prior works target reducing the precision of the model parameters (network weights). This primarily benefits the inference step only when batch sizes are small.
+
+We observe that activation maps (neuron outputs) occupy more memory compared to the model parameters for batch sizes typical during training. This observation holds even during inference when batch size is around eight or more. Based on this observation, we study schemes for training and inference using low-precision DNNs where we reduce the precision of activation maps as well as the model parameters without sacrificing network accuracy.
+
+To improve both execution efficiency and accuracy of low-precision networks, we reduce both the precision of activation maps and model parameters and increase the number of filter maps in a layer. We call networks using this scheme wide reduced-precision networks (WRPN) and find that this scheme compensates or surpasses the accuracy of the baseline full-precision network. Although the number of raw compute operations increases as we increase the number of filter maps in a layer, the compute bits required per operation is now a fraction of what is required when using full-precision operations (e.g. going from FP32 AlexNet to 4-bits precision and doubling the number of filters increases the number of compute operations by 4x, but each operation is 8x more efficient than FP32).
+
+WRPN offers better accuracies, while being computationally less expensive compared to previously reported reduced-precision networks. We report results on AlexNet (Krizhevsky et al., 2012), batch-normalized Inception (Ioffe & Szegedy, 2015), and ResNet-34 (He et al., 2015) on ILSVRC12 (Russakovsky et al., 2015) dataset. We find 4-bits to be sufficient for training deep and wide models while achieving similar or better accuracy than baseline network. With 4-bit activation and 2-bit weights, we find the accuracy to be at-par with baseline full-precision. Making the networks wider and operating with 1-bit precision, we close the accuracy gap between previously reported binary networks and show state-of-the art results for ResNet-34 $6 9 . 8 5 \%$ top-1 with $2 \mathbf { x }$ wide) and AlexNet $4 8 . 0 4 \%$ top-1 with $1 . 3 \mathrm { x }$ wide). To the best of our knowledge, our reported accuracies with binary networks and 4-bit precision are highest to date.
+
+Our reduced-precision quantization scheme is hardware friendly allowing for efficient hardware implementations. To this end, we evaluate efficiency benefits of low-precision operations (4-bits to 1-bits) on Titan X GPU, Arria-10 FPGA and ASIC. We see that FPGA and ASIC can deliver significant efficiency gain over FP32 operations $6 . 5 \mathrm { x }$ to $1 0 0 \mathrm { x }$ ), while GPU cannot take advantage of very low-precision operations.
+
+![](images/7f1ebe08f820ba2d4f6cf54598581053b3e8155aa0a62fb48e4050ad8d091021.jpg)  
+2 MOTIVATION FOR REDUCED-PRECISION ACTIVATION MAPS   
+Figure 1: Memory footprint of activations (ACTs) and weights (W) during training and inference for minibatch sizes 1 and 32.
+
+While most prior works proposing reduced-precision networks work with low precision weights (e.g. work in Courbariaux & Bengio (2016); Zhu et al. (2016); Zhou et al. (2016); Venkatesh et al. (2016); Li & Liu (2016); Courbariaux et al. (2015); Umuroglu et al. (2016)), we find that activation maps occupy a larger memory footprint when using mini-batches of inputs. Using mini-batches of inputs is typical in training of DNNs and cloud-based batched inference (Jouppi et al., 2017). Figure 1 shows memory footprint of activation maps and filter maps as batch size changes for 4 different networks (AlexNet, Inception-Resnet-v2 (Szegedy et al., 2016), ResNet-50 and ResNet-101) during the training and inference steps.
+
+![](images/95b9b13bfb188037f010166d925ca0e7b975dda5fa5f3ad20c2532785fd508d5.jpg)  
+Figure 2: Memory requirements of a feed forward convolutional deep neural network. Orange boxes denote weights (W), blue boxes are activations (ACT) and green boxes are gradient-maps (Grad).
+
+As batch-size increases, because of filter reuse across batches of inputs, activation maps occupy significantly larger fraction of memory compared to the filter weights. This aspect is illustrated in
+
+Figure 2 which shows the memory requirements of a canonical feed-forward DNN for a hardware accelerator based system (e.g. GPU, FPGA, PCIe connected ASIC device, etc.). During training, the sum of all the activation maps (ACT) and weight tensors (W) are allocated in device memory for forward pass along with memory for gradient maps during backward propagation. The total memory requirements for training phase is the sum of memory required for the activation maps, weights and the maximum of input gradient maps $( \delta \mathbf { Z } )$ and maximum of back-propagated gradients $( \delta \mathbf { X } )$ . During inference, memory is allocated for input (IFM) and output feature maps (OFM) required by a single layer, and these memory allocations are reused for other layers. The total memory allocation during inference is then the maximum of IFM and maximum of OFM required across all the layers plus the sum of all W-tensors. At batch sizes 128 and more, activations start to occupy more than $98 \%$ of total memory footprint during training.
+
+Overall, reducing precision of activations and weights reduces memory footprint, bandwidth and storage while also simplifying the requirements for hardware to efficiently support these operations.
+
+# 3 WRPN SCHEME AND STUDIES ON ALEXNET
+
+Based on the observation that activations occupy more memory footprint compared to weights, we reduce the precision of activations to speed up training and inference steps as well as cut down on memory requirements. However, a straightforward reduction in precision of activation maps leads to significant reduction in model accuracy (Zhou et al., 2016; Rastegari et al., 2016).
+
+We conduct a sensitivity study where we reduce precision of activation maps and model weights for AlexNet running ILSVRC-12 dataset and train the network from scratch. Table 1 reports our findings. Top-1 single-precision (32-bits weights and activations) accuracy is $5 7 . 2 \%$ . The accuracy with binary weights and activations is $4 4 . 2 \%$ . This is similar to what is reported in Rastegari et al. (2016). $3 2 b A$ and $2 b W$ data-point in this table is using Trained Ternary Quantization (TTQ) technique (Zhu et al., 2016). All other data points are collected using our quantization scheme (described later in Section 5), all the runs have same hyper-parameters and training is carried out for the same number of epochs as baseline network. To be consistent with results reported in prior works, we do not quantize weights and activations of the first and last layer.
+
+We find that, in general, reducing the precision of activation maps and weights hurts model accuracy. Further, reducing precision of activations hurts model accuracy much more than reducing precision of the filter parameters. We find TTQ to be quite effective on AlexNet in that one can lower the precision of weights to 2b (while activations are still FP32) and not lose accuracy. However, we did not find this scheme to be effective for other networks like ResNet or Inception.
+
+Table 1: AlexNet top-1 validation set accuracy $\%$ as precision of activations (A) and weight(W) changes. All results are with end-to-end training of the network from scratch. − is a data-point we did not experiment for.   
+
+<table><tr><td></td><td>32b A</td><td>8bA</td><td>4b A</td><td>2bA</td><td>1bA</td></tr><tr><td>32bW</td><td>57.2</td><td>54.3</td><td>54.4</td><td>52.7</td><td></td></tr><tr><td>8bW</td><td>1</td><td>54.5</td><td>53.2</td><td>51.5</td><td>1</td></tr><tr><td>4b W</td><td>1</td><td>54.2</td><td>54.4</td><td>52.4</td><td></td></tr><tr><td>2b W</td><td>57.5</td><td>50.2</td><td>50.5</td><td>51.3</td><td>1</td></tr><tr><td>1b W</td><td>56.8</td><td>1</td><td>1</td><td>1</td><td>44.2</td></tr></table>
+
+Table 2: AlexNet $2 \mathbf { x }$ -wide top-1 validation set accuracy $\%$ as precision of activations (A) and weights (W) changes.   
+
+<table><tr><td></td><td>32bA</td><td>8bA</td><td>4bA</td><td>2bA</td><td>1b A</td></tr><tr><td>32bW</td><td>60.5</td><td>58.9</td><td>58.6</td><td>57.5</td><td>52.0</td></tr><tr><td>8bW</td><td>1</td><td>59.0</td><td>58.8</td><td>57.1</td><td>50.8</td></tr><tr><td>4b W</td><td>1</td><td>58.8</td><td>58.6</td><td>57.3</td><td>/</td></tr><tr><td>2b W</td><td>1</td><td>57.6</td><td>57.2</td><td>55.8</td><td>1</td></tr><tr><td>1b W</td><td>1</td><td>1</td><td>1</td><td>1</td><td>48.3</td></tr></table>
+
+To re-gain the model accuracy while working with reduced-precision operands, we increase the number of filter maps in a layer. Although the number of raw compute operations increase with widening the filter maps in a layer, the bits required per compute operation is now a fraction of what is required when using full-precision operations. As a result, with appropriate hardware support, one can significantly reduce the dynamic memory requirements, memory bandwidth, computational energy and speed up the training and inference process.
+
+Our widening of filter maps is inspired from Wide ResNet (Zagoruyko & Komodakis, 2016) work where the depth of the network is reduced and width of each layer is increased (the operand precision is still FP32). Wide ResNet requires a re-design of the network architecture. In our work, we maintain the depth parameter same as baseline network but widen the filter maps. We call our approach WRPN - wide reduced-precision networks. In practice, we find this scheme to be very simple and effective - starting with a baseline network architecture, one can change the width of each filter map without changing any other network design parameter or hyper-parameters. Carefully reducing precision and simultaneously widening filters keeps the total compute cost of the network under or at-par with baseline cost.1
+
+Table 2 reports the accuracy of AlexNet when we double the number of filter maps in a layer. With doubling of filter maps, AlexNet with 4-bits weights and 2-bits activations exhibits accuracy at-par with full-precision networks. Operating with 4-bits weights and 4-bits activations surpasses the baseline accuracy by $1 . 4 4 \%$ . With binary weights and activations we better the accuracy of XNORNET (Rastegari et al., 2016) by $4 \%$ .
+
+When doubling the number of filter maps, AlexNet’s raw compute operations grow by $3 . 9 \mathbf { x }$ compared to the baseline full-precision network, however by using reduced-precision operands the overall compute complexity is a fraction of the baseline. For example, with 4b operands for weights and activations and $2 \mathbf { x }$ the number of filters, reduced-precision AlexNet is just $49 \%$ of the total compute cost of the full-precision baseline (compute cost comparison is shown in Table 3).
+
+Table 3: Compute cost of AlexNet $2 \mathbf { x }$ -wide vs. 1x-wide as precision of activations (A) and weights (W) changes.   
+
+<table><tr><td></td><td>32b A</td><td>8b A</td><td>4b A</td><td>2bA</td><td>1b A</td></tr><tr><td>32b W</td><td>3.9x</td><td>2.4x</td><td>2.2x</td><td>2.1x</td><td>2.0x</td></tr><tr><td>8b W</td><td>2.4x</td><td>1.0x</td><td>0.7x</td><td>0.6x</td><td>0.6x</td></tr><tr><td>4b W</td><td>2.2x</td><td>0.7x</td><td>0.5x</td><td>0.4x</td><td>0.3x</td></tr><tr><td>2b W</td><td>2.1x</td><td>0.6x</td><td>0.4x</td><td>0.2x</td><td>0.2x</td></tr><tr><td>1b W</td><td>2.0x</td><td>0.6x</td><td>0.3x</td><td>0.2x</td><td>0.1x</td></tr></table>
+
+We also experiment with other widening factors. With $1 . 3 \mathrm { x }$ widening of filters and with 4-bits of activation precision one can go as low as 8-bits of weight precision while still being at-par with baseline accuracy. With $1 . 1 \mathrm { x }$ wide filters, at least 8-bits weight and 16-bits activation precision is required for accuracy to match baseline full-precision 1x wide accuracy. Further, as Table 3 shows, when widening filters by $2 \mathbf { x }$ , one needs to lower precision to at least 8-bits so that the total compute cost is not more than baseline compute cost. Thus, there is a trade-off between widening and reducing the precision of network parameters.
+
+In our work, we trade-off higher number of raw compute operations with aggressively reducing the precision of the operands involved in these operations (activation maps and filter weights) while not sacrificing the model accuracy. Apart from other benefits of reduced precision activations as mentioned earlier, widening filter maps also improves the efficiency of underlying GEMM calls for convolution operations since compute accelerators are typically more efficient on a single kernel consisting of parallel computation on large data-structures as opposed to many small sized kernels (Zagoruyko & Komodakis, 2016).
+
+# 4 STUDIES ON DEEPER NETWORKS
+
+We study how our scheme applies to deeper networks. For this, we study ResNet-34 (He et al., 2015) and batch-normalized Inception (Ioffe & Szegedy, 2015) and find similar trends, particularly that 2-bits weight and 4-bits activations continue to provide at-par accuracy as baseline. We use TensorFlow (Abadi et al., 2015) and tensorpack for all our evaluations and use ILSVRC-12 train and val dataset for analysis.
+
+# 4.1 RESNET
+
+ResNet-34 has 3x3 filters in each of its modular layers with shortcut connections being 1x1. The filter bank width changes from 64 to 512 as depth increases. We use the pre-activation variant of ResNet and the baseline top-1 accuracy of our ResNet-34 implementation using single-precision
+
+32-bits data format is $7 3 . 5 9 \%$ . Binarizing weights and activations for all layers except the first and the last layer in this network gives top-1 accuracy of $6 0 . 5 \%$ . For binarizing ResNet we did not re-order any layer (as is done in XNOR-NET). We used the same hyper-parameters and learning rate schedule as the baseline network. As a reference, for ResNet-18, the gap between XNOR-NET (1b weights and activations) and full-precision network is $18 \%$ (Rastegari et al., 2016). It is also interesting to note that top-1 accuracy of single-precision AlexNet $( 5 7 . 2 0 \% )$ is lower than the top-1 accuracy of binarized ResNet-34 $( 6 0 . 5 \% )$ .
+
+Table 4: ResNet-34 top-1 validation accuracy $\%$ and compute cost as precision of activations (A) and weights (W) varies.   
+
+<table><tr><td>Width</td><td>Precision</td><td>Top-1 Acc. %</td><td>Compute cost</td></tr><tr><td>1x wide</td><td>32b A,32b W 1b A, 1b W</td><td>73.59 60.54</td><td>1x 0.03x</td></tr><tr><td>2x wide</td><td>4b A, 8b W 4b A, 4b W</td><td>74.48 74.52</td><td>0.74x 0.50x</td></tr><tr><td></td><td>4b A, 2b W</td><td>73.58</td><td>0.39x</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td></td><td>2b A,4b W</td><td>73.50</td><td>0.39x</td></tr><tr><td></td><td>2b A,2b W</td><td>73.32</td><td>0.27x</td></tr><tr><td></td><td>1b A, 1b W</td><td>69.85</td><td>0.15x</td></tr><tr><td>3x wide</td><td>1b A, 1b W</td><td>72.38</td><td>0.30x</td></tr></table>
+
+We experimented with doubling number of filters in each layer and reduce the precision of activations and weights. Table 4 shows the results of our analysis. Doubling the number of filters and 4-bits precision for both weights and activations beats the baseline accuracy by $0 . 9 \%$ . 4-bits activations and 2-bits (ternary) weights has top-1 accuracy at-par with baseline. Reducing precision to 2-bits for both weights and activations degrades accuracy by only $0 . 2 \%$ compared to baseline.
+
+Binarizing the weights and activations with $2 \mathbf { x }$ wide filters has a top-1 accuracy of $6 9 . 8 5 \%$ . This is just $3 . 7 \%$ worse than baseline full-precision network while being only $15 \%$ of the cost of the baseline network. Widening the filters by $3 \mathbf { x }$ and binarizing the weights and activations reduces this gap to $1 . 2 \%$ while the 3x wide network is $30 \%$ the cost of the full-precision baseline network.
+
+Although 4-bits precision seems to be enough for wide networks, we advocate for 4-bits activation precision and 2-bits weight precision. This is because with ternary weights one can get rid of the multipliers and use adders instead. Additionally, with this configuration there is no loss of accuracy. Further, if some accuracy degradation is tolerable, one can even go to binary circuits for efficient hardware implementation while saving $3 2 \mathrm { x }$ in bandwidth for each of weights and activations compared to full-precision networks. All these gains can be realized with simpler hardware implementation and lower compute cost compared to baseline networks.
+
+To the best of our knowledge, our ResNet binary and ternary (with 2-bits or 4-bits activation) top-1 accuracies are state-of-the-art results in the literature including unpublished technical reports (with similar data augmentation (Mellempudi et al., 2017)).
+
+# 4.2 BATCH-NORMALIZED INCEPTION
+
+We applied WRPN scheme to batch-normalized Inception network (Ioffe & Szegedy, 2015). This network includes batch normalization of all layers and is a variant of GoogleNet (Szegedy et al., 2014) where the 5x5 convolutional filters are replaced by two 3x3 convolutions with up to 128 wide filters. Table 5 shows the results of our analysis. Using 4-bits activations and 2-bits weight and doubling the number of filter banks in the network produces a model that is almost at-par in accuracy with the baseline single-precision network $0 . 0 2 \%$ loss in accuracy). Wide network with binary weights and activations is within $6 . 6 \%$ of the full-precision baseline network.
+
+# 5 HARDWARE FRIENDLY QUANTIZATION SCHEME
+
+We adopt the straight-through estimator (STE) approach in our work (Bengio et al., 2013). When quantizing a real number to $k$ -bits, the ordinality of the set of quantized numbers is $2 ^ { k }$ . Mathematically, this small and finite set would have zero gradients with respect to its inputs. STE method circumvents this problem by defining an operator that has arbitrary forward and backward operations.
+
+Table 5: Batch-normalized Inception top-1 validation accuracy $\%$ and compute cost as precision of activations (A) and weights (W) varies.   
+
+<table><tr><td>Width</td><td>Precision</td><td>Top-1 Acc. %</td><td>Compute cost</td></tr><tr><td>1x wide</td><td>32b A,32b W</td><td>71.64</td><td>1x</td></tr><tr><td rowspan="4">2x wide</td><td>4b A, 4b W</td><td>71.63</td><td>0.50x</td></tr><tr><td>4b A, 2b W</td><td>71.61</td><td>0.38x</td></tr><tr><td>2b A,2b W</td><td>70.75</td><td>0.25x</td></tr><tr><td>1b A,1b W</td><td>65.02</td><td>0.13x</td></tr></table>
+
+Prior works using the STE approach define operators that quantize the weights based on the expectation of the weight tensors. For instance, Ternary Weight Networks (TWN) (Li & Liu, 2016) uses a threshold and a scaling factor for each layer to quantize weights to ternary domain. In TTQ (Zhu et al., 2016), the scaling factors are learned parameters. XNOR-NET binarizes the weight tensor by computing the sign of the tensor values and then scaling by the mean of the absolute value of each output channel of weights. DoReFa uses a single scaling factor across the entire layer. For quantizing weights to $k$ -bits, where $k > 1$ , DoReFa uses:
+
+$$
+w ^ { k } = 2 * q u a n t i z e _ { k } ( \frac { t a n h ( w _ { i } ) } { 2 * m a x ( | \ t a n h ( w _ { i } ) \ | ) } + \frac { 1 } { 2 } ) - 1 )
+$$
+
+Here $w _ { k }$ is the $\mathbf { k }$ -bit quantized version of inputs $w _ { i }$ and quantize $k$ is a quantization function that quantizes a floating-point number $w _ { i }$ in the range $[ 0 , 1 ]$ to a $k$ -bit number in the same range. The transcendental tanh operation constrains the weight value to lie in between $- 1$ and $+ 1$ . The affine transformation post quantization brings the range to $[ - 1 , 1 ]$ .
+
+We build on these approaches and propose a much simpler scheme. For quantizing weight tensors we first hard constrain the values to lie within the range $[ - 1 , 1 ]$ using min-max operation (e.g. tf.clip by val when using Tensorflow (Abadi et al., 2015)). For quantizing activation tensor values, we constrain the values to lie within the range [0, 1]. This step is followed by a quantization step where a real number is quantized into a $k$ -bit number. This is given as, for $k > 1$ :
+
+$$
+w _ { k } = \frac { 1 } { 2 ^ { k - 1 } - 1 } r o u n d ( ( 2 ^ { k - 1 } - 1 ) * w _ { i } ) \qquad \mathrm { a n d } \qquad a _ { k } = \frac { 1 } { 2 ^ { k } - 1 } r o u n d ( ( 2 ^ { k } - 1 ) * a _ { i } )
+$$
+
+Here $w _ { i }$ and $a _ { i }$ are input real-valued weights and activation tensor and $w _ { k }$ and $a _ { k }$ are their quantized versions. One bit is reserved for sign-bit in case of weight values, hence the use of $2 ^ { k - 1 }$ for these quantized values. Thus, weights can be stored and interpreted using signed data-types and activations using un-signed data-types. With appropriate affine transformations, the convolution operations (the bulk of the compute operations in the network during forward pass) can be done using quantized values (integer operations in hardware) followed by scaling with floating-point constants (this scaling operation can be done in parallel with the convolution operation in hardware). When $k = 1$ , for binary weights we use the Binary Weighted Networks (BWN) approach (Courbariaux et al., 2015) where the binarized weight value is computed based on the sign of input value followed by scaling with the mean of absolute values. For binarized activations we use the formulation in Eq. 2. We do not quantize the gradients and maintain the weights in reduced precision format.
+
+For convolution operation when using WRPN, the forward pass during training (and the inference step) involves matrix multiplication of $k$ -bits signed and $k$ -bits unsigned operands. Since gradient values are in 32-bits floating-point format, the backward pass involves a matrix multiplication operation using 32-bits and $k$ -bits operand for gradient and weight update.
+
+When $k > 1$ , the hard clipping of tensors to a range maps efficiently to min-max comparator units in hardware as opposed to using transcendental operations which are long latency operations. TTQ and DoRefa (Zhou et al., 2016) schemes involve division operation and computing a maximum value in the input tensor. Floating-point division operation is expensive in hardware and computing the maximum in a tensor is an $O ( n )$ operation. Additionally, our quantization parameters are static and do not require any learning or involve back-propagation like TTQ approach. We avoid each of these costly operations and propose a simpler quantization scheme (clipping followed by rounding).
+
+5.1 EFFICIENCY IMPROVEMENTS OF REDUCED-PRECISION OPERATIONS ON GPU, FPGA AND ASIC
+
+In practice, the effective performance and energy efficiency one could achieve on a low-precision compute operation highly depends on the hardware that runs these operations. We study the efficiency of low-precision operations on various hardware targets GPU, FPGA, and ASIC.
+
+For GPU, we evaluate WRPN on Nvidia Titan X Pascal and for FPGA we use Intel Arria-10. We collect performance numbers from both previously reported analysis (Nurvitadhi et al., 2017) as well as our own experiments. For FPGA, we implement a DNN accelerator architecture shown in Figure 3(a). This is a prototypical accelerator design used in various works (e.g., on FPGA (Nurvitadhi et al., 2017) and ASIC such as TPU (Jouppi et al., 2017)). The core of the accelerator consists of a systolic array of processing elements (PEs) to perform matrix and vector operations, along with on-chip buffers, as well as off-chip memory management unit. The PEs can be configured to support different precision – (FP32, FP32), (INT4, INT4), (INT4, TER2), and (BIN1, BIN1). The (INT4, TER2) PE operates on ternary $^ { ( + 1 , 0 , - 1 ) }$ values and is optimized to include only an adder since there is no need for a multiplier in this case. The binary (BIN1, BIN1) PE is implemented using XNOR and bitcount. Our RTL design targets Arria-10 1150 FPGA. For our ASIC study, we synthesize the PE design using Intel $1 4 \mathrm { n m }$ process technology to obtain area and energy estimates.
+
+![](images/edc7cebaa1081616f22562193fbf135a3823ee6a5f5905ff4ca4bbf7144c7a10.jpg)  
+Figure 3: Efficiency improvements from low-precision operations on GPU, FPGA and ASIC.
+
+Figure 3(b) - (g) summarize our analysis. Figure 3(b) shows the efficiency improvements using first-order estimates where the efficiency is computed based on number of bits used in the operation.
+
+With this method we would expect (INT4, INT4) and (BIN1, BIN1) to be ${ 8 } \mathbf { { x } }$ and $3 2 \mathrm { x }$ more efficient, respectively, than (FP32, FP32). However, in practice the efficiency gains from reducing precision depend on whether the underlying hardware can take advantage of such low-precisions.
+
+Figure 3(c) shows performance improvement on Titan X GPU for various low-precision operations relative to FP32. In this case, GPU can only achieve up to ${ \sim } 4 \mathbf { x }$ improvements in performance over FP32 baseline. This is because GPU only provides first-class support for INT8 operations, and is not able to take advantage of the lower INT4, TER2, and BIN1 precisions. On the contrary, FPGA can take advantage of such low precisions, since they are amenable for implementations on the FPGAs reconfigurable fabric.
+
+Figure 3(d) shows that the performance improvements from (INT4, INT4), (INT4, TER2), and (BIN1, BIN1) track well with the first-order estimates from Figure 3(b). In fact, for (BIN1, BIN1), FPGA improvements exceed the first-order estimate. Reducing the precision simplifies the design of compute units and lower buffering requirements on FPGA board. Compute-precision reduction leads to significant improvement in throughput due to smaller hardware designs (allowing more parallelism) and shorter circuit delay (allowing higher frequency). Figure 3(e) shows the performance and performance/Watt of the reduced-precision operations on GPU and FPGA. FPGA performs quite well on very low precision operations. In terms of performance/watt, FPGA does better than GPU on (INT4, INT4) and lower precisions.
+
+ASIC allows for a truly customized hardware implementation. Our ASIC study provides insights to the upper bound of the efficiency benefits possible from low-precision operations. Figure 3(f) and 3(g) show improvement in performance and energy efficiency of the various low-precision ASIC PEs relative to baseline FP32 PE. As the figures show, going to lower precision offers 2 to 3 orders of magnitude efficiency improvements.
+
+In summary, FPGA and ASIC are well suited for our WRPN approach. At $2 \mathbf { x }$ wide, our WRPN approach requires $4 \mathbf { x }$ more total operations than the original network. However, for INT4 or lower precision, each operation is $6 . 5 \mathrm { x }$ or better in efficiency than FP32 for FPGA and ASIC. Hence, WRPN delivers an overall efficiency win.
+
+# 6 RELATED WORK
+
+Reduced-precision DNNs is an active research area. Reducing precision of weights for efficient inference pipeline has been very well studied. Works like Binary connect (BC) (Courbariaux et al., 2015), Ternary-weight networks (TWN) (Li & Liu, 2016), fine-grained ternary quantization (Mellempudi et al., 2017) and INQ (Zhou et al., 2017) target precision reduction of network weights while still using full-precision activations. Accuracy is almost always degraded when quantizing the weights. For AlexNet on Imagenet, TWN loses $5 \%$ top-1 accuracy. Schemes like INQ, Sung et al. (2015) and Mellempudi et al. (2017) do fine-tuning to quantize the network weights and do not sacrifice accuracy as much but are not applicable for training networks from scratch. INQ shows promising results with 5-bits of precision.
+
+XNOR-NET (Rastegari et al., 2016), BNN (Courbariaux & Bengio, 2016), DoReFa (Zhou et al., 2016) and TTQ (Zhu et al., 2016) target training as well. While TTQ targets weight quantization only, most works targeting activation quantization hurt accuracy. XNOR-NET approach reduces top-1 accuracy by $12 \%$ and DoReFa by $8 \%$ when quantizing both weights and activations to 1-bit (for AlexNet on ImageNet). Further, XNOR-NET requires re-ordering of layers for its scheme to work. Recent work in Graham (2017) targets low-precision activations and reports accuracy within $1 \%$ of baseline with 5-bits precision and logarithmic (with base $\sqrt { 2 }$ ) quantization. With fine-tuning this gap can be narrowed to be within $0 . 6 \%$ but not all layers are quantized.
+
+Non-multiples of two for operand values introduces hardware inefficiency in that memory accesses are no longer DRAM or cache-boundary aligned and end-to-end run-time performance aspect is unclear when using complicated quantization schemes. We target end-to-end training and inference, using very simple quantization method and aim for reducing precision without any loss in accuracy. To the best of our knowledge, our work is the first to study reduced-precision deep and wide networks, and show accuracy at-par with baseline for as low a precision as 4-bits activations and 2-bits weights. We report state of the art accuracy for wide binarized AlexNet and ResNet while still being lower in compute cost.
+
+Work by Gupta et al. (2015a) advocates for low precision fixed-point numbers for training. They show 16-bits to be sufficient for training on CIFAR10 dataset and find stochastic rounding to be necessary for training convergence. In our work here we focus on sub-8b training and like DoReFa scheme do not see stochastic rounding necessary when using full-precision gradients. Work by Seide et al. (2014) quantizes gradients before communication in a distributed computing setting. They use full precision gradients during the backward pass and quantize the gradients before sending them to other computation nodes (decreasing the amount of communication traffic over an interconnection network). For distributed training, we can potentially use this approach for communicating gradients across nodes.
+
+# 7 CONCLUSIONS
+
+We present the Wide Reduced-Precision Networks (WRPN) scheme for DNNs. In this scheme, the numeric precision of both weights and activations are significantly reduced without loss of network accuracy. This result is in contrast to many previous works that find reduced-precision activations to detrimentally impact accuracy; specifically, we find that 2-bit weights and 4-bit activations are sufficient to match baseline accuracy across many networks including AlexNet, ResNet-34 and batchnormalized Inception. We achieve this result with a new quantization scheme and by increasing the number of filter maps in each reduced-precision layer to compensate for the loss of information capacity induced by reducing the precision. We believe ours to be the first work to study the interplay between layer width and precision – with widening, the number of neurons in a layer increase; yet with reduced precision, we control overfitting and regularization.
+
+We motivate this work with our observation that full-precision activations contribute significantly more to the memory footprint than full-precision weight parameters when using mini-batch sizes common during training and cloud-based inference; furthermore, by reducing the precision of both activations and weights the compute complexity is greatly reduced $40 \%$ of baseline for 2-bit weights and 4-bit activations).
+
+The WRPN quantization scheme and computation on low precision activations and weights is hardware friendly making it viable for deeply-embedded system deployments as well as in cloud-based training and inference servers with compute fabrics for low-precision. We compare Titan X GPU, Arria-10 FPGA and ASIC implementations using WRPN and show our scheme increases performance and energy-efficiency for iso-accuracy across each. Overall, reducing the precision allows custom-designed compute units and lower buffering requirements to provide significant improvement in throughput.
+
+# REFERENCES
+
+tensorpack: https://github.com/ppwwyyxx/tensorpack.
+
+Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dan Mane, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, ´ Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viegas, Oriol Vinyals, Pete Warden, Martin Watten- ´ berg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. URL http://tensorflow.org/. Software available from tensorflow.org.
+
+Yoshua Bengio, Nicholas Leonard, and Aaron C. Courville. Estimating or propagating gradients ´ through stochastic neurons for conditional computation. CoRR, abs/1308.3432, 2013. URL http://arxiv.org/abs/1308.3432.
+
+Matthieu Courbariaux and Yoshua Bengio. Binarynet: Training deep neural networks with weights and activations constrained to $+ 1$ or -1. CoRR, abs/1602.02830, 2016. URL http://arxiv. org/abs/1602.02830.
+
+Matthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David. Binaryconnect: Training deep neural networks with binary weights during propagations. CoRR, abs/1511.00363, 2015. URL http: //arxiv.org/abs/1511.00363.
+
+Benjamin Graham. Low-precision batch-normalized activations. CoRR, abs/1702.08231, 2017. URL http://arxiv.org/abs/1702.08231.
+
+Suyog Gupta, Ankur Agrawal, Kailash Gopalakrishnan, and Pritish Narayanan. Deep learning with limited numerical precision. CoRR, abs/1502.02551, 2015a. URL http://arxiv.org/abs/ 1502.02551.
+
+Suyog Gupta, Ankur Agrawal, Kailash Gopalakrishnan, and Pritish Narayanan. Deep learning with limited numerical precision. CoRR, abs/1502.02551, 2015b. URL http://arxiv.org/ abs/1502.02551.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CoRR, abs/1512.03385, 2015. URL http://arxiv.org/abs/1512.03385.
+
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. CoRR, abs/1502.03167, 2015. URL http://arxiv.org/ abs/1502.03167.
+
+N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates, S. Bhatia, N. Boden, A. Borchers, R. Boyle, P.-l. Cantin, C. Chao, C. Clark, J. Coriell, M. Daley, M. Dau, J. Dean, B. Gelb, T. Vazir Ghaemmaghami, R. Gottipati, W. Gulland, R. Hagmann, C. R. Ho, D. Hogberg, J. Hu, R. Hundt, D. Hurt, J. Ibarz, A. Jaffey, A. Jaworski, A. Kaplan, H. Khaitan, A. Koch, N. Kumar, S. Lacy, J. Laudon, J. Law, D. Le, C. Leary, Z. Liu, K. Lucke, A. Lundin, G. MacKean, A. Maggiore, M. Mahony, K. Miller, R. Nagarajan, R. Narayanaswami, R. Ni, K. Nix, T. Norrie, M. Omernick, N. Penukonda, A. Phelps, J. Ross, M. Ross, A. Salek, E. Samadiani, C. Severn, G. Sizikov, M. Snelham, J. Souter, D. Steinberg, A. Swing, M. Tan, G. Thorson, B. Tian, H. Toma, E. Tuttle, V. Vasudevan, R. Walter, W. Wang, E. Wilcox, and D. H. Yoon. In-Datacenter Performance Analysis of a Tensor Processing Unit. ArXiv e-prints, April 2017.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In F. Pereira, C. J. C. Burges, L. Bottou, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 25, pp. 1097– 1105. Curran Associates, Inc., 2012. URL http://papers.nips.cc/paper/ 4824-imagenet-classification-with-deep-convolutional-neural-networks. pdf.
+
+Fengfu Li and Bin Liu. Ternary weight networks. CoRR, abs/1605.04711, 2016. URL http: //arxiv.org/abs/1605.04711.
+
+Zhouhan Lin, Matthieu Courbariaux, Roland Memisevic, and Yoshua Bengio. Neural networks with few multiplications. CoRR, abs/1510.03009, 2015. URL http://arxiv.org/abs/1510. 03009.
+
+N. Mellempudi, A. Kundu, D. Mudigere, D. Das, B. Kaul, and P. Dubey. Ternary Neural Networks with Fine-Grained Quantization. ArXiv e-prints, May 2017.
+
+Daisuke Miyashita, Edward H. Lee, and Boris Murmann. Convolutional neural networks using logarithmic data representation. CoRR, abs/1603.01025, 2016. URL http://arxiv.org/ abs/1603.01025.
+
+Eriko Nurvitadhi, Ganesh Venkatesh, Jaewoong Sim, Debbie Marr, Randy Huang, Jason Ong Gee Hock, Yeong Tat Liew, Krishnan Srivatsan, Duncan Moss, Suchit Subhaschandra, and Guy Boudoukh. Can fpgas beat gpus in accelerating next-generation deep neural networks? In Proceedings of the 2017 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays, FPGA ’17, pp. 5–14, New York, NY, USA, 2017. ACM. ISBN 978-1-4503-4354-1. doi: 10.1145/ 3020078.3021740. URL http://doi.acm.org/10.1145/3020078.3021740.
+
+Mohammad Rastegari, Vicente Ordonez, Joseph Redmon, and Ali Farhadi. Xnor-net: Imagenet classification using binary convolutional neural networks. CoRR, abs/1603.05279, 2016. URL http://arxiv.org/abs/1603.05279.
+
+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 115(3):211–252, 2015. doi: 10.1007/s11263-015-0816-y.
+
+Frank Seide, Hao Fu, Jasha Droppo, Gang Li, and Dong Yu. 1-bit stochastic gradient descent and application to data-parallel distributed training of speech dnns. In Interspeech 2014, September 2014.
+
+Wonyong Sung, Sungho Shin, and Kyuyeon Hwang. Resiliency of deep neural networks under quantization. CoRR, abs/1511.06488, 2015. URL http://arxiv.org/abs/1511.06488.
+
+Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott E. Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. CoRR, abs/1409.4842, 2014. URL http://arxiv.org/abs/1409.4842.
+
+Christian Szegedy, Sergey Ioffe, and Vincent Vanhoucke. Inception-v4, inception-resnet and the impact of residual connections on learning. CoRR, abs/1602.07261, 2016. URL http: //arxiv.org/abs/1602.07261.
+
+Yaman Umuroglu, Nicholas J. Fraser, Giulio Gambardella, Michaela Blott, Philip Heng Wai Leong, Magnus Jahre, and Kees A. Vissers. FINN: A framework for fast, scalable binarized neural network inference. CoRR, abs/1612.07119, 2016. URL http://arxiv.org/abs/1612. 07119.
+
+Vincent Vanhoucke, Andrew Senior, and Mark Z. Mao. Improving the speed of neural networks on cpus. In Deep Learning and Unsupervised Feature Learning Workshop, NIPS 2011, 2011.
+
+Ganesh Venkatesh, Eriko Nurvitadhi, and Debbie Marr. Accelerating deep convolutional networks using low-precision and sparsity. CoRR, abs/1610.00324, 2016. URL http://arxiv.org/ abs/1610.00324.
+
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. CoRR, abs/1605.07146, 2016. URL http://arxiv.org/abs/1605.07146.
+
+Aojun Zhou, Anbang Yao, Yiwen Guo, Lin Xu, and Yurong Chen. Incremental network quantization: Towards lossless cnns with low-precision weights. CoRR, abs/1702.03044, 2017. URL http://arxiv.org/abs/1702.03044.
+
+Shuchang Zhou, Zekun Ni, Xinyu Zhou, He Wen, Yuxin Wu, and Yuheng Zou. Dorefa-net: Training low bitwidth convolutional neural networks with low bitwidth gradients. CoRR, abs/1606.06160, 2016. URL http://arxiv.org/abs/1606.06160.
+
+Chenzhuo Zhu, Song Han, Huizi Mao, and William J. Dally. Trained ternary quantization. CoRR, abs/1612.01064, 2016. URL http://arxiv.org/abs/1612.01064.

parse/train/B1ZvaaeAZ/B1ZvaaeAZ_content_list.json

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+    {
+        "type": "text",
+        "text": "WRPN: WIDE REDUCED-PRECISION NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Asit Mishra, Eriko Nurvitadhi, Jeffrey J Cook & Debbie Marr   \nAccelerator Architecture Lab   \nIntel Labs   \n{asit.k.mishra,eriko.nurvitadhi,jeffrey.j.cook,debbie.marr}@intel.com ",
+        "bbox": [
+            181,
+            143,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            237,
+            544,
+            252
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For computer vision applications, prior works have shown the efficacy of reducing numeric precision of model parameters (network weights) in deep neural networks. Activation maps, however, occupy a large memory footprint during both the training and inference step when using mini-batches of inputs. One way to reduce this large memory footprint is to reduce the precision of activations. However, past works have shown that reducing the precision of activations hurts model accuracy. We study schemes to train networks from scratch using reduced-precision activations without hurting accuracy. We reduce the precision of activation maps (along with model parameters) and increase the number of filter maps in a layer, and find that this scheme matches or surpasses the accuracy of the baseline full-precision network. As a result, one can significantly improve the execution efficiency (e.g. reduce dynamic memory footprint, memory bandwidth and computational energy) and speed up the training and inference process with appropriate hardware support. We call our scheme WRPN - wide reducedprecision networks. We report results and show that WRPN scheme is better than previously reported accuracies on ILSVRC-12 dataset while being computationally less expensive compared to previously reported reduced-precision networks. ",
+        "bbox": [
+            233,
+            268,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            530,
+            334,
+            546
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A promising approach to lower the compute and memory requirements of convolutional deeplearning workloads is through the use of low numeric precision algorithms. Operating in lower precision mode reduces computation as well as data movement and storage requirements. Due to such efficiency benefits, there are many existing works which propose low-precision deep neural networks (DNNs) (Zhou et al., 2017; Lin et al., 2015; Miyashita et al., 2016; Gupta et al., 2015b; Vanhoucke et al., 2011), even down to 2-bit ternary mode (Zhu et al., 2016; Li & Liu, 2016; Venkatesh et al., 2016) and 1-bit binary mode (Zhou et al., 2016; Courbariaux & Bengio, 2016; Rastegari et al., 2016; Courbariaux et al., 2015; Umuroglu et al., 2016). However, majority of existing works in low-precision DNNs sacrifice accuracy over the baseline full-precision networks. Further, most prior works target reducing the precision of the model parameters (network weights). This primarily benefits the inference step only when batch sizes are small. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We observe that activation maps (neuron outputs) occupy more memory compared to the model parameters for batch sizes typical during training. This observation holds even during inference when batch size is around eight or more. Based on this observation, we study schemes for training and inference using low-precision DNNs where we reduce the precision of activation maps as well as the model parameters without sacrificing network accuracy. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To improve both execution efficiency and accuracy of low-precision networks, we reduce both the precision of activation maps and model parameters and increase the number of filter maps in a layer. We call networks using this scheme wide reduced-precision networks (WRPN) and find that this scheme compensates or surpasses the accuracy of the baseline full-precision network. Although the number of raw compute operations increases as we increase the number of filter maps in a layer, the compute bits required per operation is now a fraction of what is required when using full-precision operations (e.g. going from FP32 AlexNet to 4-bits precision and doubling the number of filters increases the number of compute operations by 4x, but each operation is 8x more efficient than FP32). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "WRPN offers better accuracies, while being computationally less expensive compared to previously reported reduced-precision networks. We report results on AlexNet (Krizhevsky et al., 2012), batch-normalized Inception (Ioffe & Szegedy, 2015), and ResNet-34 (He et al., 2015) on ILSVRC12 (Russakovsky et al., 2015) dataset. We find 4-bits to be sufficient for training deep and wide models while achieving similar or better accuracy than baseline network. With 4-bit activation and 2-bit weights, we find the accuracy to be at-par with baseline full-precision. Making the networks wider and operating with 1-bit precision, we close the accuracy gap between previously reported binary networks and show state-of-the art results for ResNet-34 $6 9 . 8 5 \\%$ top-1 with $2 \\mathbf { x }$ wide) and AlexNet $4 8 . 0 4 \\%$ top-1 with $1 . 3 \\mathrm { x }$ wide). To the best of our knowledge, our reported accuracies with binary networks and 4-bit precision are highest to date. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our reduced-precision quantization scheme is hardware friendly allowing for efficient hardware implementations. To this end, we evaluate efficiency benefits of low-precision operations (4-bits to 1-bits) on Titan X GPU, Arria-10 FPGA and ASIC. We see that FPGA and ASIC can deliver significant efficiency gain over FP32 operations $6 . 5 \\mathrm { x }$ to $1 0 0 \\mathrm { x }$ ), while GPU cannot take advantage of very low-precision operations. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/7f1ebe08f820ba2d4f6cf54598581053b3e8155aa0a62fb48e4050ad8d091021.jpg",
+        "image_caption": [
+            "2 MOTIVATION FOR REDUCED-PRECISION ACTIVATION MAPS ",
+            "Figure 1: Memory footprint of activations (ACTs) and weights (W) during training and inference for minibatch sizes 1 and 32. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "While most prior works proposing reduced-precision networks work with low precision weights (e.g. work in Courbariaux & Bengio (2016); Zhu et al. (2016); Zhou et al. (2016); Venkatesh et al. (2016); Li & Liu (2016); Courbariaux et al. (2015); Umuroglu et al. (2016)), we find that activation maps occupy a larger memory footprint when using mini-batches of inputs. Using mini-batches of inputs is typical in training of DNNs and cloud-based batched inference (Jouppi et al., 2017). Figure 1 shows memory footprint of activation maps and filter maps as batch size changes for 4 different networks (AlexNet, Inception-Resnet-v2 (Szegedy et al., 2016), ResNet-50 and ResNet-101) during the training and inference steps. ",
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+    {
+        "type": "image",
+        "img_path": "images/95b9b13bfb188037f010166d925ca0e7b975dda5fa5f3ad20c2532785fd508d5.jpg",
+        "image_caption": [
+            "Figure 2: Memory requirements of a feed forward convolutional deep neural network. Orange boxes denote weights (W), blue boxes are activations (ACT) and green boxes are gradient-maps (Grad). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As batch-size increases, because of filter reuse across batches of inputs, activation maps occupy significantly larger fraction of memory compared to the filter weights. This aspect is illustrated in ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 which shows the memory requirements of a canonical feed-forward DNN for a hardware accelerator based system (e.g. GPU, FPGA, PCIe connected ASIC device, etc.). During training, the sum of all the activation maps (ACT) and weight tensors (W) are allocated in device memory for forward pass along with memory for gradient maps during backward propagation. The total memory requirements for training phase is the sum of memory required for the activation maps, weights and the maximum of input gradient maps $( \\delta \\mathbf { Z } )$ and maximum of back-propagated gradients $( \\delta \\mathbf { X } )$ . During inference, memory is allocated for input (IFM) and output feature maps (OFM) required by a single layer, and these memory allocations are reused for other layers. The total memory allocation during inference is then the maximum of IFM and maximum of OFM required across all the layers plus the sum of all W-tensors. At batch sizes 128 and more, activations start to occupy more than $98 \\%$ of total memory footprint during training. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Overall, reducing precision of activations and weights reduces memory footprint, bandwidth and storage while also simplifying the requirements for hardware to efficiently support these operations. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3 WRPN SCHEME AND STUDIES ON ALEXNET ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Based on the observation that activations occupy more memory footprint compared to weights, we reduce the precision of activations to speed up training and inference steps as well as cut down on memory requirements. However, a straightforward reduction in precision of activation maps leads to significant reduction in model accuracy (Zhou et al., 2016; Rastegari et al., 2016). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We conduct a sensitivity study where we reduce precision of activation maps and model weights for AlexNet running ILSVRC-12 dataset and train the network from scratch. Table 1 reports our findings. Top-1 single-precision (32-bits weights and activations) accuracy is $5 7 . 2 \\%$ . The accuracy with binary weights and activations is $4 4 . 2 \\%$ . This is similar to what is reported in Rastegari et al. (2016). $3 2 b A$ and $2 b W$ data-point in this table is using Trained Ternary Quantization (TTQ) technique (Zhu et al., 2016). All other data points are collected using our quantization scheme (described later in Section 5), all the runs have same hyper-parameters and training is carried out for the same number of epochs as baseline network. To be consistent with results reported in prior works, we do not quantize weights and activations of the first and last layer. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We find that, in general, reducing the precision of activation maps and weights hurts model accuracy. Further, reducing precision of activations hurts model accuracy much more than reducing precision of the filter parameters. We find TTQ to be quite effective on AlexNet in that one can lower the precision of weights to 2b (while activations are still FP32) and not lose accuracy. However, we did not find this scheme to be effective for other networks like ResNet or Inception. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/68b41f414d41218fdc998e6411e9fa6fffeba26af65946108693da1db317d075.jpg",
+        "table_caption": [
+            "Table 1: AlexNet top-1 validation set accuracy $\\%$ as precision of activations (A) and weight(W) changes. All results are with end-to-end training of the network from scratch. − is a data-point we did not experiment for. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>32b A</td><td>8bA</td><td>4b A</td><td>2bA</td><td>1bA</td></tr><tr><td>32bW</td><td>57.2</td><td>54.3</td><td>54.4</td><td>52.7</td><td></td></tr><tr><td>8bW</td><td>1</td><td>54.5</td><td>53.2</td><td>51.5</td><td>1</td></tr><tr><td>4b W</td><td>1</td><td>54.2</td><td>54.4</td><td>52.4</td><td></td></tr><tr><td>2b W</td><td>57.5</td><td>50.2</td><td>50.5</td><td>51.3</td><td>1</td></tr><tr><td>1b W</td><td>56.8</td><td>1</td><td>1</td><td>1</td><td>44.2</td></tr></table>",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/cd7bc1abe74ca837190ccb83ce086162d73089e27fe7758ffeea7c05acdb6830.jpg",
+        "table_caption": [
+            "Table 2: AlexNet $2 \\mathbf { x }$ -wide top-1 validation set accuracy $\\%$ as precision of activations (A) and weights (W) changes. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>32bA</td><td>8bA</td><td>4bA</td><td>2bA</td><td>1b A</td></tr><tr><td>32bW</td><td>60.5</td><td>58.9</td><td>58.6</td><td>57.5</td><td>52.0</td></tr><tr><td>8bW</td><td>1</td><td>59.0</td><td>58.8</td><td>57.1</td><td>50.8</td></tr><tr><td>4b W</td><td>1</td><td>58.8</td><td>58.6</td><td>57.3</td><td>/</td></tr><tr><td>2b W</td><td>1</td><td>57.6</td><td>57.2</td><td>55.8</td><td>1</td></tr><tr><td>1b W</td><td>1</td><td>1</td><td>1</td><td>1</td><td>48.3</td></tr></table>",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To re-gain the model accuracy while working with reduced-precision operands, we increase the number of filter maps in a layer. Although the number of raw compute operations increase with widening the filter maps in a layer, the bits required per compute operation is now a fraction of what is required when using full-precision operations. As a result, with appropriate hardware support, one can significantly reduce the dynamic memory requirements, memory bandwidth, computational energy and speed up the training and inference process. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our widening of filter maps is inspired from Wide ResNet (Zagoruyko & Komodakis, 2016) work where the depth of the network is reduced and width of each layer is increased (the operand precision is still FP32). Wide ResNet requires a re-design of the network architecture. In our work, we maintain the depth parameter same as baseline network but widen the filter maps. We call our approach WRPN - wide reduced-precision networks. In practice, we find this scheme to be very simple and effective - starting with a baseline network architecture, one can change the width of each filter map without changing any other network design parameter or hyper-parameters. Carefully reducing precision and simultaneously widening filters keeps the total compute cost of the network under or at-par with baseline cost.1 ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Table 2 reports the accuracy of AlexNet when we double the number of filter maps in a layer. With doubling of filter maps, AlexNet with 4-bits weights and 2-bits activations exhibits accuracy at-par with full-precision networks. Operating with 4-bits weights and 4-bits activations surpasses the baseline accuracy by $1 . 4 4 \\%$ . With binary weights and activations we better the accuracy of XNORNET (Rastegari et al., 2016) by $4 \\%$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When doubling the number of filter maps, AlexNet’s raw compute operations grow by $3 . 9 \\mathbf { x }$ compared to the baseline full-precision network, however by using reduced-precision operands the overall compute complexity is a fraction of the baseline. For example, with 4b operands for weights and activations and $2 \\mathbf { x }$ the number of filters, reduced-precision AlexNet is just $49 \\%$ of the total compute cost of the full-precision baseline (compute cost comparison is shown in Table 3). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/08fc12882d7c09542ad5c72f19f19a5451c99b1bbccc43b5c8afb220cf711087.jpg",
+        "table_caption": [
+            "Table 3: Compute cost of AlexNet $2 \\mathbf { x }$ -wide vs. 1x-wide as precision of activations (A) and weights (W) changes. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>32b A</td><td>8b A</td><td>4b A</td><td>2bA</td><td>1b A</td></tr><tr><td>32b W</td><td>3.9x</td><td>2.4x</td><td>2.2x</td><td>2.1x</td><td>2.0x</td></tr><tr><td>8b W</td><td>2.4x</td><td>1.0x</td><td>0.7x</td><td>0.6x</td><td>0.6x</td></tr><tr><td>4b W</td><td>2.2x</td><td>0.7x</td><td>0.5x</td><td>0.4x</td><td>0.3x</td></tr><tr><td>2b W</td><td>2.1x</td><td>0.6x</td><td>0.4x</td><td>0.2x</td><td>0.2x</td></tr><tr><td>1b W</td><td>2.0x</td><td>0.6x</td><td>0.3x</td><td>0.2x</td><td>0.1x</td></tr></table>",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We also experiment with other widening factors. With $1 . 3 \\mathrm { x }$ widening of filters and with 4-bits of activation precision one can go as low as 8-bits of weight precision while still being at-par with baseline accuracy. With $1 . 1 \\mathrm { x }$ wide filters, at least 8-bits weight and 16-bits activation precision is required for accuracy to match baseline full-precision 1x wide accuracy. Further, as Table 3 shows, when widening filters by $2 \\mathbf { x }$ , one needs to lower precision to at least 8-bits so that the total compute cost is not more than baseline compute cost. Thus, there is a trade-off between widening and reducing the precision of network parameters. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In our work, we trade-off higher number of raw compute operations with aggressively reducing the precision of the operands involved in these operations (activation maps and filter weights) while not sacrificing the model accuracy. Apart from other benefits of reduced precision activations as mentioned earlier, widening filter maps also improves the efficiency of underlying GEMM calls for convolution operations since compute accelerators are typically more efficient on a single kernel consisting of parallel computation on large data-structures as opposed to many small sized kernels (Zagoruyko & Komodakis, 2016). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 STUDIES ON DEEPER NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We study how our scheme applies to deeper networks. For this, we study ResNet-34 (He et al., 2015) and batch-normalized Inception (Ioffe & Szegedy, 2015) and find similar trends, particularly that 2-bits weight and 4-bits activations continue to provide at-par accuracy as baseline. We use TensorFlow (Abadi et al., 2015) and tensorpack for all our evaluations and use ILSVRC-12 train and val dataset for analysis. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 RESNET ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "ResNet-34 has 3x3 filters in each of its modular layers with shortcut connections being 1x1. The filter bank width changes from 64 to 512 as depth increases. We use the pre-activation variant of ResNet and the baseline top-1 accuracy of our ResNet-34 implementation using single-precision ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "32-bits data format is $7 3 . 5 9 \\%$ . Binarizing weights and activations for all layers except the first and the last layer in this network gives top-1 accuracy of $6 0 . 5 \\%$ . For binarizing ResNet we did not re-order any layer (as is done in XNOR-NET). We used the same hyper-parameters and learning rate schedule as the baseline network. As a reference, for ResNet-18, the gap between XNOR-NET (1b weights and activations) and full-precision network is $18 \\%$ (Rastegari et al., 2016). It is also interesting to note that top-1 accuracy of single-precision AlexNet $( 5 7 . 2 0 \\% )$ is lower than the top-1 accuracy of binarized ResNet-34 $( 6 0 . 5 \\% )$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/3add07bef96aeef10c64d78f1c4f7656f89c771362ce8151713d21f74429d0e3.jpg",
+        "table_caption": [
+            "Table 4: ResNet-34 top-1 validation accuracy $\\%$ and compute cost as precision of activations (A) and weights (W) varies. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Width</td><td>Precision</td><td>Top-1 Acc. %</td><td>Compute cost</td></tr><tr><td>1x wide</td><td>32b A,32b W 1b A, 1b W</td><td>73.59 60.54</td><td>1x 0.03x</td></tr><tr><td>2x wide</td><td>4b A, 8b W 4b A, 4b W</td><td>74.48 74.52</td><td>0.74x 0.50x</td></tr><tr><td></td><td>4b A, 2b W</td><td>73.58</td><td>0.39x</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td></td><td>2b A,4b W</td><td>73.50</td><td>0.39x</td></tr><tr><td></td><td>2b A,2b W</td><td>73.32</td><td>0.27x</td></tr><tr><td></td><td>1b A, 1b W</td><td>69.85</td><td>0.15x</td></tr><tr><td>3x wide</td><td>1b A, 1b W</td><td>72.38</td><td>0.30x</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We experimented with doubling number of filters in each layer and reduce the precision of activations and weights. Table 4 shows the results of our analysis. Doubling the number of filters and 4-bits precision for both weights and activations beats the baseline accuracy by $0 . 9 \\%$ . 4-bits activations and 2-bits (ternary) weights has top-1 accuracy at-par with baseline. Reducing precision to 2-bits for both weights and activations degrades accuracy by only $0 . 2 \\%$ compared to baseline. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Binarizing the weights and activations with $2 \\mathbf { x }$ wide filters has a top-1 accuracy of $6 9 . 8 5 \\%$ . This is just $3 . 7 \\%$ worse than baseline full-precision network while being only $15 \\%$ of the cost of the baseline network. Widening the filters by $3 \\mathbf { x }$ and binarizing the weights and activations reduces this gap to $1 . 2 \\%$ while the 3x wide network is $30 \\%$ the cost of the full-precision baseline network. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Although 4-bits precision seems to be enough for wide networks, we advocate for 4-bits activation precision and 2-bits weight precision. This is because with ternary weights one can get rid of the multipliers and use adders instead. Additionally, with this configuration there is no loss of accuracy. Further, if some accuracy degradation is tolerable, one can even go to binary circuits for efficient hardware implementation while saving $3 2 \\mathrm { x }$ in bandwidth for each of weights and activations compared to full-precision networks. All these gains can be realized with simpler hardware implementation and lower compute cost compared to baseline networks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, our ResNet binary and ternary (with 2-bits or 4-bits activation) top-1 accuracies are state-of-the-art results in the literature including unpublished technical reports (with similar data augmentation (Mellempudi et al., 2017)). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 BATCH-NORMALIZED INCEPTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We applied WRPN scheme to batch-normalized Inception network (Ioffe & Szegedy, 2015). This network includes batch normalization of all layers and is a variant of GoogleNet (Szegedy et al., 2014) where the 5x5 convolutional filters are replaced by two 3x3 convolutions with up to 128 wide filters. Table 5 shows the results of our analysis. Using 4-bits activations and 2-bits weight and doubling the number of filter banks in the network produces a model that is almost at-par in accuracy with the baseline single-precision network $0 . 0 2 \\%$ loss in accuracy). Wide network with binary weights and activations is within $6 . 6 \\%$ of the full-precision baseline network. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 HARDWARE FRIENDLY QUANTIZATION SCHEME ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We adopt the straight-through estimator (STE) approach in our work (Bengio et al., 2013). When quantizing a real number to $k$ -bits, the ordinality of the set of quantized numbers is $2 ^ { k }$ . Mathematically, this small and finite set would have zero gradients with respect to its inputs. STE method circumvents this problem by defining an operator that has arbitrary forward and backward operations. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/e13d9207ef901f9c72d0c924fae9fa4f092413625cd682c0e4564ab5e48919dc.jpg",
+        "table_caption": [
+            "Table 5: Batch-normalized Inception top-1 validation accuracy $\\%$ and compute cost as precision of activations (A) and weights (W) varies. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Width</td><td>Precision</td><td>Top-1 Acc. %</td><td>Compute cost</td></tr><tr><td>1x wide</td><td>32b A,32b W</td><td>71.64</td><td>1x</td></tr><tr><td rowspan=\"4\">2x wide</td><td>4b A, 4b W</td><td>71.63</td><td>0.50x</td></tr><tr><td>4b A, 2b W</td><td>71.61</td><td>0.38x</td></tr><tr><td>2b A,2b W</td><td>70.75</td><td>0.25x</td></tr><tr><td>1b A,1b W</td><td>65.02</td><td>0.13x</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Prior works using the STE approach define operators that quantize the weights based on the expectation of the weight tensors. For instance, Ternary Weight Networks (TWN) (Li & Liu, 2016) uses a threshold and a scaling factor for each layer to quantize weights to ternary domain. In TTQ (Zhu et al., 2016), the scaling factors are learned parameters. XNOR-NET binarizes the weight tensor by computing the sign of the tensor values and then scaling by the mean of the absolute value of each output channel of weights. DoReFa uses a single scaling factor across the entire layer. For quantizing weights to $k$ -bits, where $k > 1$ , DoReFa uses: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3931dcfe2c2fef7db15078e5cf4419ef1acd1adaaf092a523d9cdccd1690c137.jpg",
+        "text": "$$\nw ^ { k } = 2 * q u a n t i z e _ { k } ( \\frac { t a n h ( w _ { i } ) } { 2 * m a x ( | \\ t a n h ( w _ { i } ) \\ | ) } + \\frac { 1 } { 2 } ) - 1 )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            424,
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+            458
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here $w _ { k }$ is the $\\mathbf { k }$ -bit quantized version of inputs $w _ { i }$ and quantize $k$ is a quantization function that quantizes a floating-point number $w _ { i }$ in the range $[ 0 , 1 ]$ to a $k$ -bit number in the same range. The transcendental tanh operation constrains the weight value to lie in between $- 1$ and $+ 1$ . The affine transformation post quantization brings the range to $[ - 1 , 1 ]$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We build on these approaches and propose a much simpler scheme. For quantizing weight tensors we first hard constrain the values to lie within the range $[ - 1 , 1 ]$ using min-max operation (e.g. tf.clip by val when using Tensorflow (Abadi et al., 2015)). For quantizing activation tensor values, we constrain the values to lie within the range [0, 1]. This step is followed by a quantization step where a real number is quantized into a $k$ -bit number. This is given as, for $k > 1$ : ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56229d6d59110f18b0d3c1a36bb5a05bf066573553347603e3b26c33f3b7b4a7.jpg",
+        "text": "$$\nw _ { k } = \\frac { 1 } { 2 ^ { k - 1 } - 1 } r o u n d ( ( 2 ^ { k - 1 } - 1 ) * w _ { i } ) \\qquad \\mathrm { a n d } \\qquad a _ { k } = \\frac { 1 } { 2 ^ { k } - 1 } r o u n d ( ( 2 ^ { k } - 1 ) * a _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            627,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here $w _ { i }$ and $a _ { i }$ are input real-valued weights and activation tensor and $w _ { k }$ and $a _ { k }$ are their quantized versions. One bit is reserved for sign-bit in case of weight values, hence the use of $2 ^ { k - 1 }$ for these quantized values. Thus, weights can be stored and interpreted using signed data-types and activations using un-signed data-types. With appropriate affine transformations, the convolution operations (the bulk of the compute operations in the network during forward pass) can be done using quantized values (integer operations in hardware) followed by scaling with floating-point constants (this scaling operation can be done in parallel with the convolution operation in hardware). When $k = 1$ , for binary weights we use the Binary Weighted Networks (BWN) approach (Courbariaux et al., 2015) where the binarized weight value is computed based on the sign of input value followed by scaling with the mean of absolute values. For binarized activations we use the formulation in Eq. 2. We do not quantize the gradients and maintain the weights in reduced precision format. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For convolution operation when using WRPN, the forward pass during training (and the inference step) involves matrix multiplication of $k$ -bits signed and $k$ -bits unsigned operands. Since gradient values are in 32-bits floating-point format, the backward pass involves a matrix multiplication operation using 32-bits and $k$ -bits operand for gradient and weight update. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "When $k > 1$ , the hard clipping of tensors to a range maps efficiently to min-max comparator units in hardware as opposed to using transcendental operations which are long latency operations. TTQ and DoRefa (Zhou et al., 2016) schemes involve division operation and computing a maximum value in the input tensor. Floating-point division operation is expensive in hardware and computing the maximum in a tensor is an $O ( n )$ operation. Additionally, our quantization parameters are static and do not require any learning or involve back-propagation like TTQ approach. We avoid each of these costly operations and propose a simpler quantization scheme (clipping followed by rounding). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 EFFICIENCY IMPROVEMENTS OF REDUCED-PRECISION OPERATIONS ON GPU, FPGA AND ASIC ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In practice, the effective performance and energy efficiency one could achieve on a low-precision compute operation highly depends on the hardware that runs these operations. We study the efficiency of low-precision operations on various hardware targets GPU, FPGA, and ASIC. ",
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+    {
+        "type": "text",
+        "text": "For GPU, we evaluate WRPN on Nvidia Titan X Pascal and for FPGA we use Intel Arria-10. We collect performance numbers from both previously reported analysis (Nurvitadhi et al., 2017) as well as our own experiments. For FPGA, we implement a DNN accelerator architecture shown in Figure 3(a). This is a prototypical accelerator design used in various works (e.g., on FPGA (Nurvitadhi et al., 2017) and ASIC such as TPU (Jouppi et al., 2017)). The core of the accelerator consists of a systolic array of processing elements (PEs) to perform matrix and vector operations, along with on-chip buffers, as well as off-chip memory management unit. The PEs can be configured to support different precision – (FP32, FP32), (INT4, INT4), (INT4, TER2), and (BIN1, BIN1). The (INT4, TER2) PE operates on ternary $^ { ( + 1 , 0 , - 1 ) }$ values and is optimized to include only an adder since there is no need for a multiplier in this case. The binary (BIN1, BIN1) PE is implemented using XNOR and bitcount. Our RTL design targets Arria-10 1150 FPGA. For our ASIC study, we synthesize the PE design using Intel $1 4 \\mathrm { n m }$ process technology to obtain area and energy estimates. ",
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+    {
+        "type": "image",
+        "img_path": "images/edc7cebaa1081616f22562193fbf135a3823ee6a5f5905ff4ca4bbf7144c7a10.jpg",
+        "image_caption": [
+            "Figure 3: Efficiency improvements from low-precision operations on GPU, FPGA and ASIC. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Figure 3(b) - (g) summarize our analysis. Figure 3(b) shows the efficiency improvements using first-order estimates where the efficiency is computed based on number of bits used in the operation. ",
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+    },
+    {
+        "type": "text",
+        "text": "With this method we would expect (INT4, INT4) and (BIN1, BIN1) to be ${ 8 } \\mathbf { { x } }$ and $3 2 \\mathrm { x }$ more efficient, respectively, than (FP32, FP32). However, in practice the efficiency gains from reducing precision depend on whether the underlying hardware can take advantage of such low-precisions. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 3(c) shows performance improvement on Titan X GPU for various low-precision operations relative to FP32. In this case, GPU can only achieve up to ${ \\sim } 4 \\mathbf { x }$ improvements in performance over FP32 baseline. This is because GPU only provides first-class support for INT8 operations, and is not able to take advantage of the lower INT4, TER2, and BIN1 precisions. On the contrary, FPGA can take advantage of such low precisions, since they are amenable for implementations on the FPGAs reconfigurable fabric. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 3(d) shows that the performance improvements from (INT4, INT4), (INT4, TER2), and (BIN1, BIN1) track well with the first-order estimates from Figure 3(b). In fact, for (BIN1, BIN1), FPGA improvements exceed the first-order estimate. Reducing the precision simplifies the design of compute units and lower buffering requirements on FPGA board. Compute-precision reduction leads to significant improvement in throughput due to smaller hardware designs (allowing more parallelism) and shorter circuit delay (allowing higher frequency). Figure 3(e) shows the performance and performance/Watt of the reduced-precision operations on GPU and FPGA. FPGA performs quite well on very low precision operations. In terms of performance/watt, FPGA does better than GPU on (INT4, INT4) and lower precisions. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ASIC allows for a truly customized hardware implementation. Our ASIC study provides insights to the upper bound of the efficiency benefits possible from low-precision operations. Figure 3(f) and 3(g) show improvement in performance and energy efficiency of the various low-precision ASIC PEs relative to baseline FP32 PE. As the figures show, going to lower precision offers 2 to 3 orders of magnitude efficiency improvements. ",
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+    },
+    {
+        "type": "text",
+        "text": "In summary, FPGA and ASIC are well suited for our WRPN approach. At $2 \\mathbf { x }$ wide, our WRPN approach requires $4 \\mathbf { x }$ more total operations than the original network. However, for INT4 or lower precision, each operation is $6 . 5 \\mathrm { x }$ or better in efficiency than FP32 for FPGA and ASIC. Hence, WRPN delivers an overall efficiency win. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Reduced-precision DNNs is an active research area. Reducing precision of weights for efficient inference pipeline has been very well studied. Works like Binary connect (BC) (Courbariaux et al., 2015), Ternary-weight networks (TWN) (Li & Liu, 2016), fine-grained ternary quantization (Mellempudi et al., 2017) and INQ (Zhou et al., 2017) target precision reduction of network weights while still using full-precision activations. Accuracy is almost always degraded when quantizing the weights. For AlexNet on Imagenet, TWN loses $5 \\%$ top-1 accuracy. Schemes like INQ, Sung et al. (2015) and Mellempudi et al. (2017) do fine-tuning to quantize the network weights and do not sacrifice accuracy as much but are not applicable for training networks from scratch. INQ shows promising results with 5-bits of precision. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "XNOR-NET (Rastegari et al., 2016), BNN (Courbariaux & Bengio, 2016), DoReFa (Zhou et al., 2016) and TTQ (Zhu et al., 2016) target training as well. While TTQ targets weight quantization only, most works targeting activation quantization hurt accuracy. XNOR-NET approach reduces top-1 accuracy by $12 \\%$ and DoReFa by $8 \\%$ when quantizing both weights and activations to 1-bit (for AlexNet on ImageNet). Further, XNOR-NET requires re-ordering of layers for its scheme to work. Recent work in Graham (2017) targets low-precision activations and reports accuracy within $1 \\%$ of baseline with 5-bits precision and logarithmic (with base $\\sqrt { 2 }$ ) quantization. With fine-tuning this gap can be narrowed to be within $0 . 6 \\%$ but not all layers are quantized. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Non-multiples of two for operand values introduces hardware inefficiency in that memory accesses are no longer DRAM or cache-boundary aligned and end-to-end run-time performance aspect is unclear when using complicated quantization schemes. We target end-to-end training and inference, using very simple quantization method and aim for reducing precision without any loss in accuracy. To the best of our knowledge, our work is the first to study reduced-precision deep and wide networks, and show accuracy at-par with baseline for as low a precision as 4-bits activations and 2-bits weights. We report state of the art accuracy for wide binarized AlexNet and ResNet while still being lower in compute cost. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Work by Gupta et al. (2015a) advocates for low precision fixed-point numbers for training. They show 16-bits to be sufficient for training on CIFAR10 dataset and find stochastic rounding to be necessary for training convergence. In our work here we focus on sub-8b training and like DoReFa scheme do not see stochastic rounding necessary when using full-precision gradients. Work by Seide et al. (2014) quantizes gradients before communication in a distributed computing setting. They use full precision gradients during the backward pass and quantize the gradients before sending them to other computation nodes (decreasing the amount of communication traffic over an interconnection network). For distributed training, we can potentially use this approach for communicating gradients across nodes. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We present the Wide Reduced-Precision Networks (WRPN) scheme for DNNs. In this scheme, the numeric precision of both weights and activations are significantly reduced without loss of network accuracy. This result is in contrast to many previous works that find reduced-precision activations to detrimentally impact accuracy; specifically, we find that 2-bit weights and 4-bit activations are sufficient to match baseline accuracy across many networks including AlexNet, ResNet-34 and batchnormalized Inception. We achieve this result with a new quantization scheme and by increasing the number of filter maps in each reduced-precision layer to compensate for the loss of information capacity induced by reducing the precision. We believe ours to be the first work to study the interplay between layer width and precision – with widening, the number of neurons in a layer increase; yet with reduced precision, we control overfitting and regularization. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We motivate this work with our observation that full-precision activations contribute significantly more to the memory footprint than full-precision weight parameters when using mini-batch sizes common during training and cloud-based inference; furthermore, by reducing the precision of both activations and weights the compute complexity is greatly reduced $40 \\%$ of baseline for 2-bit weights and 4-bit activations). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The WRPN quantization scheme and computation on low precision activations and weights is hardware friendly making it viable for deeply-embedded system deployments as well as in cloud-based training and inference servers with compute fabrics for low-precision. We compare Titan X GPU, Arria-10 FPGA and ASIC implementations using WRPN and show our scheme increases performance and energy-efficiency for iso-accuracy across each. Overall, reducing the precision allows custom-designed compute units and lower buffering requirements to provide significant improvement in throughput. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            630,
+            285,
+            645
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "tensorpack: https://github.com/ppwwyyxx/tensorpack. ",
+        "bbox": [
+            176,
+            655,
+            527,
+            670
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dan Mane, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, ´ Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viegas, Oriol Vinyals, Pete Warden, Martin Watten- ´ berg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. URL http://tensorflow.org/. Software available from tensorflow.org. ",
+        "bbox": [
+            176,
+            684,
+            825,
+            809
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yoshua Bengio, Nicholas Leonard, and Aaron C. Courville. Estimating or propagating gradients ´ through stochastic neurons for conditional computation. CoRR, abs/1308.3432, 2013. URL http://arxiv.org/abs/1308.3432. ",
+        "bbox": [
+            176,
+            824,
+            825,
+            866
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Matthieu Courbariaux and Yoshua Bengio. Binarynet: Training deep neural networks with weights and activations constrained to $+ 1$ or -1. CoRR, abs/1602.02830, 2016. URL http://arxiv. org/abs/1602.02830. ",
+        "bbox": [
+            174,
+            882,
+            823,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Matthieu Courbariaux, Yoshua Bengio, and Jean-Pierre David. Binaryconnect: Training deep neural networks with binary weights during propagations. CoRR, abs/1511.00363, 2015. URL http: //arxiv.org/abs/1511.00363. ",
+        "bbox": [
+            174,
+            103,
+            821,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Benjamin Graham. Low-precision batch-normalized activations. CoRR, abs/1702.08231, 2017. URL http://arxiv.org/abs/1702.08231. ",
+        "bbox": [
+            173,
+            154,
+            823,
+            183
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Suyog Gupta, Ankur Agrawal, Kailash Gopalakrishnan, and Pritish Narayanan. Deep learning with limited numerical precision. CoRR, abs/1502.02551, 2015a. URL http://arxiv.org/abs/ 1502.02551. ",
+        "bbox": [
+            173,
+            190,
+            823,
+            232
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Suyog Gupta, Ankur Agrawal, Kailash Gopalakrishnan, and Pritish Narayanan. Deep learning with limited numerical precision. CoRR, abs/1502.02551, 2015b. URL http://arxiv.org/ abs/1502.02551. ",
+        "bbox": [
+            173,
+            239,
+            825,
+            282
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CoRR, abs/1512.03385, 2015. URL http://arxiv.org/abs/1512.03385. ",
+        "bbox": [
+            173,
+            290,
+            823,
+            320
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. CoRR, abs/1502.03167, 2015. URL http://arxiv.org/ abs/1502.03167. ",
+        "bbox": [
+            176,
+            327,
+            823,
+            369
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates, S. Bhatia, N. Boden, A. Borchers, R. Boyle, P.-l. Cantin, C. Chao, C. Clark, J. Coriell, M. Daley, M. Dau, J. Dean, B. Gelb, T. Vazir Ghaemmaghami, R. Gottipati, W. Gulland, R. Hagmann, C. R. Ho, D. Hogberg, J. Hu, R. Hundt, D. Hurt, J. Ibarz, A. Jaffey, A. Jaworski, A. Kaplan, H. Khaitan, A. Koch, N. Kumar, S. Lacy, J. Laudon, J. Law, D. Le, C. Leary, Z. Liu, K. Lucke, A. Lundin, G. MacKean, A. Maggiore, M. Mahony, K. Miller, R. Nagarajan, R. Narayanaswami, R. Ni, K. Nix, T. Norrie, M. Omernick, N. Penukonda, A. Phelps, J. Ross, M. Ross, A. Salek, E. Samadiani, C. Severn, G. Sizikov, M. Snelham, J. Souter, D. Steinberg, A. Swing, M. Tan, G. Thorson, B. Tian, H. Toma, E. Tuttle, V. Vasudevan, R. Walter, W. Wang, E. Wilcox, and D. H. Yoon. In-Datacenter Performance Analysis of a Tensor Processing Unit. ArXiv e-prints, April 2017. ",
+        "bbox": [
+            181,
+            377,
+            825,
+            517
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In F. Pereira, C. J. C. Burges, L. Bottou, and K. Q. Weinberger (eds.), Advances in Neural Information Processing Systems 25, pp. 1097– 1105. Curran Associates, Inc., 2012. URL http://papers.nips.cc/paper/ 4824-imagenet-classification-with-deep-convolutional-neural-networks. pdf. ",
+        "bbox": [
+            173,
+            525,
+            864,
+            609
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fengfu Li and Bin Liu. Ternary weight networks. CoRR, abs/1605.04711, 2016. URL http: //arxiv.org/abs/1605.04711. ",
+        "bbox": [
+            171,
+            616,
+            820,
+            645
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhouhan Lin, Matthieu Courbariaux, Roland Memisevic, and Yoshua Bengio. Neural networks with few multiplications. CoRR, abs/1510.03009, 2015. URL http://arxiv.org/abs/1510. 03009. ",
+        "bbox": [
+            173,
+            652,
+            825,
+            695
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "N. Mellempudi, A. Kundu, D. Mudigere, D. Das, B. Kaul, and P. Dubey. Ternary Neural Networks with Fine-Grained Quantization. ArXiv e-prints, May 2017. ",
+        "bbox": [
+            169,
+            702,
+            823,
+            732
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Daisuke Miyashita, Edward H. Lee, and Boris Murmann. Convolutional neural networks using logarithmic data representation. CoRR, abs/1603.01025, 2016. URL http://arxiv.org/ abs/1603.01025. ",
+        "bbox": [
+            171,
+            739,
+            825,
+            781
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Eriko Nurvitadhi, Ganesh Venkatesh, Jaewoong Sim, Debbie Marr, Randy Huang, Jason Ong Gee Hock, Yeong Tat Liew, Krishnan Srivatsan, Duncan Moss, Suchit Subhaschandra, and Guy Boudoukh. Can fpgas beat gpus in accelerating next-generation deep neural networks? In Proceedings of the 2017 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays, FPGA ’17, pp. 5–14, New York, NY, USA, 2017. ACM. ISBN 978-1-4503-4354-1. doi: 10.1145/ 3020078.3021740. URL http://doi.acm.org/10.1145/3020078.3021740. ",
+        "bbox": [
+            173,
+            790,
+            825,
+            873
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mohammad Rastegari, Vicente Ordonez, Joseph Redmon, and Ali Farhadi. Xnor-net: Imagenet classification using binary convolutional neural networks. CoRR, abs/1603.05279, 2016. URL http://arxiv.org/abs/1603.05279. ",
+        "bbox": [
+            174,
+            882,
+            821,
+            922
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 115(3):211–252, 2015. doi: 10.1007/s11263-015-0816-y. ",
+        "bbox": [
+            173,
+            103,
+            826,
+            160
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Frank Seide, Hao Fu, Jasha Droppo, Gang Li, and Dong Yu. 1-bit stochastic gradient descent and application to data-parallel distributed training of speech dnns. In Interspeech 2014, September 2014. ",
+        "bbox": [
+            174,
+            169,
+            821,
+            210
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Wonyong Sung, Sungho Shin, and Kyuyeon Hwang. Resiliency of deep neural networks under quantization. CoRR, abs/1511.06488, 2015. URL http://arxiv.org/abs/1511.06488. ",
+        "bbox": [
+            173,
+            219,
+            823,
+            250
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott E. Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. CoRR, abs/1409.4842, 2014. URL http://arxiv.org/abs/1409.4842. ",
+        "bbox": [
+            178,
+            257,
+            823,
+            301
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Christian Szegedy, Sergey Ioffe, and Vincent Vanhoucke. Inception-v4, inception-resnet and the impact of residual connections on learning. CoRR, abs/1602.07261, 2016. URL http: //arxiv.org/abs/1602.07261. ",
+        "bbox": [
+            176,
+            309,
+            823,
+            352
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yaman Umuroglu, Nicholas J. Fraser, Giulio Gambardella, Michaela Blott, Philip Heng Wai Leong, Magnus Jahre, and Kees A. Vissers. FINN: A framework for fast, scalable binarized neural network inference. CoRR, abs/1612.07119, 2016. URL http://arxiv.org/abs/1612. 07119. ",
+        "bbox": [
+            173,
+            361,
+            825,
+            417
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Vincent Vanhoucke, Andrew Senior, and Mark Z. Mao. Improving the speed of neural networks on cpus. In Deep Learning and Unsupervised Feature Learning Workshop, NIPS 2011, 2011. ",
+        "bbox": [
+            171,
+            426,
+            823,
+            457
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ganesh Venkatesh, Eriko Nurvitadhi, and Debbie Marr. Accelerating deep convolutional networks using low-precision and sparsity. CoRR, abs/1610.00324, 2016. URL http://arxiv.org/ abs/1610.00324. ",
+        "bbox": [
+            173,
+            464,
+            823,
+            507
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. CoRR, abs/1605.07146, 2016. URL http://arxiv.org/abs/1605.07146. ",
+        "bbox": [
+            169,
+            516,
+            823,
+            545
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Aojun Zhou, Anbang Yao, Yiwen Guo, Lin Xu, and Yurong Chen. Incremental network quantization: Towards lossless cnns with low-precision weights. CoRR, abs/1702.03044, 2017. URL http://arxiv.org/abs/1702.03044. ",
+        "bbox": [
+            176,
+            554,
+            823,
+            597
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Shuchang Zhou, Zekun Ni, Xinyu Zhou, He Wen, Yuxin Wu, and Yuheng Zou. Dorefa-net: Training low bitwidth convolutional neural networks with low bitwidth gradients. CoRR, abs/1606.06160, 2016. URL http://arxiv.org/abs/1606.06160. ",
+        "bbox": [
+            176,
+            604,
+            823,
+            647
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chenzhuo Zhu, Song Han, Huizi Mao, and William J. Dally. Trained ternary quantization. CoRR, abs/1612.01064, 2016. URL http://arxiv.org/abs/1612.01064. ",
+        "bbox": [
+            174,
+            656,
+            821,
+            685
+        ],
+        "page_idx": 10
+    }
+]

parse/train/MlFcgL2AP4d/MlFcgL2AP4d.md

@@ -0,0 +1,342 @@
+# Near-Optimal Offline Reinforcement Learning via Double Variance Reduction
+
+Ming Yin 1,3, Yu Bai2, and Yu-Xiang Wang1
+
+1Department of Computer Science, UC Santa Barbara 2Salesforce Research 3Department of Statistics and Applied Probability, UC Santa Barbara ming_yin@ucsb.edu yu.bai@salesforce.com yuxiangw@cs.ucsb.edu
+
+# Abstract
+
+We consider the problem of offline reinforcement learning (RL) — a well-motivated setting of RL that aims at policy optimization using only historical data. Despite its wide applicability, theoretical understandings of offline RL, such as its optimal sample complexity, remain largely open even in basic settings such as tabular Markov Decision Processes (MDPs). In this paper, we propose Off-Policy Double Variance Reduction (OPDVR), a new variance reduction based algorithm for offline RL. Our main result shows that OPDVR provably identifies an $\epsilon$ -optimal policy with $\widetilde { O } ( H ^ { 2 } / d _ { m } \epsilon ^ { 2 } )$ episodes of offline data in the finite-horizon stationary transition setting, where $H$ is the horizon length and $d _ { m }$ is the minimal marginal stateaction distribution induced by the behavior policy. This improves over the best known upper bound by a factor of $H$ . Moreover, we establish an informationtheoretic lower bound of $\Omega ( H ^ { 2 } / d _ { m } \epsilon ^ { 2 } )$ which certifies that OPDVR is optimal up to logarithmic factors. Lastly, we show that OPDVR also achieves rate-optimal sample complexity under alternative settings such as the finite-horizon MDPs with non-stationary transitions and the infinite horizon MDPs with discounted rewards.
+
+# 1 Introduction
+
+Offline reinforcement learning (offline RL) aims at learning the near-optimal policy by using a static offline dataset that is collected by a certain behavior policy $\mu$ [Lange et al., 2012]. As offline RL agent has no access to interact with the environment, it is more widely applicable to problems where online interaction is infeasible, e.g. when trials-and-errors are expensive (robotics, education), risky (autonomous driving) or even unethical (healthcare) [see,e.g., a recent survey Levine et al., 2020].
+
+Despite its practical significance, a precise theoretical understanding of offline RL has been lacking. Previous sample complexity bounds for RL has primarily focused on the online setting [Azar et al., 2017, Jin et al., 2018, Zanette and Brunskill, 2019, Simchowitz and Jamieson, 2019, Efroni et al., 2019, Cai et al., 2019] or the generative model (simulator) setting [Azar et al., 2013, Sidford et al., 2018a,b, Yang and Wang, 2019, Agarwal et al., 2020, Wainwright, 2019, Lattimore and Szepesvari, 2019], both of which assuming interactive access to the environment and not applicable to offline RL. On the other hand, the sample complexity of offline RL remains unsettled even for environments with finitely many state and actions, a.k.a, the tabular MDPs (Markov Decision Processes). One major line of work is concerned with the off-policy evaluation (OPE) problem [Li et al., 2015, Jiang and Li, 2016, Liu et al., 2018, Kallus and Uehara, 2019a,b, Uehara and Jiang, 2019, Xie et al., 2019, Yin and Wang, 2020, Duan and Wang, 2020]. These works provide sample complexity bounds for evaluating the performance of a fixed policy, and do not imply guarantees for policy optimization. Another line of work studies the sample complexity of offline policy optimization in conjunction with function approximation [Chen and Jiang, 2019, Xie and Jiang, 2020b,a, Jin et al., 2020]. These results apply to offline RL with general function classes, but when specialized to the tabular setting, they give rather loose sample complexity bounds with suboptimal dependencies on various parameters 1
+
+Table 1: Comparison of sample complexities for tabular offline RL interpretation.   
+
+<table><tr><td>Method/Analysis</td><td>Setting</td><td>Assumptions</td><td>Sample complexitya</td></tr><tr><td>BFVT[Xie and Jiang,2020a]</td><td>Oo-horizon</td><td>onlyrealizability+MDP concentrabilityb</td><td>0(1-γ)-8C²/∈4)</td></tr><tr><td>MBS-PI/QI[Liu et al., 2020b]</td><td>oo-horizon</td><td>completeness+bounded den- sity estimation error</td><td>0(1-γ)-8C²/e²)</td></tr><tr><td>Le et al. [2019]</td><td>o-horizon</td><td>Full Concentrability</td><td>0((1-γ)-6βμ/e²）</td></tr><tr><td>FQI [Chen and Jiang, 2019]</td><td>o-horizon</td><td>Full Concentrability</td><td>0((1-γ)-6C/e2)</td></tr><tr><td>MSBO/MABO [Xie and Jiang,2020b]</td><td>oo-horizon</td><td>Full Concentrability</td><td>0((1-γ)-4Cμ/e2)</td></tr><tr><td>OPEMA [Yin et al.,2021]</td><td>H-horizon non-stationary</td><td>Full Concentrability</td><td>O(H3/dm∈²)</td></tr><tr><td>OPDVR (Section 3) OPDVR (Section 4)</td><td>H-horizon non-stationary</td><td>Weak Coverage</td><td>O(H3/dme²)</td></tr><tr><td></td><td>H-horizon stationary</td><td>Weak Coverage</td><td>O(H²/dme²)</td></tr><tr><td>OPDVR (Section 4.2)</td><td>oo-horizon</td><td>Weak Coverage</td><td>0((1-γ)-3/dm∈²)</td></tr></table>
+
+a Number of episodes in the finite horizon setting and number of steps in the infinite horizon. b $\mathrm { ~ \it ~ \beta _ { \mu } , C , C _ { \mu } , 1 \bar { / } } d _ { m }$ are the concentrability-type coefficients that measure the state-action coverage. See Assumption 2.1 and also Section F.2 for discussions.
+
+The recent work of Yin et al. [2021] showed that the optimal sample complexity for finding an $\epsilon$ -optimal policy in offline RL is $\widetilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ in the finite-horizon non-stationarysetting (with matching upper and lower bounds), where $H$ is the horizon length and $d _ { m }$ is a constant related to the data coverage of the behavior policy in the given MDP. However, the optimal sample complexity in alternative settings such as stationary transition or infinite-horizon settings remains unknown. Further, the $\widetilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ sample complexity is achieved by an off-policy evaluation $^ +$ uniform convergence type algorithm; other more practical algorithms including (stochastic) optimal planning algorithms such as Q-Learning are not well understood in offline RL.
+
+Our Contributions In this paper, we propose an algorithm OPDVR (Off-Policy Doubled Variance Reduction) for offline reinforcement learning based on an extension of the variance reduction technique initiated in [Sidford et al., 2018a, Yang and Wang, 2019]. OPDVR performs stochastic (minibatch style) value iterations using the available offline data, and can be seen as a version of stochastic optimal planning that interpolates value iteration and Q-learning. Our main contributions are summarized as follows.
+
+• We show that OPDVR finds an $\epsilon$ -optimal policy with high probability using $\widetilde { O } ( H ^ { 2 } / d _ { m } \epsilon ^ { 2 } )$ episodes of offline data (Section 4.1). This improves upon the best known sample complexity by an $H$ factor and to the best of our knowledge is the first that achieves an $\bar { O } ( H ^ { 2 } )$ horizon dependence, thus separating the stationary case with the non-stationary case for offline RL. We establish a sample (episode) complexity lower bound $\Omega ( H ^ { 2 } / d _ { m } \epsilon ^ { 2 } )$ for offline RL in the finite-horizon stationary setting (Theorem 4.2), showing that the sample complexity of OPDVR is optimal up to logarithmic factors. • In the finite-horizon non-stationary setting, and infinite horizon $\gamma$ -discounted setting, we show that OPDVR achieves $\widetilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ sample (episode) complexity (Section 3) and $\widetilde { O } ( ( 1 - \gamma ) ^ { - 3 } / d _ { m } \epsilon ^ { 2 } )$ sample complexity (Section 4.2) respectively. They are both optimal up to logarithmic factors and our infinite-horizon result improves over the best known results, e.g., those derived for the fitted Q-iteration style algorithms [Xie and Jiang, 2020b]. On the technical end, our algorithm presents a sharp analysis of offline RL with stationary transitions, and, importantly, the use of the doubling technique to resolve the initialization dependence defect which fails to make the original variance reduction algorithm of [Sidford et al., 2018a] to be optimal, see Appendix F.4. Running [Sidford et al., 2018a] may not yield the desired accuracy as they stated and our result is robust in preserving the optimality.
+
+Related work. There is a large and growing body of work on the theory of offline RL and RL in general. We could not hope to provide a comprehensive survey, thus will instead highlight the few prior work that we depend upon on the technical level. The variance reduction techniques that we use in this paper builds upon the work of [Sidford et al., 2018a] in the generative model setting, though it is nontrivial in adapting their techniques to the offline setting; and our two-stage variance reduction appears essential for obtaining optimal rate for $\epsilon > 1$ (see Section 5 and Appendix F.4 for more detailed discussions). We also used a fictitious estimator technique that originates from the OPE literature[Xie et al., 2019, Yin and Wang, 2020], but extended it to the stationary-transition case, and to the policy optimization problem. As we mentioned earlier, the optimal sample complexity in offline RL in the tabular MDPs with stationary transitions was not settled. The result of [Yin et al., 2021] is optimal in the non-stationary case, but is suboptimal by a factor of $H$ in the stationary case. Our lower bound is a variant of the construction of [Yin et al., 2021] that applies to the stationary case. Other existing work on offline RL has even weaker parameters (sometimes due to their setting being more general, see details in Table 1). We defer more detailed discussion related to the OPE literature and online RL / generative model literature to Appendix A due to space constraint.
+
+# 2 Preliminaries
+
+We consider reinforcement learning problems modeled by finite Markov Decision Processes (MDPs) (we focus on the finite-horizon episodic setting, and defer the infinite-horizon discounted setting to Section 4.2.) An MDP is denoted by a tuple $\bar { M } = ( S , A , r , T , d _ { 1 } , H )$ , where $s$ and $\mathcal { A }$ are the state and action spaces with finite cardinality $| S | = S$ and $| { \mathcal { A } } | = A$ . $P _ { t } : S \times \mathcal { A } \times \mathcal { S }  [ 0 , 1 ]$ is the transition kernel with $P _ { t } ( s ^ { \prime } | s , a )$ be the probability of entering state $s ^ { \prime }$ after taking action $a$ at state $s$ . We consider both the stationary and non-stationary transition setting: The stationary transition setting assumes $P _ { t } \equiv P$ is identical for all $t \in [ H ]$ , and the non-stationary transition setting allows $P _ { t }$ to be different for different $t$ . $r _ { t } : S \times \mathcal { A }  [ 0 , 1 ]$ is the reward function which we assume to be deterministic2. $d _ { 1 }$ is the initial state distribution, and $H$ is the time horizon. A (non-stationary) policy $\pi : S \to \mathbb { P } _ { A } ^ { H }$ assigns to each state $s _ { t } \in S$ a distribution over actions at each time $t$ . We use $d _ { t } ^ { \pi } ( s , a )$ or $d _ { t } ^ { \pi } ( s )$ to denote the marginal state-action/state distribution induced by policy $\pi$ at time $t$ i.e. $d _ { t } ^ { \pi } ( s ) : = \mathbb { P } ^ { \pi } ( s _ { t } = s )$ and $d _ { t } ^ { \pi } \bar { ( } s , a ) : = \mathbb { P } ^ { \pi } ( s _ { t } = s , a _ { t } = a )$ .
+
+$Q$ -value and Bellman operator. For any policy $\pi$ and any fixed time $t$ , the value function $\begin{array} { r } { \mathbb { E } \left[ \sum _ { i = t } ^ { H } r _ { i } \Big | s _ { t } = s \right] , Q _ { t } ^ { \pi } ( s , a ) \ = \ \mathbb { E } \left[ \sum _ { i = t } ^ { H } r _ { i } \Big | s _ { t } , a _ { t } = s , a \right] } \end{array}$ $V _ { t } ^ { \pi } ( \cdot ) ~ \in ~ \mathbb { R } ^ { S }$ and $Q$ -value function $\mathsf { \bar { \Psi } } Q _ { t } ^ { \mathsf { \bar { \pi } } } ( \cdot , \cdot ) \ \in \ \mathbb { R } ^ { S \times A } , \ \forall s , a$ . For the ease of exposition, we always is defined as: $V _ { t } ^ { \pi } ( s ) ~ =$ enumerate $Q ^ { \pi }$ as a column vector and similarly for $\bar { P } _ { t } ( \cdot | s , a )$ . Moreover, for any vector $Q \in$ $\mathbb { R } ^ { S \times A }$ , the induced value vector and policy is defined in the greedy way: $\forall s _ { t } \in \dot { S } , V _ { Q } ( s _ { t } ) =$ $\operatorname* { m a x } _ { a _ { t } \in A } Q ( s _ { t } , a _ { t } )$ , $\pi _ { Q } ( s _ { t } ) = \operatorname { a r g m a x } _ { a _ { t } \in \mathcal { A } } Q ( s _ { t } , a _ { t } )$ . Given an MDP, for any vector $V ~ \in ~ \mathbb { R } ^ { s }$ and any deterministic policy $\pi$ , $\forall t \in [ H ]$ the Bellman operator $\mathcal { T } _ { t } ^ { \pi } : \mathbb { R } ^ { S }  \mathbb { R } ^ { S }$ is defined as: $[ \mathcal T _ { t } ^ { \pi } ( V ) ] ( s ) : = r ( s , \pi _ { t } ( s ) ) + P _ { t } ^ { \top } ( \cdot | s , \pi _ { t } ( s ) ) V$ , and the corresponding Bellman optimality operator $\mathcal { T } _ { t } : \mathbb { R } ^ { S }  \mathbb { R } ^ { S }$ , $\begin{array} { r } { [ \mathcal { T } _ { t } ( V ) ] ( s ) : = \operatorname* { m a x } _ { a \in \mathcal { A } } [ r ( s , a ) + P _ { t } ^ { \top } ( \cdot | s , a ) V ] } \end{array}$ . Lastly, for a given value function $V _ { t }$ , we define backup function $z _ { t } ( s _ { t } , a _ { t } ) : = P _ { t } ^ { \top } ( \cdot | s _ { t } , a _ { t } ) V _ { t + 1 }$ and the one-step variance as $\begin{array} { r } { \sigma _ { V _ { t + 1 } } ( s _ { t } , a _ { t } ) : = \mathrm { V a r } _ { s _ { t + 1 } } [ \bar { V } _ { t + 1 } ( s _ { t + 1 } ) | s _ { t } , a _ { t } ] } \end{array}$ .
+
+Offline learning problem. In this paper we investigate the offline learning problem, where we do not have interactive access to the MDP, and can only observe a static dataset $\mathcal { D } =$ $\left\{ ( s _ { t } ^ { ( i ) } , a _ { t } ^ { ( i ) } , r _ { t } ^ { ( i ) } , s _ { t + 1 } ^ { ( i ) } ) \right\} _ { i \in [ n ] } ^ { t \in [ H ] }$ .We assume that $\mathcal { D }$ is obtained by executing a pre-specified behav$\mu$ ) for , whe $n$ episodes and collecting the trajectories each episode is rendered in the form: $\tau ^ { ( i ) } \ = \ ( s _ { 1 } ^ { ( i ) } , a _ { 1 } ^ { ( i ) } , r _ { 1 } ^ { ( i ) } , \ldots , s _ { H } ^ { ( i ) } , a _ { H } ^ { ( i ) } , r _ { H } ^ { ( i ) } , s _ { H + 1 } ^ { ( i ) } )$ $s _ { 1 } ^ { ( i ) } \sim d _ { 1 }$ , $a _ { t } ^ { ( i ) } \sim \mu _ { t } ( \cdot | s _ { t } ^ { ( i ) } )$ , $r _ { t } ^ { ( i ) } = r ( s _ { t } ^ { ( i ) } , a _ { t } ^ { ( i ) } )$ , and $s _ { t + 1 } ^ { ( i ) } \sim P _ { t } ( \cdot | s _ { t } ^ { ( i ) } , a _ { t } ^ { ( i ) } )$ . Given the dataset $\mathcal { D }$ , our goal is to find an $\epsilon$ -optimal policy $\pi _ { \mathrm { o u t } }$ , in the sense that $| | V _ { 1 } ^ { \pi ^ { \star } } - V _ { 1 } ^ { \pi _ { \mathrm { o u t } } } | | _ { \infty } < \epsilon$ .
+
+Assumption on data coverage Due to the curse of distributional shift, efficient offline RL is only possible under certain data coverage properties for the behavior policy $\mu$ . Throughout this paper we assume the following:
+
+Assumption 2.1 (Weak coverage). The behavior policy $\mu$ satisfies the following: There exists some optimal policy $\pi ^ { \star }$ such that $d _ { t ^ { \prime } } ^ { \mu } ( s _ { t ^ { \prime } } , a _ { t ^ { \prime } } ) > 0$ if there exists $t < t ^ { \prime }$ such that ${ d _ { t : t ^ { \prime } } ^ { \pi ^ { \star } } } ( s _ { t ^ { \prime } } , a _ { t ^ { \prime } } | s _ { t } , a _ { t } ) > 0$ , where $d _ { t : t ^ { \prime } } ^ { \pi ^ { \star } } \big ( s _ { t ^ { \prime } } , a _ { t ^ { \prime } } | s _ { t } , a _ { t } \big )$ is the conditional multi-step transition probability from step $t$ to $t ^ { \prime }$ .
+
+Intuitively, Assumption 2.1 requires $\mu$ to “cover” certain optimal policy $\pi ^ { \star }$ , in the sense that any $s _ { t ^ { \prime } } , a _ { t ^ { \prime } }$ is reachable by $\mu$ if it is attainable from a previous state-action pair by $\pi ^ { \star }$ . It is similar to [Liu et al., 2019, Assumption 1]. Note that this is weaker than the standard “concentrability” assumption [Munos, 2003, Le et al., 2019, Chen and Jiang, 2019]: Concentrability defines $\beta _ { \mu } : =$ $\begin{array} { r } { \operatorname* { s u p } _ { \pi \in \Pi } \vert \vert d ^ { \pi } ( s _ { t } , a _ { t } ) / d ^ { \mu } ( s _ { t } , a _ { t } ) \vert \vert _ { \infty } < \infty } \end{array}$ (cf. [Le et al., 2019, Assumption 1 & Example 4.1]), which requires the sufficient exploration for tabular case3 since we optimize over all policies (see Section F.2 for a discussion). In contrast, our assumption only requires $\mu$ to “trace” one single optimal policy.4
+
+With Assumption 2.1, we define
+
+$$
+d _ { m } : = \operatorname* { m i n } _ { t , s _ { t } , a _ { t } } \{ d _ { t } ^ { \mu } ( s _ { t } , a _ { t } ) : d _ { t } ^ { \mu } ( s _ { t } , a _ { t } ) > 0 \} ,
+$$
+
+which is decided by the behavior policy $\mu$ and is required by offline learning (see Theorem G.2 in Yin et al. [2021]). In general, $d _ { m }$ is unknown yet we assume $d _ { m }$ is known for the moment and will utilize $d _ { m }$ in our algorithms. Indeed, in Lemma 5.1, we show that estimating $d _ { m }$ up to a multiplicative factor only requires $\widetilde { O } ( 1 / d _ { m } )$ episodes of offline data; replacing the exact $d _ { m }$ with this estimator suffices for our purpose and, importantly, will not affect our downstream sample complexities.
+
+# 3 Variance reduction for offline RL
+
+In this section, we introduce our main algorithm Off-Policy Double Variance Reduction (OPDVR), and present its theoretical guarantee in the finite-horizion non-stationary setting.
+
+# 3.1 Review: variance reduction for RL
+
+We begin by briefly reviewing the variance reduction algorithm for online reinforcement learning, where we have the interactive access to the environment.
+
+Variance reduction (VR) initially emerged as a technique for obtaining fast convergence in large scale optimization problems, for example in the Stochastic Variance Reduction Gradient method (SVRG, [Johnson and Zhang, 2013, Zhang et al., 2013]). This technique is later brought into reinforcement learning for handling policy evaluation [Du et al., 2017] and policy optimization problems [Sidford et al., 2018b,a, Yang and Wang, 2019, Wainwright, 2019, Sidford et al., 2020, Li et al., 2020, Zhang et al., 2020].
+
+In the case of policy optimization, VR is an algorithm that approximately iterating the Bellman optimality equation, using an inner loop that performs an approximate value (or Q-value) iteration using fresh interactive data to estimate $\hat { V } ^ { \star }$ , and an outer loop that performs multiple steps of such iterations to refine the estimates. Concretely, to obtain an reliable $\mathsf { \bar { Q } } _ { t } ( s , a )$ for some step $t \in [ H ]$ , by the Bellman equation $Q _ { t } ( s , a ) = r ( s , a ) + P _ { t } ^ { \top } ( \cdot | s , a ) V _ { t + 1 }$ , we need to estimate $P _ { t } ^ { \top } ( \cdot | s , a ) V _ { t + 1 }$ with sufficient accuracy. VR handles this by decomposing:
+
+$$
+\begin{array} { r } { P _ { t } ^ { \top } ( \cdot | s , a ) V _ { t + 1 } = P _ { t } ^ { \top } ( \cdot | s , a ) ( V _ { t + 1 } - V _ { t + 1 } ^ { \mathrm { i n } } ) + P _ { t } ^ { \top } ( \cdot | s , a ) V _ { t + 1 } ^ { \mathrm { i n } } , } \end{array}
+$$
+
+where $V _ { t + 1 } ^ { \mathrm { i n } }$ is a reference value function obtained from previous calculation (See line 4,13 in the inner loop of Algorithm 1) and $P _ { t } ^ { \top } ( \cdot | s , a ) ( V _ { t + 1 } - V _ { t + 1 } ^ { \mathrm { i n } } )$ , $P _ { t } ^ { \top } ( \cdot | s , a ) V _ { t + 1 } ^ { \mathrm { i n } }$ are estimated separately at different stages. This technique can help in reducing the “effective variance” along the learning process (see Wainwright [2019] Section 2 for a discussion).
+
+In addition, in order to translate the guarantees from learning values to learning policies5, we build on the following “monotonicity property”: For any policy $\pi$ that satisfies the monotonicity condition $V _ { t } \le { \mathcal T } _ { \pi _ { t } } V _ { t + 1 }$ for all $t \in [ H ]$ , the performance of $\pi$ is sandwiched as $V _ { t } \leq V _ { t } ^ { \pi } \leq V _ { t } ^ { \star }$ , i.e. $\pi$ is guaranteed to perform the same or better than $V _ { t }$ . This property is first captured by [Sidford et al., 2018a] (for completeness we provide a proof in Lemma B.1), and later reused by Yang and Wang [2019], Sidford et al. [2020] under different settings. We rely on this property in our offline setting as well for providing policy optimization guarantees.
+
+1: Functional input: Integer valued function $\overline { { \mathbf { m } : \mathbb { R } _ { + } \to \mathbb { N } } }$ . Off-policy estimator $\mathbf { z } _ { t } , \mathbf { g } _ { t }$ in function forms that   
+provides lower confidence bounds (LCB) of the two terms in the bootstrapped value function (2).   
+2: Static input: Initial value function $V _ { t } ^ { ( 0 ) }$ and $\pi _ { t } ^ { ( 0 ) }$ (which satisfy $V _ { t } ^ { ( 0 ) } ~ \leq ~ { \mathcal T } _ { \pi _ { t } ^ { ( 0 ) } } V _ { t + 1 } ^ { ( 0 ) }$ and $V _ { H + 1 } ^ { ( 0 ) } \equiv 0 )$ .   
+A scalar $u ^ { ( 0 ) }$ satisfies $\begin{array} { r } { u ^ { ( 0 ) } \geq \operatorname* { s u p } _ { t } | | V _ { t } ^ { \star } - V _ { t } ^ { ( 0 ) } | | _ { \infty } } \end{array}$ . Outer loop iterations $K$ . Offline dataset $\mathcal { D } =$   
+$\{ \{ s _ { t } ^ { ( i ) } , a _ { t } ^ { ( i ) } , r _ { t } ^ { ( i ) } \} _ { t = 1 } ^ { H } \} _ { i = 1 } ^ { n }$ from $\mu$ as a data-stream where $\begin{array} { r } { n \geq \sum _ { i = 1 } ^ { K } 2 \cdot \mathbf { m } ( u ^ { ( 0 ) } \cdot 2 ^ { - ( i - 1 ) } ) } \end{array}$ .   
+3: ————————4: function QVI-VR $( \mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } ^ { ( 1 : H ) } , V _ { t } ^ { \mathrm { i n } } , \pi ^ { \mathrm { i n } } , \mathbf { z } _ { t } , \mathbf { g } _ { t } , u ^ { \mathrm { i n } } )$ —   
+5: $\diamond$ Computing reference with $\mathcal { D } _ { 1 }$ : Initialize $Q _ { t }  \mathbf { 0 } \in \mathbb { R } ^ { S \times A }$ for $t \in [ H + 1 ]$ .   
+6: for $t \in [ H ]$ and each pair $( s _ { t } , a _ { t } ) \in S \times \mathcal { A }$ do   
+7: $\diamond$ Compute an LCB of $\boldsymbol { P } _ { t } ^ { \top } ( \cdot | s _ { t } , a _ { t } ) \boldsymbol { V } _ { t + 1 } ^ { i n }$ : $z _ { t } \gets \mathbf { z } _ { t } ( \mathcal { D } _ { 1 } , V _ { t + 1 } ^ { \mathrm { i n } } , u ^ { \mathrm { i n } } )$   
+8: end for   
+9:  Value Iterations with D(1:H)2 :   
+10: for $t = H + 1 , H , . . . , 1$ do   
+11: $\diamond$ Update value function: $V _ { t } \gets \operatorname* { m a x } ( V _ { Q _ { t } } , V _ { t } ^ { \mathrm { i n } } )$ .   
+12: $\diamond$ Update policy: $\forall s _ { t }$ , if $V _ { t } ( s _ { t } ) = V _ { t } ^ { \mathrm { i n } } ( s _ { t } )$ , set $\pi _ { t } ( s _ { t } )  \pi _ { t } ^ { \mathrm { i n } } ( s _ { t } )$ ; else set $\pi _ { t } ( s _ { t } )  \pi _ { Q _ { t } } ( s _ { t } )$ .   
+13: if $t \geq 1$ then   
+14: $\diamond$ LCB of $P ^ { \top } ( \cdot | s _ { t - 1 } , a _ { t - 1 } ) [ V _ { t } - V _ { t } ^ { \mathrm { i n } } ]$ : $g _ { t - 1 } \gets \mathbf { g } _ { t - 1 } ( \mathcal { D } _ { 2 } ^ { ( t - 1 ) } , V _ { t } , V _ { t } ^ { \mathrm { i n } } , u ^ { \mathrm { i n } } ) .$   
+15: $\diamond$ Update $Q$ function: $Q _ { t - 1 }  r + z _ { t - 1 } + g _ { t - 1 }$   
+16: end if   
+17: end for   
+18: end function and Return: $V _ { 1 } , . . . , V _ { H }$ and $\pi$   
+19: OUTER LOOP —   
+20: for $i = 1 , . . . , K$ do   
+21: Set $m ^ { ( i ) } \to \mathbf { m } ( u ^ { ( i - 1 ) } )$ ; Get $\mathcal { D } _ { 1 }$ with size $\boldsymbol { m } ^ { ( i ) }$ and $\mathcal { D } _ { 2 } ^ { ( t ) }$ with size $\textstyle \frac { m ^ { ( i ) } } { H } { \left( u ^ { ( i - 1 ) } \right) } ^ { 2 }$ for all $t \in [ H ]$ from   
+the stream $\mathcal { D }$ .   
+22: Ca $V ^ { ( i ) }$ $\tilde { \mathbf { \Gamma } } _ { , \pi ^ { ( i ) } } ^ {  } \dot { \mathbf { \Gamma } } _ {  } \operatorname { Q V I } \mathbf { \mathrm { - } } \mathbf { V R } \big ( \mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } , V ^ { ( i - 1 ) } , \pi ^ { ( i - 1 ) } , \mathbf { z } _ { t } , \mathbf { g } _ { t } , u ^ { ( i - 1 ) } \big ) , \operatorname { S e t } u ^ { ( i ) }  u ^ { ( i - 1 ) } / 2 .$   
+23: end for and Output: $V ^ { ( K ) } , \pi ^ { ( K ) }$
+
+# 3.2 OPDVR: variance reduction for offline RL
+
+We now explain how we design the VR algorithm in the offline setting. Even though our primary novel contribution is for the stationary case (Theorem 4.1), we begin with non-stationary setting for the ease of explaining algorithmic design. We let $\iota : = \log ( H S A / \delta )$ as a short hand.
+
+Prototypical offline VR. We first describe a prototypical version of our offline VR algorithm in Algorithm 1, which we will instantiate with different parameters twice (hence the name“Double”) in each of the three settings of interest.
+
+Algorithm 1 takes estimators $\mathbf { z } _ { t }$ and $\mathbf { g } _ { t }$ that produce lower confidence bounds (LCB) of the two terms in (2) using offline data. Specifically, we assume $\mathbf { z } _ { t } , \mathbf { g } _ { t }$ are both available in function forms in that they take an offline dataset (with an arbitrary size), fixed value function $V _ { t + 1 } , V _ { t + 1 } ^ { \mathrm { i n } }$ and an external scalar input $u$ then return $\boldsymbol { z } _ { t } , \boldsymbol { g } _ { t } \in \mathbb { R } ^ { S \times A }$ . $z _ { t } , g _ { t }$ satisfies that
+
+$$
+\begin{array} { r } { z _ { t } ( s _ { t } , a _ { t } ) \leq P ^ { \top } ( \cdot | s _ { t } , a _ { t } ) V _ { t + 1 } ^ { \mathrm { i n } } , \quad g _ { t } ( s _ { t } , a _ { t } ) \leq P ^ { \top } ( \cdot | s _ { t } , a _ { t } ) [ V _ { t + 1 } - V _ { t + 1 } ^ { \mathrm { i n } } ] , } \end{array}
+$$
+
+uniformly for all $s _ { t } , a _ { t }$ with high probability.
+
+Algorithm 1 then proceeds by taking the input offline dataset as a stream of iid sampled trajectories and use an exponentially increasing-sized batches of independent data to pass in $\mathbf { z } _ { t }$ and $\mathbf { g } _ { t }$ while updating the estimated $Q$ value function by applying the Bellman backup operator except that the update is based on a conservative and variance reduced estimated values. Each inner loop iteration backs up from the last time-step and update all $Q _ { t }$ for $t = H , . . . , 1$ ; and each outer loop iteration passes a new batch of data into the inner loop while ensuring reducing the suboptimality gap from the optimal policy by a factor of 2 in each outer loop iteration, provided that the estimators $\mathbf { z } _ { t } + \mathbf { g } _ { t }$ are increasingly more accurate estimates of (2) as the suboptimality gap gets smaller.
+
+Plug-in estimators and high-confidence LCBs. The estimators $\mathbf { z } _ { t }$ and $\mathbf { g } _ { t }$ we use for the three different settings are provided in Figure 3.2. They are essentially the natural plug-in estimators of $P _ { t } ^ { \top } ( \cdot | s , a ) ( V _ { t + 1 } - V _ { t + 1 } ^ { \mathrm { \bar { i } n } } )$ and $P _ { t } ^ { \top } ( \cdot | s , a ) V _ { t + 1 } ^ { \mathrm { i n } }$ as well as their standard deviation by replacing $P _ { t }$ with $\widehat { P } _ { t }$ except that we use two disjoint splits $\mathcal { D } _ { 1 }$ and $\mathcal { D } _ { 2 } ^ { ( 1 ) } , \ldots , \mathcal { D } _ { 2 } ^ { ( H ) }$ for $\mathbf { z } _ { t }$ and $\mathbf { g } _ { t }$ so they remain
+
+Figure 1: The implementable “plug-in” lower confidence bound estimators $\mathbf { z } _ { t }$ and $\mathbf { g } _ { t }$   
+
+<table><tr><td>Setting</td><td>z(D,V1,u）</td><td></td><td colspan="3">gt(D,Vt+1,V1,）</td></tr><tr><td>Non-stationary</td><td>∑V(）) nst,at 1 V(</td><td rowspan="2">Wtt 1 na 1</td><td rowspan="2">∑j-V</td><td rowspan="2"></td></tr><tr><td>Stationary</td></tr><tr><td>o-Horizon</td><td>𝑛s,a ∑V(s).1s(=sa(s) 𝑛s,a</td><td>Wa 1</td><td>∑jv(s)()f</td><td></td></tr><tr><td>Setting</td><td>(st，at)</td><td>4ov</td><td>et(st,at）</td><td>ft(st，at，u)</td></tr><tr><td>Non-stationary</td><td>∑V 1 nst,at</td><td>nst,at </td><td>L 3/4 +16Vmaxnst,t L L</td><td>4u Wtat L</td></tr><tr><td>Stationary</td><td>∑ ns,a</td><td>4vin 1</td><td>)3/4 4+16Vmax ns, n5,a 3/4</td><td>4u 医</td></tr><tr><td>o-Horizon</td><td>∑V()2</td><td></td><td>+2√6·Vmax 16Vmaxt + 𝑛s,a ns，a 3𝑛𝑠,a</td><td>4u log(2RSA/8） n,a</td></tr></table>
+
+∗ $m , \iota$ are the number of episodes in $\mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } ^ { ( t ) }$ respectively. $\iota$ is a logarithmic factor in $H S A / \delta$ in the finite horizon case and $S A / \delta$ in the infinite horizon cases. ${ n } _ { s { t } , a { t } }$ is the number of times $s _ { t } , a _ { t }$ appears at time $t$ in $\mathcal { D } _ { 1 }$ ; and $n ^ { \prime } ( s _ { t } , a _ { t } )$ is the that for $\mathcal { D } _ { 2 } ^ { ( t ) }$ . In the case when $n _ { s _ { t } , a _ { t } } = 0$ , we simply output 0 for all quantities above.
+
+# Algorithm 2 (OPDVR) Off-Policy Doubled Variance Reduction
+
+input Offline Dataset $\mathcal { D }$ of size $_ n$ as a stream. Target accuracy $\epsilon , \delta$ such that the algorithm does not use up $\mathcal { D }$ .   
+input Estimators $\mathbf { z } _ { t } , \mathbf { g } _ { t }$ in function forms, $m _ { 1 } ^ { \prime } , m _ { 2 } ^ { \prime } , K _ { 1 } , K _ { 2 }$ .
+
+1: $\diamond$ Stage 1. coarse learning: a “warm-up” procedure   
+2: Set $V _ { t } ^ { ( 0 ) } : = \mathbf { 0 }$ and any policy $\pi ^ { ( 0 ) }$ . Set initial $u ^ { ( 0 ) } : = H$ . Set $\mathbf { m } ( u ) = m _ { 1 } ^ { \prime } \log ( 1 6 H S A K _ { 1 } ) / u ^ { 2 }$ .   
+3: Run Algorithm 1 with $\begin{array} { r } { \mathbf { m } , \mathbf { z } _ { t } , \mathbf { g } _ { t } , V _ { t } ^ { ( 0 ) } , \pi ^ { ( 0 ) } , u ^ { ( 0 ) } , K _ { 1 } , \mathcal { D } } \end{array}$ and return $V _ { t } ^ { \mathrm { i } }$ ntermediate, $\pi$ intermediate.   
+4: $\diamond$ Stage 2. fine learning: reduce error to given accuracy   
+5: Reset initial values $V _ { t } ^ { ( 0 ) } : = V _ { t } ^ { \mathrm { i n t e r m e d i a t e } }$ and policy $\pi ^ { ( 0 ) } : = \pi ^ { \mathrm { i n t e r m e d i a t e } }$ . Set $u ^ { ( 0 ) } : = \sqrt { H }$ .   
+6: Reset $\mathbf { m } ( u )$ by replacing $m _ { 1 } ^ { \prime }$ with $m _ { 2 } ^ { \prime }$ , $K _ { 1 }$ with $K _ { 2 }$ .   
+7: Run Algorithm 1 with $\mathbf { m } , \mathbf { z } _ { t } , \mathbf { g } _ { t } , V _ { t } ^ { ( 0 ) } , \pi ^ { ( 0 ) } , u ^ { ( 0 ) } , K _ { 2 } , \mathcal { D }$ and return $V _ { t } ^ { \mathrm { f i n a l } } , \pi ^ { \mathrm { f i n a l } }$ .   
+output $V _ { t } ^ { \mathrm { f i n a l } }$ , $\pi$ final
+
+statistically independent. We also remark that $P _ { t }$ is of dimension $S ^ { 2 } A$ , and we operates in the sparse regime where the number of observed samples could be drastically smaller than the total number of coordinates there are. In other words, $\widehat { P } _ { t }$ will be sparse as most of its coordinates are 0. The saving grace is that when we fix $V _ { t + 1 }$ and $V _ { t + 1 } ^ { \mathrm { i n } }$ , the output space collapsed to only $_ { S A }$ dimensional, which ensures that we have sufficient number of data points to produce accurate estimates.
+
+A key difference from the generative model setting is that these estimators are dependent across $t$ , thus it requires new technical steps to establish the convergence of these estimators as well as putting them together to show that Algorithm 1 works.
+
+The doubling procedure. It turns out that Algorithm 1 alone does not yield a tight sample complexity guarantee, due to its suboptimal dependence on the initial optimality gap $u ^ { ( 0 ) } \geq \operatorname* { s u p } _ { t } \| V _ { t } ^ { \star } - V _ { t } ^ { ( 0 ) } \| _ { \infty }$ (recall $u ^ { ( 0 ) }$ is the initial parameter in the outer loop of Algorithm 1). This is captured in the following (for the non-stationary case):
+
+Proposition 3.1 (Informal version of Lemma B.10). Suppose $\epsilon \in ( 0 , 1 ]$ is the final target accuracy. Algorithm 1 outputs the $\epsilon$ -optimal policy with episode complexity:
+
+$$
+\bullet \tilde { O } ( H ^ { 4 } / d _ { m } \epsilon ^ { 2 } ) , \quad I f u ^ { ( 0 ) } > \sqrt { H } ; \quad \bullet \tilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } ) , \quad I f u ^ { ( 0 ) } \leq \sqrt { H } .
+$$
+
+Proposition 3.1 suggests that Algorithm 1 may have a suboptimal sample complexity when the initial optimality gap $u ^ { ( 0 ) } > \sqrt { H }$ . Unfortunately, this is precisely the case for standard initializations such as $V _ { t } ^ { ( 0 ) } : = \mathbf { 0 }$ , for which we must take $u ^ { ( 0 ) } = H$ . We overcome this issue by designing a two-stage doubling procedure: At stage 1, we use Algorithm 1 to obtain $V _ { t } ^ { 1 }$ ntermediate, $\pi$ intermediate that are $\epsilon ^ { \prime } = \sqrt { H } \epsilon$ accurate; At stage 2, we then use Algorithm 1 again with $V _ { t } ^ { \mathrm { i } }$ ntermediate, $\pi$ intermediate as the input and further reduce the error from $\epsilon ^ { \prime }$ to $\epsilon$ . The main take-away of this doubling procedure is that the episode complexity of both stage is only $\tilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ , therefore the total sample complexity optimality is preserved. The pseudo-code of the two-stage procedure OPDVR is summarized in Algorithm 2.
+
+# 3.3 OPDVR for non-stationary transition settings
+
+We now state our main theoretical guarantee for OPDVR in the finite-horizon non-stationary transition setting.
+
+Theorem 3.2 (OPDVR in episodic non-stationary setting). For the $H$ -horizon non-stationary setting, there exist universal constants $c _ { 1 } , c _ { 2 } , c _ { 3 } > 0$ such that if we set √ $m _ { 1 } ^ { \prime } = c _ { 1 } H ^ { 4 } / d _ { m }$ for Stage 1, $m _ { 2 } ^ { \prime } =$ $c _ { 2 } H ^ { 3 } / d _ { m }$ for Stage 2, set $K _ { 1 } = K _ { 2 } = \log _ { 2 } ( \sqrt { H } / \epsilon )$ , take $\mathbf { g } _ { t }$ and $\mathbf { z } _ { t }$ according to Figure 3.2, then OPDVR (Algorithm 2) with probability $1 - \delta$ outputs an $\epsilon$ -optimal policy $\hat { \pi }$ provided that the number of episodes in the offline data $\mathcal { D }$ exceeds (below can be readily simplified as $\widetilde { O } \left( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } \right) )$ :
+
+$$
+\frac { c _ { 3 } \operatorname* { m a x } [ \frac { m _ { 1 } ^ { \prime } } { H } , m _ { 2 } ^ { \prime } ] } { \epsilon ^ { 2 } } \big ( \iota + \log \log _ { 2 } ( \frac { \sqrt { H } } { \epsilon } ) \big ) \log _ { 2 } ( \frac { \sqrt { H } } { \epsilon } ) ,
+$$
+
+Optimality of sample complexity. Theorem 3.2 shows that our OPDVR algorithm can find an $\epsilon$ -optimal policy with $\widetilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ episodes of offline data. Compared with the sample complexity lower bound $\dot { \Omega ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } ) }$ for offline learning (Theorem G.2. in Yin et al. [2021]), we see that our OPDVR algorithm matches the lower bound up to logarithmic factors. The same rate was achieved previously by the local uniform convergence argument of Yin et al. [2021] under a stronger assumption of full data coverage.
+
+Proof sketch of Theorem 3.2. One challenge in analyzing OPDVR is that the number of statetransition samples are random and dependent. The idea of the proof is to first construct fictitious versions of estimators $\mathbf { z } _ { t }$ (or $\mathbf { g } _ { t }$ ), that uses the empirical plug-in formula only if the event $\begin{array} { r } { E _ { m , t } = \{ n _ { s _ { t } , a _ { t } } > \frac { 1 } { 2 } m \cdot d _ { t } ^ { \mu } ( s _ { t } , \dot { a _ { t } } ) \} } \end{array}$ (or $E _ { l , t } )$ are true, i.e., there are sufficient number of samples at $s _ { t } , a _ { t }$ . When there aren’t, we plug in the ground truth transition kernel $P _ { t }$ instead. These fictitious estimators are not implementable in practice, but they simplify the analysis and are central to our extension of the Variance Reduction framework previously used in the generative model setting [Sidford et al., 2018a] to the offline setting. The following proposition shows that practical implementations (summarized in Figure 3.2) are identical to these fictitious estimators with high probability:
+
+Proposition 3.3 (Summary of Section B.4). Under the condition of Theorem 3.2, we have
+
+$$
+\mathbb { P } \Bigg [ \bigcup _ { \stackrel { i \in [ K _ { 1 } ] , } { t \in [ H ] } } \left( E _ { l , t } ^ { ( i ) c } \cup E _ { m , t } ^ { ( i ) c } \right) \bigcup _ { \stackrel { j \in [ K _ { 2 } ] , } { t \in [ H ] } } \left( E _ { l , t } ^ { ( j ) c } \cup E _ { m , t } ^ { ( j ) c } \right) \Bigg ] \leq \delta / 2 .
+$$
+
+This means with probability $1 - \delta / 2$ , fictitious estimators $\tilde { z } _ { t } , \tilde { g } _ { t } , \tilde { \sigma }$ are all identical to their practical versions (summarized in Figure 3.2).
+
+By Proposition 3.3, it suffices to analyze the performance of OPDVR instantiated with fictitious estimators (60) and (59). Theorem 3.2 then relies on analyzing the the prototypical OPDVR (Algorithm 1) with the fictitious estimators. In particular, both off-policy estimators $z _ { t }$ and $g _ { t }$ use lower confidence update to avoid over-optimism and the max operator in $V _ { t } = \operatorname* { m a x } ( V _ { Q _ { t } } , V _ { t } ^ { \mathrm { i n } } )$ helps prevent pessimism. By doing so the update $V _ { t }$ in Algorithm 1 always satisfies $0 \leq V _ { t } \leq V _ { t } ^ { \star }$ , which is always within valid range. The doubling procedure of Algorithm 2 then first decreases the accuracy to a coarse level√ $\epsilon ^ { \prime } = \sqrt { H } \epsilon$ , and further lowers it to the given accuracy $\epsilon$ . The key technical lemma for achieving optimal dependence in $H$ is Lemma G.5, which bounds the term $\begin{array} { r } { \sum _ { u = t } ^ { \bar { H } } \mathbb { E } _ { s _ { u } , a _ { u } } ^ { \pi ^ { \star } } \left[ \mathrm { V a r } [ V _ { u + 1 } ^ { \star } ( s _ { u + 1 } ) | s _ { u } , \bar { a _ { u } } ] \right] } \end{array}$ by $O ( H ^ { 2 } )$ instead of the naive $O ( H ^ { 3 } )$ . The full proof of Theorem 3.2 can be found in Appendix B.
+
+# 4 OPDVR for stationary transition settings
+
+In this section, we switch gears to the stationary transition setting, in which the transition probabilities are identical at all time steps: $P _ { t } ( s ^ { \prime } | s , a ) : \equiv P ( s ^ { \prime } | s , a )$ . We will consider both the (a) finite-horizon case where each episode is consist of $H$ steps; and (b) the infinite-horizon case where the reward at the $t$ -th step is discounted by $\gamma ^ { t }$ , where $\gamma \in ( 0 , 1 )$ is a discount factor. These settings encompass additional challenges compared with the non-stationary case, as in theory the transition probabilities can now be estimated more accurately due to the shared information across time steps.
+
+# 4.1 Finite-horizon stationary setting
+
+We begin by considering the finite-horizon stationary setting. As this is a special case of the nonstationary setting, Theorem 3.2 implies that OPDVR achieves $\widetilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ sample complexity. However, similar as in online RL [Azar et al., 2017], this result may be potentially loose by an $O ( H )$ factor, as the algorithm does not take into account the stationarity of the transitions. This motivates us to design an algorithm that better leverages the stationarity by aggregating state-action pairs across different time steps (see the second rows of the two tables in Figure 3.2).
+
+Theorem 4.1 (Sample complexity of OPDVR in finite-horizon stationary setting). In the $H$ -horizon stationary transition setting, there exists universal constants $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , c _ { 3 } ^ { \prime }$ such that $i f$ we set $m _ { 1 } ^ { \prime } =$ $c _ { 1 } ^ { \prime } H ^ { 3 } / d _ { m }$ , $m _ { 2 } ^ { \prime } = { c _ { 2 } ^ { \prime } H ^ { 2 } } / { d _ { m } }$ for Stage 1 and 2, set $K _ { 1 } = K _ { 2 } = \log _ { 2 } ( \sqrt { H } / \epsilon )$ , and take $\mathbf { z } _ { t }$ and $\mathbf { g } _ { t }$ according to Figure 3.2, then with probability $1 - \delta$ , Practical OPDVR finds an $\epsilon$ -optimal policy provided that the number of episodes in the offline data $\mathcal { D }$ exceeds (which is of order $\begin{array} { r } { \widetilde { O } \left( \frac { H ^ { 2 } } { d _ { m } \epsilon ^ { 2 } } \right) } \end{array}$ :
+
+$$
+\frac { c _ { 3 } ^ { \prime } \operatorname* { m a x } [ \frac { m _ { 1 } ^ { \prime } } { H } , m _ { 2 } ^ { \prime } ] } { \epsilon ^ { 2 } } \Bigl ( \iota + \log \log _ { 2 } ( \frac { \sqrt { H } } { \epsilon } ) \Bigr ) \log _ { 2 } ( \frac { \sqrt { H } } { \epsilon } ) ,
+$$
+
+Proof sketch. Similar to the proof of the non-stationary case, we start by reducing the problem to a fictitious version that replace the pathological events that happen with lowprobability with ground truth. Then the key challenge is how to analyze the stronger fictitious estimators that pools over steps within each roll-out. We design a martingale $\begin{array} { r l } { X _ { k } } & { { } = } \end{array}$ $\begin{array} { r l r } { \sum _ { i = 1 } ^ { m } \sum _ { u = 1 } ^ { k - 1 } \Big ( V _ { t + 1 } ^ { \mathrm { i n } } \big ( s _ { u + 1 } ^ { ( i ) } \big ) - P ^ { \top } \big ( \cdot | s , a \big ) \bar { V } _ { t + 1 } ^ { \mathrm { i n } } \Big ) \cdot \mathbf { 1 } \big [ s _ { u } ^ { ( i ) } = s , a _ { u } ^ { ( i ) } = a \big ] . } \end{array}$ under the filtration $\begin{array} { r l } { \mathcal { F } _ { k } } & { { } : = } \end{array}$ {s(i)u , a(i)u }u∈[k]i∈[m] for bounding $z _ { t } ( s _ { t } , a _ { t } ) ~ \leq ~ P ^ { \top } ( \cdot | s _ { t } , a _ { t } ) V _ { t + 1 } ^ { \mathrm { i n } }$ . The conditional variance sum $\begin{array} { r } { \sum _ { k = 1 } ^ { H } \operatorname { V a r } \big [ X _ { k + 1 } \ \big | \ \mathcal { F } _ { k } \big ] \ = \ \sum _ { k = 1 } ^ { H } \sum _ { i = 1 } ^ { m } \mathbf { 1 } \Big [ s _ { k } ^ { ( i ) } , a _ { k } ^ { ( i ) } = s , a \Big ] \operatorname { V a r } \Big [ V _ { t + 1 } ^ { \operatorname* { i n } } \ \Big ( s _ { k + 1 } ^ { ( i ) } \Big ) \ \big | \ s _ { k } ^ { ( i ) } , a _ { k } ^ { ( i ) } = s , a \Big ] } \end{array}$ . For stationary case, $s _ { k + 1 } ^ { ( i ) } \sim P ( \cdot | s _ { k } ^ { ( i ) } , a _ { k } ^ { ( i ) } = s , a )$ is irrelevant to time $k$ so above equals $\begin{array} { r } { \sum _ { k = 1 } ^ { H } \sum _ { i = 1 } ^ { m } \mathbf { 1 } \ : \biggl [ s _ { k } ^ { ( i ) } , a _ { k } ^ { ( i ) } = s , a \biggr ] } \end{array}$ $\sigma _ { V _ { t + 1 } ^ { \mathrm { i n } } } ( s , a ) = n _ { s , a } \cdot \sigma _ { V _ { t + 1 } ^ { \mathrm { i n } } } ( s , a ) .$ , where $V _ { t + 1 } ^ { \mathrm { i n } }$ is later replaced by $V _ { t + 1 } ^ { \star }$ and $\begin{array} { r l } { ~ } & { { } \sum _ { t = 1 } ^ { H } \mathbb { E } _ { s , a } ^ { \pi ^ { \star } } [ \sigma _ { V _ { t } ^ { \star } } ( s , a ) ] } \end{array}$ can be bounded by $H ^ { 2 }$ which is tight. In contrast, in non-stationary regime $P _ { t }$ is varying across time so we can only obtain $\begin{array} { r } { \sum _ { k = 1 } ^ { H } \operatorname { V a r } \left[ X _ { k + 1 } ~ | ~ \mathcal { F } _ { k } \right] \leq n _ { s , a } \operatorname* { m a x } _ { t } \sigma _ { V _ { t } ^ { \mathrm { i n } } } ( s , a ) } \end{array}$ , which is later translated into $\begin{array} { r } { \sum _ { t = 1 } ^ { H } \mathbb { E } _ { s , a } ^ { \pi ^ { \star } } [ \operatorname* { m a x } _ { t } \sigma _ { V _ { t } ^ { \star } } ( s , a ) ] } \end{array}$ and are of order $H ^ { 3 }$ tgeneral. To sum, the fact that $P$ is identical is carefully leveraged multiple times for obtaining $\widetilde { O } ( H ^ { 2 } / d _ { m } \epsilon ^ { 2 } )$ rate. The detailed proof of Theorem 4.1 can be found in Appendix C.
+
+Theorem 4.1 encompasses our main technical contribution, as the compact data aggregation among different time steps make analyzing the estimators (61) and (62) knotty due to data-dependence (unlike the non-stationary transition setting where estimators are designed using data at specific time). In particular, we need to fully exploit the property that transition $P$ is identical across different times in a pinpoint way to obtain the $\Dot { H } ^ { 2 }$ dependence in the sample complexity bound.
+
+Improved dependence on $H$ . Theorem 4.1 shows that OPDVR achieves a sample complexity upper bound $\widetilde { O } ( H ^ { 2 } / d _ { m } \epsilon ^ { 2 } )$ in the stationary setting. To the best of our knowledge, this is the first result that achieves an $H ^ { 2 }$ dependence for offline RL with stationary transitions, and improves over the $H ^ { 3 }$ dependence in either the (non-stationary) OPDVR (Theorem 3.2) or the “off-policy evaluation $^ +$ uniform convergence” algorithm of Yin et al. [2021]. We exploit specific properties of $O P D V R$ in our techniques for knocking off a factor of $H$ and there seems to be no direct ways in applying the same techniques in improving the uniform convergence-style results for the stationary-transition setting.
+
+Optimality of $\widetilde { O } ( H ^ { 2 } / d _ { m } \epsilon ^ { 2 } )$ . We accompany Theorem 4.1 by a establishing a sample complexity lower bound for this setting, showing that our algorithm achieves the optimal dependence of all parameters up to logarithmic factors. The proof of Theorem 4.2 builds on modifying the $\widetilde { O } ( H ^ { 3 } / d _ { m } \epsilon ^ { 2 } )$ sample complexity lower bound in Yin et al. [2021], and can be found in Appendix E.
+
+Theorem 4.2 (Information-theoretic lower bound). For all $0 ~ < ~ d _ { m } ~ \le ~ \frac { 1 } { S A }$ , let the family of problem be $\begin{array} { r } { \mathcal { M } _ { d _ { m } } \ : = \ \big \{ ( \mu , M ) \big | \ \operatorname* { m i n } _ { t , s _ { t } , a _ { t } } d _ { t } ^ { \mu } \big ( s _ { t } , a _ { t } \big ) \ \geq \ d _ { m } \big \} . } \end{array}$ . There exists universal constants $c _ { 1 } , c _ { 2 } , c , p$ (with $H , S , A \ \geq \ c _ { 1 }$ and $0 ~ < ~ \epsilon ~ < ~ c _ { 2 } ,$ ) such that when $n ~ \le ~ c H ^ { 2 } / d _ { m } \epsilon ^ { 2 }$ , $\begin{array} { r } { \operatorname* { i n f } _ { v ^ { \pi _ { a l g } } } \operatorname* { s u p } _ { ( \mu , M ) \in \mathcal { M } _ { d _ { m } } } \mathbb { P } _ { \mu , M } \left( v ^ { * } - v ^ { \pi _ { a l g } } \geq \epsilon \right) \geq p } \end{array}$ .
+
+# 4.2 Infinite-horizon discounted setting
+
+Finally, we consider the infinite-horizon discounted setting. The setting is slightly different to the finite horizon case as we adopt the same assumption of Chen and Jiang [2019], Xie and Jiang [2020b] that data $\mathcal { D } = \{ s ^ { ( i ) } , \bar { a } ^ { ( i ) } , r ^ { ( i ) } , s ^ { \prime ( i ) } \} _ { i \in [ n ] }$ are i.i.d off-policy pieces with $( s , a ) \sim d ^ { \mu }$ and $s ^ { \prime } \sim P ( \cdot | s , a )$ . The infinite horizon-versions of OPDVR (Algorithm 3 and 4) are stated in the Appendix due to the space limit.
+
+Theorem 4.3 (Sampe complexity of OPDVR in infinite-horizon discounted setting). Consider Algorithm 4. There are constants $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , c _ { 3 } ^ { \prime }$ , such that if we set $m _ { 1 } ^ { \prime } = O ( ( 1 - \gamma ) ^ { - \bar { 4 } } / d _ { m } ) , m _ { 2 } ^ { \prime } =$ ${ \check { O ( ( 1 - \gamma ) ^ { - 3 } / d _ { m } ) } }$ (see more precise expressions in Lemma $D . 7 ,$ ), $K _ { 1 } = \log _ { 2 } ( ( 1 - \gamma ) ^ { - 1 } / \epsilon ) , K _ { 2 } =$ $\log _ { 2 } ( \sqrt { ( 1 - \gamma ) ^ { - 1 } } / \epsilon )$ , $R = \log ( 4 / \epsilon ( 1 - \gamma ) )$ , and choose LCB estimators z and $\mathbf { g }$ as in Figure 3.2, then with probability $1 - \delta$ , the infinite horizon version of OPDVR (Algorithm 4) outputs an -optimal policy provided that in offline data $\mathcal { D }$ has number of samples exceeding
+
+$$
+\begin{array} { c } { { \frac { c _ { 3 } ^ { \prime } \operatorname* { m a x } [ \frac { m _ { 1 } ^ { \prime } } { ( 1 - \gamma ) ^ { - 1 } } , m _ { 2 } ^ { \prime } ] } { \epsilon ^ { 2 } } \cdot \iota ^ { \prime } = \widetilde O \left[ ( 1 - \gamma ) ^ { - 3 } / d _ { m } \epsilon ^ { 2 } \right] . } } \\ { { \cdot \iota ^ { \prime } : = R \cdot ( \log ( 3 2 ( 1 - \gamma ) ^ { - 1 } R S A / \delta ) + \log \log _ { 2 } ( \sqrt { ( 1 - \gamma ) ^ { - 1 } } / \epsilon ) ) \cdot \log _ { 2 } ( \sqrt { ( 1 - \gamma ) ^ { - 1 } } / \epsilon ) . } } \end{array}
+$$
+
+We note that for the infinite horizon case, the sample-complexity measures the number of steps, thus $( 1 - \gamma ) ^ { - 3 }$ is comparable to the $H ^ { 2 }$ dependence. To the best of our knowledge, Theorem 4.1 and Theorem 4.3 are the first results that achieve $H ^ { 2 }$ , $( 1 - \gamma ) ^ { - 3 }$ dependence in the offline regime respectively for stationary transition and infinite horizon setting, see Table 1.
+
+# 5 Discussions
+
+Estimating $d _ { m }$ . It is worth mentioning that the input of OPDVR depends on unknown system quantity $d _ { m }$ . Nevertheless, $d _ { m }$ is only one-dimensional scalar and thus it is plausible (from a statistical perspective) to leverage standard parameter-tuning tools (e.g. cross validation [Varma and Simon, 2006]) for obtaining a reliable estimate in practice. On the theoretical side, we provide the following result to show plug-in on-policy estimator $\widehat { d } _ { t } ^ { \mu } ( s _ { t } , a _ { t } ) = n _ { s _ { t } , a _ { t } } / n$ and $\begin{array} { r } { \widehat { d } _ { m } : = \operatorname* { m i n } _ { t , s _ { t } , a _ { t } } \{ n _ { s _ { t } , a _ { t } } / n : } \end{array}$ $n _ { s _ { t } , a _ { t } } > 0 \}$ , is sufficient for accurately estimating $d _ { t } ^ { \mu } , d _ { m }$ simultaneously.
+
+Lemma 5.1. For the finite-horizon setting (either stationary or non-stationary), there exists universal constant c, s.t. when $n \geq c \cdot 1 / d _ { m } \cdot \log ( H S A / \delta )$ , then w.p. $1 - \delta$ , we have $\forall t$ , $s _ { t } , a _ { t }$ , $\textstyle { \frac { 1 } { 2 } } d _ { t } ^ { \mu } ( s _ { t } , a _ { t } ) \leq$ $\begin{array} { r } { \widehat { d } _ { t } ^ { \mu } ( s _ { t } , a _ { t } ) \leq \frac { 3 } { 2 } d _ { t } ^ { \mu } ( s _ { t } , a _ { t } ) } \end{array}$ and, in particular, $\textstyle { \frac { 1 } { 2 } } d _ { m } \leq { \widehat { d } } _ { m } \leq { \frac { 3 } { 2 } } d _ { m }$ . See Appendix F.1 for the proof.
+
+Lemma 5.1 ensures one can replace $d _ { t } ^ { \mu }$ by $\widehat { d } _ { t } ^ { \mu }$ $d _ { m }$ by $\widehat { d } _ { m }$ ) in OPDVR and we obtain a fully dataadaptive algorithm. Note that the requirement on $n$ does not affect our near-minimax complexity bound in either Theorem 3.2 and 4.1—we only require $n \approx \tilde { \Theta } ( 1 / d _ { m } )$ additional episodes to estimate $d _ { m }$ and it is of lower order compared to our upper bound $\widetilde { O } ( H ^ { 3 } / d _ { m } )$ or $\widetilde { \cal O } ( H ^ { 2 } / d _ { m } ) )$ .
+
+Computational and memory cost. OPDVR can be implemented as a streaming algorithm that uses only one pass of the dataset. Its computational cost is $\widetilde { O } ( H ^ { 4 } / d _ { m } \epsilon ^ { 2 } )$ — the same as its sample complexity in steps ( $H$ steps is an episode), and the memory cost is $O ( H S A )$ for the episodic case and $O ( S A )$ for the stationary or infinite horizon case. In particular, the double variance reduction technique does not introduce additional overhead beyond constant factors.
+
+Improvement over VR in the generative model setting. This work extends of the variance reduction framework for RL in the generative model setting [Sidford et al., 2018a, Yang and Wang, 2019]. However, we make two improvements to these works. First, the data collected in the offline case are highly dependent (in contrast in the generative model setting each simulator call is independent), and disentangling the dependent structure makes the offline setting inherently more challenging. Second, our doubling mechanism always guarantee the minimax rate with any initialization and the single VR procedure does not have this property (see Appendix F.4 for a more detailed discussion), which could be a critical issue when Sidford et al. [2018a] claims the optimality.
+
+# 6 Conclusion
+
+This paper proposes OPDVR (off-policy double variance reduction), a new variance reduction algorithm for offline reinforcement learning. We show that OPDVR achieves tight sample complexity for offline RL in tabular MDPs; in particular, OPDVR is the first algorithm that achieves the optimal sample complexity for offline RL in the stationary transition setting. On the technical end, we present a sharp analysis under stationary transitions, and use the doubling technique to resolve the initialization dependence in variance reduction, both of which could be of broader interest. There are several interesting next directions. For example, can our analysis shed light on other commonly used algorithms (such as Q-Learning) for offline RL? How can we better deal with insufficient data coverage? Can we go beyond the tabular setting? We would like to leave these as future work.
+
+# Acknowledgment
+
+The authors would like to thank Lin F. Yang for the discussions about [Sidford et al., 2018a]. Ming Yin and Yu-Xiang Wang are partially supported by NSF Awards #2007117 and #2003257. Yu Bai is partly funded through the employment with Salesforce.
+
+# References
+
+Alekh Agarwal, Nan Jiang, and Sham M Kakade. Reinforcement learning: Theory and algorithms. CS Dept., UW Seattle, Seattle, WA, USA, Tech. Rep, 2019.
+
+Alekh Agarwal, Sham Kakade, and Lin F Yang. Model-based reinforcement learning with a generative model is minimax optimal. In Conference on Learning Theory, pages 67–83. PMLR, 2020.
+
+András Antos, Csaba Szepesvári, and Rémi Munos. Fitted q-iteration in continuous action-space mdps. In Advances in neural information processing systems, pages 9–16, 2008a.
+
+András Antos, Csaba Szepesvári, and Rémi Munos. Learning near-optimal policies with bellmanresidual minimization based fitted policy iteration and a single sample path. Machine Learning, 71 (1):89–129, 2008b.
+
+Mohammad Gheshlaghi Azar, Rémi Munos, and Hilbert J Kappen. Minimax pac bounds on the sample complexity of reinforcement learning with a generative model. Machine learning, 91(3): 325–349, 2013.
+
+Mohammad Gheshlaghi Azar, Ian Osband, and Rémi Munos. Minimax regret bounds for reinforcement learning. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 263–272. JMLR. org, 2017.
+
+Qi Cai, Zhuoran Yang, Chi Jin, and Zhaoran Wang. Provably efficient exploration in policy optimization. arXiv preprint arXiv:1912.05830, 2019.
+
+Jinglin Chen and Nan Jiang. Information-theoretic considerations in batch reinforcement learning. arXiv preprint arXiv:1905.00360, 2019.
+
+Herman Chernoff et al. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. The Annals of Mathematical Statistics, 23(4):493–507, 1952.
+
+Fan Chung and Linyuan Lu. Concentration inequalities and martingale inequalities: a survey. Internet Mathematics, 3(1):79–127, 2006.
+
+Christoph Dann and Emma Brunskill. Sample complexity of episodic fixed-horizon reinforcement learning. In Advances in Neural Information Processing Systems, pages 2818–2826, 2015.
+
+Christoph Dann, Lihong Li, Wei Wei, and Emma Brunskill. Policy certificates: Towards accountable reinforcement learning. arXiv preprint arXiv:1811.03056, 2018.
+
+Omar Darwiche Domingues, Pierre Ménard, Emilie Kaufmann, and Michal Valko. Episodic reinforcement learning in finite mdps: Minimax lower bounds revisited. arXiv preprint arXiv:2010.03531, 2020.
+
+Simon S Du, Jianshu Chen, Lihong Li, Lin Xiao, and Dengyong Zhou. Stochastic variance reduction methods for policy evaluation. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 1049–1058. JMLR. org, 2017.
+
+Yaqi Duan and Mengdi Wang. Minimax-optimal off-policy evaluation with linear function approximation. arXiv preprint arXiv:2002.09516, 2020.
+
+Yonathan Efroni, Nadav Merlis, Mohammad Ghavamzadeh, and Shie Mannor. Tight regret bounds for model-based reinforcement learning with greedy policies. In Advances in Neural Information Processing Systems, pages 12203–12213, 2019.
+
+Yihao Feng, Tongzheng Ren, Ziyang Tang, and Qiang Liu. Accountable off-policy evaluation with kernel bellman statistics. In International Conference on Machine Learning, pages 3102–3111. PMLR, 2020.
+
+Nan Jiang and Lihong Li. Doubly robust off-policy value evaluation for reinforcement learning. In Proceedings of the 33rd International Conference on International Conference on Machine Learning-Volume 48, pages 652–661. JMLR. org, 2016.
+
+Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, John Langford, and Robert E Schapire. Contextual decision processes with low bellman rank are pac-learnable. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 1704–1713. JMLR. org, 2017.
+
+Chi Jin, Zeyuan Allen-Zhu, Sebastien Bubeck, and Michael I Jordan. Is q-learning provably efficient? In Advances in Neural Information Processing Systems, pages 4863–4873, 2018.
+
+Ying Jin, Zhuoran Yang, and Zhaoran Wang. Is pessimism provably efficient for offline rl? arXiv preprint arXiv:2012.15085, 2020.
+
+Rie Johnson and Tong Zhang. Accelerating stochastic gradient descent using predictive variance reduction. In Advances in neural information processing systems, pages 315–323, 2013.
+
+Nathan Kallus and Masatoshi Uehara. Double reinforcement learning for efficient off-policy evaluation in markov decision processes. arXiv preprint arXiv:1908.08526, 2019a.
+
+Nathan Kallus and Masatoshi Uehara. Efficiently breaking the curse of horizon: Double reinforcement learning in infinite-horizon processes. arXiv preprint arXiv:1909.05850, 2019b.
+
+Akshay Krishnamurthy, Alekh Agarwal, and John Langford. Pac reinforcement learning with rich observations. In Advances in Neural Information Processing Systems, pages 1840–1848, 2016.
+
+Sascha Lange, Thomas Gabel, and Martin Riedmiller. Batch reinforcement learning. In Reinforcement learning, pages 45–73. Springer, 2012.
+
+Tor Lattimore and Csaba Szepesvari. Learning with good feature representations in bandits and in rl with a generative model. arXiv preprint arXiv:1911.07676, 2019.
+
+Hoang M Le, Cameron Voloshin, and Yisong Yue. Batch policy learning under constraints. arXiv preprint arXiv:1903.08738, 2019.
+
+Sergey Levine, Aviral Kumar, George Tucker, and Justin Fu. Offline reinforcement learning: Tutorial, review, and perspectives on open problems. arXiv preprint arXiv:2005.01643, 2020.
+
+Gen Li, Yuting Wei, Yuejie Chi, Yuantao Gu, and Yuxin Chen. Sample complexity of asynchronous q-learning: Sharper analysis and variance reduction. arXiv preprint arXiv:2006.03041, 2020.
+
+Lihong Li, Rémi Munos, and Csaba Szepesvári. Toward minimax off-policy value estimation. 2015.
+
+Qiang Liu, Lihong Li, Ziyang Tang, and Dengyong Zhou. Breaking the curse of horizon: Infinitehorizon off-policy estimation. In Advances in Neural Information Processing Systems, pages 5361–5371, 2018.
+
+Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Off-policy policy gradient with state distribution correction. In Uncertainty in Artificial Intelligence, 2019.
+
+Yao Liu, Pierre-Luc Bacon, and Emma Brunskill. Understanding the curse of horizon in off-policy evaluation via conditional importance sampling. In International Conference on Machine Learning, pages 6184–6193. PMLR, 2020a.
+
+Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Provably good batch reinforcement learning without great exploration. arXiv preprint arXiv:2007.08202, 2020b.
+
+Rémi Munos. Error bounds for approximate policy iteration. In ICML, volume 3, pages 560–567, 2003.
+
+Aaron Sidford, Mengdi Wang, Xian Wu, Lin Yang, and Yinyu Ye. Near-optimal time and sample complexities for solving markov decision processes with a generative model. In Advances in Neural Information Processing Systems, pages 5186–5196, 2018a.
+
+Aaron Sidford, Mengdi Wang, Xian Wu, and Yinyu Ye. Variance reduced value iteration and faster algorithms for solving markov decision processes. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770–787. SIAM, 2018b.
+
+Aaron Sidford, Mengdi Wang, Lin Yang, and Yinyu Ye. Solving discounted stochastic two-player games with near-optimal time and sample complexity. In International Conference on Artificial Intelligence and Statistics, pages 2992–3002. PMLR, 2020.
+
+Max Simchowitz and Kevin G Jamieson. Non-asymptotic gap-dependent regret bounds for tabular mdps. In Advances in Neural Information Processing Systems, pages 1151–1160, 2019.
+
+Joel Tropp et al. Freedman’s inequality for matrix martingales. Electronic Communications in Probability, 16:262–270, 2011.
+
+Masatoshi Uehara and Nan Jiang. Minimax weight and q-function learning for off-policy evaluation. arXiv preprint arXiv:1910.12809, 2019.
+
+Sudhir Varma and Richard Simon. Bias in error estimation when using cross-validation for model selection. BMC bioinformatics, 7(1):91, 2006.
+
+Martin J Wainwright. Variance-reduced $q$ -learning is minimax optimal. arXiv preprint arXiv:1906.04697, 2019.
+
+Ruosong Wang, Dean P Foster, and Sham M Kakade. What are the statistical limits of offline rl with linear function approximation? arXiv preprint arXiv:2010.11895, 2020.
+
+Tengyang Xie and Nan Jiang. Batch value-function approximation with only realizability. arXiv preprint arXiv:2008.04990, 2020a.
+
+Tengyang Xie and Nan Jiang. $\mathbf { Q } ^ { * }$ approximation schemes for batch reinforcement learning: A theoretical comparison. arXiv preprint arXiv:2003.03924, 2020b.
+
+Tengyang Xie, Yifei Ma, and Yu-Xiang. Wang. Towards optimal off-policy evaluation for reinforcement learning with marginalized importance sampling. In Advances in Neural Information Processing Systems, 2019.
+
+Lin Yang and Mengdi Wang. Sample-optimal parametric q-learning using linearly additive features. In International Conference on Machine Learning, pages 6995–7004, 2019.
+
+Ming Yin and Yu-Xiang Wang. Asymptotically efficient off-policy evaluation for tabular reinforcement learning. In AISTATS-20, 2020.
+
+Ming Yin, Yu Bai, and Yu-Xiang Wang. Near optimal provable uniform convergence in off-policy evaluation for reinforcement learning. In AISTATS-21, 2021.
+
+Andrea Zanette. Exponential lower bounds for batch reinforcement learning: Batch rl can be exponentially harder than online rl. arXiv preprint arXiv:2012.08005, 2020.
+
+Andrea Zanette and Emma Brunskill. Tighter problem-dependent regret bounds in reinforcement learning without domain knowledge using value function bounds. arXiv preprint arXiv:1901.00210, 2019.
+
+Lijun Zhang, Mehrdad Mahdavi, and Rong Jin. Linear convergence with condition number independent access of full gradients. Advances in Neural Information Processing Systems, 26:980–988, 2013.
+
+Zihan Zhang, Yuan Zhou, and Xiangyang Ji. Almost optimal model-free reinforcement learning via reference-advantage decomposition. arXiv preprint arXiv:2004.10019, 2020.

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+        "text": "Near-Optimal Offline Reinforcement Learning via Double Variance Reduction ",
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+        "text": "Ming Yin 1,3, Yu Bai2, and Yu-Xiang Wang1 ",
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+        "type": "text",
+        "text": "1Department of Computer Science, UC Santa Barbara 2Salesforce Research 3Department of Statistics and Applied Probability, UC Santa Barbara ming_yin@ucsb.edu yu.bai@salesforce.com yuxiangw@cs.ucsb.edu ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "We consider the problem of offline reinforcement learning (RL) — a well-motivated setting of RL that aims at policy optimization using only historical data. Despite its wide applicability, theoretical understandings of offline RL, such as its optimal sample complexity, remain largely open even in basic settings such as tabular Markov Decision Processes (MDPs). In this paper, we propose Off-Policy Double Variance Reduction (OPDVR), a new variance reduction based algorithm for offline RL. Our main result shows that OPDVR provably identifies an $\\epsilon$ -optimal policy with $\\widetilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )$ episodes of offline data in the finite-horizon stationary transition setting, where $H$ is the horizon length and $d _ { m }$ is the minimal marginal stateaction distribution induced by the behavior policy. This improves over the best known upper bound by a factor of $H$ . Moreover, we establish an informationtheoretic lower bound of $\\Omega ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )$ which certifies that OPDVR is optimal up to logarithmic factors. Lastly, we show that OPDVR also achieves rate-optimal sample complexity under alternative settings such as the finite-horizon MDPs with non-stationary transitions and the infinite horizon MDPs with discounted rewards. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Offline reinforcement learning (offline RL) aims at learning the near-optimal policy by using a static offline dataset that is collected by a certain behavior policy $\\mu$ [Lange et al., 2012]. As offline RL agent has no access to interact with the environment, it is more widely applicable to problems where online interaction is infeasible, e.g. when trials-and-errors are expensive (robotics, education), risky (autonomous driving) or even unethical (healthcare) [see,e.g., a recent survey Levine et al., 2020]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Despite its practical significance, a precise theoretical understanding of offline RL has been lacking. Previous sample complexity bounds for RL has primarily focused on the online setting [Azar et al., 2017, Jin et al., 2018, Zanette and Brunskill, 2019, Simchowitz and Jamieson, 2019, Efroni et al., 2019, Cai et al., 2019] or the generative model (simulator) setting [Azar et al., 2013, Sidford et al., 2018a,b, Yang and Wang, 2019, Agarwal et al., 2020, Wainwright, 2019, Lattimore and Szepesvari, 2019], both of which assuming interactive access to the environment and not applicable to offline RL. On the other hand, the sample complexity of offline RL remains unsettled even for environments with finitely many state and actions, a.k.a, the tabular MDPs (Markov Decision Processes). One major line of work is concerned with the off-policy evaluation (OPE) problem [Li et al., 2015, Jiang and Li, 2016, Liu et al., 2018, Kallus and Uehara, 2019a,b, Uehara and Jiang, 2019, Xie et al., 2019, Yin and Wang, 2020, Duan and Wang, 2020]. These works provide sample complexity bounds for evaluating the performance of a fixed policy, and do not imply guarantees for policy optimization. Another line of work studies the sample complexity of offline policy optimization in conjunction with function approximation [Chen and Jiang, 2019, Xie and Jiang, 2020b,a, Jin et al., 2020]. These results apply to offline RL with general function classes, but when specialized to the tabular setting, they give rather loose sample complexity bounds with suboptimal dependencies on various parameters 1 ",
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+        "type": "table",
+        "img_path": "images/9f4d58c5db91d883dad056bf7216c439652183f5cae9decac3079defe9752264.jpg",
+        "table_caption": [
+            "Table 1: Comparison of sample complexities for tabular offline RL interpretation. "
+        ],
+        "table_footnote": [
+            "a Number of episodes in the finite horizon setting and number of steps in the infinite horizon. b $\\mathrm { ~ \\it ~ \\beta _ { \\mu } , C , C _ { \\mu } , 1 \\bar { / } } d _ { m }$ are the concentrability-type coefficients that measure the state-action coverage. See Assumption 2.1 and also Section F.2 for discussions. "
+        ],
+        "table_body": "<table><tr><td>Method/Analysis</td><td>Setting</td><td>Assumptions</td><td>Sample complexitya</td></tr><tr><td>BFVT[Xie and Jiang,2020a]</td><td>Oo-horizon</td><td>onlyrealizability+MDP concentrabilityb</td><td>0(1-γ)-8C²/∈4)</td></tr><tr><td>MBS-PI/QI[Liu et al., 2020b]</td><td>oo-horizon</td><td>completeness+bounded den- sity estimation error</td><td>0(1-γ)-8C²/e²)</td></tr><tr><td>Le et al. [2019]</td><td>o-horizon</td><td>Full Concentrability</td><td>0((1-γ)-6βμ/e²）</td></tr><tr><td>FQI [Chen and Jiang, 2019]</td><td>o-horizon</td><td>Full Concentrability</td><td>0((1-γ)-6C/e2)</td></tr><tr><td>MSBO/MABO [Xie and Jiang,2020b]</td><td>oo-horizon</td><td>Full Concentrability</td><td>0((1-γ)-4Cμ/e2)</td></tr><tr><td>OPEMA [Yin et al.,2021]</td><td>H-horizon non-stationary</td><td>Full Concentrability</td><td>O(H3/dm∈²)</td></tr><tr><td>OPDVR (Section 3) OPDVR (Section 4)</td><td>H-horizon non-stationary</td><td>Weak Coverage</td><td>O(H3/dme²)</td></tr><tr><td></td><td>H-horizon stationary</td><td>Weak Coverage</td><td>O(H²/dme²)</td></tr><tr><td>OPDVR (Section 4.2)</td><td>oo-horizon</td><td>Weak Coverage</td><td>0((1-γ)-3/dm∈²)</td></tr></table>",
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+    {
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+        "text": "The recent work of Yin et al. [2021] showed that the optimal sample complexity for finding an $\\epsilon$ -optimal policy in offline RL is $\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ in the finite-horizon non-stationarysetting (with matching upper and lower bounds), where $H$ is the horizon length and $d _ { m }$ is a constant related to the data coverage of the behavior policy in the given MDP. However, the optimal sample complexity in alternative settings such as stationary transition or infinite-horizon settings remains unknown. Further, the $\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ sample complexity is achieved by an off-policy evaluation $^ +$ uniform convergence type algorithm; other more practical algorithms including (stochastic) optimal planning algorithms such as Q-Learning are not well understood in offline RL. ",
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+        "text": "Our Contributions In this paper, we propose an algorithm OPDVR (Off-Policy Doubled Variance Reduction) for offline reinforcement learning based on an extension of the variance reduction technique initiated in [Sidford et al., 2018a, Yang and Wang, 2019]. OPDVR performs stochastic (minibatch style) value iterations using the available offline data, and can be seen as a version of stochastic optimal planning that interpolates value iteration and Q-learning. Our main contributions are summarized as follows. ",
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+        "text": "• We show that OPDVR finds an $\\epsilon$ -optimal policy with high probability using $\\widetilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )$ episodes of offline data (Section 4.1). This improves upon the best known sample complexity by an $H$ factor and to the best of our knowledge is the first that achieves an $\\bar { O } ( H ^ { 2 } )$ horizon dependence, thus separating the stationary case with the non-stationary case for offline RL. We establish a sample (episode) complexity lower bound $\\Omega ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )$ for offline RL in the finite-horizon stationary setting (Theorem 4.2), showing that the sample complexity of OPDVR is optimal up to logarithmic factors. • In the finite-horizon non-stationary setting, and infinite horizon $\\gamma$ -discounted setting, we show that OPDVR achieves $\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ sample (episode) complexity (Section 3) and $\\widetilde { O } ( ( 1 - \\gamma ) ^ { - 3 } / d _ { m } \\epsilon ^ { 2 } )$ sample complexity (Section 4.2) respectively. They are both optimal up to logarithmic factors and our infinite-horizon result improves over the best known results, e.g., those derived for the fitted Q-iteration style algorithms [Xie and Jiang, 2020b]. On the technical end, our algorithm presents a sharp analysis of offline RL with stationary transitions, and, importantly, the use of the doubling technique to resolve the initialization dependence defect which fails to make the original variance reduction algorithm of [Sidford et al., 2018a] to be optimal, see Appendix F.4. Running [Sidford et al., 2018a] may not yield the desired accuracy as they stated and our result is robust in preserving the optimality. ",
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+        "type": "text",
+        "text": "Related work. There is a large and growing body of work on the theory of offline RL and RL in general. We could not hope to provide a comprehensive survey, thus will instead highlight the few prior work that we depend upon on the technical level. The variance reduction techniques that we use in this paper builds upon the work of [Sidford et al., 2018a] in the generative model setting, though it is nontrivial in adapting their techniques to the offline setting; and our two-stage variance reduction appears essential for obtaining optimal rate for $\\epsilon > 1$ (see Section 5 and Appendix F.4 for more detailed discussions). We also used a fictitious estimator technique that originates from the OPE literature[Xie et al., 2019, Yin and Wang, 2020], but extended it to the stationary-transition case, and to the policy optimization problem. As we mentioned earlier, the optimal sample complexity in offline RL in the tabular MDPs with stationary transitions was not settled. The result of [Yin et al., 2021] is optimal in the non-stationary case, but is suboptimal by a factor of $H$ in the stationary case. Our lower bound is a variant of the construction of [Yin et al., 2021] that applies to the stationary case. Other existing work on offline RL has even weaker parameters (sometimes due to their setting being more general, see details in Table 1). We defer more detailed discussion related to the OPE literature and online RL / generative model literature to Appendix A due to space constraint. ",
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+        "text": "2 Preliminaries ",
+        "text_level": 1,
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+        "text": "We consider reinforcement learning problems modeled by finite Markov Decision Processes (MDPs) (we focus on the finite-horizon episodic setting, and defer the infinite-horizon discounted setting to Section 4.2.) An MDP is denoted by a tuple $\\bar { M } = ( S , A , r , T , d _ { 1 } , H )$ , where $s$ and $\\mathcal { A }$ are the state and action spaces with finite cardinality $| S | = S$ and $| { \\mathcal { A } } | = A$ . $P _ { t } : S \\times \\mathcal { A } \\times \\mathcal { S }  [ 0 , 1 ]$ is the transition kernel with $P _ { t } ( s ^ { \\prime } | s , a )$ be the probability of entering state $s ^ { \\prime }$ after taking action $a$ at state $s$ . We consider both the stationary and non-stationary transition setting: The stationary transition setting assumes $P _ { t } \\equiv P$ is identical for all $t \\in [ H ]$ , and the non-stationary transition setting allows $P _ { t }$ to be different for different $t$ . $r _ { t } : S \\times \\mathcal { A }  [ 0 , 1 ]$ is the reward function which we assume to be deterministic2. $d _ { 1 }$ is the initial state distribution, and $H$ is the time horizon. A (non-stationary) policy $\\pi : S \\to \\mathbb { P } _ { A } ^ { H }$ assigns to each state $s _ { t } \\in S$ a distribution over actions at each time $t$ . We use $d _ { t } ^ { \\pi } ( s , a )$ or $d _ { t } ^ { \\pi } ( s )$ to denote the marginal state-action/state distribution induced by policy $\\pi$ at time $t$ i.e. $d _ { t } ^ { \\pi } ( s ) : = \\mathbb { P } ^ { \\pi } ( s _ { t } = s )$ and $d _ { t } ^ { \\pi } \\bar { ( } s , a ) : = \\mathbb { P } ^ { \\pi } ( s _ { t } = s , a _ { t } = a )$ . ",
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+        "type": "text",
+        "text": "$Q$ -value and Bellman operator. For any policy $\\pi$ and any fixed time $t$ , the value function $\\begin{array} { r } { \\mathbb { E } \\left[ \\sum _ { i = t } ^ { H } r _ { i } \\Big | s _ { t } = s \\right] , Q _ { t } ^ { \\pi } ( s , a ) \\ = \\ \\mathbb { E } \\left[ \\sum _ { i = t } ^ { H } r _ { i } \\Big | s _ { t } , a _ { t } = s , a \\right] } \\end{array}$ $V _ { t } ^ { \\pi } ( \\cdot ) ~ \\in ~ \\mathbb { R } ^ { S }$ and $Q$ -value function $\\mathsf { \\bar { \\Psi } } Q _ { t } ^ { \\mathsf { \\bar { \\pi } } } ( \\cdot , \\cdot ) \\ \\in \\ \\mathbb { R } ^ { S \\times A } , \\ \\forall s , a$ . For the ease of exposition, we always is defined as: $V _ { t } ^ { \\pi } ( s ) ~ =$ enumerate $Q ^ { \\pi }$ as a column vector and similarly for $\\bar { P } _ { t } ( \\cdot | s , a )$ . Moreover, for any vector $Q \\in$ $\\mathbb { R } ^ { S \\times A }$ , the induced value vector and policy is defined in the greedy way: $\\forall s _ { t } \\in \\dot { S } , V _ { Q } ( s _ { t } ) =$ $\\operatorname* { m a x } _ { a _ { t } \\in A } Q ( s _ { t } , a _ { t } )$ , $\\pi _ { Q } ( s _ { t } ) = \\operatorname { a r g m a x } _ { a _ { t } \\in \\mathcal { A } } Q ( s _ { t } , a _ { t } )$ . Given an MDP, for any vector $V ~ \\in ~ \\mathbb { R } ^ { s }$ and any deterministic policy $\\pi$ , $\\forall t \\in [ H ]$ the Bellman operator $\\mathcal { T } _ { t } ^ { \\pi } : \\mathbb { R } ^ { S }  \\mathbb { R } ^ { S }$ is defined as: $[ \\mathcal T _ { t } ^ { \\pi } ( V ) ] ( s ) : = r ( s , \\pi _ { t } ( s ) ) + P _ { t } ^ { \\top } ( \\cdot | s , \\pi _ { t } ( s ) ) V$ , and the corresponding Bellman optimality operator $\\mathcal { T } _ { t } : \\mathbb { R } ^ { S }  \\mathbb { R } ^ { S }$ , $\\begin{array} { r } { [ \\mathcal { T } _ { t } ( V ) ] ( s ) : = \\operatorname* { m a x } _ { a \\in \\mathcal { A } } [ r ( s , a ) + P _ { t } ^ { \\top } ( \\cdot | s , a ) V ] } \\end{array}$ . Lastly, for a given value function $V _ { t }$ , we define backup function $z _ { t } ( s _ { t } , a _ { t } ) : = P _ { t } ^ { \\top } ( \\cdot | s _ { t } , a _ { t } ) V _ { t + 1 }$ and the one-step variance as $\\begin{array} { r } { \\sigma _ { V _ { t + 1 } } ( s _ { t } , a _ { t } ) : = \\mathrm { V a r } _ { s _ { t + 1 } } [ \\bar { V } _ { t + 1 } ( s _ { t + 1 } ) | s _ { t } , a _ { t } ] } \\end{array}$ . ",
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+        "text": "Offline learning problem. In this paper we investigate the offline learning problem, where we do not have interactive access to the MDP, and can only observe a static dataset $\\mathcal { D } =$ $\\left\\{ ( s _ { t } ^ { ( i ) } , a _ { t } ^ { ( i ) } , r _ { t } ^ { ( i ) } , s _ { t + 1 } ^ { ( i ) } ) \\right\\} _ { i \\in [ n ] } ^ { t \\in [ H ] }$ .We assume that $\\mathcal { D }$ is obtained by executing a pre-specified behav$\\mu$ ) for , whe $n$ episodes and collecting the trajectories each episode is rendered in the form: $\\tau ^ { ( i ) } \\ = \\ ( s _ { 1 } ^ { ( i ) } , a _ { 1 } ^ { ( i ) } , r _ { 1 } ^ { ( i ) } , \\ldots , s _ { H } ^ { ( i ) } , a _ { H } ^ { ( i ) } , r _ { H } ^ { ( i ) } , s _ { H + 1 } ^ { ( i ) } )$ $s _ { 1 } ^ { ( i ) } \\sim d _ { 1 }$ , $a _ { t } ^ { ( i ) } \\sim \\mu _ { t } ( \\cdot | s _ { t } ^ { ( i ) } )$ , $r _ { t } ^ { ( i ) } = r ( s _ { t } ^ { ( i ) } , a _ { t } ^ { ( i ) } )$ , and $s _ { t + 1 } ^ { ( i ) } \\sim P _ { t } ( \\cdot | s _ { t } ^ { ( i ) } , a _ { t } ^ { ( i ) } )$ . Given the dataset $\\mathcal { D }$ , our goal is to find an $\\epsilon$ -optimal policy $\\pi _ { \\mathrm { o u t } }$ , in the sense that $| | V _ { 1 } ^ { \\pi ^ { \\star } } - V _ { 1 } ^ { \\pi _ { \\mathrm { o u t } } } | | _ { \\infty } < \\epsilon$ . ",
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+        "type": "text",
+        "text": "Assumption on data coverage Due to the curse of distributional shift, efficient offline RL is only possible under certain data coverage properties for the behavior policy $\\mu$ . Throughout this paper we assume the following: ",
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+        "text": "Assumption 2.1 (Weak coverage). The behavior policy $\\mu$ satisfies the following: There exists some optimal policy $\\pi ^ { \\star }$ such that $d _ { t ^ { \\prime } } ^ { \\mu } ( s _ { t ^ { \\prime } } , a _ { t ^ { \\prime } } ) > 0$ if there exists $t < t ^ { \\prime }$ such that ${ d _ { t : t ^ { \\prime } } ^ { \\pi ^ { \\star } } } ( s _ { t ^ { \\prime } } , a _ { t ^ { \\prime } } | s _ { t } , a _ { t } ) > 0$ , where $d _ { t : t ^ { \\prime } } ^ { \\pi ^ { \\star } } \\big ( s _ { t ^ { \\prime } } , a _ { t ^ { \\prime } } | s _ { t } , a _ { t } \\big )$ is the conditional multi-step transition probability from step $t$ to $t ^ { \\prime }$ . ",
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+        "text": "Intuitively, Assumption 2.1 requires $\\mu$ to “cover” certain optimal policy $\\pi ^ { \\star }$ , in the sense that any $s _ { t ^ { \\prime } } , a _ { t ^ { \\prime } }$ is reachable by $\\mu$ if it is attainable from a previous state-action pair by $\\pi ^ { \\star }$ . It is similar to [Liu et al., 2019, Assumption 1]. Note that this is weaker than the standard “concentrability” assumption [Munos, 2003, Le et al., 2019, Chen and Jiang, 2019]: Concentrability defines $\\beta _ { \\mu } : =$ $\\begin{array} { r } { \\operatorname* { s u p } _ { \\pi \\in \\Pi } \\vert \\vert d ^ { \\pi } ( s _ { t } , a _ { t } ) / d ^ { \\mu } ( s _ { t } , a _ { t } ) \\vert \\vert _ { \\infty } < \\infty } \\end{array}$ (cf. [Le et al., 2019, Assumption 1 & Example 4.1]), which requires the sufficient exploration for tabular case3 since we optimize over all policies (see Section F.2 for a discussion). In contrast, our assumption only requires $\\mu$ to “trace” one single optimal policy.4 ",
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+        "type": "text",
+        "text": "With Assumption 2.1, we define ",
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+        "text": "$$\nd _ { m } : = \\operatorname* { m i n } _ { t , s _ { t } , a _ { t } } \\{ d _ { t } ^ { \\mu } ( s _ { t } , a _ { t } ) : d _ { t } ^ { \\mu } ( s _ { t } , a _ { t } ) > 0 \\} ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which is decided by the behavior policy $\\mu$ and is required by offline learning (see Theorem G.2 in Yin et al. [2021]). In general, $d _ { m }$ is unknown yet we assume $d _ { m }$ is known for the moment and will utilize $d _ { m }$ in our algorithms. Indeed, in Lemma 5.1, we show that estimating $d _ { m }$ up to a multiplicative factor only requires $\\widetilde { O } ( 1 / d _ { m } )$ episodes of offline data; replacing the exact $d _ { m }$ with this estimator suffices for our purpose and, importantly, will not affect our downstream sample complexities. ",
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+        "type": "text",
+        "text": "3 Variance reduction for offline RL ",
+        "text_level": 1,
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+        "text": "In this section, we introduce our main algorithm Off-Policy Double Variance Reduction (OPDVR), and present its theoretical guarantee in the finite-horizion non-stationary setting. ",
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+        "text": "3.1 Review: variance reduction for RL ",
+        "text_level": 1,
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+        "text": "We begin by briefly reviewing the variance reduction algorithm for online reinforcement learning, where we have the interactive access to the environment. ",
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+        "text": "Variance reduction (VR) initially emerged as a technique for obtaining fast convergence in large scale optimization problems, for example in the Stochastic Variance Reduction Gradient method (SVRG, [Johnson and Zhang, 2013, Zhang et al., 2013]). This technique is later brought into reinforcement learning for handling policy evaluation [Du et al., 2017] and policy optimization problems [Sidford et al., 2018b,a, Yang and Wang, 2019, Wainwright, 2019, Sidford et al., 2020, Li et al., 2020, Zhang et al., 2020]. ",
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+        "text": "In the case of policy optimization, VR is an algorithm that approximately iterating the Bellman optimality equation, using an inner loop that performs an approximate value (or Q-value) iteration using fresh interactive data to estimate $\\hat { V } ^ { \\star }$ , and an outer loop that performs multiple steps of such iterations to refine the estimates. Concretely, to obtain an reliable $\\mathsf { \\bar { Q } } _ { t } ( s , a )$ for some step $t \\in [ H ]$ , by the Bellman equation $Q _ { t } ( s , a ) = r ( s , a ) + P _ { t } ^ { \\top } ( \\cdot | s , a ) V _ { t + 1 }$ , we need to estimate $P _ { t } ^ { \\top } ( \\cdot | s , a ) V _ { t + 1 }$ with sufficient accuracy. VR handles this by decomposing: ",
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+        "text": "$$\n\\begin{array} { r } { P _ { t } ^ { \\top } ( \\cdot | s , a ) V _ { t + 1 } = P _ { t } ^ { \\top } ( \\cdot | s , a ) ( V _ { t + 1 } - V _ { t + 1 } ^ { \\mathrm { i n } } ) + P _ { t } ^ { \\top } ( \\cdot | s , a ) V _ { t + 1 } ^ { \\mathrm { i n } } , } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "where $V _ { t + 1 } ^ { \\mathrm { i n } }$ is a reference value function obtained from previous calculation (See line 4,13 in the inner loop of Algorithm 1) and $P _ { t } ^ { \\top } ( \\cdot | s , a ) ( V _ { t + 1 } - V _ { t + 1 } ^ { \\mathrm { i n } } )$ , $P _ { t } ^ { \\top } ( \\cdot | s , a ) V _ { t + 1 } ^ { \\mathrm { i n } }$ are estimated separately at different stages. This technique can help in reducing the “effective variance” along the learning process (see Wainwright [2019] Section 2 for a discussion). ",
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+        "text": "In addition, in order to translate the guarantees from learning values to learning policies5, we build on the following “monotonicity property”: For any policy $\\pi$ that satisfies the monotonicity condition $V _ { t } \\le { \\mathcal T } _ { \\pi _ { t } } V _ { t + 1 }$ for all $t \\in [ H ]$ , the performance of $\\pi$ is sandwiched as $V _ { t } \\leq V _ { t } ^ { \\pi } \\leq V _ { t } ^ { \\star }$ , i.e. $\\pi$ is guaranteed to perform the same or better than $V _ { t }$ . This property is first captured by [Sidford et al., 2018a] (for completeness we provide a proof in Lemma B.1), and later reused by Yang and Wang [2019], Sidford et al. [2020] under different settings. We rely on this property in our offline setting as well for providing policy optimization guarantees. ",
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+        "text": "1: Functional input: Integer valued function $\\overline { { \\mathbf { m } : \\mathbb { R } _ { + } \\to \\mathbb { N } } }$ . Off-policy estimator $\\mathbf { z } _ { t } , \\mathbf { g } _ { t }$ in function forms that   \nprovides lower confidence bounds (LCB) of the two terms in the bootstrapped value function (2).   \n2: Static input: Initial value function $V _ { t } ^ { ( 0 ) }$ and $\\pi _ { t } ^ { ( 0 ) }$ (which satisfy $V _ { t } ^ { ( 0 ) } ~ \\leq ~ { \\mathcal T } _ { \\pi _ { t } ^ { ( 0 ) } } V _ { t + 1 } ^ { ( 0 ) }$ and $V _ { H + 1 } ^ { ( 0 ) } \\equiv 0 )$ .   \nA scalar $u ^ { ( 0 ) }$ satisfies $\\begin{array} { r } { u ^ { ( 0 ) } \\geq \\operatorname* { s u p } _ { t } | | V _ { t } ^ { \\star } - V _ { t } ^ { ( 0 ) } | | _ { \\infty } } \\end{array}$ . Outer loop iterations $K$ . Offline dataset $\\mathcal { D } =$   \n$\\{ \\{ s _ { t } ^ { ( i ) } , a _ { t } ^ { ( i ) } , r _ { t } ^ { ( i ) } \\} _ { t = 1 } ^ { H } \\} _ { i = 1 } ^ { n }$ from $\\mu$ as a data-stream where $\\begin{array} { r } { n \\geq \\sum _ { i = 1 } ^ { K } 2 \\cdot \\mathbf { m } ( u ^ { ( 0 ) } \\cdot 2 ^ { - ( i - 1 ) } ) } \\end{array}$ .   \n3: ————————4: function QVI-VR $( \\mathcal { D } _ { 1 } , \\mathcal { D } _ { 2 } ^ { ( 1 : H ) } , V _ { t } ^ { \\mathrm { i n } } , \\pi ^ { \\mathrm { i n } } , \\mathbf { z } _ { t } , \\mathbf { g } _ { t } , u ^ { \\mathrm { i n } } )$ —   \n5: $\\diamond$ Computing reference with $\\mathcal { D } _ { 1 }$ : Initialize $Q _ { t }  \\mathbf { 0 } \\in \\mathbb { R } ^ { S \\times A }$ for $t \\in [ H + 1 ]$ .   \n6: for $t \\in [ H ]$ and each pair $( s _ { t } , a _ { t } ) \\in S \\times \\mathcal { A }$ do   \n7: $\\diamond$ Compute an LCB of $\\boldsymbol { P } _ { t } ^ { \\top } ( \\cdot | s _ { t } , a _ { t } ) \\boldsymbol { V } _ { t + 1 } ^ { i n }$ : $z _ { t } \\gets \\mathbf { z } _ { t } ( \\mathcal { D } _ { 1 } , V _ { t + 1 } ^ { \\mathrm { i n } } , u ^ { \\mathrm { i n } } )$   \n8: end for   \n9: \u0005 Value Iterations with D(1:H)2 :   \n10: for $t = H + 1 , H , . . . , 1$ do   \n11: $\\diamond$ Update value function: $V _ { t } \\gets \\operatorname* { m a x } ( V _ { Q _ { t } } , V _ { t } ^ { \\mathrm { i n } } )$ .   \n12: $\\diamond$ Update policy: $\\forall s _ { t }$ , if $V _ { t } ( s _ { t } ) = V _ { t } ^ { \\mathrm { i n } } ( s _ { t } )$ , set $\\pi _ { t } ( s _ { t } )  \\pi _ { t } ^ { \\mathrm { i n } } ( s _ { t } )$ ; else set $\\pi _ { t } ( s _ { t } )  \\pi _ { Q _ { t } } ( s _ { t } )$ .   \n13: if $t \\geq 1$ then   \n14: $\\diamond$ LCB of $P ^ { \\top } ( \\cdot | s _ { t - 1 } , a _ { t - 1 } ) [ V _ { t } - V _ { t } ^ { \\mathrm { i n } } ]$ : $g _ { t - 1 } \\gets \\mathbf { g } _ { t - 1 } ( \\mathcal { D } _ { 2 } ^ { ( t - 1 ) } , V _ { t } , V _ { t } ^ { \\mathrm { i n } } , u ^ { \\mathrm { i n } } ) .$   \n15: $\\diamond$ Update $Q$ function: $Q _ { t - 1 }  r + z _ { t - 1 } + g _ { t - 1 }$   \n16: end if   \n17: end for   \n18: end function and Return: $V _ { 1 } , . . . , V _ { H }$ and $\\pi$   \n19: OUTER LOOP —   \n20: for $i = 1 , . . . , K$ do   \n21: Set $m ^ { ( i ) } \\to \\mathbf { m } ( u ^ { ( i - 1 ) } )$ ; Get $\\mathcal { D } _ { 1 }$ with size $\\boldsymbol { m } ^ { ( i ) }$ and $\\mathcal { D } _ { 2 } ^ { ( t ) }$ with size $\\textstyle \\frac { m ^ { ( i ) } } { H } { \\left( u ^ { ( i - 1 ) } \\right) } ^ { 2 }$ for all $t \\in [ H ]$ from   \nthe stream $\\mathcal { D }$ .   \n22: Ca $V ^ { ( i ) }$ $\\tilde { \\mathbf { \\Gamma } } _ { , \\pi ^ { ( i ) } } ^ {  } \\dot { \\mathbf { \\Gamma } } _ {  } \\operatorname { Q V I } \\mathbf { \\mathrm { - } } \\mathbf { V R } \\big ( \\mathcal { D } _ { 1 } , \\mathcal { D } _ { 2 } , V ^ { ( i - 1 ) } , \\pi ^ { ( i - 1 ) } , \\mathbf { z } _ { t } , \\mathbf { g } _ { t } , u ^ { ( i - 1 ) } \\big ) , \\operatorname { S e t } u ^ { ( i ) }  u ^ { ( i - 1 ) } / 2 .$   \n23: end for and Output: $V ^ { ( K ) } , \\pi ^ { ( K ) }$ ",
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+        "type": "text",
+        "text": "3.2 OPDVR: variance reduction for offline RL ",
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+        "text": "We now explain how we design the VR algorithm in the offline setting. Even though our primary novel contribution is for the stationary case (Theorem 4.1), we begin with non-stationary setting for the ease of explaining algorithmic design. We let $\\iota : = \\log ( H S A / \\delta )$ as a short hand. ",
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+        "text": "Prototypical offline VR. We first describe a prototypical version of our offline VR algorithm in Algorithm 1, which we will instantiate with different parameters twice (hence the name“Double”) in each of the three settings of interest. ",
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+        "text": "Algorithm 1 takes estimators $\\mathbf { z } _ { t }$ and $\\mathbf { g } _ { t }$ that produce lower confidence bounds (LCB) of the two terms in (2) using offline data. Specifically, we assume $\\mathbf { z } _ { t } , \\mathbf { g } _ { t }$ are both available in function forms in that they take an offline dataset (with an arbitrary size), fixed value function $V _ { t + 1 } , V _ { t + 1 } ^ { \\mathrm { i n } }$ and an external scalar input $u$ then return $\\boldsymbol { z } _ { t } , \\boldsymbol { g } _ { t } \\in \\mathbb { R } ^ { S \\times A }$ . $z _ { t } , g _ { t }$ satisfies that ",
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+        "text": "$$\n\\begin{array} { r } { z _ { t } ( s _ { t } , a _ { t } ) \\leq P ^ { \\top } ( \\cdot | s _ { t } , a _ { t } ) V _ { t + 1 } ^ { \\mathrm { i n } } , \\quad g _ { t } ( s _ { t } , a _ { t } ) \\leq P ^ { \\top } ( \\cdot | s _ { t } , a _ { t } ) [ V _ { t + 1 } - V _ { t + 1 } ^ { \\mathrm { i n } } ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "uniformly for all $s _ { t } , a _ { t }$ with high probability. ",
+        "bbox": [
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+            713,
+            468,
+            728
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 then proceeds by taking the input offline dataset as a stream of iid sampled trajectories and use an exponentially increasing-sized batches of independent data to pass in $\\mathbf { z } _ { t }$ and $\\mathbf { g } _ { t }$ while updating the estimated $Q$ value function by applying the Bellman backup operator except that the update is based on a conservative and variance reduced estimated values. Each inner loop iteration backs up from the last time-step and update all $Q _ { t }$ for $t = H , . . . , 1$ ; and each outer loop iteration passes a new batch of data into the inner loop while ensuring reducing the suboptimality gap from the optimal policy by a factor of 2 in each outer loop iteration, provided that the estimators $\\mathbf { z } _ { t } + \\mathbf { g } _ { t }$ are increasingly more accurate estimates of (2) as the suboptimality gap gets smaller. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Plug-in estimators and high-confidence LCBs. The estimators $\\mathbf { z } _ { t }$ and $\\mathbf { g } _ { t }$ we use for the three different settings are provided in Figure 3.2. They are essentially the natural plug-in estimators of $P _ { t } ^ { \\top } ( \\cdot | s , a ) ( V _ { t + 1 } - V _ { t + 1 } ^ { \\mathrm { \\bar { i } n } } )$ and $P _ { t } ^ { \\top } ( \\cdot | s , a ) V _ { t + 1 } ^ { \\mathrm { i n } }$ as well as their standard deviation by replacing $P _ { t }$ with $\\widehat { P } _ { t }$ except that we use two disjoint splits $\\mathcal { D } _ { 1 }$ and $\\mathcal { D } _ { 2 } ^ { ( 1 ) } , \\ldots , \\mathcal { D } _ { 2 } ^ { ( H ) }$ for $\\mathbf { z } _ { t }$ and $\\mathbf { g } _ { t }$ so they remain ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/ff70f9ecdd7d5c4b814a68484016f26bef4d0d6055e9d72b3a0e8a7893f3dead.jpg",
+        "table_caption": [
+            "Figure 1: The implementable “plug-in” lower confidence bound estimators $\\mathbf { z } _ { t }$ and $\\mathbf { g } _ { t }$ "
+        ],
+        "table_footnote": [
+            "∗ $m , \\iota$ are the number of episodes in $\\mathcal { D } _ { 1 } , \\mathcal { D } _ { 2 } ^ { ( t ) }$ respectively. $\\iota$ is a logarithmic factor in $H S A / \\delta$ in the finite horizon case and $S A / \\delta$ in the infinite horizon cases. ${ n } _ { s { t } , a { t } }$ is the number of times $s _ { t } , a _ { t }$ appears at time $t$ in $\\mathcal { D } _ { 1 }$ ; and $n ^ { \\prime } ( s _ { t } , a _ { t } )$ is the that for $\\mathcal { D } _ { 2 } ^ { ( t ) }$ . In the case when $n _ { s _ { t } , a _ { t } } = 0$ , we simply output 0 for all quantities above. ",
+            ""
+        ],
+        "table_body": "<table><tr><td>Setting</td><td>z(D,V1,u）</td><td></td><td colspan=\"3\">gt(D,Vt+1,V1,）</td></tr><tr><td>Non-stationary</td><td>∑V(）) nst,at 1 V(</td><td rowspan=\"2\">Wtt 1 na 1</td><td rowspan=\"2\">∑j-V</td><td rowspan=\"2\"></td></tr><tr><td>Stationary</td></tr><tr><td>o-Horizon</td><td>𝑛s,a ∑V(s).1s(=sa(s) 𝑛s,a</td><td>Wa 1</td><td>∑jv(s)()f</td><td></td></tr><tr><td>Setting</td><td>(st，at)</td><td>4ov</td><td>et(st,at）</td><td>ft(st，at，u)</td></tr><tr><td>Non-stationary</td><td>∑V 1 nst,at</td><td>nst,at </td><td>L 3/4 +16Vmaxnst,t L L</td><td>4u Wtat L</td></tr><tr><td>Stationary</td><td>∑ ns,a</td><td>4vin 1</td><td>)3/4 4+16Vmax ns, n5,a 3/4</td><td>4u 医</td></tr><tr><td>o-Horizon</td><td>∑V()2</td><td></td><td>+2√6·Vmax 16Vmaxt + 𝑛s,a ns，a 3𝑛𝑠,a</td><td>4u log(2RSA/8） n,a</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 (OPDVR) Off-Policy Doubled Variance Reduction ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            299,
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+            314
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "input Offline Dataset $\\mathcal { D }$ of size $_ n$ as a stream. Target accuracy $\\epsilon , \\delta$ such that the algorithm does not use up $\\mathcal { D }$ .   \ninput Estimators $\\mathbf { z } _ { t } , \\mathbf { g } _ { t }$ in function forms, $m _ { 1 } ^ { \\prime } , m _ { 2 } ^ { \\prime } , K _ { 1 } , K _ { 2 }$ . ",
+        "bbox": [
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+            316,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: $\\diamond$ Stage 1. coarse learning: a “warm-up” procedure   \n2: Set $V _ { t } ^ { ( 0 ) } : = \\mathbf { 0 }$ and any policy $\\pi ^ { ( 0 ) }$ . Set initial $u ^ { ( 0 ) } : = H$ . Set $\\mathbf { m } ( u ) = m _ { 1 } ^ { \\prime } \\log ( 1 6 H S A K _ { 1 } ) / u ^ { 2 }$ .   \n3: Run Algorithm 1 with $\\begin{array} { r } { \\mathbf { m } , \\mathbf { z } _ { t } , \\mathbf { g } _ { t } , V _ { t } ^ { ( 0 ) } , \\pi ^ { ( 0 ) } , u ^ { ( 0 ) } , K _ { 1 } , \\mathcal { D } } \\end{array}$ and return $V _ { t } ^ { \\mathrm { i } }$ ntermediate, $\\pi$ intermediate.   \n4: $\\diamond$ Stage 2. fine learning: reduce error to given accuracy   \n5: Reset initial values $V _ { t } ^ { ( 0 ) } : = V _ { t } ^ { \\mathrm { i n t e r m e d i a t e } }$ and policy $\\pi ^ { ( 0 ) } : = \\pi ^ { \\mathrm { i n t e r m e d i a t e } }$ . Set $u ^ { ( 0 ) } : = \\sqrt { H }$ .   \n6: Reset $\\mathbf { m } ( u )$ by replacing $m _ { 1 } ^ { \\prime }$ with $m _ { 2 } ^ { \\prime }$ , $K _ { 1 }$ with $K _ { 2 }$ .   \n7: Run Algorithm 1 with $\\mathbf { m } , \\mathbf { z } _ { t } , \\mathbf { g } _ { t } , V _ { t } ^ { ( 0 ) } , \\pi ^ { ( 0 ) } , u ^ { ( 0 ) } , K _ { 2 } , \\mathcal { D }$ and return $V _ { t } ^ { \\mathrm { f i n a l } } , \\pi ^ { \\mathrm { f i n a l } }$ .   \noutput $V _ { t } ^ { \\mathrm { f i n a l } }$ , $\\pi$ final ",
+        "bbox": [
+            178,
+            343,
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+            452
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "statistically independent. We also remark that $P _ { t }$ is of dimension $S ^ { 2 } A$ , and we operates in the sparse regime where the number of observed samples could be drastically smaller than the total number of coordinates there are. In other words, $\\widehat { P } _ { t }$ will be sparse as most of its coordinates are 0. The saving grace is that when we fix $V _ { t + 1 }$ and $V _ { t + 1 } ^ { \\mathrm { i n } }$ , the output space collapsed to only $_ { S A }$ dimensional, which ensures that we have sufficient number of data points to produce accurate estimates. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "A key difference from the generative model setting is that these estimators are dependent across $t$ , thus it requires new technical steps to establish the convergence of these estimators as well as putting them together to show that Algorithm 1 works. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The doubling procedure. It turns out that Algorithm 1 alone does not yield a tight sample complexity guarantee, due to its suboptimal dependence on the initial optimality gap $u ^ { ( 0 ) } \\geq \\operatorname* { s u p } _ { t } \\| V _ { t } ^ { \\star } - V _ { t } ^ { ( 0 ) } \\| _ { \\infty }$ (recall $u ^ { ( 0 ) }$ is the initial parameter in the outer loop of Algorithm 1). This is captured in the following (for the non-stationary case): ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.1 (Informal version of Lemma B.10). Suppose $\\epsilon \\in ( 0 , 1 ]$ is the final target accuracy. Algorithm 1 outputs the $\\epsilon$ -optimal policy with episode complexity: ",
+        "bbox": [
+            171,
+            690,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aae7b592d56efa8bf373f9d12c2e319eee0a5190c0b910b88f391f42e85d364f.jpg",
+        "text": "$$\n\\bullet \\tilde { O } ( H ^ { 4 } / d _ { m } \\epsilon ^ { 2 } ) , \\quad I f u ^ { ( 0 ) } > \\sqrt { H } ; \\quad \\bullet \\tilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } ) , \\quad I f u ^ { ( 0 ) } \\leq \\sqrt { H } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            263,
+            734,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.1 suggests that Algorithm 1 may have a suboptimal sample complexity when the initial optimality gap $u ^ { ( 0 ) } > \\sqrt { H }$ . Unfortunately, this is precisely the case for standard initializations such as $V _ { t } ^ { ( 0 ) } : = \\mathbf { 0 }$ , for which we must take $u ^ { ( 0 ) } = H$ . We overcome this issue by designing a two-stage doubling procedure: At stage 1, we use Algorithm 1 to obtain $V _ { t } ^ { 1 }$ ntermediate, $\\pi$ intermediate that are $\\epsilon ^ { \\prime } = \\sqrt { H } \\epsilon$ accurate; At stage 2, we then use Algorithm 1 again with $V _ { t } ^ { \\mathrm { i } }$ ntermediate, $\\pi$ intermediate as the input and further reduce the error from $\\epsilon ^ { \\prime }$ to $\\epsilon$ . The main take-away of this doubling procedure is that the episode complexity of both stage is only $\\tilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ , therefore the total sample complexity optimality is preserved. The pseudo-code of the two-stage procedure OPDVR is summarized in Algorithm 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 OPDVR for non-stationary transition settings ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            90,
+            526,
+            106
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now state our main theoretical guarantee for OPDVR in the finite-horizon non-stationary transition setting. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2 (OPDVR in episodic non-stationary setting). For the $H$ -horizon non-stationary setting, there exist universal constants $c _ { 1 } , c _ { 2 } , c _ { 3 } > 0$ such that if we set √ $m _ { 1 } ^ { \\prime } = c _ { 1 } H ^ { 4 } / d _ { m }$ for Stage 1, $m _ { 2 } ^ { \\prime } =$ $c _ { 2 } H ^ { 3 } / d _ { m }$ for Stage 2, set $K _ { 1 } = K _ { 2 } = \\log _ { 2 } ( \\sqrt { H } / \\epsilon )$ , take $\\mathbf { g } _ { t }$ and $\\mathbf { z } _ { t }$ according to Figure 3.2, then OPDVR (Algorithm 2) with probability $1 - \\delta$ outputs an $\\epsilon$ -optimal policy $\\hat { \\pi }$ provided that the number of episodes in the offline data $\\mathcal { D }$ exceeds (below can be readily simplified as $\\widetilde { O } \\left( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } \\right) )$ : ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/66253805447d94bb8fc3cee8b21bc6bb3a48331c67cc660a54439f51515dcf17.jpg",
+        "text": "$$\n\\frac { c _ { 3 } \\operatorname* { m a x } [ \\frac { m _ { 1 } ^ { \\prime } } { H } , m _ { 2 } ^ { \\prime } ] } { \\epsilon ^ { 2 } } \\big ( \\iota + \\log \\log _ { 2 } ( \\frac { \\sqrt { H } } { \\epsilon } ) \\big ) \\log _ { 2 } ( \\frac { \\sqrt { H } } { \\epsilon } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Optimality of sample complexity. Theorem 3.2 shows that our OPDVR algorithm can find an $\\epsilon$ -optimal policy with $\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ episodes of offline data. Compared with the sample complexity lower bound $\\dot { \\Omega ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } ) }$ for offline learning (Theorem G.2. in Yin et al. [2021]), we see that our OPDVR algorithm matches the lower bound up to logarithmic factors. The same rate was achieved previously by the local uniform convergence argument of Yin et al. [2021] under a stronger assumption of full data coverage. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proof sketch of Theorem 3.2. One challenge in analyzing OPDVR is that the number of statetransition samples are random and dependent. The idea of the proof is to first construct fictitious versions of estimators $\\mathbf { z } _ { t }$ (or $\\mathbf { g } _ { t }$ ), that uses the empirical plug-in formula only if the event $\\begin{array} { r } { E _ { m , t } = \\{ n _ { s _ { t } , a _ { t } } > \\frac { 1 } { 2 } m \\cdot d _ { t } ^ { \\mu } ( s _ { t } , \\dot { a _ { t } } ) \\} } \\end{array}$ (or $E _ { l , t } )$ are true, i.e., there are sufficient number of samples at $s _ { t } , a _ { t }$ . When there aren’t, we plug in the ground truth transition kernel $P _ { t }$ instead. These fictitious estimators are not implementable in practice, but they simplify the analysis and are central to our extension of the Variance Reduction framework previously used in the generative model setting [Sidford et al., 2018a] to the offline setting. The following proposition shows that practical implementations (summarized in Figure 3.2) are identical to these fictitious estimators with high probability: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.3 (Summary of Section B.4). Under the condition of Theorem 3.2, we have ",
+        "bbox": [
+            171,
+            520,
+            764,
+            535
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/757a682cc941ebfe6bdb470b88ec05647dfd1909f17d5f032cf54af5958a387e.jpg",
+        "text": "$$\n\\mathbb { P } \\Bigg [ \\bigcup _ { \\stackrel { i \\in [ K _ { 1 } ] , } { t \\in [ H ] } } \\left( E _ { l , t } ^ { ( i ) c } \\cup E _ { m , t } ^ { ( i ) c } \\right) \\bigcup _ { \\stackrel { j \\in [ K _ { 2 } ] , } { t \\in [ H ] } } \\left( E _ { l , t } ^ { ( j ) c } \\cup E _ { m , t } ^ { ( j ) c } \\right) \\Bigg ] \\leq \\delta / 2 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            544,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This means with probability $1 - \\delta / 2$ , fictitious estimators $\\tilde { z } _ { t } , \\tilde { g } _ { t } , \\tilde { \\sigma }$ are all identical to their practical versions (summarized in Figure 3.2). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "By Proposition 3.3, it suffices to analyze the performance of OPDVR instantiated with fictitious estimators (60) and (59). Theorem 3.2 then relies on analyzing the the prototypical OPDVR (Algorithm 1) with the fictitious estimators. In particular, both off-policy estimators $z _ { t }$ and $g _ { t }$ use lower confidence update to avoid over-optimism and the max operator in $V _ { t } = \\operatorname* { m a x } ( V _ { Q _ { t } } , V _ { t } ^ { \\mathrm { i n } } )$ helps prevent pessimism. By doing so the update $V _ { t }$ in Algorithm 1 always satisfies $0 \\leq V _ { t } \\leq V _ { t } ^ { \\star }$ , which is always within valid range. The doubling procedure of Algorithm 2 then first decreases the accuracy to a coarse level√ $\\epsilon ^ { \\prime } = \\sqrt { H } \\epsilon$ , and further lowers it to the given accuracy $\\epsilon$ . The key technical lemma for achieving optimal dependence in $H$ is Lemma G.5, which bounds the term $\\begin{array} { r } { \\sum _ { u = t } ^ { \\bar { H } } \\mathbb { E } _ { s _ { u } , a _ { u } } ^ { \\pi ^ { \\star } } \\left[ \\mathrm { V a r } [ V _ { u + 1 } ^ { \\star } ( s _ { u + 1 } ) | s _ { u } , \\bar { a _ { u } } ] \\right] } \\end{array}$ by $O ( H ^ { 2 } )$ instead of the naive $O ( H ^ { 3 } )$ . The full proof of Theorem 3.2 can be found in Appendix B. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 OPDVR for stationary transition settings ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            795,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we switch gears to the stationary transition setting, in which the transition probabilities are identical at all time steps: $P _ { t } ( s ^ { \\prime } | s , a ) : \\equiv P ( s ^ { \\prime } | s , a )$ . We will consider both the (a) finite-horizon case where each episode is consist of $H$ steps; and (b) the infinite-horizon case where the reward at the $t$ -th step is discounted by $\\gamma ^ { t }$ , where $\\gamma \\in ( 0 , 1 )$ is a discount factor. These settings encompass additional challenges compared with the non-stationary case, as in theory the transition probabilities can now be estimated more accurately due to the shared information across time steps. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Finite-horizon stationary setting ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We begin by considering the finite-horizon stationary setting. As this is a special case of the nonstationary setting, Theorem 3.2 implies that OPDVR achieves $\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ sample complexity. However, similar as in online RL [Azar et al., 2017], this result may be potentially loose by an $O ( H )$ factor, as the algorithm does not take into account the stationarity of the transitions. This motivates us to design an algorithm that better leverages the stationarity by aggregating state-action pairs across different time steps (see the second rows of the two tables in Figure 3.2). ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.1 (Sample complexity of OPDVR in finite-horizon stationary setting). In the $H$ -horizon stationary transition setting, there exists universal constants $c _ { 1 } ^ { \\prime } , c _ { 2 } ^ { \\prime } , c _ { 3 } ^ { \\prime }$ such that $i f$ we set $m _ { 1 } ^ { \\prime } =$ $c _ { 1 } ^ { \\prime } H ^ { 3 } / d _ { m }$ , $m _ { 2 } ^ { \\prime } = { c _ { 2 } ^ { \\prime } H ^ { 2 } } / { d _ { m } }$ for Stage 1 and 2, set $K _ { 1 } = K _ { 2 } = \\log _ { 2 } ( \\sqrt { H } / \\epsilon )$ , and take $\\mathbf { z } _ { t }$ and $\\mathbf { g } _ { t }$ according to Figure 3.2, then with probability $1 - \\delta$ , Practical OPDVR finds an $\\epsilon$ -optimal policy provided that the number of episodes in the offline data $\\mathcal { D }$ exceeds (which is of order $\\begin{array} { r } { \\widetilde { O } \\left( \\frac { H ^ { 2 } } { d _ { m } \\epsilon ^ { 2 } } \\right) } \\end{array}$ : ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c87071c7e36b78e824455e9e94dffd55c1921c359ba444198a4e126419f2fbbb.jpg",
+        "text": "$$\n\\frac { c _ { 3 } ^ { \\prime } \\operatorname* { m a x } [ \\frac { m _ { 1 } ^ { \\prime } } { H } , m _ { 2 } ^ { \\prime } ] } { \\epsilon ^ { 2 } } \\Bigl ( \\iota + \\log \\log _ { 2 } ( \\frac { \\sqrt { H } } { \\epsilon } ) \\Bigr ) \\log _ { 2 } ( \\frac { \\sqrt { H } } { \\epsilon } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Proof sketch. Similar to the proof of the non-stationary case, we start by reducing the problem to a fictitious version that replace the pathological events that happen with lowprobability with ground truth. Then the key challenge is how to analyze the stronger fictitious estimators that pools over steps within each roll-out. We design a martingale $\\begin{array} { r l } { X _ { k } } & { { } = } \\end{array}$ $\\begin{array} { r l r } { \\sum _ { i = 1 } ^ { m } \\sum _ { u = 1 } ^ { k - 1 } \\Big ( V _ { t + 1 } ^ { \\mathrm { i n } } \\big ( s _ { u + 1 } ^ { ( i ) } \\big ) - P ^ { \\top } \\big ( \\cdot | s , a \\big ) \\bar { V } _ { t + 1 } ^ { \\mathrm { i n } } \\Big ) \\cdot \\mathbf { 1 } \\big [ s _ { u } ^ { ( i ) } = s , a _ { u } ^ { ( i ) } = a \\big ] . } \\end{array}$ under the filtration $\\begin{array} { r l } { \\mathcal { F } _ { k } } & { { } : = } \\end{array}$ {s(i)u , a(i)u }u∈[k]i∈[m] for bounding $z _ { t } ( s _ { t } , a _ { t } ) ~ \\leq ~ P ^ { \\top } ( \\cdot | s _ { t } , a _ { t } ) V _ { t + 1 } ^ { \\mathrm { i n } }$ . The conditional variance sum $\\begin{array} { r } { \\sum _ { k = 1 } ^ { H } \\operatorname { V a r } \\big [ X _ { k + 1 } \\ \\big | \\ \\mathcal { F } _ { k } \\big ] \\ = \\ \\sum _ { k = 1 } ^ { H } \\sum _ { i = 1 } ^ { m } \\mathbf { 1 } \\Big [ s _ { k } ^ { ( i ) } , a _ { k } ^ { ( i ) } = s , a \\Big ] \\operatorname { V a r } \\Big [ V _ { t + 1 } ^ { \\operatorname* { i n } } \\ \\Big ( s _ { k + 1 } ^ { ( i ) } \\Big ) \\ \\big | \\ s _ { k } ^ { ( i ) } , a _ { k } ^ { ( i ) } = s , a \\Big ] } \\end{array}$ . For stationary case, $s _ { k + 1 } ^ { ( i ) } \\sim P ( \\cdot | s _ { k } ^ { ( i ) } , a _ { k } ^ { ( i ) } = s , a )$ is irrelevant to time $k$ so above equals $\\begin{array} { r } { \\sum _ { k = 1 } ^ { H } \\sum _ { i = 1 } ^ { m } \\mathbf { 1 } \\ : \\biggl [ s _ { k } ^ { ( i ) } , a _ { k } ^ { ( i ) } = s , a \\biggr ] } \\end{array}$ $\\sigma _ { V _ { t + 1 } ^ { \\mathrm { i n } } } ( s , a ) = n _ { s , a } \\cdot \\sigma _ { V _ { t + 1 } ^ { \\mathrm { i n } } } ( s , a ) .$ , where $V _ { t + 1 } ^ { \\mathrm { i n } }$ is later replaced by $V _ { t + 1 } ^ { \\star }$ and $\\begin{array} { r l } { ~ } & { { } \\sum _ { t = 1 } ^ { H } \\mathbb { E } _ { s , a } ^ { \\pi ^ { \\star } } [ \\sigma _ { V _ { t } ^ { \\star } } ( s , a ) ] } \\end{array}$ can be bounded by $H ^ { 2 }$ which is tight. In contrast, in non-stationary regime $P _ { t }$ is varying across time so we can only obtain $\\begin{array} { r } { \\sum _ { k = 1 } ^ { H } \\operatorname { V a r } \\left[ X _ { k + 1 } ~ | ~ \\mathcal { F } _ { k } \\right] \\leq n _ { s , a } \\operatorname* { m a x } _ { t } \\sigma _ { V _ { t } ^ { \\mathrm { i n } } } ( s , a ) } \\end{array}$ , which is later translated into $\\begin{array} { r } { \\sum _ { t = 1 } ^ { H } \\mathbb { E } _ { s , a } ^ { \\pi ^ { \\star } } [ \\operatorname* { m a x } _ { t } \\sigma _ { V _ { t } ^ { \\star } } ( s , a ) ] } \\end{array}$ and are of order $H ^ { 3 }$ tgeneral. To sum, the fact that $P$ is identical is carefully leveraged multiple times for obtaining $\\widetilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )$ rate. The detailed proof of Theorem 4.1 can be found in Appendix C. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.1 encompasses our main technical contribution, as the compact data aggregation among different time steps make analyzing the estimators (61) and (62) knotty due to data-dependence (unlike the non-stationary transition setting where estimators are designed using data at specific time). In particular, we need to fully exploit the property that transition $P$ is identical across different times in a pinpoint way to obtain the $\\Dot { H } ^ { 2 }$ dependence in the sample complexity bound. ",
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+    {
+        "type": "text",
+        "text": "Improved dependence on $H$ . Theorem 4.1 shows that OPDVR achieves a sample complexity upper bound $\\widetilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )$ in the stationary setting. To the best of our knowledge, this is the first result that achieves an $H ^ { 2 }$ dependence for offline RL with stationary transitions, and improves over the $H ^ { 3 }$ dependence in either the (non-stationary) OPDVR (Theorem 3.2) or the “off-policy evaluation $^ +$ uniform convergence” algorithm of Yin et al. [2021]. We exploit specific properties of $O P D V R$ in our techniques for knocking off a factor of $H$ and there seems to be no direct ways in applying the same techniques in improving the uniform convergence-style results for the stationary-transition setting. ",
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+    {
+        "type": "text",
+        "text": "Optimality of $\\widetilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )$ . We accompany Theorem 4.1 by a establishing a sample complexity lower bound for this setting, showing that our algorithm achieves the optimal dependence of all parameters up to logarithmic factors. The proof of Theorem 4.2 builds on modifying the $\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ sample complexity lower bound in Yin et al. [2021], and can be found in Appendix E. ",
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+        "type": "text",
+        "text": "Theorem 4.2 (Information-theoretic lower bound). For all $0 ~ < ~ d _ { m } ~ \\le ~ \\frac { 1 } { S A }$ , let the family of problem be $\\begin{array} { r } { \\mathcal { M } _ { d _ { m } } \\ : = \\ \\big \\{ ( \\mu , M ) \\big | \\ \\operatorname* { m i n } _ { t , s _ { t } , a _ { t } } d _ { t } ^ { \\mu } \\big ( s _ { t } , a _ { t } \\big ) \\ \\geq \\ d _ { m } \\big \\} . } \\end{array}$ . There exists universal constants $c _ { 1 } , c _ { 2 } , c , p$ (with $H , S , A \\ \\geq \\ c _ { 1 }$ and $0 ~ < ~ \\epsilon ~ < ~ c _ { 2 } ,$ ) such that when $n ~ \\le ~ c H ^ { 2 } / d _ { m } \\epsilon ^ { 2 }$ , $\\begin{array} { r } { \\operatorname* { i n f } _ { v ^ { \\pi _ { a l g } } } \\operatorname* { s u p } _ { ( \\mu , M ) \\in \\mathcal { M } _ { d _ { m } } } \\mathbb { P } _ { \\mu , M } \\left( v ^ { * } - v ^ { \\pi _ { a l g } } \\geq \\epsilon \\right) \\geq p } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 Infinite-horizon discounted setting ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finally, we consider the infinite-horizon discounted setting. The setting is slightly different to the finite horizon case as we adopt the same assumption of Chen and Jiang [2019], Xie and Jiang [2020b] that data $\\mathcal { D } = \\{ s ^ { ( i ) } , \\bar { a } ^ { ( i ) } , r ^ { ( i ) } , s ^ { \\prime ( i ) } \\} _ { i \\in [ n ] }$ are i.i.d off-policy pieces with $( s , a ) \\sim d ^ { \\mu }$ and $s ^ { \\prime } \\sim P ( \\cdot | s , a )$ . The infinite horizon-versions of OPDVR (Algorithm 3 and 4) are stated in the Appendix due to the space limit. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.3 (Sampe complexity of OPDVR in infinite-horizon discounted setting). Consider Algorithm 4. There are constants $c _ { 1 } ^ { \\prime } , c _ { 2 } ^ { \\prime } , c _ { 3 } ^ { \\prime }$ , such that if we set $m _ { 1 } ^ { \\prime } = O ( ( 1 - \\gamma ) ^ { - \\bar { 4 } } / d _ { m } ) , m _ { 2 } ^ { \\prime } =$ ${ \\check { O ( ( 1 - \\gamma ) ^ { - 3 } / d _ { m } ) } }$ (see more precise expressions in Lemma $D . 7 ,$ ), $K _ { 1 } = \\log _ { 2 } ( ( 1 - \\gamma ) ^ { - 1 } / \\epsilon ) , K _ { 2 } =$ $\\log _ { 2 } ( \\sqrt { ( 1 - \\gamma ) ^ { - 1 } } / \\epsilon )$ , $R = \\log ( 4 / \\epsilon ( 1 - \\gamma ) )$ , and choose LCB estimators z and $\\mathbf { g }$ as in Figure 3.2, then with probability $1 - \\delta$ , the infinite horizon version of OPDVR (Algorithm 4) outputs an \u000f-optimal policy provided that in offline data $\\mathcal { D }$ has number of samples exceeding ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/47e9d1c3dea2ae825e5a20d5f9357528cb6ced6483bcccd8c18d1c1630d55ccf.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\frac { c _ { 3 } ^ { \\prime } \\operatorname* { m a x } [ \\frac { m _ { 1 } ^ { \\prime } } { ( 1 - \\gamma ) ^ { - 1 } } , m _ { 2 } ^ { \\prime } ] } { \\epsilon ^ { 2 } } \\cdot \\iota ^ { \\prime } = \\widetilde O \\left[ ( 1 - \\gamma ) ^ { - 3 } / d _ { m } \\epsilon ^ { 2 } \\right] . } } \\\\ { { \\cdot \\iota ^ { \\prime } : = R \\cdot ( \\log ( 3 2 ( 1 - \\gamma ) ^ { - 1 } R S A / \\delta ) + \\log \\log _ { 2 } ( \\sqrt { ( 1 - \\gamma ) ^ { - 1 } } / \\epsilon ) ) \\cdot \\log _ { 2 } ( \\sqrt { ( 1 - \\gamma ) ^ { - 1 } } / \\epsilon ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We note that for the infinite horizon case, the sample-complexity measures the number of steps, thus $( 1 - \\gamma ) ^ { - 3 }$ is comparable to the $H ^ { 2 }$ dependence. To the best of our knowledge, Theorem 4.1 and Theorem 4.3 are the first results that achieve $H ^ { 2 }$ , $( 1 - \\gamma ) ^ { - 3 }$ dependence in the offline regime respectively for stationary transition and infinite horizon setting, see Table 1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 Discussions ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Estimating $d _ { m }$ . It is worth mentioning that the input of OPDVR depends on unknown system quantity $d _ { m }$ . Nevertheless, $d _ { m }$ is only one-dimensional scalar and thus it is plausible (from a statistical perspective) to leverage standard parameter-tuning tools (e.g. cross validation [Varma and Simon, 2006]) for obtaining a reliable estimate in practice. On the theoretical side, we provide the following result to show plug-in on-policy estimator $\\widehat { d } _ { t } ^ { \\mu } ( s _ { t } , a _ { t } ) = n _ { s _ { t } , a _ { t } } / n$ and $\\begin{array} { r } { \\widehat { d } _ { m } : = \\operatorname* { m i n } _ { t , s _ { t } , a _ { t } } \\{ n _ { s _ { t } , a _ { t } } / n : } \\end{array}$ $n _ { s _ { t } , a _ { t } } > 0 \\}$ , is sufficient for accurately estimating $d _ { t } ^ { \\mu } , d _ { m }$ simultaneously. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.1. For the finite-horizon setting (either stationary or non-stationary), there exists universal constant c, s.t. when $n \\geq c \\cdot 1 / d _ { m } \\cdot \\log ( H S A / \\delta )$ , then w.p. $1 - \\delta$ , we have $\\forall t$ , $s _ { t } , a _ { t }$ , $\\textstyle { \\frac { 1 } { 2 } } d _ { t } ^ { \\mu } ( s _ { t } , a _ { t } ) \\leq$ $\\begin{array} { r } { \\widehat { d } _ { t } ^ { \\mu } ( s _ { t } , a _ { t } ) \\leq \\frac { 3 } { 2 } d _ { t } ^ { \\mu } ( s _ { t } , a _ { t } ) } \\end{array}$ and, in particular, $\\textstyle { \\frac { 1 } { 2 } } d _ { m } \\leq { \\widehat { d } } _ { m } \\leq { \\frac { 3 } { 2 } } d _ { m }$ . See Appendix F.1 for the proof. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lemma 5.1 ensures one can replace $d _ { t } ^ { \\mu }$ by $\\widehat { d } _ { t } ^ { \\mu }$ $d _ { m }$ by $\\widehat { d } _ { m }$ ) in OPDVR and we obtain a fully dataadaptive algorithm. Note that the requirement on $n$ does not affect our near-minimax complexity bound in either Theorem 3.2 and 4.1—we only require $n \\approx \\tilde { \\Theta } ( 1 / d _ { m } )$ additional episodes to estimate $d _ { m }$ and it is of lower order compared to our upper bound $\\widetilde { O } ( H ^ { 3 } / d _ { m } )$ or $\\widetilde { \\cal O } ( H ^ { 2 } / d _ { m } ) )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Computational and memory cost. OPDVR can be implemented as a streaming algorithm that uses only one pass of the dataset. Its computational cost is $\\widetilde { O } ( H ^ { 4 } / d _ { m } \\epsilon ^ { 2 } )$ — the same as its sample complexity in steps ( $H$ steps is an episode), and the memory cost is $O ( H S A )$ for the episodic case and $O ( S A )$ for the stationary or infinite horizon case. In particular, the double variance reduction technique does not introduce additional overhead beyond constant factors. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Improvement over VR in the generative model setting. This work extends of the variance reduction framework for RL in the generative model setting [Sidford et al., 2018a, Yang and Wang, 2019]. However, we make two improvements to these works. First, the data collected in the offline case are highly dependent (in contrast in the generative model setting each simulator call is independent), and disentangling the dependent structure makes the offline setting inherently more challenging. Second, our doubling mechanism always guarantee the minimax rate with any initialization and the single VR procedure does not have this property (see Appendix F.4 for a more detailed discussion), which could be a critical issue when Sidford et al. [2018a] claims the optimality. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper proposes OPDVR (off-policy double variance reduction), a new variance reduction algorithm for offline reinforcement learning. We show that OPDVR achieves tight sample complexity for offline RL in tabular MDPs; in particular, OPDVR is the first algorithm that achieves the optimal sample complexity for offline RL in the stationary transition setting. On the technical end, we present a sharp analysis under stationary transitions, and use the doubling technique to resolve the initialization dependence in variance reduction, both of which could be of broader interest. There are several interesting next directions. For example, can our analysis shed light on other commonly used algorithms (such as Q-Learning) for offline RL? How can we better deal with insufficient data coverage? Can we go beyond the tabular setting? We would like to leave these as future work. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgment ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            205,
+            297,
+            219
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Lin F. Yang for the discussions about [Sidford et al., 2018a]. Ming Yin and Yu-Xiang Wang are partially supported by NSF Awards #2007117 and #2003257. Yu Bai is partly funded through the employment with Salesforce. ",
+        "bbox": [
+            174,
+            231,
+            825,
+            272
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            292,
+            266,
+            309
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alekh Agarwal, Nan Jiang, and Sham M Kakade. Reinforcement learning: Theory and algorithms. CS Dept., UW Seattle, Seattle, WA, USA, Tech. Rep, 2019. ",
+        "bbox": [
+            171,
+            316,
+            823,
+            345
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alekh Agarwal, Sham Kakade, and Lin F Yang. Model-based reinforcement learning with a generative model is minimax optimal. In Conference on Learning Theory, pages 67–83. PMLR, 2020. ",
+        "bbox": [
+            173,
+            354,
+            823,
+            383
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "András Antos, Csaba Szepesvári, and Rémi Munos. Fitted q-iteration in continuous action-space mdps. In Advances in neural information processing systems, pages 9–16, 2008a. ",
+        "bbox": [
+            171,
+            392,
+            821,
+            421
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "András Antos, Csaba Szepesvári, and Rémi Munos. Learning near-optimal policies with bellmanresidual minimization based fitted policy iteration and a single sample path. Machine Learning, 71 (1):89–129, 2008b. ",
+        "bbox": [
+            173,
+            431,
+            823,
+            473
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mohammad Gheshlaghi Azar, Rémi Munos, and Hilbert J Kappen. Minimax pac bounds on the sample complexity of reinforcement learning with a generative model. Machine learning, 91(3): 325–349, 2013. ",
+        "bbox": [
+            173,
+            483,
+            826,
+            525
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mohammad Gheshlaghi Azar, Ian Osband, and Rémi Munos. Minimax regret bounds for reinforcement learning. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 263–272. JMLR. org, 2017. ",
+        "bbox": [
+            173,
+            535,
+            826,
+            578
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Qi Cai, Zhuoran Yang, Chi Jin, and Zhaoran Wang. Provably efficient exploration in policy optimization. arXiv preprint arXiv:1912.05830, 2019. ",
+        "bbox": [
+            173,
+            587,
+            823,
+            616
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jinglin Chen and Nan Jiang. Information-theoretic considerations in batch reinforcement learning. arXiv preprint arXiv:1905.00360, 2019. ",
+        "bbox": [
+            173,
+            626,
+            825,
+            654
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Herman Chernoff et al. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. The Annals of Mathematical Statistics, 23(4):493–507, 1952. ",
+        "bbox": [
+            169,
+            664,
+            825,
+            693
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fan Chung and Linyuan Lu. Concentration inequalities and martingale inequalities: a survey. Internet Mathematics, 3(1):79–127, 2006. ",
+        "bbox": [
+            173,
+            702,
+            825,
+            731
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Christoph Dann and Emma Brunskill. Sample complexity of episodic fixed-horizon reinforcement learning. In Advances in Neural Information Processing Systems, pages 2818–2826, 2015. ",
+        "bbox": [
+            173,
+            739,
+            823,
+            770
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Christoph Dann, Lihong Li, Wei Wei, and Emma Brunskill. Policy certificates: Towards accountable reinforcement learning. arXiv preprint arXiv:1811.03056, 2018. ",
+        "bbox": [
+            171,
+            779,
+            823,
+            808
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Omar Darwiche Domingues, Pierre Ménard, Emilie Kaufmann, and Michal Valko. Episodic reinforcement learning in finite mdps: Minimax lower bounds revisited. arXiv preprint arXiv:2010.03531, 2020. ",
+        "bbox": [
+            176,
+            816,
+            823,
+            859
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Simon S Du, Jianshu Chen, Lihong Li, Lin Xiao, and Dengyong Zhou. Stochastic variance reduction methods for policy evaluation. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 1049–1058. JMLR. org, 2017. ",
+        "bbox": [
+            176,
+            868,
+            825,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yaqi Duan and Mengdi Wang. Minimax-optimal off-policy evaluation with linear function approximation. arXiv preprint arXiv:2002.09516, 2020. ",
+        "bbox": [
+            171,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yonathan Efroni, Nadav Merlis, Mohammad Ghavamzadeh, and Shie Mannor. Tight regret bounds for model-based reinforcement learning with greedy policies. In Advances in Neural Information Processing Systems, pages 12203–12213, 2019. ",
+        "bbox": [
+            178,
+            128,
+            823,
+            172
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yihao Feng, Tongzheng Ren, Ziyang Tang, and Qiang Liu. Accountable off-policy evaluation with kernel bellman statistics. In International Conference on Machine Learning, pages 3102–3111. PMLR, 2020. ",
+        "bbox": [
+            176,
+            180,
+            823,
+            223
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Nan Jiang and Lihong Li. Doubly robust off-policy value evaluation for reinforcement learning. In Proceedings of the 33rd International Conference on International Conference on Machine Learning-Volume 48, pages 652–661. JMLR. org, 2016. ",
+        "bbox": [
+            174,
+            232,
+            823,
+            276
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, John Langford, and Robert E Schapire. Contextual decision processes with low bellman rank are pac-learnable. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 1704–1713. JMLR. org, 2017. ",
+        "bbox": [
+            174,
+            284,
+            823,
+            328
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chi Jin, Zeyuan Allen-Zhu, Sebastien Bubeck, and Michael I Jordan. Is q-learning provably efficient? In Advances in Neural Information Processing Systems, pages 4863–4873, 2018. ",
+        "bbox": [
+            174,
+            335,
+            821,
+            366
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ying Jin, Zhuoran Yang, and Zhaoran Wang. Is pessimism provably efficient for offline rl? arXiv preprint arXiv:2012.15085, 2020. ",
+        "bbox": [
+            176,
+            373,
+            823,
+            404
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Rie Johnson and Tong Zhang. Accelerating stochastic gradient descent using predictive variance reduction. In Advances in neural information processing systems, pages 315–323, 2013. ",
+        "bbox": [
+            174,
+            411,
+            823,
+            443
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Nathan Kallus and Masatoshi Uehara. Double reinforcement learning for efficient off-policy evaluation in markov decision processes. arXiv preprint arXiv:1908.08526, 2019a. ",
+        "bbox": [
+            174,
+            450,
+            823,
+            479
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Nathan Kallus and Masatoshi Uehara. Efficiently breaking the curse of horizon: Double reinforcement learning in infinite-horizon processes. arXiv preprint arXiv:1909.05850, 2019b. ",
+        "bbox": [
+            173,
+            487,
+            821,
+            518
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Akshay Krishnamurthy, Alekh Agarwal, and John Langford. Pac reinforcement learning with rich observations. In Advances in Neural Information Processing Systems, pages 1840–1848, 2016. ",
+        "bbox": [
+            173,
+            526,
+            823,
+            556
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sascha Lange, Thomas Gabel, and Martin Riedmiller. Batch reinforcement learning. In Reinforcement learning, pages 45–73. Springer, 2012. ",
+        "bbox": [
+            173,
+            564,
+            823,
+            594
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Tor Lattimore and Csaba Szepesvari. Learning with good feature representations in bandits and in rl with a generative model. arXiv preprint arXiv:1911.07676, 2019. ",
+        "bbox": [
+            171,
+            602,
+            825,
+            632
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Hoang M Le, Cameron Voloshin, and Yisong Yue. Batch policy learning under constraints. arXiv preprint arXiv:1903.08738, 2019. ",
+        "bbox": [
+            173,
+            640,
+            823,
+            670
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Sergey Levine, Aviral Kumar, George Tucker, and Justin Fu. Offline reinforcement learning: Tutorial, review, and perspectives on open problems. arXiv preprint arXiv:2005.01643, 2020. ",
+        "bbox": [
+            173,
+            678,
+            825,
+            708
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Gen Li, Yuting Wei, Yuejie Chi, Yuantao Gu, and Yuxin Chen. Sample complexity of asynchronous q-learning: Sharper analysis and variance reduction. arXiv preprint arXiv:2006.03041, 2020. ",
+        "bbox": [
+            171,
+            717,
+            825,
+            746
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Lihong Li, Rémi Munos, and Csaba Szepesvári. Toward minimax off-policy value estimation. 2015. ",
+        "bbox": [
+            173,
+            753,
+            825,
+            771
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Qiang Liu, Lihong Li, Ziyang Tang, and Dengyong Zhou. Breaking the curse of horizon: Infinitehorizon off-policy estimation. In Advances in Neural Information Processing Systems, pages 5361–5371, 2018. ",
+        "bbox": [
+            174,
+            779,
+            825,
+            821
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Off-policy policy gradient with state distribution correction. In Uncertainty in Artificial Intelligence, 2019. ",
+        "bbox": [
+            171,
+            830,
+            823,
+            859
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yao Liu, Pierre-Luc Bacon, and Emma Brunskill. Understanding the curse of horizon in off-policy evaluation via conditional importance sampling. In International Conference on Machine Learning, pages 6184–6193. PMLR, 2020a. ",
+        "bbox": [
+            176,
+            868,
+            826,
+            911
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Provably good batch reinforcement learning without great exploration. arXiv preprint arXiv:2007.08202, 2020b. ",
+        "bbox": [
+            171,
+            90,
+            825,
+            121
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Rémi Munos. Error bounds for approximate policy iteration. In ICML, volume 3, pages 560–567, 2003. ",
+        "bbox": [
+            173,
+            126,
+            825,
+            155
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Aaron Sidford, Mengdi Wang, Xian Wu, Lin Yang, and Yinyu Ye. Near-optimal time and sample complexities for solving markov decision processes with a generative model. In Advances in Neural Information Processing Systems, pages 5186–5196, 2018a. ",
+        "bbox": [
+            174,
+            161,
+            823,
+            204
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Aaron Sidford, Mengdi Wang, Xian Wu, and Yinyu Ye. Variance reduced value iteration and faster algorithms for solving markov decision processes. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770–787. SIAM, 2018b. ",
+        "bbox": [
+            174,
+            210,
+            821,
+            253
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Aaron Sidford, Mengdi Wang, Lin Yang, and Yinyu Ye. Solving discounted stochastic two-player games with near-optimal time and sample complexity. In International Conference on Artificial Intelligence and Statistics, pages 2992–3002. PMLR, 2020. ",
+        "bbox": [
+            173,
+            258,
+            823,
+            303
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Max Simchowitz and Kevin G Jamieson. Non-asymptotic gap-dependent regret bounds for tabular mdps. In Advances in Neural Information Processing Systems, pages 1151–1160, 2019. ",
+        "bbox": [
+            174,
+            309,
+            821,
+            338
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Joel Tropp et al. Freedman’s inequality for matrix martingales. Electronic Communications in Probability, 16:262–270, 2011. ",
+        "bbox": [
+            174,
+            343,
+            823,
+            373
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Masatoshi Uehara and Nan Jiang. Minimax weight and q-function learning for off-policy evaluation. arXiv preprint arXiv:1910.12809, 2019. ",
+        "bbox": [
+            169,
+            378,
+            825,
+            409
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Sudhir Varma and Richard Simon. Bias in error estimation when using cross-validation for model selection. BMC bioinformatics, 7(1):91, 2006. ",
+        "bbox": [
+            171,
+            415,
+            825,
+            444
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Martin J Wainwright. Variance-reduced $q$ -learning is minimax optimal. arXiv preprint arXiv:1906.04697, 2019. ",
+        "bbox": [
+            174,
+            449,
+            823,
+            479
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ruosong Wang, Dean P Foster, and Sham M Kakade. What are the statistical limits of offline rl with linear function approximation? arXiv preprint arXiv:2010.11895, 2020. ",
+        "bbox": [
+            173,
+            484,
+            823,
+            515
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tengyang Xie and Nan Jiang. Batch value-function approximation with only realizability. arXiv preprint arXiv:2008.04990, 2020a. ",
+        "bbox": [
+            174,
+            520,
+            821,
+            550
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tengyang Xie and Nan Jiang. $\\mathbf { Q } ^ { * }$ approximation schemes for batch reinforcement learning: A theoretical comparison. arXiv preprint arXiv:2003.03924, 2020b. ",
+        "bbox": [
+            171,
+            556,
+            823,
+            585
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tengyang Xie, Yifei Ma, and Yu-Xiang. Wang. Towards optimal off-policy evaluation for reinforcement learning with marginalized importance sampling. In Advances in Neural Information Processing Systems, 2019. ",
+        "bbox": [
+            173,
+            590,
+            823,
+            633
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Lin Yang and Mengdi Wang. Sample-optimal parametric q-learning using linearly additive features. In International Conference on Machine Learning, pages 6995–7004, 2019. ",
+        "bbox": [
+            169,
+            640,
+            825,
+            670
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ming Yin and Yu-Xiang Wang. Asymptotically efficient off-policy evaluation for tabular reinforcement learning. In AISTATS-20, 2020. ",
+        "bbox": [
+            173,
+            676,
+            825,
+            705
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Ming Yin, Yu Bai, and Yu-Xiang Wang. Near optimal provable uniform convergence in off-policy evaluation for reinforcement learning. In AISTATS-21, 2021. ",
+        "bbox": [
+            174,
+            710,
+            823,
+            741
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Andrea Zanette. Exponential lower bounds for batch reinforcement learning: Batch rl can be exponentially harder than online rl. arXiv preprint arXiv:2012.08005, 2020. ",
+        "bbox": [
+            173,
+            746,
+            821,
+            776
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Andrea Zanette and Emma Brunskill. Tighter problem-dependent regret bounds in reinforcement learning without domain knowledge using value function bounds. arXiv preprint arXiv:1901.00210, 2019. ",
+        "bbox": [
+            174,
+            781,
+            825,
+            824
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Lijun Zhang, Mehrdad Mahdavi, and Rong Jin. Linear convergence with condition number independent access of full gradients. Advances in Neural Information Processing Systems, 26:980–988, 2013. ",
+        "bbox": [
+            176,
+            830,
+            825,
+            873
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Zihan Zhang, Yuan Zhou, and Xiangyang Ji. Almost optimal model-free reinforcement learning via reference-advantage decomposition. arXiv preprint arXiv:2004.10019, 2020. ",
+        "bbox": [
+            173,
+            880,
+            823,
+            910
+        ],
+        "page_idx": 11
+    }
+]

parse/train/TgSVWXw22FQ/TgSVWXw22FQ.md

@@ -0,0 +1,311 @@
+# IMPROVING ZERO-SHOT VOICE STYLE TRANSFER VIADISENTANGLED REPRESENTATION LEARNING
+
+Siyang Yuan1∗, Pengyu Cheng1∗, Ruiyi Zhang1, Weituo $\mathbf { H a o } ^ { 1 }$ , Zhe $\mathbf { G a n ^ { 2 } }$ and Lawrence Carin
+
+1Duke University, Durham, North Carolina, USA 2Microsoft, Redmond, Washington, USA {siyang.yuan,pengyu.cheng}@duke.edu
+
+# ABSTRACT
+
+Voice style transfer, also called voice conversion, seeks to modify one speaker’s voice to generate speech as if it came from another (target) speaker. Previous works have made progress on voice conversion with parallel training data and pre-known speakers. However, zero-shot voice style transfer, which learns from non-parallel data and generates voices for previously unseen speakers, remains a challenging problem. We propose a novel zero-shot voice transfer method via disentangled representation learning. The proposed method first encodes speakerrelated style and voice content of each input voice into separated low-dimensional embedding spaces, and then transfers to a new voice by combining the source content embedding and target style embedding through a decoder. With informationtheoretic guidance, the style and content embedding spaces are representative and (ideally) independent of each other. On real-world VCTK datasets, our method outperforms other baselines and obtains state-of-the-art results in terms of transfer accuracy and voice naturalness for voice style transfer experiments under both many-to-many and zero-shot setups.
+
+# 1 INTRODUCTION
+
+Style transfer, which automatically converts a data instance into a target style, while preserving its content information, has attracted considerable attention in various machine learning domains, including computer vision (Gatys et al., 2016; Luan et al., 2017; Huang & Belongie, 2017), video processing (Huang et al., 2017; Chen et al., 2017), and natural language processing (Shen et al., 2017; Yang et al., 2018; Lample et al., 2019; Cheng et al., 2020b). In speech processing, style transfer was earlier recognized as voice conversion (VC) (Muda et al., 2010), which converts one speaker’s utterance, as if it was from another speaker but with the same semantic meaning. Voice style transfer (VST) has received long-term research interest, due to its potential for applications in security (Sisman et al., 2018), medicine (Nakamura et al., 2006), entertainment (Villavicencio & Bonada, 2010) and education (Mohammadi & Kain, 2017), among others.
+
+Although widely investigated, VST remains challenging when applied to more general application scenarios. Most of the traditional VST methods require parallel training data, i.e., paired voices from two speakers uttering the same sentence. This constraint limits the application of such models in the real world, where data are often not pair-wise available. Among the few existing models that address non-parallel data (Hsu et al., 2016; Lee & Wu, 2006; Godoy et al., 2011), most methods cannot handle many-to-many transfer (Saito et al., 2018; Kaneko & Kameoka, 2018; Kameoka et al., 2018), which prevents them from converting multiple source voices to multiple target speaker styles. Even among the few non-parallel many-to-many transfer models, to the best of our knowledge, only two models (Qian et al., 2019; Chou & Lee, 2019) allow zero-shot transfer, i.e., conversion from/to newly-coming speakers (unseen during training) without re-training the model.
+
+The only two zero-shot VST models (AUTOVC (Qian et al., 2019) and AdaIN-VC (Chou & Lee, 2019)) share a common weakness. Both methods construct encoder-decoder frameworks, which extract the style and the content information into style and content embeddings, and generate a voice sample by combining a style embedding and a content embedding through the decoder. With the combination of the source content embedding and the target style embedding, the models generate the transferred voice, based only on source and target voice samples. AUTOVC (Qian et al., 2019) uses a GE2E (Wan et al., 2018) pre-trained style encoder to ensure rich speaker-related information in style embeddings. However, AUTOVC has no regularizer to guarantee that the content encoder does not encode any style information. AdaIN-VC (Chou & Lee, 2019) applies instance normalization (Ulyanov et al., 2016) to the feature map of content representations, which helps to eliminate the style information from content embeddings. However, AdaIN-VC fails to prevent content information from being revealed in the style embeddings. Both methods cannot assure that the style and content embeddings are disentangled without information revealed from each other.
+
+With information-theoretic guidance, we propose a disentangled-representation-learning method to enhance the encoder-decoder zero-shot VST framework, for both style and content information preservation. We call the proposed method Information-theoretic Disentangled Embedding for Voice Conversion (IDE-VC). Our model successfully induces the style and content of voices into independent representation spaces by minimizing the mutual information between style and content embeddings. We also derive two new multi-group mutual information lower bounds, to further improve the representativeness of the latent embeddings. Experiments demonstrate that our method outperforms previous works under both many-to-many and zero-shot transfer setups on two objective metrics and two subjective metrics.
+
+# 2 BACKGROUND
+
+In information theory, mutual information (MI) is a crucial concept that measures the dependence between two random variables. Mathematically, the MI between two variables $_ { \textbf { \em x } }$ and $\textbf {  { y } }$ is
+
+$$
+\mathcal { T } ( \pmb { x } ; \pmb { y } ) : = \mathbb { E } _ { p ( \pmb { x } , \pmb { y } ) } \Big [ \log \frac { p ( \pmb { x } , \pmb { y } ) } { p ( \pmb { x } ) p ( \pmb { y } ) } \Big ] ,
+$$
+
+where $p ( { \pmb x } )$ and $p ( \pmb { y } )$ are marginal distributions of $_ { \textbf { \em x } }$ and $\textbf {  { y } }$ , and $p ( { \pmb x } , { \pmb y } )$ is the joint distribution. Recently, MI has attracted considerable interest in machine learning as a criterion to minimize or maximize the dependence between different parts of a model (Chen et al., 2016; Alemi et al., 2016; Hjelm et al., 2018; Velickovi ˇ c et al., 2018; Song et al., 2019). However, the calculation of exact MI ´ values is challenging in practice, since the closed form of joint distribution $p ( { \pmb x } , { \pmb y } )$ in equation (1) is generally unknown. To solve this problem, several MI estimators have been proposed. For MI maximization tasks, Nguyen, Wainwright and Jordan (NWJ) (Nguyen et al., 2010) propose a lower bound by representing (1) as an $f$ -divergence (Moon & Hero, 2014):
+
+$$
+\begin{array} { r } { \mathcal { T } _ { \mathrm { N W J } } : = \mathbb { E } _ { p ( \pmb { x } , \pmb { y } ) } [ f ( \pmb { x } , \pmb { y } ) ] - e ^ { - 1 } \mathbb { E } _ { p ( \pmb { x } ) p ( \pmb { y } ) } [ e ^ { f ( \pmb { x } , \pmb { y } ) } ] , } \end{array}
+$$
+
+with a score function $f ( { \pmb x } , { \pmb y } )$ . Another widely-used sample-based MI lower bound is InfoNCE (Oord et al., 2018), which is derived with Noise Contrastive Estimation (NCE) (Gutmann & Hyvarinen, 2010). With sample pairs ¨ $\{ ( \pmb { x } _ { i } , \pmb { y } _ { i } ) \} _ { i = 1 } ^ { N }$ drawn from the joint distribution $p ( { \pmb x } , { \pmb y } )$ , the InfoNCE lower bound is defined as
+
+$$
+\mathcal { T } _ { \mathrm { N C E } } : = \mathbb { E } \Big [ \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \log \frac { e ^ { f ( { \pmb x } _ { i } , { \pmb y } _ { i } ) } } { \frac { 1 } { N } \sum _ { j = 1 } ^ { N } e ^ { f ( { \pmb x } _ { i } , { \pmb y } _ { j } ) } } \Big ] .
+$$
+
+For MI minimization tasks, Cheng et al. (2020a) proposed a contrastively learned upper bound that requires the conditional distribution $p ( { \pmb x } | { \pmb y } )$ :
+
+$$
+\mathcal { T } ( \pmb { x } ; \pmb { y } ) \leq \mathbb { E } \Big [ \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \Big [ \log p ( \pmb { x } _ { i } | \pmb { y } _ { i } ) - \frac { 1 } { N } \sum _ { j = 1 } ^ { N } \log p ( \pmb { x } _ { j } | \pmb { y } _ { i } ) \Big ] \Big ] .
+$$
+
+where the MI is bounded by the log-ratio of conditional distribution $p ( { \pmb x } | { \pmb y } )$ between positive and negative sample pairs. In the following, we derive our information-theoretic disentangled representation learning framework for voice style transfer based on the MI estimators described above.
+
+# 3 PROPOSED MODEL
+
+We assume access to $N$ audio (voice) recordings from $M$ speakers, where speaker $u$ has $N _ { u }$ voice samples $\mathcal { X } _ { u } = \{ \pmb { x } _ { u i } \} _ { i = 1 } ^ { N _ { u } }$ . The proposed approach encodes each voice input $\pmb { x } \in \mathcal { X } = \cup _ { u = 1 } ^ { M } \pmb { \chi } _ { u }$ into a speaker-related (style) embedding $\pmb { s } = E _ { s } ( \pmb { x } )$ and a content-related embedding ${ \pmb c } = E _ { c } ( { \pmb x } )$ , using respectively a style encoder $E _ { s } ( \cdot )$ and a content encoder $E _ { c } ( \cdot )$ . To transfer a source ${ \pmb x } _ { u i }$ from speaker $u$ to the target style of the voice of speaker $v$ , $\mathbf { \boldsymbol { x } } _ { v j }$ , we combine the content embedding $\bar { \mathbf { c } _ { u i } } = E _ { c } ( \mathbf { x } _ { u i } )$ and the style embedding ${ \pmb s } _ { v j } = E _ { s } ( { \pmb x } _ { v j } )$ to generate the transferred voice $\hat { \pmb x } _ { u  v , i } = D ( \pmb s _ { v j } , \pmb c _ { u i } )$ with a decoder $D ( s , c )$ . To implement this two-step transfer process, we introduce a novel mutual information (MI)-based learning objective, that induces the style embedding $\pmb { s }$ and content embedding $^ c$ into independent representation spaces (i.e., ideally, $\pmb { s }$ contains rich style information of $_ { \textbf { \em x } }$ with no content information, and vice versa). In the following, we first describe our MI-based training objective in Section 3.1, and then discuss the practical estimation of the objective in Sections 3.2 and 3.3.
+
+# 3.1 MI-BASED DISENTANGLING OBJECTIVE
+
+From an information-theoretic perspective, to learn representative latent embedding $( s , c )$ , it is desirable to maximize the mutual information between the embedding pair $( s , c )$ and the input $_ { \textbf { \em x } }$ . Meanwhile, the style embedding $\pmb { s }$ and the content $^ c$ are desired to be independent, so that we can control the style transfer process with different style and content attributes. Therefore, we minimize the mutual information $\bar { \mathcal { T } } ( s ; c )$ to disentangle the style embedding and content embedding spaces. Consequently, our overall disentangled-representation-learning objective seeks to minimize
+
+$$
+\mathcal { L } = \mathcal { T } ( \pmb { s } ; \pmb { c } ) - \mathcal { T } ( \pmb { x } ; \pmb { s } , \pmb { c } ) = \mathcal { T } ( \pmb { s } ; \pmb { c } ) - \mathcal { T } ( \pmb { x } ; \pmb { c } | \pmb { s } ) - \mathcal { T } ( \pmb { x } ; \pmb { s } ) .
+$$
+
+As discussed in Locatello et al. (Locatello et al., 2019), without inductive bias for supervision, the learned representation can be meaningless. To address this problem, we use the speaker identity $\textbf { \em u }$ as a variable with values $\{ 1 , \dots , M \}$ to learn representative style embedding $\pmb { s }$ for speaker-related attributes. Noting that the process from speaker $u$ to his/her voice ${ \pmb x } _ { u i }$ to the style embedding $s _ { u i }$ (as $\mathbf { \delta } \mathbf { \boldsymbol { u } } \to \mathbf { \mathcal { x } } \to \mathbf { \mathcal { s } }$ ) is a Markov Chain, we conclude ${ \mathcal { T } } ( s ; x ) \geq { \mathcal { T } } ( s ; u )$ based on the MI data-processing inequality (Cover & Thomas, 2012) (as stated in the Supplementary Material). Therefore, we replace $\scriptstyle { \mathcal { T } } ( s ; x )$ in $\mathcal { L }$ with $\pmb { \mathcal { T } } ( \pmb { s } ; \pmb { u } )$ and minimize an upper bound instead:
+
+$$
+\bar { \mathcal { L } } = \mathcal { T } ( \pmb { s } ; \pmb { c } ) - \mathcal { T } ( \pmb { x } ; \pmb { c } | \pmb { s } ) - \mathcal { T } ( \pmb { u } ; \pmb { s } ) \geq \mathcal { T } ( \pmb { s } ; \pmb { c } ) - \mathcal { T } ( \pmb { x } ; \pmb { c } | \pmb { s } ) - \mathcal { T } ( \pmb { x } ; \pmb { s } ) ,
+$$
+
+In practice, calculating the MI is challenging, as we typically only have access to samples, and lack the required distributions (Chen et al., 2016). To solve this problem, below we provide several MI estimates to the objective terms $\mathcal { T } ( s ; c )$ , $\pmb { \mathcal { T } } ( \pmb { x } ; \pmb { c } | \pmb { s } )$ and $\mathcal { T } ( u ; s )$ .
+
+# 3.2 MI LOWER BOUND ESTIMATION
+
+To maximize $\scriptstyle { \mathcal { T } } ( u ; s )$ , we derive the following multi-group MI lower bound (Theorem 3.1) based on the NWJ bound developed in Nguyen et al. (Nguyen et al., 2010). The detailed proof is provided in the Supplementary Material. Let $\dot { \mu } _ { v } ^ { ( - u i ) } = \mu _ { v }$ represent the mean of all style embeddings in group $\mathcal { X } _ { v }$ , constituting the style centroid of speaker $v$ ; $\mu _ { u } ^ { ( - u i ) }$ is the mean of all style embeddings in group $\mathcal { X } _ { u }$ except data point ${ \pmb x } _ { u i }$ , representing a leave- ${ \bf { x } } _ { u i }$ -out style centroid of speaker $u$ . Intuitively, we minimize $\| s _ { u i } - \mu _ { u } ^ { ( - u i ) } \|$ to encourage the style embedding of voice ${ \pmb x } _ { u i }$ to be more similar to the style centroid of speaker $u$ , while maximizing $\| s _ { u i } - \mu _ { v } ^ { ( - u i ) } \|$ to enlarge the margin between $s _ { u i }$ and the other speakers’ style centroids $\pmb { \mu } _ { v }$ . We denote the right-hand side of (7) as $\hat { \mathcal { I } } _ { 1 }$ .
+
+Theorem 3.1. Let $\begin{array} { r } { \pmb { \mu } _ { v } ^ { ( - u i ) } = \frac { 1 } { N _ { v } } \sum _ { k = 1 } ^ { N _ { v } } \pmb { s } _ { v k } i f u \neq } \end{array}$ $u \ne v$ ; and $\begin{array} { r } { \pmb { \mu } _ { u } ^ { ( - u i ) } = \frac { 1 } { N _ { u } - 1 } \sum _ { j \neq i } \pmb { s } _ { u j } } \end{array}$
+
+$$
+\mathcal { Z } ( u ; s ) \geq \mathbb { E } \Big [ \frac { 1 } { N } \sum _ { u = 1 } ^ { M } \sum _ { i = 1 } ^ { N _ { u } } \Big [ - \| s _ { u i } - \mu _ { u } ^ { ( - u i ) } \| ^ { 2 } - \frac { e ^ { - 1 } } { N } \sum _ { v = 1 } ^ { M } N _ { v } \exp \{ - \| s _ { u i } - \mu _ { v } ^ { ( - u i ) } \| ^ { 2 } \} \Big ] \Big ] .
+$$
+
+To maximize $\pmb { \mathcal { T } } ( \pmb { x } ; \pmb { c } | \pmb { s } )$ , we derive a conditional mutual information lower bound below:
+
+Theorem 3.2. Assume that given $\begin{array} { r } { \boldsymbol { s } = \boldsymbol { s } _ { u } , } \end{array}$ , samples $\{ ( \pmb { x } _ { u i } , \pmb { c } _ { u i } ) \} _ { i = 1 } ^ { N _ { u } }$ are observed. With a variational distribution $q _ { \phi } ( { \pmb x } | { \pmb s } , { \pmb c } )$ , we have $\mathcal { T } ( \pmb { x } ; \pmb { c } | \pmb { s } ) \geq \mathbb { E } [ \hat { \mathcal { T } } ]$ , where
+
+$$
+\hat { \mathcal { T } } = \frac { 1 } { N } \sum _ { u = 1 } ^ { M } \sum _ { i = 1 } ^ { N _ { u } } \left[ \log q _ { \phi } ( \pmb { x } _ { u i } | \pmb { c } _ { u i } , \pmb { s } _ { u } ) - \log \left( \frac { 1 } { N _ { u } } \sum _ { j = 1 } ^ { N _ { u } } q _ { \phi } ( \pmb { x } _ { u j } | \pmb { c } _ { u i } , \pmb { s } _ { u } ) \right) \right] .
+$$
+
+![](images/49eb8d52383f2f877a1546711e5b0f7b4d073ca4a8800001f2aee488249c822c.jpg)  
+Figure 1: Training and transfer processes. (a) Training style encoder $E _ { s }$ with objective $\hat { \mathcal { I } } _ { 1 }$ : All voice samples are encoded into style embedding space. For style embedding $s _ { u i }$ of ${ \pmb x } _ { u i }$ , we minimize its distance with speaker $u$ ’s style centroid $\pmb { \mu } _ { u }$ , and maximize its distance to other speaker style centroids $\pmb { \mu } _ { v }$ . (b) Training for content encoder $E _ { c }$ and decoder $D$ as objectives $\hat { \mathcal { T } } _ { 2 } , \tilde { \mathcal { T } } _ { 3 }$ : We encode content $\mathbf { \mathit { c } } _ { u i }$ from voice $\mathbf { \mathcal { x } } _ { u i }$ from speaker $u$ . The style of speaker $u$ is encoded from another speaker $u$ ’s voice $\mathbf { \boldsymbol { x } } _ { u j }$ . The dependency of style and content embedding is minimized with $\hat { \mathcal { T } } _ { 3 }$ . With $\mathbf { \mathit { c } } _ { u i }$ and $\mathbf { } _ { s \alpha }$ , the decoder reconstructs the voice ${ \pmb x } _ { u i }$ as $\hat { \pmb { x } } _ { u i } = D ( \pmb { s } _ { u } , \pmb { c } _ { u i } )$ . Then $\hat { \mathcal { I } } _ { 2 }$ is calculated based on the original voice $\mathbf { \phi } _ { c u i }$ and the reconstruction $\hat { c } _ { u i }$ . (c) Transfer process: for zero-shot voice style transfer, with ${ \pmb x } _ { u i }$ from speaker $u$ and $\boldsymbol { x } _ { v j }$ from speaker $v$ , we encode content $c _ { u i }$ and style $\scriptstyle { \pmb { s } } _ { v }$ , and combine them together to generate a transferred voice $\hat { \pmb { x } } _ { u  v , i } = D ( \pmb { s } _ { v } , \pmb { c } _ { u i } )$ .
+
+Based on the criterion for $\pmb { s }$ in equation (7), a well-learned style encoder $E _ { s }$ pulls all style embeddings $s _ { u i }$ from speaker $u$ together. Suppose $\mathbf { } _ { s \mu }$ is representative of the style embeddings of set $\mathcal { X } _ { u }$ . If we parameterize the distribution $q _ { \phi } ( \bar { \pmb { x } } | s , \pmb { c } ) \propto \mathrm { e x p } ( - \| \pmb { x } - D ( \pmb { s } , \pmb { c } ) \| ^ { 2 } )$ with decoder $D ( s , c )$ , then based on Theorem 3.2, we can estimate the lower bound of $\pmb { \mathcal { T } } ( \pmb { x } ; \pmb { c } | \pmb { s } )$ with the following objective:
+
+$$
+\hat { \mathcal { Z } } _ { 2 } : = \frac { 1 } { N } \sum _ { u = 1 } ^ { M } \sum _ { i = 1 } ^ { N _ { u } } \Big [ - \| \pmb { x } _ { u i } - D \big ( \boldsymbol { c } _ { u i } , \boldsymbol { s } _ { u } \big ) \| ^ { 2 } - \log \Big ( \frac { 1 } { N _ { u } } \sum _ { j = 1 } ^ { N _ { u } } \exp \{ - \| \pmb { x } _ { u j } - D \big ( \boldsymbol { c } _ { u i } , \boldsymbol { s } _ { u } \big ) \| ^ { 2 } \} \Big ) \Big ] .
+$$
+
+When maximizing $\hat { \mathcal { I } } _ { 2 }$ , for speaker $u$ with his/her given voice style $\mathbf { } _ { s _ { u } }$ , we encourage the content embedding $c _ { u i }$ to well reconstruct the original voice ${ \pmb x } _ { u i }$ , with small $\| \pmb { x } _ { u i } - D ( \pmb { c } _ { u i } , \pmb { s } _ { u } ) \|$ . Additionally, the distance $\| \pmb { x } _ { u j } - D ( \pmb { c } _ { u i } , \pmb { s } _ { u } ) \|$ is minimized, ensuring $\mathbf { \mathit { c } } _ { u i }$ does not contain information to reconstruct other voices $\mathbf { \boldsymbol { x } } _ { u j }$ from speaker $u$ . With $\hat { \mathcal { I } } _ { 2 }$ , the correlation between ${ \pmb x } _ { u i }$ and $c _ { u i }$ is amplified, which improves $\mathbf { \mathit { c } } _ { u i }$ in preserving the content information.
+
+# 3.3 MI UPPER BOUND ESTIMATION
+
+The crucial part of our framework is disentangling the style and the content embedding spaces, which imposes (ideally) that the style embedding $\pmb { s }$ excludes any content information and vice versa. Therefore, the mutual information between $\pmb { s }$ and $^ c$ is expected to be minimized. To estimate $\mathcal { T } ( s ; c )$ , we derive a sample-based MI upper bound in Theorem 3.3 base on (4).
+
+Theorem 3.3. If $p ( \boldsymbol { s } | \boldsymbol { c } )$ provides the conditional distribution between variables s and $^ c$ , then
+
+$$
+\mathcal { T } ( s ; c ) \leq \mathbb { E } \Big [ \frac { 1 } { N } \sum _ { u = 1 } ^ { M } \sum _ { i = 1 } ^ { N _ { u } } \Big [ \log p ( s _ { u i } | c _ { u i } ) - \frac { 1 } { N } \sum _ { v = 1 } ^ { M } \sum _ { j = 1 } ^ { N _ { v } } \log p ( s _ { u i } | c _ { v j } ) \Big ] \Big ] .
+$$
+
+The upper bound in (9) requires the ground-truth conditional distribution $p ( \boldsymbol { s } | \boldsymbol { c } )$ , whose closed form is unknown. Therefore, we use a probabilistic neural network $q _ { \theta } ( s | \boldsymbol { c } )$ to approximate $p ( \boldsymbol { s } | \boldsymbol { c } )$ by maximizing the log-likelihood $\begin{array} { r } { \mathcal { F } ( \theta ) = \sum _ { u = 1 } ^ { M } \sum _ { i = 1 } ^ { N _ { u } } \log q _ { \theta } ( s _ { u i } | \boldsymbol { c } _ { u i } ) } \end{array}$ . With the learned $q _ { \theta } ( s | c )$ , the
+
+objective for minimizing $\mathcal { T } ( s ; c )$ becomes:
+
+$$
+\hat { \mathcal { Z } } _ { 3 } : = \frac { 1 } { N } \sum _ { u = 1 } ^ { M } \sum _ { i = 1 } ^ { N _ { u } } \left[ \log q _ { \theta } ( \boldsymbol s _ { u i } | \boldsymbol c _ { u i } ) - \frac { 1 } { N } \sum _ { v = 1 } ^ { M } \sum _ { j = 1 } ^ { N _ { v } } \log q _ { \theta } ( \boldsymbol s _ { u i } | \boldsymbol c _ { v j } ) \right] .
+$$
+
+When weights of encoders $E _ { c } , E _ { s }$ are updated, the embedding spaces $s , c$ change, which leads to the changing of conditional distribution $p ( \boldsymbol { s } | \boldsymbol { c } )$ . Therefore, the neural approximation $q _ { \theta } ( s | c )$ must be updated again. Consequently, during training, the encoders $E _ { c } , E _ { s }$ and the approximation $q _ { \theta } ( s | c )$ are updated iteratively. In the Supplementary Material, we further discuss that with a good approximation $q _ { \theta } ( s | \boldsymbol { c } ) , \hat { Z } _ { 3 }$ remains an MI upper bound.
+
+# 3.4 ENCODER-DECODER FRAMEWORK
+
+With the aforementioned MI estimates $\hat { \mathcal { T } } _ { 1 } , \hat { \mathcal { T } } _ { 2 }$ , and $\hat { \mathcal { T } } _ { 3 }$ , the final training loss of our method is
+
+$$
+\mathcal { L } ^ { * } = [ \hat { \mathcal { T } } _ { 3 } - \hat { \mathcal { T } } _ { 1 } - \hat { \mathcal { T } } _ { 2 } ] - \beta \mathcal { F } ( \boldsymbol { \theta } ) ,
+$$
+
+where $\beta$ is a positive number re-weighting the two objective terms. Term $\hat { \mathcal { T } } _ { 3 } - \hat { \mathcal { T } } _ { 1 } - \hat { \mathcal { T } } _ { 2 }$ is minimized w.r.t the parameters in encoders $E _ { c } , E _ { s }$ and decoder $D$ ; term ${ \mathcal { F } } ( \theta )$ as the likelihood function of $q _ { \theta } ( s | c )$ is maximized w.r.t the parameter $\theta$ . In practice, the two terms are updated iteratively with gradient descent (by fixing one and updating another). The training and transfer processes of our model are shown in Figure 1. We name this MI-guided learning framework as Information-theoretic Disentangled Embedding for Voice Conversion (IDE-VC).
+
+# 4 RELATED WORK
+
+Many-to-many Voice Conversion Traditional voice style transfer methods mainly focus on one-toone and many-to-one conversion tasks, which can only transfer voices into one target speaking style. This constraint limits the applicability of the methods. Recently, several many-to-many voice conversion methods have been proposed, to convert voices in broader application scenarios. StarGANVC (Kameoka et al., 2018) uses StarGAN (Choi et al., 2018) to enable many-to-many transfer, in which voices are fed into a unique generator conditioned on the target speaker identity. A discriminator is also used to evaluate generation quality and transfer accuracy. Blow (Serra et al., 2019) is a \` flow-based generative model (Kingma & Dhariwal, 2018), that maps voices from different speakers into the same latent space via normalizing flow (Rezende & Mohamed, 2015). The conversion is accomplished by transforming the latent representation back to the observation space with the target speaker’s identifier. Two other many-to-many conversion models, AUTOVC (Qian et al., 2019) and AdaIN-VC (Chou & Lee, 2019), extend applications into zero-shot scenarios, i.e., conversion from/to a new speaker (unseen during training), based on only a few utterances. Both AUTOVC and AdaIN-VC construct an encoder-decoder framework, which extracts the style and content of one speech sample into separate latent embeddings. Then when a new voice from an unseen speaker comes, both its style and content embeddings can be extracted directly. However, as discussed in the Introduction, both methods do not have explicit regularizers to reduce the correlation between style and content embeddings, which limits their performance.
+
+Disentangled Representation Learning Disentangled representation learning (DRL) aims to encode data points into separate independent embedding subspaces, where different subspaces represent different data attributes. DRL methods can be classified into unsupervised and supervised approaches. Under unsupervised setups, Burgess et al. (2018), Higgins et al. (2016) and Kim & Mnih (2018) use latent embeddings to reconstruct the original data while keeping each dimension of the embeddings independent with correlation regularizers. This has been challenged by Locatello et al. (2019), in that each part of the learned embeddings may not be mapped to a meaningful data attribute. In contrast, supervised DRL methods effectively learn meaningful disentangled embedding parts by adding different supervision to different embedding components. Between the two embedding parts, the correlation is still required to be reduced to prevent the revealing of information to each other. The correlation-reducing methods mainly focus on adversarial training between embedding parts (Hjelm et al., 2018; Kim & Mnih, 2018), and mutual information minimization (Chen et al., 2018; Cheng et al., 2020b). By applying operations such as switching and combining, one can use disentangled representations to improve empirical performance on downstream tasks, e.g. conditional generation (Burgess et al., 2018), domain adaptation (Gholami et al., 2020), and few-shot learning (Higgins et al., 2017).
+
+# 5 EXPERIMENTS
+
+We evaluate our IDE-VC on real-world voice a dataset under both many-to-many and zero-shot VST setups. The selected dataset is CSTR Voice Cloning Toolkit (VCTK) (Yamagishi et al., 2019), which includes 46 hours of audio from 109 speakers. Each speaker reads a different sets of utterances, and the training voices are provided in a non-parallel manner. The audios are downsampled at $1 6 \mathrm { k H z }$ . In the following, we first describe the evaluation metrics and the implementation details, and then analyze our model’s performance relative to other baselines under many-to-many and zero-shot VST settings.
+
+# 5.1 EVALUATION METRICS
+
+Objective Metrics We consider two objective metrics: Speaker verification accuracy (Verification) and the Mel-Cepstral Distance (Distance) (Kubichek, 1993). The speaker verification accuracy measures whether the transferred voice belongs to the target speaker. For fair comparison, we used a third-party pre-trained speaker encoder Resemblyzer1 to classify the speaker identity from the transferred voices. Specifically, style centroids for speakers are learned with ground-truth voice samples. For a transferred voice, we encode it via the pre-trained speaker encoder and find the speaker with the closest style centroid as the identity prediction. For the Distance, the vanilla Mel-Cepstral Distance (MCD) cannot handle the time alignment issue described in Section 2. To make reasonable comparisons between the generation and ground truth, we apply the Dynamic Time Warping (DTW) algorithm (Berndt & Clifford, 1994) to automatically align the time-evolving sequences before calculating MCD. This DTW-MCD distance measures the similarity of the transferred voice and the real voice from the target speaker. Since the calculation of DTW-MCD requires parallel data, we select voices with the same content from the VCTK dataset as testing pairs. Then we transfer one voice in the pair and calculate DTW-MCD with the other voice as reference.
+
+Subjective Metrics Following Wester et al. (Wester et al., 2016), we use the naturalness of the speech (Naturalness), and the similarity of the transferred speech to target identity (Similarity) as subjective metrics. For Naturalness, annotators are asked to rate the score from 1-5 for each transferred speech.For the Similarity, the annotators are presented with two audios (the converted speech and the corresponding reference), and are asked to rate the score from 1 to 4. For both scores, the higher the better. Following the setting in Blow (Serra et al., 2019), we report Similarity defined \` as a total percentage from the binary rating. The evaluation of both subjective metrics is conducted on Amazon Mechanical Turk (MTurk)2. More details about evaluation metrics are provided in the Supplementary Material.
+
+# 5.2 IMPLEMENTATION DETAILS
+
+Following AUTOVC (Qian et al., 2019), our model inputs are represented via mel-spectrogram. The number of mel-frequency bins is set as 80. When voices are generated, we adopt the WaveNet vocoder (Oord et al., 2016) pre-trained on the VCTK corpus to invert the spectrogram signal back to a waveform. The spectrogram is first upsampled with deconvolutional layers to match the sampling rate, and then a standard 40-layer WaveNet is applied to generate speech waveforms. Our model is implememted with Pytorch and takes 1 GPU day on an Nvidia Xp to train.
+
+Encoder Architecture The speaker encoder consists of a 2-layer long short-term memory (LSTM) with cell size of 768, and a fully-connected layer with output dimension 256. The speaker encoder is initialized with weights from a pretrained GE2E (Wan et al., 2018) encoder. The input of the content encoder is the concatenation of the mel-spectrogram signal and the corresponding speaker embedding. The content encoder consists of three convolutional layers with 512 channels, and two layers of a bidirectional LSTM with cell dimension 32. Following the setup in AUTOVC (Qian et al., 2019), the forward and backward outputs of the bi-directional LSTM are downsampled by 16.
+
+Decoder Architecture Following AUTOVC (Qian et al., 2019), the initial decoder consists of a three-layer convolutional neural network (CNN) with 512 channels, three LSTM layers with cell dimension 1024, and another convolutional layer to project the output of the LSTM to dimension of 80. To enhance the quality of the spectrogram, following AUTOVC (Qian et al., 2019), we use a post-network consisting of five convolutional layers with 512 channels for the first four layers, and
+
+Table 1: Many-to-many VST evaluation results. For all metrics except Distance, higher is better.   
+
+<table><tr><td rowspan="2">Metric</td><td colspan="2">Objective</td><td colspan="2">Subjective</td></tr><tr><td>Distance</td><td>Verification[%]</td><td>Naturalness [1-5]</td><td>Similarity [%]</td></tr><tr><td>StarGAN</td><td>6.73</td><td>71.1</td><td>2.77</td><td>51.5</td></tr><tr><td>AdaIN-VC</td><td>6.98</td><td>85.5</td><td>2.19</td><td>50.8</td></tr><tr><td>AUTOVC</td><td>6.73</td><td>89.9</td><td>3.25</td><td>55.0</td></tr><tr><td>Blow</td><td>8.08</td><td>=</td><td>2.11</td><td>10.8</td></tr><tr><td>IDE-VC (Ours)</td><td>6.70</td><td>92.2</td><td>3.26</td><td>68.5</td></tr></table>
+
+Table 2: Zero-Shot VST evaluation results. For all metrics except Distance, higher is better.   
+
+<table><tr><td rowspan="2">Metric</td><td colspan="2">Objective</td><td colspan="2"> Subjective</td></tr><tr><td>Distance</td><td>Verification[%]</td><td>Naturalness [1-5]</td><td>Similarity [%]</td></tr><tr><td>AdaIN-VC</td><td>6.37</td><td>76.7</td><td>2.67</td><td>68.4</td></tr><tr><td>AUTOVC</td><td>6.68</td><td>60.0</td><td>2.19</td><td>58.6</td></tr><tr><td>IDE-VC (Ours)</td><td>6.31</td><td>81.1</td><td>3.33</td><td>76.4</td></tr></table>
+
+80 channels for the last layer. The output of the post-network can be viewed as a residual signal. The final conversion signal is computed by directly adding the output of the initial decoder and the post-network. The reconstruction loss is applied to both the output of the initial decoder and the final conversion signal.
+
+Approximation Network Architecture As described in Section 3.3, minimizing the mutual information between style and content embeddings requires an auxiliary variational approximation $q _ { \theta } ( s | c )$ . For implementation, we parameterize the variational distribution in the Gaussian distribution family $q _ { \theta } ( \bar { \mathbf { \rho } } _ { s } | \mathbf { \rho } _ { c } ) = \mathcal { N } ( \mu _ { \theta } ( \mathbf { \rho } _ { c } ) , \mathbf { \bar { \alpha } } \sigma _ { \theta } ^ { 2 } ( \mathbf { \rho } _ { c } ) \cdot \mathbf { I } )$ , where mean $\mu _ { \theta } ( \cdot )$ and variance $\sigma _ { \theta } ^ { 2 } ( \cdot )$ are two-layer fully-connected networks with $\operatorname { t a n h } ( { \cdot } )$ as the activation function. With the Gaussian parameterization, the likelihoods in objective $\hat { \mathcal { T } } _ { 3 }$ can be calculated in closed form.
+
+# 5.3 STYLE TRANSFER PERFORMANCE
+
+For the many-to-many VST task, we randomly select $10 \%$ of the sentences for validation and $10 \%$ of the sentences for testing from the VCTK dataset, following the setting in Blow (Serra et al., 2019). \` The rest of the data are used for training in a non-parallel scheme. For evaluation, we select voice pairs from the testing set, in which each pair of voices have the same content but come from different speakers. In each testing pair, we conduct transfer from one voice to the other voice’s speaking style, and then we compare the transferred voice and the other voice as evaluating the model performance. We test our model with four competitive baselines: Blow (Serra et al., 2019) \` 3, AUTOVC (Qian et al., 2019), AdaIN-VC (Chou & Lee, 2019) and StarGAN-VC (Kameoka et al., 2018). The detailed implementation of these four methods are provided in the Supplementary Material. Table 1 shows the subjective and objective evaluation for the many-to-many VST task. Both methods with the encoder-decoder framework, AdaIN-VC and AUTOVC, have competitive results. However, our IDE-VC outperforms the other baselines on all metrics, demonstrating that the style-content disentanglement in the latent space improves the performance of the encoder-decoder framework.
+
+For the zero-shot VST task, we use the same train-validation dataset split as in the many-to-many setup. The testing data are selected to guarantee that no test speaker has any utterance in the training set. We compare our model with the only two baselines, AUTOVC (Qian et al., 2019) and AdaINVC (Chou & Lee, 2019), that are able to handle voice transfer for newly-coming unseen speakers. We used the same implementations of AUTOVC and AdaIN-VC as in the many-to-many VST. The evaluation results of zero-shot VST are shown in Table 2, among the two baselines AdaIN-VC performs better than AUTOVC overall.Our IDE-VC outperforms both baseline methods, on all metrics. All three tested models have encoder-decoder transfer frameworks, the superior performance of IDE-VC indicates the effectiveness of our disentangled representation learning scheme. More evaluation details are provided in the supplementary material.
+
+![](images/6b582b1d9dadfa47cab2f94b7090d8878092a61b53bc79d2a18b689eb1b3f0ee.jpg)  
+Figure 2: Left: t-SNE visualization for speaker embeddings. Right: t-SNE visualization for content embedding. The embeddings are extracted from the voice samples of 10 different speakers.
+
+# 5.4 DISENTANGLEMENT DISCUSSION
+
+Besides the performance comparison with other VST baselines, we demonstrate the capability of our information-theoretic disentangled representation learning scheme. First, we conduct a tSNE (Maaten & Hinton, 2008) visualization of the latent spaces of the IDE-VC model. As shown in the left of Figure 2, style embeddings from the same speaker are well clustered, and style embeddings from different speakers separate in a clean manner. The clear pattern indicates our style encoder $E _ { s }$ can verify the speakers’ identity from the voice samples. In contrast, the content embeddings (in the right of Figure 2) are indistinguishable for different speakers, which means our content encoder $E _ { c }$ successfully eliminates speaker-related information and extracts rich semantic content from the data.
+
+We also empirically evaluate the disentanglement, by predicting the speakers’ identity based on only the content embeddings. A two-layer fully-connected network is trained on the testing set with a content embedding as input, and the corresponding speaker identity as output. We compare our IDE-VC with AUTOVC and AdaIN-VC, which also output content embeddings. The classification results are shown in Table 3. Our IDE-VC reaches the lowest classification accuracy, indicating that the content embeddings learned by IDE-VC contains the least speaker-related information. Therefore, our IDE-VC learns disentangled representations with high quality compared with other baselines.
+
+Table 3: Speaker identity prediction accuracy on content embedding.   
+
+<table><tr><td></td><td>Accuracy[%]</td></tr><tr><td>AUTOVC</td><td>9.5</td></tr><tr><td>AdaIN-VC</td><td>19.0</td></tr><tr><td>IDE-VC</td><td>8.1</td></tr></table>
+
+# 5.5 ABLATION STUDY
+
+Moreover, we have considered an ablation study that addresses performance effects from different learning losses in (11), with results shown in Table 4. We compare our model with two models trained by part of the loss function in (11), while keeping the other training setups unchanged, including the model structure. From the results, when the model is trained without the style encoder loss term $\hat { \mathcal { I } } _ { 1 }$ , a transferred voice still is generated, but with a large distance to the ground truth. The verification accuracy also significantly decreases with no speaker-related information utilized. When the disentangling term $\hat { \mathcal { T } } _ { 3 }$ is removed, the model still reaches competitive performance, because the style encoder $E _ { s }$ and decoder $D$ are well trained by $\hat { \mathcal { I } } _ { 1 }$ and $\hat { \mathcal { I } } _ { 2 }$ . However, when adding term $\hat { \mathcal { T } } _ { 3 }$ , we disentangle the style and content spaces, and improve the transfer quality with higher verification accuracy and less distortion. The performance without term $\hat { \mathcal { I } } _ { 2 }$ is not reported, because the model cannot even generate fluent speech without the reconstruction loss.
+
+Table 4: Ablation study with different training losses. Performance is measured by objective metrics.   
+
+<table><tr><td colspan="2">Distance Verification[%]</td></tr><tr><td>Without I1</td><td>9.81 11.1</td></tr><tr><td>Without I3 6.73</td><td>89.4</td></tr><tr><td>IDE-VC 5.66</td><td>92.2</td></tr></table>
+
+# 6 CONCLUSIONS
+
+We have improved the encoder-decoder voice style transfer framework by disentangled latent representation learning. To effectively induce the style and content information of speech into independent embedding latent spaces, we minimize a sample-based mutual information upper bound between style and content embeddings. The disentanglement of the two embedding spaces ensures the voice transfer accuracy without information revealed from each other. We have also derived two new multi-group mutual information lower bounds, which are maximized during training to enhance the representativeness of the latent embeddings. On the real-world VCTK dataset, our model outperforms previous works under both many-to-many and zero-shot voice style transfer setups. Our model can be naturally extended to other style transfer tasks modeling time-evolving sequences, e.g., video and music style transfer. Moreover, our general multi-group mutual information lower bounds have broader potential applications in other representation learning tasks.
+
+# ACKNOWLEDGEMENTS
+
+This research was supported in part by the DOE, NSF and ONR.
+
+# REFERENCES
+
+Alexander A Alemi, Ian Fischer, Joshua V Dillon, and Kevin Murphy. Deep variational information bottleneck. arXiv preprint arXiv:1612.00410, 2016.
+
+Donald J Berndt and James Clifford. Using dynamic time warping to find patterns in time series. In KDD workshop, pp. 359–370. Seattle, WA, 1994.
+
+Christopher P Burgess, Irina Higgins, Arka Pal, Loic Matthey, Nick Watters, Guillaume Desjardins, and Alexander Lerchner. Understanding disentangling in beta-vae. arXiv preprint arXiv:1804.03599, 2018.
+
+Santosh V Chapaneri. Spoken digits recognition using weighted mfcc and improved features for dynamic time warping. International Journal of Computer Applications, 40(3):6–12, 2012.
+
+Dongdong Chen, Jing Liao, Lu Yuan, Nenghai Yu, and Gang Hua. Coherent online video style transfer. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1105– 1114, 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.
+
+Pengyu Cheng, Weituo Hao, Shuyang Dai, Jiachang Liu, Zhe Gan, and Lawrence Carin. Club: A contrastive log-ratio upper bound of mutual information. In International Conference on Machine Learning, pp. 1779–1788. PMLR, 2020a.
+
+Pengyu Cheng, Renqiang Min, Shen Dinghan, Christopher Malon, Yizhe Zhang, Li Yitong, and Lawrence Carin. Improving disentangled text representation learning with information-theoretic guidance. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, 2020b.
+
+Yunjey Choi, Minje Choi, Munyoung Kim, Jung-Woo Ha, Sunghun Kim, and Jaegul Choo. Stargan: Unified generative adversarial networks for multi-domain image-to-image translation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8789–8797, 2018.
+
+Ju-chieh Chou and Hung-Yi Lee. One-shot voice conversion by separating speaker and content representations with instance normalization. Proc. Interspeech 2019, pp. 664–668, 2019.
+
+Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012.
+
+Shivanker Dev Dhingra, Geeta Nijhawan, and Poonam Pandit. Isolated speech recognition using mfcc and dtw. International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering, 2(8):4085–4092, 2013.
+
+Leon A Gatys, Alexander S Ecker, and Matthias Bethge. Image style transfer using convolutional neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2414–2423, 2016.
+
+Behnam Gholami, Pritish Sahu, Ognjen Rudovic, Konstantinos Bousmalis, and Vladimir Pavlovic. Unsupervised multi-target domain adaptation: An information theoretic approach. IEEE Transactions on Image Processing, 2020.
+
+Elizabeth Godoy, Olivier Rosec, and Thierry Chonavel. Voice conversion using dynamic frequency warping with amplitude scaling, for parallel or nonparallel corpora. IEEE Transactions on Audio, Speech, and Language Processing, 20(4):1313–1323, 2011.
+
+Michael Gutmann and Aapo Hyvarinen. Noise-contrastive estimation: A new estimation principle ¨ for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pp. 297–304, 2010.
+
+Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. In International Conference on Learning Representations, 2016.
+
+Irina Higgins, Arka Pal, Andrei Rusu, Loic Matthey, Christopher Burgess, Alexander Pritzel, Matthew Botvinick, Charles Blundell, and Alexander Lerchner. Darla: Improving zero-shot transfer in reinforcement learning. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1480–1490. JMLR. org, 2017.
+
+R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. arXiv preprint arXiv:1808.06670, 2018.
+
+Chin-Cheng Hsu, Hsin-Te Hwang, Yi-Chiao Wu, Yu Tsao, and Hsin-Min Wang. Voice conversion from non-parallel corpora using variational auto-encoder. In 2016 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA), pp. 1–6. IEEE, 2016.
+
+Haozhi Huang, Hao Wang, Wenhan Luo, Lin Ma, Wenhao Jiang, Xiaolong Zhu, Zhifeng Li, and Wei Liu. Real-time neural style transfer for videos. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 783–791, 2017.
+
+Xun Huang and Serge Belongie. Arbitrary style transfer in real-time with adaptive instance normalization. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1501– 1510, 2017.
+
+Hirokazu Kameoka, Takuhiro Kaneko, Kou Tanaka, and Nobukatsu Hojo. Stargan-vc: Non-parallel many-to-many voice conversion using star generative adversarial networks. In 2018 IEEE Spoken Language Technology Workshop (SLT), pp. 266–273. IEEE, 2018.
+
+Takuhiro Kaneko and Hirokazu Kameoka. Cyclegan-vc: Non-parallel voice conversion using cycleconsistent adversarial networks. In 2018 26th European Signal Processing Conference (EUSIPCO), pp. 2100–2104. IEEE, 2018.
+
+Hyunjik Kim and Andriy Mnih. Disentangling by factorising. In International Conference on Machine Learning, pp. 2649–2658, 2018.
+
+Durk P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. In Advances in Neural Information Processing Systems, pp. 10215–10224, 2018.
+
+Robert Kubichek. Mel-cepstral distance measure for objective speech quality assessment. In Proceedings of IEEE Pacific Rim Conference on Communications Computers and Signal Processing, volume 1, pp. 125–128. IEEE, 1993.
+
+Guillaume Lample, Sandeep Subramanian, Eric Smith, Ludovic Denoyer, Marc’Aurelio Ranzato, and Y-Lan Boureau. Multiple-attribute text rewriting. In International Conference on Learning Representations, 2019.
+
+Chung-Han Lee and Chung-Hsien Wu. Map-based adaptation for speech conversion using adaptation data selection and non-parallel training. In Ninth International Conference on Spoken Language Processing, 2006.
+
+Francesco Locatello, Stefan Bauer, Mario Lucic, Gunnar Raetsch, Sylvain Gelly, Bernhard Scholkopf, and Olivier Bachem. Challenging common assumptions in the unsupervised learning ¨ of disentangled representations. In International Conference on Machine Learning, pp. 4114– 4124, 2019.
+
+Fujun Luan, Sylvain Paris, Eli Shechtman, and Kavita Bala. Deep photo style transfer. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4990–4998, 2017.
+
+Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(Nov):2579–2605, 2008.
+
+Seyed Hamidreza Mohammadi and Alexander Kain. An overview of voice conversion systems. Speech Communication, 88:65–82, 2017.
+
+Kevin R Moon and Alfred O Hero. Ensemble estimation of multivariate f-divergence. In 2014 IEEE International Symposium on Information Theory, pp. 356–360. IEEE, 2014.
+
+Lindasalwa Muda, Mumtaj Begam, and Irraivan Elamvazuthi. Voice recognition algorithms using mel frequency cepstral coefficient (mfcc) and dynamic time warping (dtw) techniques. arXiv preprint arXiv:1003.4083, 2010.
+
+Arsha Nagrani, Joon Son Chung, and Andrew Zisserman. Voxceleb: a large-scale speaker identification dataset. arXiv preprint arXiv:1706.08612, 2017.
+
+Keigo Nakamura, Tomoki Toda, Hiroshi Saruwatari, and Kiyohiro Shikano. Speaking aid system for total laryngectomees using voice conversion of body transmitted artificial speech. In Ninth International Conference on Spoken Language Processing, 2006.
+
+XuanLong Nguyen, Martin J Wainwright, and Michael I Jordan. Estimating divergence functionals and the likelihood ratio by convex risk minimization. IEEE Transactions on Information Theory, 56(11):5847–5861, 2010.
+
+Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016.
+
+Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.
+
+Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev Khudanpur. Librispeech: an asr corpus based on public domain audio books. In 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5206–5210. IEEE, 2015.
+
+Kaizhi Qian, Yang Zhang, Shiyu Chang, Xuesong Yang, and Mark Hasegawa-Johnson. Autovc: Zero-shot voice style transfer with only autoencoder loss. In International Conference on Machine Learning, pp. 5210–5219, 2019.
+
+Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In International Conference on Machine Learning, pp. 1530–1538, 2015.
+
+Yuki Saito, Yusuke Ijima, Kyosuke Nishida, and Shinnosuke Takamichi. Non-parallel voice conversion using variational autoencoders conditioned by phonetic posteriorgrams and d-vectors. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5274–5278. IEEE, 2018.
+
+Joan Serra, Santiago Pascual, and Carlos Segura Perales. Blow: a single-scale hyperconditioned \` flow for non-parallel raw-audio voice conversion. In Advances in Neural Information Processing Systems, pp. 6790–6800, 2019.
+
+Tianxiao Shen, Tao Lei, Regina Barzilay, and Tommi Jaakkola. Style transfer from non-parallel text by cross-alignment. In Advances in neural information processing systems, pp. 6830–6841, 2017.
+
+Berrak Sisman, Mingyang Zhang, Sakriani Sakti, Haizhou Li, and Satoshi Nakamura. Adaptive wavenet vocoder for residual compensation in gan-based voice conversion. In 2018 IEEE Spoken Language Technology Workshop (SLT), pp. 282–289. IEEE, 2018.
+
+Jiaming Song, Pratyusha Kalluri, Aditya Grover, Shengjia Zhao, and Stefano Ermon. Learning controllable fair representations. In The 22nd International Conference on Artificial Intelligence and Statistics, pp. 2164–2173, 2019.
+
+Dmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky. Instance normalization: The missing ingredient for fast stylization. arXiv preprint arXiv:1607.08022, 2016.
+
+Petar Velickovi ˇ c, William Fedus, William L Hamilton, Pietro Li ´ o, Yoshua Bengio, and R Devon \` Hjelm. Deep graph infomax. arXiv preprint arXiv:1809.10341, 2018.
+
+Fernando Villavicencio and Jordi Bonada. Applying voice conversion to concatenative singing-voice synthesis. In Eleventh annual conference of the international speech communication association, 2010.
+
+Li Wan, Quan Wang, Alan Papir, and Ignacio Lopez Moreno. Generalized end-to-end loss for speaker verification. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4879–4883. IEEE, 2018.
+
+Mirjam Wester, Zhizheng Wu, and Junichi Yamagishi. Analysis of the voice conversion challenge 2016 evaluation results. In Interspeech, pp. 1637–1641, 2016.
+
+Junichi Yamagishi, Christophe Veaux, Kirsten MacDonald, et al. Cstr vctk corpus: English multispeaker corpus for cstr voice cloning toolkit (version 0.92). 2019.
+
+Zichao Yang, Zhiting Hu, Chris Dyer, Eric P Xing, and Taylor Berg-Kirkpatrick. Unsupervised text style transfer using language models as discriminators. In Advances in Neural Information Processing Systems, pp. 7287–7298, 2018.

parse/train/TgSVWXw22FQ/TgSVWXw22FQ_content_list.json

@@ -0,0 +1,1646 @@
+[
+    {
+        "type": "text",
+        "text": "IMPROVING ZERO-SHOT VOICE STYLE TRANSFER VIADISENTANGLED REPRESENTATION LEARNING",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Siyang Yuan1∗, Pengyu Cheng1∗, Ruiyi Zhang1, Weituo $\\mathbf { H a o } ^ { 1 }$ , Zhe $\\mathbf { G a n ^ { 2 } }$ and Lawrence Carin ",
+        "bbox": [
+            187,
+            169,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Duke University, Durham, North Carolina, USA 2Microsoft, Redmond, Washington, USA {siyang.yuan,pengyu.cheng}@duke.edu ",
+        "bbox": [
+            184,
+            186,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            257,
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+            272
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Voice style transfer, also called voice conversion, seeks to modify one speaker’s voice to generate speech as if it came from another (target) speaker. Previous works have made progress on voice conversion with parallel training data and pre-known speakers. However, zero-shot voice style transfer, which learns from non-parallel data and generates voices for previously unseen speakers, remains a challenging problem. We propose a novel zero-shot voice transfer method via disentangled representation learning. The proposed method first encodes speakerrelated style and voice content of each input voice into separated low-dimensional embedding spaces, and then transfers to a new voice by combining the source content embedding and target style embedding through a decoder. With informationtheoretic guidance, the style and content embedding spaces are representative and (ideally) independent of each other. On real-world VCTK datasets, our method outperforms other baselines and obtains state-of-the-art results in terms of transfer accuracy and voice naturalness for voice style transfer experiments under both many-to-many and zero-shot setups. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            515,
+            336,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Style transfer, which automatically converts a data instance into a target style, while preserving its content information, has attracted considerable attention in various machine learning domains, including computer vision (Gatys et al., 2016; Luan et al., 2017; Huang & Belongie, 2017), video processing (Huang et al., 2017; Chen et al., 2017), and natural language processing (Shen et al., 2017; Yang et al., 2018; Lample et al., 2019; Cheng et al., 2020b). In speech processing, style transfer was earlier recognized as voice conversion (VC) (Muda et al., 2010), which converts one speaker’s utterance, as if it was from another speaker but with the same semantic meaning. Voice style transfer (VST) has received long-term research interest, due to its potential for applications in security (Sisman et al., 2018), medicine (Nakamura et al., 2006), entertainment (Villavicencio & Bonada, 2010) and education (Mohammadi & Kain, 2017), among others. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although widely investigated, VST remains challenging when applied to more general application scenarios. Most of the traditional VST methods require parallel training data, i.e., paired voices from two speakers uttering the same sentence. This constraint limits the application of such models in the real world, where data are often not pair-wise available. Among the few existing models that address non-parallel data (Hsu et al., 2016; Lee & Wu, 2006; Godoy et al., 2011), most methods cannot handle many-to-many transfer (Saito et al., 2018; Kaneko & Kameoka, 2018; Kameoka et al., 2018), which prevents them from converting multiple source voices to multiple target speaker styles. Even among the few non-parallel many-to-many transfer models, to the best of our knowledge, only two models (Qian et al., 2019; Chou & Lee, 2019) allow zero-shot transfer, i.e., conversion from/to newly-coming speakers (unseen during training) without re-training the model. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The only two zero-shot VST models (AUTOVC (Qian et al., 2019) and AdaIN-VC (Chou & Lee, 2019)) share a common weakness. Both methods construct encoder-decoder frameworks, which extract the style and the content information into style and content embeddings, and generate a voice sample by combining a style embedding and a content embedding through the decoder. With the combination of the source content embedding and the target style embedding, the models generate the transferred voice, based only on source and target voice samples. AUTOVC (Qian et al., 2019) uses a GE2E (Wan et al., 2018) pre-trained style encoder to ensure rich speaker-related information in style embeddings. However, AUTOVC has no regularizer to guarantee that the content encoder does not encode any style information. AdaIN-VC (Chou & Lee, 2019) applies instance normalization (Ulyanov et al., 2016) to the feature map of content representations, which helps to eliminate the style information from content embeddings. However, AdaIN-VC fails to prevent content information from being revealed in the style embeddings. Both methods cannot assure that the style and content embeddings are disentangled without information revealed from each other. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "With information-theoretic guidance, we propose a disentangled-representation-learning method to enhance the encoder-decoder zero-shot VST framework, for both style and content information preservation. We call the proposed method Information-theoretic Disentangled Embedding for Voice Conversion (IDE-VC). Our model successfully induces the style and content of voices into independent representation spaces by minimizing the mutual information between style and content embeddings. We also derive two new multi-group mutual information lower bounds, to further improve the representativeness of the latent embeddings. Experiments demonstrate that our method outperforms previous works under both many-to-many and zero-shot transfer setups on two objective metrics and two subjective metrics. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In information theory, mutual information (MI) is a crucial concept that measures the dependence between two random variables. Mathematically, the MI between two variables $_ { \\textbf { \\em x } }$ and $\\textbf {  { y } }$ is ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d01569d7c9b9320b9546c16159cb93a055b2741162a50bcc63b2e874baa6e960.jpg",
+        "text": "$$\n\\mathcal { T } ( \\pmb { x } ; \\pmb { y } ) : = \\mathbb { E } _ { p ( \\pmb { x } , \\pmb { y } ) } \\Big [ \\log \\frac { p ( \\pmb { x } , \\pmb { y } ) } { p ( \\pmb { x } ) p ( \\pmb { y } ) } \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "where $p ( { \\pmb x } )$ and $p ( \\pmb { y } )$ are marginal distributions of $_ { \\textbf { \\em x } }$ and $\\textbf {  { y } }$ , and $p ( { \\pmb x } , { \\pmb y } )$ is the joint distribution. Recently, MI has attracted considerable interest in machine learning as a criterion to minimize or maximize the dependence between different parts of a model (Chen et al., 2016; Alemi et al., 2016; Hjelm et al., 2018; Velickovi ˇ c et al., 2018; Song et al., 2019). However, the calculation of exact MI ´ values is challenging in practice, since the closed form of joint distribution $p ( { \\pmb x } , { \\pmb y } )$ in equation (1) is generally unknown. To solve this problem, several MI estimators have been proposed. For MI maximization tasks, Nguyen, Wainwright and Jordan (NWJ) (Nguyen et al., 2010) propose a lower bound by representing (1) as an $f$ -divergence (Moon & Hero, 2014): ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a850aeb915fbe7f1de1c35009aee2ad7471f8c60eabf196b348110b45df9ee5a.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { T } _ { \\mathrm { N W J } } : = \\mathbb { E } _ { p ( \\pmb { x } , \\pmb { y } ) } [ f ( \\pmb { x } , \\pmb { y } ) ] - e ^ { - 1 } \\mathbb { E } _ { p ( \\pmb { x } ) p ( \\pmb { y } ) } [ e ^ { f ( \\pmb { x } , \\pmb { y } ) } ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "with a score function $f ( { \\pmb x } , { \\pmb y } )$ . Another widely-used sample-based MI lower bound is InfoNCE (Oord et al., 2018), which is derived with Noise Contrastive Estimation (NCE) (Gutmann & Hyvarinen, 2010). With sample pairs ¨ $\\{ ( \\pmb { x } _ { i } , \\pmb { y } _ { i } ) \\} _ { i = 1 } ^ { N }$ drawn from the joint distribution $p ( { \\pmb x } , { \\pmb y } )$ , the InfoNCE lower bound is defined as ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/58190c929edee4858d038e7a4dd70bec12a44ed1025380cd548ac054e79d2e83.jpg",
+        "text": "$$\n\\mathcal { T } _ { \\mathrm { N C E } } : = \\mathbb { E } \\Big [ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\log \\frac { e ^ { f ( { \\pmb x } _ { i } , { \\pmb y } _ { i } ) } } { \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } e ^ { f ( { \\pmb x } _ { i } , { \\pmb y } _ { j } ) } } \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "For MI minimization tasks, Cheng et al. (2020a) proposed a contrastively learned upper bound that requires the conditional distribution $p ( { \\pmb x } | { \\pmb y } )$ : ",
+        "bbox": [
+            171,
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c23818841ae94bc2c9ad823996791022922eb9806a7ce7141a8abc55bba47948.jpg",
+        "text": "$$\n\\mathcal { T } ( \\pmb { x } ; \\pmb { y } ) \\leq \\mathbb { E } \\Big [ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\Big [ \\log p ( \\pmb { x } _ { i } | \\pmb { y } _ { i } ) - \\frac { 1 } { N } \\sum _ { j = 1 } ^ { N } \\log p ( \\pmb { x } _ { j } | \\pmb { y } _ { i } ) \\Big ] \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            297,
+            748,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where the MI is bounded by the log-ratio of conditional distribution $p ( { \\pmb x } | { \\pmb y } )$ between positive and negative sample pairs. In the following, we derive our information-theoretic disentangled representation learning framework for voice style transfer based on the MI estimators described above. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 PROPOSED MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
+            362,
+            872
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We assume access to $N$ audio (voice) recordings from $M$ speakers, where speaker $u$ has $N _ { u }$ voice samples $\\mathcal { X } _ { u } = \\{ \\pmb { x } _ { u i } \\} _ { i = 1 } ^ { N _ { u } }$ . The proposed approach encodes each voice input $\\pmb { x } \\in \\mathcal { X } = \\cup _ { u = 1 } ^ { M } \\pmb { \\chi } _ { u }$ into a speaker-related (style) embedding $\\pmb { s } = E _ { s } ( \\pmb { x } )$ and a content-related embedding ${ \\pmb c } = E _ { c } ( { \\pmb x } )$ , using respectively a style encoder $E _ { s } ( \\cdot )$ and a content encoder $E _ { c } ( \\cdot )$ . To transfer a source ${ \\pmb x } _ { u i }$ from speaker $u$ to the target style of the voice of speaker $v$ , $\\mathbf { \\boldsymbol { x } } _ { v j }$ , we combine the content embedding $\\bar { \\mathbf { c } _ { u i } } = E _ { c } ( \\mathbf { x } _ { u i } )$ and the style embedding ${ \\pmb s } _ { v j } = E _ { s } ( { \\pmb x } _ { v j } )$ to generate the transferred voice $\\hat { \\pmb x } _ { u  v , i } = D ( \\pmb s _ { v j } , \\pmb c _ { u i } )$ with a decoder $D ( s , c )$ . To implement this two-step transfer process, we introduce a novel mutual information (MI)-based learning objective, that induces the style embedding $\\pmb { s }$ and content embedding $^ c$ into independent representation spaces (i.e., ideally, $\\pmb { s }$ contains rich style information of $_ { \\textbf { \\em x } }$ with no content information, and vice versa). In the following, we first describe our MI-based training objective in Section 3.1, and then discuss the practical estimation of the objective in Sections 3.2 and 3.3. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 MI-BASED DISENTANGLING OBJECTIVE ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "From an information-theoretic perspective, to learn representative latent embedding $( s , c )$ , it is desirable to maximize the mutual information between the embedding pair $( s , c )$ and the input $_ { \\textbf { \\em x } }$ . Meanwhile, the style embedding $\\pmb { s }$ and the content $^ c$ are desired to be independent, so that we can control the style transfer process with different style and content attributes. Therefore, we minimize the mutual information $\\bar { \\mathcal { T } } ( s ; c )$ to disentangle the style embedding and content embedding spaces. Consequently, our overall disentangled-representation-learning objective seeks to minimize ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/526a936a0c3f2fac2c684f2ab6b1a70c0adc1236e57dafe77b41af58dcf053a3.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathcal { T } ( \\pmb { s } ; \\pmb { c } ) - \\mathcal { T } ( \\pmb { x } ; \\pmb { s } , \\pmb { c } ) = \\mathcal { T } ( \\pmb { s } ; \\pmb { c } ) - \\mathcal { T } ( \\pmb { x } ; \\pmb { c } | \\pmb { s } ) - \\mathcal { T } ( \\pmb { x } ; \\pmb { s } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            381
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As discussed in Locatello et al. (Locatello et al., 2019), without inductive bias for supervision, the learned representation can be meaningless. To address this problem, we use the speaker identity $\\textbf { \\em u }$ as a variable with values $\\{ 1 , \\dots , M \\}$ to learn representative style embedding $\\pmb { s }$ for speaker-related attributes. Noting that the process from speaker $u$ to his/her voice ${ \\pmb x } _ { u i }$ to the style embedding $s _ { u i }$ (as $\\mathbf { \\delta } \\mathbf { \\boldsymbol { u } } \\to \\mathbf { \\mathcal { x } } \\to \\mathbf { \\mathcal { s } }$ ) is a Markov Chain, we conclude ${ \\mathcal { T } } ( s ; x ) \\geq { \\mathcal { T } } ( s ; u )$ based on the MI data-processing inequality (Cover & Thomas, 2012) (as stated in the Supplementary Material). Therefore, we replace $\\scriptstyle { \\mathcal { T } } ( s ; x )$ in $\\mathcal { L }$ with $\\pmb { \\mathcal { T } } ( \\pmb { s } ; \\pmb { u } )$ and minimize an upper bound instead: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/59d21668ffd521aed11e0603aa88d08069fae713c70a93f717f8019ba776aa1f.jpg",
+        "text": "$$\n\\bar { \\mathcal { L } } = \\mathcal { T } ( \\pmb { s } ; \\pmb { c } ) - \\mathcal { T } ( \\pmb { x } ; \\pmb { c } | \\pmb { s } ) - \\mathcal { T } ( \\pmb { u } ; \\pmb { s } ) \\geq \\mathcal { T } ( \\pmb { s } ; \\pmb { c } ) - \\mathcal { T } ( \\pmb { x } ; \\pmb { c } | \\pmb { s } ) - \\mathcal { T } ( \\pmb { x } ; \\pmb { s } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            493,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In practice, calculating the MI is challenging, as we typically only have access to samples, and lack the required distributions (Chen et al., 2016). To solve this problem, below we provide several MI estimates to the objective terms $\\mathcal { T } ( s ; c )$ , $\\pmb { \\mathcal { T } } ( \\pmb { x } ; \\pmb { c } | \\pmb { s } )$ and $\\mathcal { T } ( u ; s )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 MI LOWER BOUND ESTIMATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To maximize $\\scriptstyle { \\mathcal { T } } ( u ; s )$ , we derive the following multi-group MI lower bound (Theorem 3.1) based on the NWJ bound developed in Nguyen et al. (Nguyen et al., 2010). The detailed proof is provided in the Supplementary Material. Let $\\dot { \\mu } _ { v } ^ { ( - u i ) } = \\mu _ { v }$ represent the mean of all style embeddings in group $\\mathcal { X } _ { v }$ , constituting the style centroid of speaker $v$ ; $\\mu _ { u } ^ { ( - u i ) }$ is the mean of all style embeddings in group $\\mathcal { X } _ { u }$ except data point ${ \\pmb x } _ { u i }$ , representing a leave- ${ \\bf { x } } _ { u i }$ -out style centroid of speaker $u$ . Intuitively, we minimize $\\| s _ { u i } - \\mu _ { u } ^ { ( - u i ) } \\|$ to encourage the style embedding of voice ${ \\pmb x } _ { u i }$ to be more similar to the style centroid of speaker $u$ , while maximizing $\\| s _ { u i } - \\mu _ { v } ^ { ( - u i ) } \\|$ to enlarge the margin between $s _ { u i }$ and the other speakers’ style centroids $\\pmb { \\mu } _ { v }$ . We denote the right-hand side of (7) as $\\hat { \\mathcal { I } } _ { 1 }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1. Let $\\begin{array} { r } { \\pmb { \\mu } _ { v } ^ { ( - u i ) } = \\frac { 1 } { N _ { v } } \\sum _ { k = 1 } ^ { N _ { v } } \\pmb { s } _ { v k } i f u \\neq } \\end{array}$ $u \\ne v$ ; and $\\begin{array} { r } { \\pmb { \\mu } _ { u } ^ { ( - u i ) } = \\frac { 1 } { N _ { u } - 1 } \\sum _ { j \\neq i } \\pmb { s } _ { u j } } \\end{array}$ ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/89fe967f1f4f9ae4a77bec2528a7b0c730e4b35a8554564c14746c99e1121729.jpg",
+        "text": "$$\n\\mathcal { Z } ( u ; s ) \\geq \\mathbb { E } \\Big [ \\frac { 1 } { N } \\sum _ { u = 1 } ^ { M } \\sum _ { i = 1 } ^ { N _ { u } } \\Big [ - \\| s _ { u i } - \\mu _ { u } ^ { ( - u i ) } \\| ^ { 2 } - \\frac { e ^ { - 1 } } { N } \\sum _ { v = 1 } ^ { M } N _ { v } \\exp \\{ - \\| s _ { u i } - \\mu _ { v } ^ { ( - u i ) } \\| ^ { 2 } \\} \\Big ] \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To maximize $\\pmb { \\mathcal { T } } ( \\pmb { x } ; \\pmb { c } | \\pmb { s } )$ , we derive a conditional mutual information lower bound below: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2. Assume that given $\\begin{array} { r } { \\boldsymbol { s } = \\boldsymbol { s } _ { u } , } \\end{array}$ , samples $\\{ ( \\pmb { x } _ { u i } , \\pmb { c } _ { u i } ) \\} _ { i = 1 } ^ { N _ { u } }$ are observed. With a variational distribution $q _ { \\phi } ( { \\pmb x } | { \\pmb s } , { \\pmb c } )$ , we have $\\mathcal { T } ( \\pmb { x } ; \\pmb { c } | \\pmb { s } ) \\geq \\mathbb { E } [ \\hat { \\mathcal { T } } ]$ , where ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b27891ce3d07448729e6549df6f2879916a773bd1fa446549efcf469c29e8b93.jpg",
+        "text": "$$\n\\hat { \\mathcal { T } } = \\frac { 1 } { N } \\sum _ { u = 1 } ^ { M } \\sum _ { i = 1 } ^ { N _ { u } } \\left[ \\log q _ { \\phi } ( \\pmb { x } _ { u i } | \\pmb { c } _ { u i } , \\pmb { s } _ { u } ) - \\log \\left( \\frac { 1 } { N _ { u } } \\sum _ { j = 1 } ^ { N _ { u } } q _ { \\phi } ( \\pmb { x } _ { u j } | \\pmb { c } _ { u i } , \\pmb { s } _ { u } ) \\right) \\right] .\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/49eb8d52383f2f877a1546711e5b0f7b4d073ca4a8800001f2aee488249c822c.jpg",
+        "image_caption": [
+            "Figure 1: Training and transfer processes. (a) Training style encoder $E _ { s }$ with objective $\\hat { \\mathcal { I } } _ { 1 }$ : All voice samples are encoded into style embedding space. For style embedding $s _ { u i }$ of ${ \\pmb x } _ { u i }$ , we minimize its distance with speaker $u$ ’s style centroid $\\pmb { \\mu } _ { u }$ , and maximize its distance to other speaker style centroids $\\pmb { \\mu } _ { v }$ . (b) Training for content encoder $E _ { c }$ and decoder $D$ as objectives $\\hat { \\mathcal { T } } _ { 2 } , \\tilde { \\mathcal { T } } _ { 3 }$ : We encode content $\\mathbf { \\mathit { c } } _ { u i }$ from voice $\\mathbf { \\mathcal { x } } _ { u i }$ from speaker $u$ . The style of speaker $u$ is encoded from another speaker $u$ ’s voice $\\mathbf { \\boldsymbol { x } } _ { u j }$ . The dependency of style and content embedding is minimized with $\\hat { \\mathcal { T } } _ { 3 }$ . With $\\mathbf { \\mathit { c } } _ { u i }$ and $\\mathbf { } _ { s \\alpha }$ , the decoder reconstructs the voice ${ \\pmb x } _ { u i }$ as $\\hat { \\pmb { x } } _ { u i } = D ( \\pmb { s } _ { u } , \\pmb { c } _ { u i } )$ . Then $\\hat { \\mathcal { I } } _ { 2 }$ is calculated based on the original voice $\\mathbf { \\phi } _ { c u i }$ and the reconstruction $\\hat { c } _ { u i }$ . (c) Transfer process: for zero-shot voice style transfer, with ${ \\pmb x } _ { u i }$ from speaker $u$ and $\\boldsymbol { x } _ { v j }$ from speaker $v$ , we encode content $c _ { u i }$ and style $\\scriptstyle { \\pmb { s } } _ { v }$ , and combine them together to generate a transferred voice $\\hat { \\pmb { x } } _ { u  v , i } = D ( \\pmb { s } _ { v } , \\pmb { c } _ { u i } )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Based on the criterion for $\\pmb { s }$ in equation (7), a well-learned style encoder $E _ { s }$ pulls all style embeddings $s _ { u i }$ from speaker $u$ together. Suppose $\\mathbf { } _ { s \\mu }$ is representative of the style embeddings of set $\\mathcal { X } _ { u }$ . If we parameterize the distribution $q _ { \\phi } ( \\bar { \\pmb { x } } | s , \\pmb { c } ) \\propto \\mathrm { e x p } ( - \\| \\pmb { x } - D ( \\pmb { s } , \\pmb { c } ) \\| ^ { 2 } )$ with decoder $D ( s , c )$ , then based on Theorem 3.2, we can estimate the lower bound of $\\pmb { \\mathcal { T } } ( \\pmb { x } ; \\pmb { c } | \\pmb { s } )$ with the following objective: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/035e5666e4a4592d315b6e6c392af030e5ae1b29ecf37e5bbae09eb661379500.jpg",
+        "text": "$$\n\\hat { \\mathcal { Z } } _ { 2 } : = \\frac { 1 } { N } \\sum _ { u = 1 } ^ { M } \\sum _ { i = 1 } ^ { N _ { u } } \\Big [ - \\| \\pmb { x } _ { u i } - D \\big ( \\boldsymbol { c } _ { u i } , \\boldsymbol { s } _ { u } \\big ) \\| ^ { 2 } - \\log \\Big ( \\frac { 1 } { N _ { u } } \\sum _ { j = 1 } ^ { N _ { u } } \\exp \\{ - \\| \\pmb { x } _ { u j } - D \\big ( \\boldsymbol { c } _ { u i } , \\boldsymbol { s } _ { u } \\big ) \\| ^ { 2 } \\} \\Big ) \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When maximizing $\\hat { \\mathcal { I } } _ { 2 }$ , for speaker $u$ with his/her given voice style $\\mathbf { } _ { s _ { u } }$ , we encourage the content embedding $c _ { u i }$ to well reconstruct the original voice ${ \\pmb x } _ { u i }$ , with small $\\| \\pmb { x } _ { u i } - D ( \\pmb { c } _ { u i } , \\pmb { s } _ { u } ) \\|$ . Additionally, the distance $\\| \\pmb { x } _ { u j } - D ( \\pmb { c } _ { u i } , \\pmb { s } _ { u } ) \\|$ is minimized, ensuring $\\mathbf { \\mathit { c } } _ { u i }$ does not contain information to reconstruct other voices $\\mathbf { \\boldsymbol { x } } _ { u j }$ from speaker $u$ . With $\\hat { \\mathcal { I } } _ { 2 }$ , the correlation between ${ \\pmb x } _ { u i }$ and $c _ { u i }$ is amplified, which improves $\\mathbf { \\mathit { c } } _ { u i }$ in preserving the content information. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 MI UPPER BOUND ESTIMATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The crucial part of our framework is disentangling the style and the content embedding spaces, which imposes (ideally) that the style embedding $\\pmb { s }$ excludes any content information and vice versa. Therefore, the mutual information between $\\pmb { s }$ and $^ c$ is expected to be minimized. To estimate $\\mathcal { T } ( s ; c )$ , we derive a sample-based MI upper bound in Theorem 3.3 base on (4). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.3. If $p ( \\boldsymbol { s } | \\boldsymbol { c } )$ provides the conditional distribution between variables s and $^ c$ , then ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7f3ca2b008418ad21b9e28d0de5e15bd13c76b3f292ea5c7e0f8feef1f8e0207.jpg",
+        "text": "$$\n\\mathcal { T } ( s ; c ) \\leq \\mathbb { E } \\Big [ \\frac { 1 } { N } \\sum _ { u = 1 } ^ { M } \\sum _ { i = 1 } ^ { N _ { u } } \\Big [ \\log p ( s _ { u i } | c _ { u i } ) - \\frac { 1 } { N } \\sum _ { v = 1 } ^ { M } \\sum _ { j = 1 } ^ { N _ { v } } \\log p ( s _ { u i } | c _ { v j } ) \\Big ] \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The upper bound in (9) requires the ground-truth conditional distribution $p ( \\boldsymbol { s } | \\boldsymbol { c } )$ , whose closed form is unknown. Therefore, we use a probabilistic neural network $q _ { \\theta } ( s | \\boldsymbol { c } )$ to approximate $p ( \\boldsymbol { s } | \\boldsymbol { c } )$ by maximizing the log-likelihood $\\begin{array} { r } { \\mathcal { F } ( \\theta ) = \\sum _ { u = 1 } ^ { M } \\sum _ { i = 1 } ^ { N _ { u } } \\log q _ { \\theta } ( s _ { u i } | \\boldsymbol { c } _ { u i } ) } \\end{array}$ . With the learned $q _ { \\theta } ( s | c )$ , the ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "objective for minimizing $\\mathcal { T } ( s ; c )$ becomes: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1481b85f3f8409b371afc4a08d4e594be1e612a036a6df94d7d87300b412afbc.jpg",
+        "text": "$$\n\\hat { \\mathcal { Z } } _ { 3 } : = \\frac { 1 } { N } \\sum _ { u = 1 } ^ { M } \\sum _ { i = 1 } ^ { N _ { u } } \\left[ \\log q _ { \\theta } ( \\boldsymbol s _ { u i } | \\boldsymbol c _ { u i } ) - \\frac { 1 } { N } \\sum _ { v = 1 } ^ { M } \\sum _ { j = 1 } ^ { N _ { v } } \\log q _ { \\theta } ( \\boldsymbol s _ { u i } | \\boldsymbol c _ { v j } ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "When weights of encoders $E _ { c } , E _ { s }$ are updated, the embedding spaces $s , c$ change, which leads to the changing of conditional distribution $p ( \\boldsymbol { s } | \\boldsymbol { c } )$ . Therefore, the neural approximation $q _ { \\theta } ( s | c )$ must be updated again. Consequently, during training, the encoders $E _ { c } , E _ { s }$ and the approximation $q _ { \\theta } ( s | c )$ are updated iteratively. In the Supplementary Material, we further discuss that with a good approximation $q _ { \\theta } ( s | \\boldsymbol { c } ) , \\hat { Z } _ { 3 }$ remains an MI upper bound. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 ENCODER-DECODER FRAMEWORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With the aforementioned MI estimates $\\hat { \\mathcal { T } } _ { 1 } , \\hat { \\mathcal { T } } _ { 2 }$ , and $\\hat { \\mathcal { T } } _ { 3 }$ , the final training loss of our method is ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8c796ef0057306b8549d60b22f4c370b973973273fad317a0a529bbb97731174.jpg",
+        "text": "$$\n\\mathcal { L } ^ { * } = [ \\hat { \\mathcal { T } } _ { 3 } - \\hat { \\mathcal { T } } _ { 1 } - \\hat { \\mathcal { T } } _ { 2 } ] - \\beta \\mathcal { F } ( \\boldsymbol { \\theta } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\beta$ is a positive number re-weighting the two objective terms. Term $\\hat { \\mathcal { T } } _ { 3 } - \\hat { \\mathcal { T } } _ { 1 } - \\hat { \\mathcal { T } } _ { 2 }$ is minimized w.r.t the parameters in encoders $E _ { c } , E _ { s }$ and decoder $D$ ; term ${ \\mathcal { F } } ( \\theta )$ as the likelihood function of $q _ { \\theta } ( s | c )$ is maximized w.r.t the parameter $\\theta$ . In practice, the two terms are updated iteratively with gradient descent (by fixing one and updating another). The training and transfer processes of our model are shown in Figure 1. We name this MI-guided learning framework as Information-theoretic Disentangled Embedding for Voice Conversion (IDE-VC). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Many-to-many Voice Conversion Traditional voice style transfer methods mainly focus on one-toone and many-to-one conversion tasks, which can only transfer voices into one target speaking style. This constraint limits the applicability of the methods. Recently, several many-to-many voice conversion methods have been proposed, to convert voices in broader application scenarios. StarGANVC (Kameoka et al., 2018) uses StarGAN (Choi et al., 2018) to enable many-to-many transfer, in which voices are fed into a unique generator conditioned on the target speaker identity. A discriminator is also used to evaluate generation quality and transfer accuracy. Blow (Serra et al., 2019) is a \\` flow-based generative model (Kingma & Dhariwal, 2018), that maps voices from different speakers into the same latent space via normalizing flow (Rezende & Mohamed, 2015). The conversion is accomplished by transforming the latent representation back to the observation space with the target speaker’s identifier. Two other many-to-many conversion models, AUTOVC (Qian et al., 2019) and AdaIN-VC (Chou & Lee, 2019), extend applications into zero-shot scenarios, i.e., conversion from/to a new speaker (unseen during training), based on only a few utterances. Both AUTOVC and AdaIN-VC construct an encoder-decoder framework, which extracts the style and content of one speech sample into separate latent embeddings. Then when a new voice from an unseen speaker comes, both its style and content embeddings can be extracted directly. However, as discussed in the Introduction, both methods do not have explicit regularizers to reduce the correlation between style and content embeddings, which limits their performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Disentangled Representation Learning Disentangled representation learning (DRL) aims to encode data points into separate independent embedding subspaces, where different subspaces represent different data attributes. DRL methods can be classified into unsupervised and supervised approaches. Under unsupervised setups, Burgess et al. (2018), Higgins et al. (2016) and Kim & Mnih (2018) use latent embeddings to reconstruct the original data while keeping each dimension of the embeddings independent with correlation regularizers. This has been challenged by Locatello et al. (2019), in that each part of the learned embeddings may not be mapped to a meaningful data attribute. In contrast, supervised DRL methods effectively learn meaningful disentangled embedding parts by adding different supervision to different embedding components. Between the two embedding parts, the correlation is still required to be reduced to prevent the revealing of information to each other. The correlation-reducing methods mainly focus on adversarial training between embedding parts (Hjelm et al., 2018; Kim & Mnih, 2018), and mutual information minimization (Chen et al., 2018; Cheng et al., 2020b). By applying operations such as switching and combining, one can use disentangled representations to improve empirical performance on downstream tasks, e.g. conditional generation (Burgess et al., 2018), domain adaptation (Gholami et al., 2020), and few-shot learning (Higgins et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We evaluate our IDE-VC on real-world voice a dataset under both many-to-many and zero-shot VST setups. The selected dataset is CSTR Voice Cloning Toolkit (VCTK) (Yamagishi et al., 2019), which includes 46 hours of audio from 109 speakers. Each speaker reads a different sets of utterances, and the training voices are provided in a non-parallel manner. The audios are downsampled at $1 6 \\mathrm { k H z }$ . In the following, we first describe the evaluation metrics and the implementation details, and then analyze our model’s performance relative to other baselines under many-to-many and zero-shot VST settings. ",
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+    {
+        "type": "text",
+        "text": "5.1 EVALUATION METRICS ",
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+        "type": "text",
+        "text": "Objective Metrics We consider two objective metrics: Speaker verification accuracy (Verification) and the Mel-Cepstral Distance (Distance) (Kubichek, 1993). The speaker verification accuracy measures whether the transferred voice belongs to the target speaker. For fair comparison, we used a third-party pre-trained speaker encoder Resemblyzer1 to classify the speaker identity from the transferred voices. Specifically, style centroids for speakers are learned with ground-truth voice samples. For a transferred voice, we encode it via the pre-trained speaker encoder and find the speaker with the closest style centroid as the identity prediction. For the Distance, the vanilla Mel-Cepstral Distance (MCD) cannot handle the time alignment issue described in Section 2. To make reasonable comparisons between the generation and ground truth, we apply the Dynamic Time Warping (DTW) algorithm (Berndt & Clifford, 1994) to automatically align the time-evolving sequences before calculating MCD. This DTW-MCD distance measures the similarity of the transferred voice and the real voice from the target speaker. Since the calculation of DTW-MCD requires parallel data, we select voices with the same content from the VCTK dataset as testing pairs. Then we transfer one voice in the pair and calculate DTW-MCD with the other voice as reference. ",
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+        "type": "text",
+        "text": "Subjective Metrics Following Wester et al. (Wester et al., 2016), we use the naturalness of the speech (Naturalness), and the similarity of the transferred speech to target identity (Similarity) as subjective metrics. For Naturalness, annotators are asked to rate the score from 1-5 for each transferred speech.For the Similarity, the annotators are presented with two audios (the converted speech and the corresponding reference), and are asked to rate the score from 1 to 4. For both scores, the higher the better. Following the setting in Blow (Serra et al., 2019), we report Similarity defined \\` as a total percentage from the binary rating. The evaluation of both subjective metrics is conducted on Amazon Mechanical Turk (MTurk)2. More details about evaluation metrics are provided in the Supplementary Material. ",
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+        "type": "text",
+        "text": "5.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Following AUTOVC (Qian et al., 2019), our model inputs are represented via mel-spectrogram. The number of mel-frequency bins is set as 80. When voices are generated, we adopt the WaveNet vocoder (Oord et al., 2016) pre-trained on the VCTK corpus to invert the spectrogram signal back to a waveform. The spectrogram is first upsampled with deconvolutional layers to match the sampling rate, and then a standard 40-layer WaveNet is applied to generate speech waveforms. Our model is implememted with Pytorch and takes 1 GPU day on an Nvidia Xp to train. ",
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+        "type": "text",
+        "text": "Encoder Architecture The speaker encoder consists of a 2-layer long short-term memory (LSTM) with cell size of 768, and a fully-connected layer with output dimension 256. The speaker encoder is initialized with weights from a pretrained GE2E (Wan et al., 2018) encoder. The input of the content encoder is the concatenation of the mel-spectrogram signal and the corresponding speaker embedding. The content encoder consists of three convolutional layers with 512 channels, and two layers of a bidirectional LSTM with cell dimension 32. Following the setup in AUTOVC (Qian et al., 2019), the forward and backward outputs of the bi-directional LSTM are downsampled by 16. ",
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+    {
+        "type": "text",
+        "text": "Decoder Architecture Following AUTOVC (Qian et al., 2019), the initial decoder consists of a three-layer convolutional neural network (CNN) with 512 channels, three LSTM layers with cell dimension 1024, and another convolutional layer to project the output of the LSTM to dimension of 80. To enhance the quality of the spectrogram, following AUTOVC (Qian et al., 2019), we use a post-network consisting of five convolutional layers with 512 channels for the first four layers, and ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/dcc2b6604d142214a7019ba17af80a4e4e66b71fb68fe549292fe1a772c73614.jpg",
+        "table_caption": [
+            "Table 1: Many-to-many VST evaluation results. For all metrics except Distance, higher is better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Metric</td><td colspan=\"2\">Objective</td><td colspan=\"2\">Subjective</td></tr><tr><td>Distance</td><td>Verification[%]</td><td>Naturalness [1-5]</td><td>Similarity [%]</td></tr><tr><td>StarGAN</td><td>6.73</td><td>71.1</td><td>2.77</td><td>51.5</td></tr><tr><td>AdaIN-VC</td><td>6.98</td><td>85.5</td><td>2.19</td><td>50.8</td></tr><tr><td>AUTOVC</td><td>6.73</td><td>89.9</td><td>3.25</td><td>55.0</td></tr><tr><td>Blow</td><td>8.08</td><td>=</td><td>2.11</td><td>10.8</td></tr><tr><td>IDE-VC (Ours)</td><td>6.70</td><td>92.2</td><td>3.26</td><td>68.5</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/14b6fadc12a091d2aa1d552b0e199445b730272f733a35f3b3c182f50da8e70e.jpg",
+        "table_caption": [
+            "Table 2: Zero-Shot VST evaluation results. For all metrics except Distance, higher is better. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Metric</td><td colspan=\"2\">Objective</td><td colspan=\"2\"> Subjective</td></tr><tr><td>Distance</td><td>Verification[%]</td><td>Naturalness [1-5]</td><td>Similarity [%]</td></tr><tr><td>AdaIN-VC</td><td>6.37</td><td>76.7</td><td>2.67</td><td>68.4</td></tr><tr><td>AUTOVC</td><td>6.68</td><td>60.0</td><td>2.19</td><td>58.6</td></tr><tr><td>IDE-VC (Ours)</td><td>6.31</td><td>81.1</td><td>3.33</td><td>76.4</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "80 channels for the last layer. The output of the post-network can be viewed as a residual signal. The final conversion signal is computed by directly adding the output of the initial decoder and the post-network. The reconstruction loss is applied to both the output of the initial decoder and the final conversion signal. ",
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+    {
+        "type": "text",
+        "text": "Approximation Network Architecture As described in Section 3.3, minimizing the mutual information between style and content embeddings requires an auxiliary variational approximation $q _ { \\theta } ( s | c )$ . For implementation, we parameterize the variational distribution in the Gaussian distribution family $q _ { \\theta } ( \\bar { \\mathbf { \\rho } } _ { s } | \\mathbf { \\rho } _ { c } ) = \\mathcal { N } ( \\mu _ { \\theta } ( \\mathbf { \\rho } _ { c } ) , \\mathbf { \\bar { \\alpha } } \\sigma _ { \\theta } ^ { 2 } ( \\mathbf { \\rho } _ { c } ) \\cdot \\mathbf { I } )$ , where mean $\\mu _ { \\theta } ( \\cdot )$ and variance $\\sigma _ { \\theta } ^ { 2 } ( \\cdot )$ are two-layer fully-connected networks with $\\operatorname { t a n h } ( { \\cdot } )$ as the activation function. With the Gaussian parameterization, the likelihoods in objective $\\hat { \\mathcal { T } } _ { 3 }$ can be calculated in closed form. ",
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+    {
+        "type": "text",
+        "text": "5.3 STYLE TRANSFER PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the many-to-many VST task, we randomly select $10 \\%$ of the sentences for validation and $10 \\%$ of the sentences for testing from the VCTK dataset, following the setting in Blow (Serra et al., 2019). \\` The rest of the data are used for training in a non-parallel scheme. For evaluation, we select voice pairs from the testing set, in which each pair of voices have the same content but come from different speakers. In each testing pair, we conduct transfer from one voice to the other voice’s speaking style, and then we compare the transferred voice and the other voice as evaluating the model performance. We test our model with four competitive baselines: Blow (Serra et al., 2019) \\` 3, AUTOVC (Qian et al., 2019), AdaIN-VC (Chou & Lee, 2019) and StarGAN-VC (Kameoka et al., 2018). The detailed implementation of these four methods are provided in the Supplementary Material. Table 1 shows the subjective and objective evaluation for the many-to-many VST task. Both methods with the encoder-decoder framework, AdaIN-VC and AUTOVC, have competitive results. However, our IDE-VC outperforms the other baselines on all metrics, demonstrating that the style-content disentanglement in the latent space improves the performance of the encoder-decoder framework. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the zero-shot VST task, we use the same train-validation dataset split as in the many-to-many setup. The testing data are selected to guarantee that no test speaker has any utterance in the training set. We compare our model with the only two baselines, AUTOVC (Qian et al., 2019) and AdaINVC (Chou & Lee, 2019), that are able to handle voice transfer for newly-coming unseen speakers. We used the same implementations of AUTOVC and AdaIN-VC as in the many-to-many VST. The evaluation results of zero-shot VST are shown in Table 2, among the two baselines AdaIN-VC performs better than AUTOVC overall.Our IDE-VC outperforms both baseline methods, on all metrics. All three tested models have encoder-decoder transfer frameworks, the superior performance of IDE-VC indicates the effectiveness of our disentangled representation learning scheme. More evaluation details are provided in the supplementary material. ",
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+    {
+        "type": "image",
+        "img_path": "images/6b582b1d9dadfa47cab2f94b7090d8878092a61b53bc79d2a18b689eb1b3f0ee.jpg",
+        "image_caption": [
+            "Figure 2: Left: t-SNE visualization for speaker embeddings. Right: t-SNE visualization for content embedding. The embeddings are extracted from the voice samples of 10 different speakers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.4 DISENTANGLEMENT DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Besides the performance comparison with other VST baselines, we demonstrate the capability of our information-theoretic disentangled representation learning scheme. First, we conduct a tSNE (Maaten & Hinton, 2008) visualization of the latent spaces of the IDE-VC model. As shown in the left of Figure 2, style embeddings from the same speaker are well clustered, and style embeddings from different speakers separate in a clean manner. The clear pattern indicates our style encoder $E _ { s }$ can verify the speakers’ identity from the voice samples. In contrast, the content embeddings (in the right of Figure 2) are indistinguishable for different speakers, which means our content encoder $E _ { c }$ successfully eliminates speaker-related information and extracts rich semantic content from the data. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also empirically evaluate the disentanglement, by predicting the speakers’ identity based on only the content embeddings. A two-layer fully-connected network is trained on the testing set with a content embedding as input, and the corresponding speaker identity as output. We compare our IDE-VC with AUTOVC and AdaIN-VC, which also output content embeddings. The classification results are shown in Table 3. Our IDE-VC reaches the lowest classification accuracy, indicating that the content embeddings learned by IDE-VC contains the least speaker-related information. Therefore, our IDE-VC learns disentangled representations with high quality compared with other baselines. ",
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+    {
+        "type": "table",
+        "img_path": "images/4726e224d38f87bfaf48076f1d7800912001181db37ea84ec0726b47880eb7ab.jpg",
+        "table_caption": [
+            "Table 3: Speaker identity prediction accuracy on content embedding. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Accuracy[%]</td></tr><tr><td>AUTOVC</td><td>9.5</td></tr><tr><td>AdaIN-VC</td><td>19.0</td></tr><tr><td>IDE-VC</td><td>8.1</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.5 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Moreover, we have considered an ablation study that addresses performance effects from different learning losses in (11), with results shown in Table 4. We compare our model with two models trained by part of the loss function in (11), while keeping the other training setups unchanged, including the model structure. From the results, when the model is trained without the style encoder loss term $\\hat { \\mathcal { I } } _ { 1 }$ , a transferred voice still is generated, but with a large distance to the ground truth. The verification accuracy also significantly decreases with no speaker-related information utilized. When the disentangling term $\\hat { \\mathcal { T } } _ { 3 }$ is removed, the model still reaches competitive performance, because the style encoder $E _ { s }$ and decoder $D$ are well trained by $\\hat { \\mathcal { I } } _ { 1 }$ and $\\hat { \\mathcal { I } } _ { 2 }$ . However, when adding term $\\hat { \\mathcal { T } } _ { 3 }$ , we disentangle the style and content spaces, and improve the transfer quality with higher verification accuracy and less distortion. The performance without term $\\hat { \\mathcal { I } } _ { 2 }$ is not reported, because the model cannot even generate fluent speech without the reconstruction loss. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4055a23250caa3130af7478d6117eaae67b3726f5bc35c7aa62b46c9d67c5c10.jpg",
+        "table_caption": [
+            "Table 4: Ablation study with different training losses. Performance is measured by objective metrics. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Distance Verification[%]</td></tr><tr><td>Without I1</td><td>9.81 11.1</td></tr><tr><td>Without I3 6.73</td><td>89.4</td></tr><tr><td>IDE-VC 5.66</td><td>92.2</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have improved the encoder-decoder voice style transfer framework by disentangled latent representation learning. To effectively induce the style and content information of speech into independent embedding latent spaces, we minimize a sample-based mutual information upper bound between style and content embeddings. The disentanglement of the two embedding spaces ensures the voice transfer accuracy without information revealed from each other. We have also derived two new multi-group mutual information lower bounds, which are maximized during training to enhance the representativeness of the latent embeddings. On the real-world VCTK dataset, our model outperforms previous works under both many-to-many and zero-shot voice style transfer setups. Our model can be naturally extended to other style transfer tasks modeling time-evolving sequences, e.g., video and music style transfer. Moreover, our general multi-group mutual information lower bounds have broader potential applications in other representation learning tasks. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This research was supported in part by the DOE, NSF and ONR. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            367,
+            285,
+            382
+        ],
+        "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": [
+            173,
+            390,
+            823,
+            419
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Donald J Berndt and James Clifford. Using dynamic time warping to find patterns in time series. In KDD workshop, pp. 359–370. Seattle, WA, 1994. ",
+        "bbox": [
+            173,
+            428,
+            823,
+            457
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Christopher P Burgess, Irina Higgins, Arka Pal, Loic Matthey, Nick Watters, Guillaume Desjardins, and Alexander Lerchner. Understanding disentangling in beta-vae. arXiv preprint arXiv:1804.03599, 2018. ",
+        "bbox": [
+            176,
+            467,
+            823,
+            508
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Santosh V Chapaneri. Spoken digits recognition using weighted mfcc and improved features for dynamic time warping. International Journal of Computer Applications, 40(3):6–12, 2012. ",
+        "bbox": [
+            171,
+            518,
+            823,
+            547
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Dongdong Chen, Jing Liao, Lu Yuan, Nenghai Yu, and Gang Hua. Coherent online video style transfer. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1105– 1114, 2017. ",
+        "bbox": [
+            174,
+            555,
+            823,
+            598
+        ],
+        "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": [
+            173,
+            607,
+            825,
+            650
+        ],
+        "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": [
+            174,
+            660,
+            825,
+            703
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Pengyu Cheng, Weituo Hao, Shuyang Dai, Jiachang Liu, Zhe Gan, and Lawrence Carin. Club: A contrastive log-ratio upper bound of mutual information. In International Conference on Machine Learning, pp. 1779–1788. PMLR, 2020a. ",
+        "bbox": [
+            173,
+            712,
+            825,
+            755
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Pengyu Cheng, Renqiang Min, Shen Dinghan, Christopher Malon, Yizhe Zhang, Li Yitong, and Lawrence Carin. Improving disentangled text representation learning with information-theoretic guidance. In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, 2020b. ",
+        "bbox": [
+            174,
+            763,
+            825,
+            819
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yunjey Choi, Minje Choi, Munyoung Kim, Jung-Woo Ha, Sunghun Kim, and Jaegul Choo. Stargan: Unified generative adversarial networks for multi-domain image-to-image translation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 8789–8797, 2018. ",
+        "bbox": [
+            174,
+            829,
+            825,
+            885
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ju-chieh Chou and Hung-Yi Lee. One-shot voice conversion by separating speaker and content representations with instance normalization. Proc. Interspeech 2019, pp. 664–668, 2019. ",
+        "bbox": [
+            174,
+            895,
+            820,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012. ",
+        "bbox": [
+            169,
+            103,
+            810,
+            119
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Shivanker Dev Dhingra, Geeta Nijhawan, and Poonam Pandit. Isolated speech recognition using mfcc and dtw. International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering, 2(8):4085–4092, 2013. ",
+        "bbox": [
+            176,
+            127,
+            821,
+            171
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Leon A Gatys, Alexander S Ecker, and Matthias Bethge. Image style transfer using convolutional neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2414–2423, 2016. ",
+        "bbox": [
+            174,
+            181,
+            823,
+            223
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Behnam Gholami, Pritish Sahu, Ognjen Rudovic, Konstantinos Bousmalis, and Vladimir Pavlovic. Unsupervised multi-target domain adaptation: An information theoretic approach. IEEE Transactions on Image Processing, 2020. ",
+        "bbox": [
+            173,
+            234,
+            825,
+            276
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Elizabeth Godoy, Olivier Rosec, and Thierry Chonavel. Voice conversion using dynamic frequency warping with amplitude scaling, for parallel or nonparallel corpora. IEEE Transactions on Audio, Speech, and Language Processing, 20(4):1313–1323, 2011. ",
+        "bbox": [
+            173,
+            286,
+            825,
+            330
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Michael Gutmann and Aapo Hyvarinen. Noise-contrastive estimation: A new estimation principle ¨ for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pp. 297–304, 2010. ",
+        "bbox": [
+            173,
+            339,
+            825,
+            383
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Irina Higgins, Loic Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. In International Conference on Learning Representations, 2016. ",
+        "bbox": [
+            173,
+            392,
+            825,
+            448
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Irina Higgins, Arka Pal, Andrei Rusu, Loic Matthey, Christopher Burgess, Alexander Pritzel, Matthew Botvinick, Charles Blundell, and Alexander Lerchner. Darla: Improving zero-shot transfer in reinforcement learning. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1480–1490. JMLR. org, 2017. ",
+        "bbox": [
+            173,
+            458,
+            825,
+            516
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. arXiv preprint arXiv:1808.06670, 2018. ",
+        "bbox": [
+            176,
+            525,
+            825,
+            569
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chin-Cheng Hsu, Hsin-Te Hwang, Yi-Chiao Wu, Yu Tsao, and Hsin-Min Wang. Voice conversion from non-parallel corpora using variational auto-encoder. In 2016 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA), pp. 1–6. IEEE, 2016. ",
+        "bbox": [
+            174,
+            578,
+            826,
+            635
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Haozhi Huang, Hao Wang, Wenhan Luo, Lin Ma, Wenhao Jiang, Xiaolong Zhu, Zhifeng Li, and Wei Liu. Real-time neural style transfer for videos. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 783–791, 2017. ",
+        "bbox": [
+            174,
+            645,
+            826,
+            688
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xun Huang and Serge Belongie. Arbitrary style transfer in real-time with adaptive instance normalization. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1501– 1510, 2017. ",
+        "bbox": [
+            173,
+            696,
+            825,
+            741
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hirokazu Kameoka, Takuhiro Kaneko, Kou Tanaka, and Nobukatsu Hojo. Stargan-vc: Non-parallel many-to-many voice conversion using star generative adversarial networks. In 2018 IEEE Spoken Language Technology Workshop (SLT), pp. 266–273. IEEE, 2018. ",
+        "bbox": [
+            173,
+            751,
+            823,
+            794
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Takuhiro Kaneko and Hirokazu Kameoka. Cyclegan-vc: Non-parallel voice conversion using cycleconsistent adversarial networks. In 2018 26th European Signal Processing Conference (EUSIPCO), pp. 2100–2104. IEEE, 2018. ",
+        "bbox": [
+            174,
+            803,
+            823,
+            847
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hyunjik Kim and Andriy Mnih. Disentangling by factorising. In International Conference on Machine Learning, pp. 2649–2658, 2018. ",
+        "bbox": [
+            169,
+            856,
+            823,
+            886
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Durk P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. In Advances in Neural Information Processing Systems, pp. 10215–10224, 2018. ",
+        "bbox": [
+            173,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Robert Kubichek. Mel-cepstral distance measure for objective speech quality assessment. In Proceedings of IEEE Pacific Rim Conference on Communications Computers and Signal Processing, volume 1, pp. 125–128. IEEE, 1993. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Guillaume Lample, Sandeep Subramanian, Eric Smith, Ludovic Denoyer, Marc’Aurelio Ranzato, and Y-Lan Boureau. Multiple-attribute text rewriting. In International Conference on Learning Representations, 2019. ",
+        "bbox": [
+            173,
+            156,
+            825,
+            199
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chung-Han Lee and Chung-Hsien Wu. Map-based adaptation for speech conversion using adaptation data selection and non-parallel training. In Ninth International Conference on Spoken Language Processing, 2006. ",
+        "bbox": [
+            173,
+            209,
+            825,
+            252
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Francesco Locatello, Stefan Bauer, Mario Lucic, Gunnar Raetsch, Sylvain Gelly, Bernhard Scholkopf, and Olivier Bachem. Challenging common assumptions in the unsupervised learning ¨ of disentangled representations. In International Conference on Machine Learning, pp. 4114– 4124, 2019. ",
+        "bbox": [
+            173,
+            262,
+            825,
+            318
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Fujun Luan, Sylvain Paris, Eli Shechtman, and Kavita Bala. Deep photo style transfer. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4990–4998, 2017. ",
+        "bbox": [
+            174,
+            328,
+            823,
+            371
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(Nov):2579–2605, 2008. ",
+        "bbox": [
+            173,
+            382,
+            821,
+            411
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Seyed Hamidreza Mohammadi and Alexander Kain. An overview of voice conversion systems. Speech Communication, 88:65–82, 2017. ",
+        "bbox": [
+            173,
+            420,
+            820,
+            450
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Kevin R Moon and Alfred O Hero. Ensemble estimation of multivariate f-divergence. In 2014 IEEE International Symposium on Information Theory, pp. 356–360. IEEE, 2014. ",
+        "bbox": [
+            173,
+            459,
+            823,
+            489
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Lindasalwa Muda, Mumtaj Begam, and Irraivan Elamvazuthi. Voice recognition algorithms using mel frequency cepstral coefficient (mfcc) and dynamic time warping (dtw) techniques. arXiv preprint arXiv:1003.4083, 2010. ",
+        "bbox": [
+            173,
+            498,
+            826,
+            541
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Arsha Nagrani, Joon Son Chung, and Andrew Zisserman. Voxceleb: a large-scale speaker identification dataset. arXiv preprint arXiv:1706.08612, 2017. ",
+        "bbox": [
+            171,
+            551,
+            823,
+            582
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Keigo Nakamura, Tomoki Toda, Hiroshi Saruwatari, and Kiyohiro Shikano. Speaking aid system for total laryngectomees using voice conversion of body transmitted artificial speech. In Ninth International Conference on Spoken Language Processing, 2006. ",
+        "bbox": [
+            176,
+            590,
+            823,
+            633
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "XuanLong Nguyen, Martin J Wainwright, and Michael I Jordan. Estimating divergence functionals and the likelihood ratio by convex risk minimization. IEEE Transactions on Information Theory, 56(11):5847–5861, 2010. ",
+        "bbox": [
+            174,
+            643,
+            825,
+            686
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016. ",
+        "bbox": [
+            174,
+            696,
+            825,
+            739
+        ],
+        "page_idx": 10
+    },
+    {
+        "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": [
+            171,
+            750,
+            823,
+            779
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev Khudanpur. Librispeech: an asr corpus based on public domain audio books. In 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5206–5210. IEEE, 2015. ",
+        "bbox": [
+            174,
+            789,
+            823,
+            833
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Kaizhi Qian, Yang Zhang, Shiyu Chang, Xuesong Yang, and Mark Hasegawa-Johnson. Autovc: Zero-shot voice style transfer with only autoencoder loss. In International Conference on Machine Learning, pp. 5210–5219, 2019. ",
+        "bbox": [
+            174,
+            842,
+            823,
+            885
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In International Conference on Machine Learning, pp. 1530–1538, 2015. ",
+        "bbox": [
+            173,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yuki Saito, Yusuke Ijima, Kyosuke Nishida, and Shinnosuke Takamichi. Non-parallel voice conversion using variational autoencoders conditioned by phonetic posteriorgrams and d-vectors. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 5274–5278. IEEE, 2018. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            160
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Joan Serra, Santiago Pascual, and Carlos Segura Perales. Blow: a single-scale hyperconditioned \\` flow for non-parallel raw-audio voice conversion. In Advances in Neural Information Processing Systems, pp. 6790–6800, 2019. ",
+        "bbox": [
+            174,
+            169,
+            821,
+            210
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Tianxiao Shen, Tao Lei, Regina Barzilay, and Tommi Jaakkola. Style transfer from non-parallel text by cross-alignment. In Advances in neural information processing systems, pp. 6830–6841, 2017. ",
+        "bbox": [
+            173,
+            219,
+            821,
+            250
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Berrak Sisman, Mingyang Zhang, Sakriani Sakti, Haizhou Li, and Satoshi Nakamura. Adaptive wavenet vocoder for residual compensation in gan-based voice conversion. In 2018 IEEE Spoken Language Technology Workshop (SLT), pp. 282–289. IEEE, 2018. ",
+        "bbox": [
+            174,
+            257,
+            823,
+            301
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Jiaming Song, Pratyusha Kalluri, Aditya Grover, Shengjia Zhao, and Stefano Ermon. Learning controllable fair representations. In The 22nd International Conference on Artificial Intelligence and Statistics, pp. 2164–2173, 2019. ",
+        "bbox": [
+            173,
+            309,
+            823,
+            352
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Dmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky. Instance normalization: The missing ingredient for fast stylization. arXiv preprint arXiv:1607.08022, 2016. ",
+        "bbox": [
+            173,
+            361,
+            823,
+            390
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Petar Velickovi ˇ c, William Fedus, William L Hamilton, Pietro Li ´ o, Yoshua Bengio, and R Devon \\` Hjelm. Deep graph infomax. arXiv preprint arXiv:1809.10341, 2018. ",
+        "bbox": [
+            174,
+            398,
+            823,
+            428
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Fernando Villavicencio and Jordi Bonada. Applying voice conversion to concatenative singing-voice synthesis. In Eleventh annual conference of the international speech communication association, 2010. ",
+        "bbox": [
+            174,
+            436,
+            825,
+            479
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Li Wan, Quan Wang, Alan Papir, and Ignacio Lopez Moreno. Generalized end-to-end loss for speaker verification. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4879–4883. IEEE, 2018. ",
+        "bbox": [
+            173,
+            488,
+            823,
+            531
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Mirjam Wester, Zhizheng Wu, and Junichi Yamagishi. Analysis of the voice conversion challenge 2016 evaluation results. In Interspeech, pp. 1637–1641, 2016. ",
+        "bbox": [
+            169,
+            540,
+            823,
+            569
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Junichi Yamagishi, Christophe Veaux, Kirsten MacDonald, et al. Cstr vctk corpus: English multispeaker corpus for cstr voice cloning toolkit (version 0.92). 2019. ",
+        "bbox": [
+            169,
+            577,
+            823,
+            607
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Zichao Yang, Zhiting Hu, Chris Dyer, Eric P Xing, and Taylor Berg-Kirkpatrick. Unsupervised text style transfer using language models as discriminators. In Advances in Neural Information Processing Systems, pp. 7287–7298, 2018. ",
+        "bbox": [
+            176,
+            614,
+            823,
+            659
+        ],
+        "page_idx": 11
+    }
+]

parse/train/rJ5C67-C-/rJ5C67-C-.md

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+# HYPEREDGE2VEC: DISTRIBUTED REPRESENTATIONS FOR HYPEREDGES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Data structured in form of overlapping or non-overlapping sets is found in a variety of domains, sometimes explicitly but often subtly. For example, teams, which are of prime importance in social science studies are “sets of individuals”; “item sets” in pattern mining are sets; and for various types of analysis in language studies a sentence can be considered as a “set or bag of words”. Although building models and inference algorithms for structured data has been an important task in the fields of machine learning and statistics, research on “set-like” data still remains less explored. Relationships between pairs of elements can be modeled as edges in a graph. However, modeling relationships that involve all members of a set, a hyperedge is a more natural representation for the set. In this work, we focus on the problem of embedding hyperedges in a hypergraph (a network of overlapping sets) to a low dimensional vector space. We propose a probabilistic deep-learning based method as well as a tensor-based algebraic model, both of which capture the hypergraph structure in a principled manner without loosing set-level information. Our central focus is to highlight the connection between hypergraphs (topology), tensors (algebra) and probabilistic models. We present a number of interesting baselines, some of which adapt existing node-level embedding models to the hyperedge-level, as well as sequence based language techniques which are adapted for set structured hypergraph topology. The performance is evaluated with a network of social groups and a network of word phrases. Our experiments show that accuracy wise our methods perform similar to those of baselines which are not designed for hypergraphs. Moreover, our tensor based method is quiet efficient as compared to deep-learning based auto-encoder method. We therefore, argue that we have proposed more general methods which are suited for hypergraphs (and therefore also for graphs) while maintaining accuracy and efficiency.
+
+# 1 INTRODUCTION
+
+In group structured data we have multiple entities related by some form of group relationships. In fact, such data is more abundantly found in the real world than has been usually studied (Estrada & Rodriguez-Velazquez, 2005). There are increasing number of fields where such data is being found more than ever before. In social networks domain: team data from massive online multiplayer games (Ahmed et al., 2011) such as World of Warcraft, group communication tools such as Skype and Google Docs and research collaborations (Subbian et al., 2013; pub). There are other fields in which structural relationships between entities is important as well, and large datasets capturing them exist. Examples include Natural Language Processing (Bengio & Bengio, 2000), Biology (Hwang et al., 2008; Klamt et al., 2009), e-commerce (Deshpande & Karypis, 2004; Christakopoulou & Karypis, 2014) and Chemistry (Bartholomay, 1960). Figure 1 shows such examples for networks of groups, sentences, and item sets.
+
+Hypergraph (Berge, 1984), which is a generalization of graphs, is a popular model to naturally capture higher-order relationships between sets of objects (Figure 2) (Estrada & Rodriguez-Velazquez, 2005). Within machine learning, algorithms guided by the structure of such higher order networks (Zhou et al., 2006) have found applications in a variety of domains (Tian et al., 2009; Gao et al., 2013; Li & Li, 2013; Sharma et al., 2015; 2017).
+
+![](images/5673e5d3ff2a680a4163ef84a34218dd8d11264827cca7b99e4de5cc00e81a4b.jpg)  
+Figure 1: Example illustrating “set-like” structures from various domains.
+
+More recently, the interest of representation learning in NLP (Mikolov et al., 2013a) has stimulated its application in graph embedding (learning low dimensional representation for graph nodes) (Perozzi et al., 2014; Tang et al., 2015b; Grover & Leskovec, 2016; Cai et al., 2017). However, this new line of research is limited to simple graphs and we are unaware of any work that considers hyperedge embeddings (learning representations for hyperedges). In this paper we aim to develop methods which (1) learn hypergraph embeddings directly, (2) leverage the hypergraph topology and (3) not loose the hyperedge-level joint information. These learned embeddings can then be employed by a supervised or semi-supervised algorithm to perform various predictive tasks pertaining to hyperedges. For example performance prediction of a team (set of individuals) engaged in a collaborative task or sentiment analysis of a sentence (set of words).
+
+Although, there have been a variety of attempts to learn node embeddings for hypergraphs by extending traditional graph embedding methods for hypergraph setting. These approaches differ in the extent they address the aims (2) and (3) as pointed previously. In first category, there are a number of methods that incorporate the hypergraph topology using proxy graphs (Zhou et al., 2006; Hwang et al., 2008), therefore, incur loss of information. Also, Agarwal et al. (2006) squarely criticize that such representations can be learned by constructing graphs, which are proxies for the hypergraph structure. Second, are methods which actually, take into account the hyperedge-level information, like the classical Neural Language Model which maintains the higher order sequence information (Bengio & Bengio, 2000). But their model is designed for sequences and also simply ignores the topology (sequences are linked by common words forming a network of sequences). However recently, there has been interest in modeling “set-like” structures within deep-learning community (Vinyals et al., 2016; Rezatofighi et al., 2016). But they do not consider the hypergraph structure between the sets and therefore, not model hypergraphs in a principled manner. Third category are those methods that do not ignore the topology completely and also retain the hyperedge-level information. Among this are $k$ -way tensor based methods that explicitly do not work on hypergraph (Shashua et al., 2006; Bulo & Pelillo, 2009), but the connection exists in the form that \` $k$ -way tensor represents a $\mathbf { k }$ -uniform hypergraph (Qi, 2005). But they are restricted to uniform hypergraphs.
+
+We therefore, propose two methods, both of which directly learn hyperedge embeddings for general hypergraphs and capture the hypergraph structure in a principled manner. First, is an auto-encoder based deep-learning method and second, is a tensor-based algebraic method. We also highlight the connection between hypergraphs (topology), tensors (algebra) and probabilistic models. A number of interesting baselines are also developed, some of which adapt existing node-level embedding models to hypergraph setting and also sequence based NLP techniques are adapted for set structured hypergraphs. We have experimented with social group dataset (network of teams as a hypergraph) from an online game for team (hyperedge) performance prediction as well as language networks (sentence/phrase hypergraphs) for phrase-level sentiment analysis. Our experiments show that accuracy wise our methods perform similar to those of baselines which are not designed for hypergraphs. Moreover, our tensor based method is quiet efficient as compared to deep-learning based auto-encoder method. We therefore, argue that we have proposed more general methods which are suited for hypergraphs (and therefore also for graphs) while maintaining accuracy and efficiency. In summary the main contributions of our paper are:
+
+![](images/d9529e3eec2c18a9da98aa3b1dc1936c4df10b7d8828075081b7cb5ebd941335.jpg)  
+Figure 2: Example illustrating network of groups (NOG) which is the hypergraph (HG) (top left), the corresponding simplicial complex $( S C )$ (top right) and the hasse diagram $( H D )$ (bottom) corresponding to the simplicial complex, for a scenario where the actors $\{ 1 , 2 , 3 , 4 , 5 \}$ have collaberated in past as groups: $g _ { 1 } = \{ 1 , 2 \}$ , $g _ { 2 } = \{ 1 , 2 , 3 , 4 \}$ and $g _ { 3 } = \{ 3 , 4 , 5 \}$ .
+
+• We propose the novel concept of a dual tensor corresponding to the hypergraph dual and allows us to get hyperedge embedding directly. We propose a general hypergraph tensor decomposition method designed for general hypergraphs (containing different cardinality hyperedges) unlike simple uniform hypergraph tensor decomposition which is restricted to fixed cardinality hyperedges (i.e. uniform hypergraph). We are unaware of any such works or applications employing this approach. We propose the novel idea to use of de-noising auto-encoder in a hypergraph setting. Moreover, we also develop techniques of creating noise using random-walks over hasse diagram topology, which is original and unique.   
+• We highlight and argue that embeddings from tensor based methods have a natural hypergraph theoretic interpretation unlike the deep learning based black-box approaches.   
+• Our proposed methods learn hypergraph embeddings based on the unique approach of leveraging the existing structure present in network data as context, in contrast to context generation based existing graph embedding techniques.
+
+Following is the outline for the rest of the paper. In section 2 we describe the problem definition and statement followed by Section 3 where we describe in detail the various methods proposed in this paper. Section 4 describes the datasets, experimental tasks & settings for the models. Section 5 provides an overview of the related literature followed by conclusion and appendix.
+
+# 2 PRELIMINARIES
+
+In this paper we consider the scenario where we have a collection of elements. These elements can represent individual actors in case of social groups or words in sentences or items in itemsets within a transaction database. In other words a social group or a sentence or an item-set are sets which contain these elements. Let $V = \{ v _ { 1 } , v _ { 2 } , . . . , v _ { n } \}$ represents $n$ elements and we have $m$ sets defined over these elements, denoted by $G = \{ g _ { 1 } , g _ { 2 } , . . . , g _ { m } \}$ , where $g _ { i } \subseteq V$ represents the $i ^ { t h }$ set. The cardinality $\left| g _ { i } \right|$ represents the number of elements in the set. Also each set $g _ { i } \in$ $G$ has an occurrence number $\dot { R } ( g _ { i } )$ , which denotes the number of times it has occurred. Such overlapping or non-overlapping sets can be modeled as a hypergraph Berge (1984), where the nodes and hyperedges, represent the elements and sets, respectively. This hypergraph is represented as $N _ { g } = ( V , G )$ with $G$ as the collection of hyperedges over the nodes $V$ . The incidence matrix $\mathbf { H } \in$ $\{ 0 , 1 \} ^ { | G | \times | V | }$ for $N _ { g }$ represents the presence of nodes in different hyperedges with $\mathbf { H } ( g _ { i } , v ) = 1$ if $v \in g _ { i }$ else 0. We also define degree $d ( v )$ of a vertex $v$ as the number of hyperedges incident on this vertex i.e. $\begin{array} { r } { d ( v ) = \sum _ { g _ { i } \in G } \mathbf { H } ( g _ { i } , v ) } \end{array}$ . A specialization of hypergraph model is the simplicial complex (Munkres, 1984) (Figure 2), in which additionally each hyperedge has the subset closure property, i.e., each subset of hyperedge is also a valid hyperedge.
+
+Problem Statement: Given this setting, our goal is to learn the mapping $\phi : G \to \mathbb { R } ^ { d }$ from hyperedges to feature representations (i.e., embeddings) that can be used to build predictive models involving sets. Here $d$ is a parameter specifying the number of dimensions of the embedding vector. Equivalently, $\phi$ can be thought of as a look-up matrix of size $| G | \times d$ , where $| G |$ is the total number of sets or hyperedges.
+
+# 3 METHODOLOGY
+
+As we had mentioned before there are several methods to learn representation of nodes in a graph. Given that a hyperedge is a set of nodes, a natural question arises is that, can we combine the node level embeddings (learned using existing methods) within a given hyperedge to find a suitable representation of that hyperedge? However, there are large number of possible ways one can combine the node embeddings. Therefore, in the following two subsections, we develop two methods that learn the embeddings for hyperedges directly in a more principled manner. Our tensor based method is focused on retaining the set-level information intact while harnessing the hypergraph network structure. Where the auto-encoder based method harnesses the non-linearity of representation offered by deep learning to generate crisp and powerful embeddings. Tensor based method has a natural hasse diagram based topological interpretation and can generate both node and hyperedge embeddings. Whereas deep learning model is more black-box based approach and can only generate hyperedge embeddings.
+
+# 3.1 HYPEREDGE2VEC USING HYPERGRAPH TENSOR DECOMPOSITION
+
+In this section, we develop tensor (higher-order matrices) based linear algebraic methods that learn node as well as hyperedge embedding by taking into account the joint probability over a hyperedge. The idea behind using tensors is that they retain the set-level information contained in a hypergraph, unlike the proxy graphs (corresponding to hypergraphs) based techniques (used as baselines in our experiments), which approximate hyperedge or set-level information with dyadic edge-level information. Therefore, we design an algorithm which is principally suited for hypergraph structured data. For a given hypergraph we can extract a sub-hypergraph that only consists of the hyperedges with cardinality $k$ . This sub-hypergraph is a $k$ -uniform hypergraph or $k$ -graph (Cooper $\&$ Dutle, 2012). Corresponding to this $k$ -uniform hypergraph, we can define a $k ^ { t h }$ order $n$ -dimensional symmetric tensor (Qi, 2005) $\mathcal { A } _ { \mathbf { h } \mathbf { y } \mathbf { p } } ^ { \mathbf { k } } = ( a _ { p _ { 1 } , p _ { 2 } , . . , p _ { k } } ) \in \mathbb { R } ^ { [ k , n ] }$ whose elements are initialized as follows:
+
+$$
+a _ { p _ { 1 } , p _ { 2 } , . . , p _ { k } } = R ( g _ { i } )
+$$
+
+where $\{ v _ { p _ { 1 } } , v _ { p _ { 2 } } , . . . , v _ { p _ { k } } \} \in g _ { i }$ and $| g _ { i } | = k , \forall i \in \left\{ 1 , . . . , m \right\}$ . Note that symmetry here implies that value of element $a _ { p _ { 1 } , p _ { 2 } , \dots , p _ { k } }$ is invariable under any permutation of its indices $\left( p _ { 1 } , p _ { 2 } , . . , p _ { k } \right)$ . Rest all the elements in the tensor are zeros.
+
+In a similar manner we can also define a dual tensor, corresponding to hypergraph dual where the roles of nodes and hyperedges are interchanged. We consider all the hyperedges in the hypergraph dual that are of cardinality $k$ . This basically corresponds to all the vertices in the original hypergraph which have a degree of $k$ , i.e., they are part of exactly $k$ hyperedges in the original hypergraph. Corresponding to this $k$ -uniform hypergraph dual, we can define a $\check { k } ^ { t h }$ order $m$ -dimensional symmetric dual tensor $\mathcal { A } _ { \bf d u a l } ^ { \bf k } = ( a _ { q _ { 1 } , q _ { 2 } , . . , q _ { k } } ) \in \mathbb { R } ^ { [ k , m ] }$ whose elements are initialized as follows:
+
+$$
+a _ { q 1 , q 2 , . . , q _ { k } } = 1
+$$
+
+where $\left\{ g _ { q _ { 1 } } , g _ { q _ { 2 } } , . . . , g _ { q _ { k } } \right\} \ni v _ { j }$ and $d ( v _ { j } ) = k , \forall j \in \{ 1 , . . . , n \}$ . Note that this tensor is also symmetric and rest all the elements in the tensor are zeros. Moreover, we can also have other meaningful initialization schemes, for that we refer to Appendix B.1. To realize our aim of learning node and hyperedge embeddings we perform joint CP Tensor Decomposition (Kolda & Bader, 2009) (of the tensors we just described) across different cardinality hyperedges simultaneously. Specifically, for the node embeddings we solve the following optimization problem:
+
+$$
+\operatorname* { m i n } _ { \mathbf { Z } } \sum _ { k = c _ { m i n } } ^ { c _ { m a x } } \| \mathcal { A } _ { \mathbf { h y p } } ^ { \mathbf { k } } - \widehat { \mathcal { A } _ { \mathbf { h y p } } ^ { \mathbf { k } } } \|
+$$
+
+where,
+
+$$
+\widehat { \mathcal { A } _ { \mathbf { h y p } } ^ { \mathbf { k } } } = \sum _ { r = 1 } ^ { d } \lambda _ { r } \underbrace { \mathbf { z } _ { r } \circ \mathbf { z } _ { r } . . . \circ \mathbf { z } _ { r } } _ { k \mathrm { t i m e s } } \equiv \langle \lambda | \underbrace { \mathbf { Z } , \mathbf { Z } , . . . . , \mathbf { Z } } _ { k \mathrm { t i m e s } } \rangle
+$$
+
+with $\mathbf { z } _ { r } \in \mathbb { R } ^ { n }$ , $\lambda _ { r } \in \mathbb { R } _ { + }$ , $\textbf { Z } \in \ \mathbb { R } ^ { n \times d }$ and $\mathbf { Z } ( : , r ) ~ = ~ \mathbf { z } _ { r }$ . Notice, that equation 4 is the standard symmetric CP decomposition but equation 3 is summation of reconstruction errors in different tensor decompositions for different cardinality hyperedges. Each error term learns a common latent factor matrix $\mathbf { Z }$ using the empirical observed $k$ -uniform sub-hypergraph stored in $\mathcal { A } _ { \mathbf { h y p } } ^ { \mathbf { k } }$ tensor. We perform this joint decomposition by augmenting the standard CP Decomposition into the Hypergraph-CP-ALS algorithm 1. Lines (4-8) is the standard (non-symmetric) CP Decomposition (see reference Kolda & Bader (2009) for details), which solves the following optimization:
+
+$$
+\widehat { \mathcal { A } _ { \mathbf { h y p } } ^ { \mathbf { k } } } = \sum _ { r = 1 } ^ { d } \lambda _ { r k } \mathbf { u } _ { r } ^ { ( 1 ) } \circ _ { k } \mathbf { u } _ { r } ^ { ( 2 ) } . . . \circ _ { k } \mathbf { u } _ { r } ^ { ( k ) } \equiv \langle \lambda | _ { k } \mathbf { U } ^ { ( 1 ) } , _ { k } \mathbf { U } ^ { ( 2 ) } , . . . . , _ { k } \mathbf { U } ^ { ( \mathbf { k } ) } \rangle
+$$
+
+with $\mathbf { \Psi } _ { k } \mathbf { u } _ { r } ^ { ( j ) } \in \mathbb { R } ^ { n }$ , $\lambda _ { r } \in \mathbb { R } _ { + }$ , $\mathbf { \Psi } _ { k } \mathbf { U } ^ { ( \mathbf { j } ) } \in \mathbb { R } ^ { n \times d }$ , $\mathbf { \nabla } _ { k } \mathbf { U } ^ { ( \mathbf { j } ) } ( : , r ) = \mathbf { \nabla } _ { k } \mathbf { u } _ { r } ^ { ( j ) }$ and $j \in \{ 1 , . . , k \}$ . As we want to learn common representations $\mathbf { \rho } ( \mathbf { Z } )$ for all the nodes, we add additional constraint that enforce that $\mathbf { \Sigma } _ { k } \mathbf { U } ^ { ( \mathbf { j } ) }$ are same for all $j \in \{ 1 , . . , k \}$ and for all $k \in [ c _ { m i n } , c _ { m a x } ]$ . Lines (9-15) is where we force all representations for all cardinalities to be average of the $\mathbf { \Sigma } _ { k } \mathbf { U } ^ { ( \mathbf { j } ) }$ achieved by the last decomposition iteration that occurred in lines (4-8). Averaging can be interpreted as equal contribution from the latent factors learned from different cardinality (uniform) sub-hypergraphs. We make repeated pass through the entire hypergraph (by learning via different $k$ -uniform sub-hypergraph (line 3)) until the objective (equation 3) converges. In our implementation we empirically observe that using the above mentioned unbiased averaging heuristic, our algorithm converges successfully. The same algorithm 1 is used to get the hyperedge embeddings, by just passing $\mathcal { A } _ { \bf d u a l }$ instead of $\mathcal { A } _ { \bf h y p }$ . We shall jointly refer to the embeddings achieved for nodes and hyperedges via the above tensor decomposition techniques as t2v.
+
+# Algorithm 1 Hypergraph-CP-ALS $( \mathcal { A } _ { \bf h y p } , c _ { m i n } , c _ { m a x } )$
+
+1: randomly initialize $\mathbf { \Phi } _ { k } \mathbf { U ^ { ( j ) } } , \forall k \in [ c _ { m i n } , c _ { m a x } ] , \forall j \in \{ 1 , . . . , k \}$   
+2: repeat   
+3: for $k = c _ { m i n }$ to $c _ { m a x }$ do   
+4: for $j = 1$ to $k$ do   
+5: $\begin{array} { r l } & { \overset { \mathrm {  ~ \psi ~ } } { \mathbf { V } } \longleftarrow _ { k } { \mathbf { U } } ^ { ( 1 ) \top } _ { k } { \mathbf { U } } ^ { ( 1 ) } * \ldots * _ { k } { \mathbf { U } } ^ { ( \mathrm { j - 1 } ) \top } _ { k } { \mathbf { U } } ^ { ( \mathrm { j - 1 } ) } * _ { k } { \mathbf { U } } ^ { ( \mathrm { j + 1 } ) \top } _ { k } { \mathbf { U } } ^ { ( \mathrm { j + 1 } ) } * \ldots * _ { k } { \mathbf { U } } ^ { ( \mathrm { j } ) \top } _ { k } { \mathbf { U } } ^ { ( \mathrm { j } ) } } \\ & { _ { k } { \mathbf { U } } ^ { ( \mathrm { j } ) } \longleftarrow ( \mathcal { A } _ { \mathbf { h y p } } ^ { \mathrm { k } } ) ^ { ( j ) } \big ( _ { k } { \mathbf { U } } ^ { ( \mathrm { k } ) } \odot \ldots \odot _ { k } { \mathbf { U } } ^ { ( \mathrm { j + 1 } ) } \odot _ { k } { \mathbf { U } } ^ { ( \mathrm { j - 1 } ) } \odot \ldots \odot _ { k } { \mathbf { U } } ^ { ( \mathrm { i } ) } \big ) { \mathbf { V } } ^ { \dagger } } \end{array}$   
+6:   
+7: normalize columns of $\mathbf { \Sigma } _ { k } \mathbf { U } ^ { ( \mathbf { j } ) }$ (and store norms as $\lambda$ )   
+8: end for   
+9: $\begin{array} { r } { \mathbf { Z }  \frac { 1 } { k } \sum _ { j = 1 } ^ { k } \mathbf { \sigma } _ { k } \mathbf { U } ^ { ( \mathbf { j } ) } } \end{array}$   
+10: for $p = c _ { m i n }$ to $c _ { m a x }$ do   
+11: for $j = 1$ to $p$ do   
+12: $\mathbf { \Sigma } _ { p } \mathbf { U } ^ { ( \mathbf { j } ) }  \mathbf { Z }$   
+13: end for   
+14: end for   
+15: end for   
+16: until fit criteria achieved or maximum number of iterations exceeded   
+17: return Z
+
+We would like to highlight a few points regarding the tensor methods. The tensors that we have employed are super-symmetric and hence able to capture distribution over sets rather than sequence. But in general we can employ a $k$ -way tensor which is not symmetric to even capture sequence. In this sense tensors are the hypergraph tensor given hyperedge (Eq. $\mathcal { A } _ { \mathbf { h y p } } ^ { \mathbf { k } }$ general purpose. Another point one can observe that when we initialize, we have initialized all the permutations of vertices corresponding to aoreover, we initialize it by the repetition counts.
+
+# 3.2 HYPEREDGE2VEC USING HASSE DE-NOISING AUTOENCODER
+
+An autoencoder (Bengio et al., 2009), takes an input vector $\mathbf { x } \in [ 0 , 1 ] ^ { n }$ and maps it to a latent representation $\mathbf { y } ~ \in ~ [ 0 , 1 ] ^ { d }$ . This is typically done using an affine mapping followed by a nonlinearity (more so when the input, like in our case, is binary (Vincent et al., 2010)): $f _ { e n } ( { \bf x } ) =$ $s ( \mathbf { W } \mathbf { x } + \mathbf { b } )$ , with parameters $\bar { \boldsymbol { \theta } _ { } } = \{ \mathbf { W } , \mathbf { b } \}$ . Here, $\mathbf { W }$ is a $n \times d$ weight matrix and $\mathbf { b }$ is the offset.
+
+![](images/303929718f8854ff19c1941e1022db299d6f7534561c53d75a8c5943e5769b42.jpg)  
+Figure 3: For the given hypergraph between four nodes (A,B,C,D) we consider the complete hasse lattice. For a given hyperedge $\{ { \mathrm { B } } , { \mathrm { C } } \}$ (square box) we then construct the sub-lattice made of hyperedges with distance $h = 2$ from $\{ { \mathrm { B } } , { \mathrm { C } } \}$ . We perform random walk (with $\tau = 0 . 2$ ) starting from the node corresponding to hyperedge $\{ { \mathrm { B } } , { \mathrm { C } } \}$ and sample $p = 3$ hyperedges (nodes visited by the random walk; shown with a check-mark $\checkmark$ ). Finally, we train the autoencoder to reconstruct the original hyperedge from these $p$ noisy hyperedges.
+
+This latent representation is reconstructed into a vector $\mathbf { z } = f _ { d e } ( \mathbf { y } ) = s ( \mathbf { W } ^ { \prime } \mathbf { y } + \mathbf { b } ^ { \prime } )$ , in the input space, $\mathbf { z } \in [ 0 , 1 ] ^ { n }$ and parameters $\theta ^ { \prime } = \{ \mathbf { W } ^ { \prime } , \mathbf { b } ^ { \prime } \}$ . The mappings $f _ { e n }$ and $f _ { d e }$ are referred as the encoder and decoder, respectively. The representation $\mathbf { y }$ is learned by minimizing the following reconstruction error:
+
+$$
+\theta ^ { * } , \theta ^ { \prime * } = \underset { \theta ^ { * } , \theta ^ { \prime * } } { \arg \operatorname* { m i n } } \frac { 1 } { m } \sum _ { i = 1 } ^ { m } L ( \mathbf { x } _ { i } , \mathbf { z } _ { i } ) = \underset { \theta ^ { * } , \theta ^ { \prime * } } { \arg \operatorname* { m i n } } \frac { 1 } { m } \sum _ { i = 1 } ^ { m } L ( \mathbf { x } _ { i } , g _ { \theta ^ { \prime } } ( f _ { \theta } ( \mathbf { x } _ { i } ) )
+$$
+
+where $L$ is a loss function, which in case of binary or bit probabilities is often chosen as the crossentropy loss:
+
+$$
+L ( \mathbf { x } , \mathbf { z } ) = \sum _ { j = 1 } ^ { n } [ \mathbf { x } ( j ) \log \mathbf { z } ( j ) + ( 1 - \mathbf { x } ( j ) ) \log ( 1 - \mathbf { z } ( j ) ) ]
+$$
+
+In their paper, Vincent et al. (2010) have shown that minimizing reconstruction amounts to maximizing the lower bound on the mutual information between input $( x )$ and the representation $\mathbf { y }$ . However, they have further argued (Vincent et al. (2008)) that $\mathbf { y }$ retaining information about input $\mathbf { x }$ is insufficient. They further, propose the idea that the learned representation should be able to recover (denoising) the original input even after being trained with corrupted input (adding noise). They generate the corrupted input $(  { { \widetilde { \mathbf { x } } } } )$ , using a stochastic mapping $q ( \tilde { \mathbf { x } } | \mathbf { x } )$ . Choice of noise is usually either Gaussian for real inputs and Salt-and-pepper noise for discrete inputs. The denoising autoencoder then learns the representation for each input, $\mathbf { x }$ , same as Equation 6, but with the following modified loss function: $L ( \mathbf { x } ( i ) , g _ { \boldsymbol { \theta } ^ { \prime } } ( f _ { \boldsymbol { \theta } } ( \tilde { \mathbf { x } } ( i ) ) ) )$ .
+
+We leverage the denoising autoencoder for learning representation for $j ^ { t h }$ hyperedge, by treating each hyperedge as an input, $\mathbf { x } = \mathbf { H } ( j , : )$ . The size of this input vector for each hyperedge is $n$ , which is the number of vertices in the hypergraph. In most natural hypergraphs, specially social networks, $n$ can be quiet high ranging from thousands to millions or even billions (like Facebook for example). Therefore, randomly using a discrete noise like salt-and-pepper, might not be reasonable, as there are large number of possible permutations (as size $n$ is large) and not all of them are related. Random addition of 1s or deletion of existing 1s from $\mathbf { x }$ , amounts to randomly adding or deleting vertices to the hyperedge corresponding to $\mathbf { x }$ . This might end up in new hyperedges that are completely unrelated to the given hyperedge $\mathbf { \tau } ( \mathbf { x } )$ . For example, users (nodes) in a social network from completely different regions of the network suddenly form a group (hyperedge). Such anomalous scenarios rarely happen in practice and social groups evolve in a gradual fashion via simple processes (Sharma et al., 2017; 2015).
+
+Rather, we take advantage of the hypergraph structure to systematically guide us in generating this noise. A hypergraph can be defined by its corresponding hasse lattice (Sharma et al., 2017). For a given hyperedge $\mathbf { \tau } ( \mathbf { x } )$ , we consider the sub-lattice consisting of only those hyperedges that are a distance $h$ from it in the complete lattice. On this sub-lattice we sample $p$ hyperedges (nodes in sublattice) by performing random walk starting at the given hyperedge’s node (see Figure 3). We assume that all the nodes in the sub-lattice which correspond to previously occurred hyperedge $g _ { i }$ have weight $R ( g _ { i } )$ and rest all nodes have a constant weight $\tau$ . During random walk from a given node we choose a neighboring node (as the next node) in proportion to this neighboring node’s weight as compared to the other neighboring nodes. Our stochastic mapping $q ( { \tilde { \mathbf { x } } } | \mathbf { x } )$ is therefore, a random walk on the sub-lattice of hyperdge $\mathbf { \tau } ( \mathbf { x } )$ containing hyperedges at distance $h$ from it. Intuitively, the hyperedges coming within a reasonable distance will affect each others representations and will have more similar representations. We will refer to the hyperedge representations learned by the above autoencoder technique, as h2v-auto.
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASET DESCRIPTION
+
+As the first dataset, we use group interaction log-data of the Sony’s Online multi-player game EverQuest II $( \mathbf { E Q } \mathbf { \Lambda } \mathbf { U } )$ (www.everquest2.com) for a time period of nine months. In this game, several players work together as a team to perform various tasks. Each team earn points after completion of each task, and as the teams progress by earning points, they are escalated to different levels of game play. The interestingness of the game increases with each level. The points earned by the teams are treated as a measure of group performance. Each set of players who played together is treated as a hyperedge. We treat the number of times same set of players play together again as hyperedge occurrence number $( R ( g _ { i } ) )$ . Players can participate in several teams over time, therefore, resulting in a hypergraph with overlapping hyperedges. We consider hyperedges of cardinality $\in \ [ 2 , 6 ]$ as almost $9 0 \%$ of our hyperedges lie within this range. The resulting dataset contains a total of 5964 hyperedges (teams) among 6536 nodes (players).
+
+Second dataset, is the fully labeled and publicly available sentiment analysis corpus of Stanford Sentiment Treebank (LangNet) (Socher et al., 2013). This dataset is based on the reviews from a movie review website (www.rottentomatoes.com) and contains 215,154 unique phrases. Each of the phrases are labeled with a sentiment score (a real number $\in \ [ 0 , 1 ]$ , larger value indicates positive sentiment) by human annotators. Each phrase is a set or hyperedge of words. As there is no occurrence information for a phrase hyperedge we consider $R ( \bar { g _ { i } } \bar { ) } = \bar { 1 , } \forall i \in \{ 1 , . . . , m \}$ . Given that words are shared across various phrases, these common words connect the phrase hyperedges, resulting in a phrase hypergraph with overlapping phrase hyperedges. Again, we only consider hyperedges of cardinality $\in \ [ 2 , 6 ]$ . After applying this cardinality filter we are left with 141,410 hyperedges (phrases) and 21,122 nodes (words).
+
+# .2 EVALUATION METHODOLOGY AND EXPERIMENTAL SETUP
+
+# 4.2.1 METHODS COMPARED
+
+As mentioned before we refer to our proposed methods: tensor based hypergraph tensor decomposition and deep auto-encoder based hypergraph auto-encoder, as t2v and h2v-auto, respectively. We compare our proposed methods against six baselines which are listed below. Methods (1-2) are adapted from sequence based language models for set structured hypergraph data. Methods (3-6) are various kind of dyadic graph based embedding methods adapted for hypergraph setting.
+
+1. h2v-DM: Hyperedge and node embeddings obtained using sentence embedding using distributed memory (DM) model (refer section A.1)   
+2. h2v-DBOW: Hyperedge and node embeddings obtained using sentence embedding using distributed memory (DBOW) model (refer section A.1)
+
+3. h2v-inv: Hyperedge and node embeddings obtained using node2vec on inverted hypergraph and simple graph, respectively (refer section A.2) 4. h2v-dual: Hyperedge and node embeddings obtained using node2vec on hypergraph dual and the proxy graph for hypergraph, respectively (refer section A.2) 5. e2v: Hyperedge and node spectral embeddings obtained using eigen decomposition of inverted hypergraph laplacian and graph laplacian, respectively (refer section A.3) 6. e2v-hyp: Hyperedge and node spectral embeddings obtained using eigen decomposition of dual laplacian and hypergraph laplacian, respectively (refer section A.3)
+
+The details of these baseline can be found in Appendix A. Except for h2v-auto, each of the baselines as well as $\mathbf { t } 2 \mathbf { v }$ outputs both node as well as hyperedge embeddings of dimension $d = 1 2 8$ . We further combine the node and hyperedge embedding using five different strategies: (i) node embedding summation (dimension $d = 1 2 8 )$ , (ii) node embedding summation and concatenation with hyperedge embedding (dimension $2 \times d = 2 5 6 $ ), (iii) node embedding averaging (dimension $d = 1 2 8$ ), (iv) node embedding averaging and concatenation with hyperedge embedding (dimension $2 \times d = 2 5 6$ ), and (v) only hyperedge embedding (dimension $d = 1 2 8$ ). h2v-auto only produces hyperedge embeddings of dimension $d = 1 2 8$ . But it builds the embeddings using three different scenarios as mentioned in next section. Therefore, in total we have 3 $\mathrm { 8 } \ : ( = 3 5 + 3$ ) different scenarios each resulting in a different hyperedge embedding.
+
+# 4.2.2 EVALUATION TASKS AND SETUP
+
+We perform two regression based tasks for the two datasets. In EQII dataset each team (hyperedge) has a team performance score associated with it. This team performance score is a real number, equal to the number of points earned by the team while performing one or more tasks within a gaming session. We treat the embedding learned for a given team (hyperedge) as its feature vector which is associated with a real number (team performance). We therefore, perform on regression over all the hyperedges (teams) with team performance as the dependent variable.
+
+Similarly, in LangNet dataset each phrase (hyperedge) has a sentiment score associated with it, which again is a real number. Similar to the team dataset above, we treat the embedding learned for a given phrase (hyperedge) as its feature vector which is associated with a real number (sentiment score). We therefore, treat this as a regression task with sentiment score as the dependent variable and perform regression using the feature matrix containing embeddings of all the phrases.
+
+For both the tasks we just described, we perform several evaluation runs. In each run we randomly choose $3 0 \%$ of hyperedges (teams or phrases) as the test set and learn ridge regression parameters using the remaining $7 0 \%$ training hyperedges for each of the 38 different embedding scenarios. Root mean squared error (RMSE) was chosen as the evaluation metric (the lower, the better). RMSE was calculated for each of the 38 scenarios and for each run. Final RMSE score was taken as the average RMSE score across five runs. Ridge regression’s hyper-parameter was chosen by 5-fold cross-validation.
+
+For the auto-encoder method (h2v-auto) we consider three scenarios: (1) single hidden layer $( L 1 )$ of $d = 1 2 8$ ; (2) two hidden layers ( $L 1$ & $L 2$ ) with size of $L 1 : d = 9 6$ and of $L 2 : d = 3 2$ . We concatenate these embedding to get a single $d = 1 2 8$ size embedding; and (3) two hidden layers ( $\mathrm { . } L 1$ & $L 2$ ) with size of $L 1 : d = 5 1 2$ and of $L 2 : d = 1 2 8$ . We use the output of $L 2$ , which is of dimension $d = 1 2 8$ , as the embedding. For sampling, we use the distance parameter $h = 2$ for generating the sub-lattice and $\tau = 0 . 2$ for both datasets. Also, $p = 1 0 \ : \& \ : p = 5$ number of hyperedges are sampled (corresponding to each hyperedge) from EQII and LangNet, respectively.
+
+# 4.3 RESULTS AND DISCUSSION
+
+Tables 1 & 2 show the RMSE scores of $\mathbf { t } 2 \mathbf { v }$ (as compared with baselines) for the tasks of team performance prediction and sentiment score prediction, respectively. These tables contain scores for all the 35 different scenarios: columns represent 7 (6 baselines and the proposed hypergraph tensor decomposition (t2v)) different models while rows represent combination strategies. The scores for hypergraph auto-encoder (h2v-auto) are shown in the separate Table 4 as auto-encoder only generates hyperedge embeddings.
+
+Accuracy & Run-times As we can observe that for our datasets and for both the regression tasks, almost all the embeddings (baselines and proposed) are performing very similarly in terms of accuracy. However, we can observer in Table 3 that tensor based method (t2v) have significant less time than the expensive auto-encoder technique (h2v-auto) (Table 4). Among the baselines e2v-hyp is the fastest. We also observer that sentence based techniques run faster on text-based LangNet dataset as compared to node2vec based methods which are not designed for text data and vice-versa.
+
+Interpretability All the matrix or tensor based algebraic techniques: e2v, e2v-hyp and t2v, have a natural graph theoretic interpretation. e2v and e2v-hyp are both well studied spectral techniques with several eigen-value based interpretations. t2v has a hierarchical hasse diagram based interpretation. In contrast the sentence and node2vec based techniques can be understood only intuitively in terms of the cost function. Apart from the noise which has a random walk based interpretation, h2v-auto is a deep-learning based method which exploits multiple level of non-linearity and is over all a black-box approach.
+
+Information Loss To reiterate one of the primary aims of this study is to design methods that retain the hyperedge-level joint information. The proposed tensor-based t2v principally capture the joint distribution over various cardinality hyperedges unlike conditional distribution like sentence embedding (h2v-dm) (which are more appropriate for sequences). In comparison the other method proposed h2v-auto, although are not directly designed to retain hyperedge-level joint distribution, but we hypothesize that the deep layered neural network should output highly informative nonlinear representations. Several other baselines only retain pair-wise information. For example, the methods developed using Node2Vec in Section A.2 are based on the skip-gram model, which learns embedding of nodes while maximizing the conditional probability of a node given another node in a context (Eq. 17). Similarly, the spectral methods (Section A.3) are inherently two dimensional as they are based on matrix. Same is the case with skip-gram based sentence embedding (h2v-dbow). However, sentence embedding based on the DM architecture (h2v-dm) maximizes the conditional probability of a word given the previous set of words (context) without breaking this context (by concatenating or averaging the embedding of the previous words).
+
+Leveraging Hypergraph Topology is another feature that we stress as a desirable property in a method which aims to build hyperedge embedding. Hypergraph topology is an important auxiliary information, if left unused is wastage of resources at hand. We observe that except the sentencebased baselines, both proposed methods as well as other baselines, which are adapted for the hypergraph setting, leverage hypergraph topology using various hypergraph representations. Proposed h2v-auto generated noise using the hasse lattice and t2v directly models the hasse lattice by storing hyperedge level joint distribution using super-symmetric tensors. Node2vec based and spectral baselines use various kinds of matrices (that capture hypergraph topology up to varying degrees) as enlisted in (3-6) in Section 4.2.1.
+
+Ability to generate hyperedge embeddings directly is another critical aim as was highlighted in the introduction. We emphasize that none of the baselines are designed to give hyperedge embeddings in a principled manner. However, both tensor based (t2v using $\mathbf { \mathcal { A } _ { d u a l } ^ { k } } )$ as well as auto-encoder (h2v-auto) methods proposed in this paper are specifically designed to give hyperedge representations directly.
+
+Choice of Method Both tensor based (t2v using $\mathcal { A } _ { \mathbf { d u a l } } ^ { \mathbf { k } } )$ as well as auto-encoder (h2v-auto) methods proposed in this paper are able to generate embeddings for all hyperedges of various cardinalities present in data. However, the auto-encoder method cannot generate node embeddings unlike tensor method (using $\mathcal { A } _ { \mathbf { h y p } } ^ { \mathbf { k } } .$ ). On contrary, in auto-encoder technique we don’t need to figure out $\alpha$ and $\beta$ parameters (as discussed in the next subsection 4.4). Another, thing to note is that tensors are inherently capable of retaining joint information because of their higher-order structure. Auto-encoder method implicitly (possibly lossy manner) leverage the higher order information embedded in network in form of node weights $( R ( g _ { i } ) )$ while performing the random walk sampling. Main advantage of auto-encoders is the crisp embedding achieved via multiple levels of non-linearity offered by deep neural networks. Although this non-linearity improves the accuracy of auto-encoder methods very slightly (Table 4) as compared to tensor method (bottom right corner of (Table 1 & 2), the computation cost of tensor is far less than the auto-encoder method (Table 3 & 4). Intuitively, it seems that depending upon the task at hand, in some tasks the retention of joint hyperedge-level information is more important while modeling and data has lesser non-linear structure; or vice versa. But the low computational cost (as reflected in our tasks) makes tensor method definitely more lucrative.
+
+Table 1: RMSE Scores of (t2v) compared to baselines for EQ II Team Performance Analysis   
+
+<table><tr><td rowspan="2"></td><td colspan="6">Baselines</td><td rowspan="2">Hypergraph Tensor Decomp.</td></tr><tr><td>Sentence Embed based</td><td></td><td></td><td>Node2Vec based</td><td>Spectral methods</td><td></td></tr><tr><td>Embed Combination</td><td>h2v-DM</td><td>h2v-DBOW</td><td>h2v-inv</td><td>h2v-dual</td><td>e2v</td><td>e2v-hyp</td><td>(t2v)</td></tr><tr><td>Node Embed Sum</td><td>0.79308</td><td>0.79567</td><td>0.80418</td><td>0.79956</td><td>0.81183</td><td>0.81405</td><td>0.81341</td></tr><tr><td>Node Embed Sum+Hyperedge Embed</td><td>0.79651</td><td>0.80241</td><td>0.81362</td><td>0.80636</td><td>0.8113</td><td>0.81652</td><td>0.81299</td></tr><tr><td>Node Embed Average</td><td>0.81584</td><td>0.81733</td><td>0.82407</td><td>0.82281</td><td>0.81234</td><td>0.81369</td><td>0.81303</td></tr><tr><td>Node Embed Avg +Hyperedge Embed</td><td>0.8182</td><td>0.82077</td><td>0.83378</td><td>0.82896</td><td>0.81223</td><td>0.81608</td><td>0.8127</td></tr><tr><td>OnlyHyperedge Embed</td><td>0.81203</td><td>0.81522</td><td>0.82189</td><td>0.81984</td><td>0.81233</td><td>0.81608</td><td>0.81341</td></tr></table>
+
+Table 2: RMSE Scores of (t2v) compared to baselines for LangNet Sentiment Analysis   
+
+<table><tr><td rowspan="2"></td><td colspan="6">Baselines</td><td rowspan="2">Hypergraph Tensor Decomp.</td></tr><tr><td>Sentence Embed based</td><td></td><td></td><td>Node2Vec based</td><td></td><td>Spectral methods</td></tr><tr><td>Embed Combination</td><td>h2v-DM</td><td>h2v-DBOW</td><td>h2v-inv</td><td>h2v-dual</td><td>e2v</td><td>e2v-hyp</td><td>(t2v)</td></tr><tr><td>Node Embed Sum</td><td>0.14081</td><td>0.14029</td><td>N/A</td><td>N/A</td><td>0.14633</td><td>0.14854</td><td>0.14194</td></tr><tr><td>Node Embed Sum + Hyperedge Embed</td><td>0.14028</td><td>0.13883</td><td>N/A</td><td>N/A</td><td>0.14627</td><td>0.14845</td><td>0.14144</td></tr><tr><td>Node Embed Average</td><td>0.14245</td><td>0.14115</td><td>N/A</td><td>N/A</td><td>0.14665</td><td>0.14852</td><td>0.14381</td></tr><tr><td>Node Embed Avg +Hyperedge Embed</td><td>0.14178</td><td>0.14007</td><td>N/A</td><td>N/A</td><td>0.14661</td><td>0.14845</td><td>0.14333</td></tr><tr><td>OnlyHyperedge Embed</td><td>0.14194</td><td>0.14147</td><td>N/A</td><td>N/A</td><td>0.14744</td><td>0.14844</td><td>0.1482</td></tr></table>
+
+Table 3: Average Runtime (seconds) of (t2v) compared to baselines across datasets   
+
+<table><tr><td rowspan="2"></td><td colspan="5">Baselines</td><td rowspan="2">Hypergraph Tensor Decomp.</td></tr><tr><td>Sentence Embed based</td><td></td><td>Node2Vec based</td><td></td><td>Spectral methods</td></tr><tr><td>Dataset</td><td>h2v-DM</td><td>h2v-DBOW</td><td>h2v-inv</td><td>h2v-dual</td><td>e2v</td><td>e2v-hyp (t2v)</td></tr><tr><td>EQ2</td><td>455.84</td><td>103.47</td><td>90.05</td><td>93.41</td><td>128.03 12.01</td><td>213.37</td></tr><tr><td>LangNet</td><td>80.61</td><td>62.31</td><td>211.97*</td><td>207.86*</td><td>221.46 47.12</td><td>483.81</td></tr></table>
+
+\* these are average time taken for learning vertex embeddings only
+
+Table 4: RMSE Scores & Run-times for EQ II (Team Performance) & LangNet (Sentiment Analysis) using Hypergraph Auto-encoder (h2v-auto)   
+
+<table><tr><td rowspan="2">Layer Sizes</td><td colspan="3">EQ II</td><td colspan="3">LangNet</td></tr><tr><td>L1:128</td><td>L1:96/L2:32</td><td>L1:512/L2:128</td><td>L1:128</td><td>L1:96/L2:32</td><td>L1:512/L2:128</td></tr><tr><td>RMSE</td><td>0.81104</td><td>0.81512</td><td>0.81635</td><td>0.14568</td><td>0.14529</td><td>0.14784</td></tr><tr><td>Run Time</td><td>52 min</td><td>40 min</td><td>1 hr 20 min</td><td>2 hr 10 min</td><td>3 hr 20 min</td><td>6hr</td></tr></table>
+
+Together with the natural graph theoretic interpretation, tensor methods in comparison to black-box approach of deep learning based auto-encoder, makes them even more viable.
+
+Highlight Another distinguishing feature of our work is that we leverage the existing structure (group structure) within the data. Recent attempts along these lines in network literature (Grover & Leskovec, 2016; Tang et al., 2015b; Perozzi et al., 2014) argue that language models have readymade context in the form of sentences or paragraphs to train the model, which are not available in networks, and therefore, they propose different ways to generate this context. In contrast, we focus on networks where the context is already present, e.g. collaboration networks where collaborative teams are hyperedges, or language hyper-networks where sentences are hyperedges.
+
+# 4.4 SCALABILITY
+
+Both tensor based (t2v using $\mathcal { A } _ { \mathbf { d u a l } } ^ { \mathbf { k } } )$ as well as auto-encoder (h2v-auto) methods proposed in this paper are able to generate embeddings for all hyperedges of various cardinalities present in data.
+
+Scalability for t2v The scalability issue in tensor based approach arise on two fronts: (1) the increase in number of hyperedges $( m )$ or number of vertices $( n )$ and (2) increase in parameter $\alpha$ (maximum cardinality hyperedge considered in the cost function) or $\beta$ (maximum degree vertex considered in the cost function). Also there are two kinds of cost: (1) enumeration cost associated associated while building hypergraph and dual tensors and (2) cost for hypergraph tensor decomposition.
+
+In case of hyperedge embeddings, for each vertex $v$ of degree (i.e. number of hyperedges it is part of) $d ( v )$ there are $d ( v )$ ! entries (permutations of hyperedges incident on $v$ ) to be enumerated to fill the tensor Ad(v)dual . This amounts to a worst case enumeration cost of $\textstyle \sum _ { v \in V } d ( v ) !$ . However, $d ( v )$ ! can grow prohibitively large for vertices with large vertex degree $d ( v )$ . We therefore, propose to restrict ourselves to vertices of degree $d ( v ) \leq \alpha$ . In fact we propose the following augmented initialization scheme for the dual tensors. For degree $k \left( \leq \alpha \right)$ vertex, $v _ { j } \in \{ g _ { q _ { 1 } } , g _ { q _ { 2 } } , . . . , g _ { q _ { k } } \}$ and $d ( v _ { j } ) = k \leq$ $\alpha , \forall j \in \left\{ 1 , . . . , n \right\}$ , we initialize the following elements of $\mathcal { A } _ { \bf d u a l } ^ { \bf k } = ( a _ { q _ { 1 } , q _ { 2 } , . . , q _ { k } } ) \in \mathbb { R } ^ { [ k , m ] }$ , same as Equation 29. For degree $k$ $\mathbf { \chi } ( > \alpha )$ vertex, $v _ { j } \in \{ g _ { q _ { 1 } } , g _ { q _ { 2 } } , . . . , g _ { q _ { k } } \} , \forall j \in \{ 1 , . . . , n \}$ , we initialize elements of $\mathcal { A } _ { \bf d u a l } ^ { \alpha }$ as,
+
+$$
+\begin{array} { r } { a _ { q _ { 1 } , q _ { 2 } , . . . , q _ { \alpha } } = 1 } \\ { a _ { q _ { ( \alpha + 1 ) } , q _ { ( \alpha + 2 ) } , . . . . , q _ { 2 \alpha } } = 1 } \\ { . . . . . . } \\ { . . . . . . } \\ { a _ { q _ { ( ( \left\lfloor \frac { k } { \alpha } \right\rfloor - 1 ) \alpha + 1 ) } , q _ { ( ( \left\lfloor \frac { k } { \alpha } \right\rfloor - 1 ) \alpha + 2 ) } , . . . . , q _ { ( ( \left\lfloor \frac { k } { \alpha } \right\rfloor ) \alpha ) } } = 1 } \end{array}
+$$
+
+and if $\delta = ( k - ( \textstyle \left\lfloor { \frac { k } { \alpha } } \right\rfloor \alpha ) ) \neq 0$ , then we also initialize the following element of $\mathcal { A } _ { \mathbf { d u a l } } ^ { \delta }$
+
+$$
+a _ { q _ { ( ( \left\lfloor \frac { k } { \alpha } \right\rfloor ) \alpha + 1 ) } , q _ { ( ( \left\lfloor \frac { k } { \alpha } \right\rfloor ) \alpha + 2 ) } , \dots , q _ { k } } = 1 .
+$$
+
+The above initialization basically partitions the $k$ hyperedges incident on a vertex into $\textstyle { \left( \left\lfloor { \frac { k } { \alpha } } \right\rfloor + 1 \right) }$ (or $\textstyle \left( \left\lfloor { \frac { k } { \alpha } } \right\rfloor \right)$ , if $\alpha$ is a divisor of $k$ ) partitions. There are several techniques to obtain such a partition, but in this paper we simply sort the incident hyperedges by their cardinality and then sequentially make sets of $\alpha$ elements each (except the last partition which is of size $\textstyle ( k - ( \left\lfloor { \frac { k } { \alpha } } \right\rfloor \alpha ) )$ is not an integer) as shown in Equations( 8- 9). This augmentation shall decrease the enumeration cost to: $\pmb { \Delta } _ { \mathbf { d u a l } } =$ $\begin{array} { r l } & { \sum _ { v \in V _ { a } } d ( v ) ! + \sum _ { v \in V _ { b } } \left( \left\lfloor \frac { d ( v ) } { \alpha } \right\rfloor \alpha ! + \left( d ( v ) - \left( \left\lfloor \frac { d ( v ) } { \alpha } \right\rfloor \alpha \right) \right) ! \right) } \end{array}$ , where $V _ { a } = \{ v | v \in V , d ( v ) \leq \alpha \}$ and $V _ { b } = \left( V - V _ { a } \right)$ . Depending upon the availability of resources (single machine or a distributed computing resources) we can choose the cut-off degree $( \alpha )$ accordingly. Notice that although we perform this degree thresholding we still get embedding for all the hyperedges. However, what we sacrifice is the higher order information that a given vertex connects a set of incident hyperedges jointly. But for higher order vertices we argue that as the vertex degree get too high our belief in the inference that hyperedges incident are related diminishes. These high degree vertices can be interpreted as highly popular person participating in a huge number of teams, but should the embeddings of these teams be related to each other is difficult to infer. In text data these high degree nodes correspond to words occurring in a large number of phrases, but this doesn’t mean the sentences are related, rather its just that this word is quiet common.
+
+In case of vertex embeddings, for each hyperedge $g _ { i } \in G$ , we need to initialize $| g _ { i } | !$ elements of $\mathcal { A } _ { \bf h y p } ^ { \left| \bf g _ { i } \right| }$ , with a worst case enumeration cost of $\sum _ { g _ { i } \in G } | g _ { i } | !$ . Given this cost can be prohibitive as the hyperedge cardinalities increase, we propose an augmented initialization scheme for hypergraph e we shall reswe initialize f cardinality same as Equ $\le \beta$ . For hyperedges n 26. For hypered $g _ { i }$ with with $| g _ { i } | = k \le \beta$ $\mathcal { A } _ { \mathbf { h } \mathbf { y } \mathbf { p } } ^ { \mathbf { k } } = ( a _ { p _ { 1 } , p _ { 2 } , . . , p _ { k } } ) \in \bar { \mathbb { R } } ^ { [ k , n ] }$ cardinality $| g _ { i } | > \beta$ , we initialize the elements of $\mathcal { A } _ { \mathbf { h y p } } ^ { \beta } \in \mathbb { R } ^ { [ \beta , n ] }$ as follows:
+
+$$
+a _ { p _ { 1 } ^ { l } , p _ { 2 } ^ { l } , . . , p _ { \beta } ^ { l } } = \frac { R ( g _ { i } ) } { \gamma }
+$$
+
+where $( p _ { 1 } ^ { l } , p _ { 2 } ^ { l } , . . , p _ { \beta } ^ { l } )$ is the $l$ -th permutation among $l = \{ 1 , 2 , . . . , \gamma \}$ , where $\gamma = \binom { \left| g _ { i } \right| } { \beta }$ . Enumeration cost now decreases to: $\begin{array} { r } { \pmb { \Delta } _ { \mathbf { h y p } } = \sum _ { g _ { i } \in G _ { a } } | g _ { i } | ! + \sum _ { g _ { i } \in G _ { b } } \binom { | g _ { i } | } { \beta } \beta ! } \end{array}$ , where $G _ { a } = \{ g _ { i } | g _ { i } \in G , | g _ { i } | = \beta \}$ and $G _ { b } = ( G - G _ { a } )$ . The choice of cut-off $( \beta )$ is therefore, dictated by the computation resources available. Notice that although we perform this cardinality thresholding we still get embedding for all the vertices. However, what we sacrifice is the higher order information that a given hyperedge connects a set of incident vertices jointly. Now, as the $k$ increases, the number of $k$ cardinality hyperedge decrease as well as the $\mathcal { A } _ { \bf h y p } ^ { \bf k }$ tensor grows exponentially making it increasingly sparse and less informative (in information theoretic sense). Therefore, restricting to $\beta$ cardinality hyperedges is a trade-off between enumeration cost and hyperedge level joint information loss. We observer that enumeration involved in both hyperedge and vertex embeddings, Equations( 8- 10), are either vertex or hyperedge centric computations and can be performed in a scalable manner using the generic hyperedge-centric distributed computation libraries like MESH (Heintz & Chandra, 2015; Heintz et al., 2017) and HyperX (Huang et al., 2015).
+
+Now we consider the tensor decomposition complexity and its scalability. For hyperedge embeddings, the per iteration complexity of the algorithm is $O ( \sum _ { \delta = 1 } ^ { \alpha } \mathrm { { n n } } \mathbf { \dot { z } } ( \mathbf { \mathcal { A } } _ { \mathbf { d u a l } } ^ { \delta } ) )$ . Notice, $\begin{array} { r } { \big ( \sum _ { \delta = 1 } ^ { \alpha } \mathtt { n n z } ( \mathcal { A } _ { \bf d u a l } ^ { \delta } ) \big ) \ \le \ \Delta _ { \bf d u a l } } \end{array}$ of vertex . Notice, -. $\scriptstyle O ( \sum _ { k = c _ { m i n } } ^ { \beta } { \mathrm { n n z } } ( { \mathcal { A } } _ { \mathbf { h y p } } ^ { \mathbf { k } } ) )$ $\big ( \sum _ { k = c _ { m i n } } ^ { \beta } \Pi \mathbf { n } \mathbf { n } \mathbf { z } ( \mathcal { A } _ { \mathbf { h } \mathbf { y } \mathbf { p } } ^ { \mathbf { k } } ) \big ) \ \leq \ \Delta _ { \mathbf { h } \mathbf { y } \mathbf { p } }$ Hypergraph Tensor Decomposition basically involves learning vertex or hyperedge embeddings, i.e. parameters which are vertex or hyperedge centric. We can therefore, convert the tensor decompositions into equivalent hyperedge or vertex centric message passing algorithm and use the generic hyperedge or vertex centric distributed computation libraries like MESH or HyperX, as mentioned previously. Although, we highlight the possible directions for scalability, but we propose them as a future work.
+
+Scalability without partitioning for t2v As discussed previously, as we are performing hyperedge cardinality as well as vertex degree thresholding and for that reason we developed augmented enumeration schemes in Equation ( 8- 10). The partitioning involved in this augmentation ensures that all the hyperedges and all the vertices are linked by some vertex of degree $\leq \alpha$ and some hyperedge of cardinality $\le \beta$ . If this augmentation was not performed, and we simply use the enumeration of Equations ( 26- 29), it was possible for example, that some hyperedge which has no vertex whose degree is $\leq \alpha$ , and therefore, would be left out and no embedding would be learned for this hyperedge (because of cold start in tensor decomposition). Rather than performing the augmentation we can use semi-supervised learning to learn the embeddings for critical hyperedges (hyperedges with all vertices with degree $> \alpha \backslash$ ) using the embeddings of the non-critical hyperedges ((hyperedges with at least one vertex with degree $\leq \alpha )$ ). As all the hyperedges (critical or not) are linked using the hypergraph topology, we can perform semi-supervised learning using topology based regularization while performing tensor decomposition. This can be done using a graph regularization term $( \mathbf { \Psi } _ { k } \mathbf { U ^ { ( j ) } } ^ { T } ) \mathbf { L } ( \mathbf { \Psi } _ { k } \mathbf { U ^ { ( j ) } } ) )$ in our tensor decomposition objective function 5. Here, L is the graph laplacian of our choice. For example we can use $\mathbf { L _ { h y p } }$ or $\mathbf { L } _ { \mathbf { g r a p h } }$ when we perform tensor decomposition for vertex embeddings and use $\mathbf { L _ { i n v } }$ or $\mathbf { L _ { d u a l } }$ for hyperedge embeddings. Appendix A.3 mentions the details about these laplacians.
+
+Scalability for h2v-auto In case of auto-encoder method the main scalability challenge lies in generating the noisy hyperedges. For a given hyperedge the intermediate sub-lattice from which $p$ samples are drawn is of size $O ( n ^ { h } )$ , where $h$ is max distance from the hyperedge considered. Once this sampling is performed we have total $m p$ hyperedges as input for the auto-encoder, which is linear in the number of hyperedge as $p$ will be a constant. The main challenge therefore, is to generate the noisy samples where we have to tackle exponential size sub-lattice per hyperedge. Again, as this sampling is done for each hyperedge separately (hyperedge-centric), we can therefore, use distributed computing for hypergraphs using MESH and HyperX, as mentioned before.
+
+Issues with aggregating vertex embedding Furthermore, we also observe that even simple element-wise summation or averaging of node embeddings for the nodes (in a given hyperedge) also perform comparably when compared to hyperedge embedding alone. From this we can infer that depending upon the dataset, if we have less hyperedges and more nodes, than we would rather prefer to simply learn the hyperedge embedding directly rather than learning node embeddings and then performing aggregating operation over them. Aggregation might turn out be costly specially if average hyperedge size is large and the choice of aggregation function is an issue. Therefore, learning hyperedge embeddings directly seems to be escape the problem of choosing the aggregation function all together.
+
+Scalability issues for baselines Another thing we notice, is that in case of LangNet dataset, that node2vec based h2v-inv and h2v-dual methods are simply unable to run (see N/A in Table 2). It seems that in LangNet (and possibly in text data in general) the hypergraph dual, $\mathbf { A } _ { \mathbf { d u a l } }$ , which contain phrase to phrase edges turns out be containing significantly more edges than the number of node to node edges in $\mathbf { A _ { h y p } }$ . Therefore, performing context generation (using node2vec) over $\mathbf { A _ { i n v } }$ and $\mathbf { A _ { d u a l } }$ graphs turns out be very costly and we were unable to get hyperedge embeddings for h2v-inv and h2v-dual methods. Note, we however do get vertex embedding (as mentioned with a \* mark below Table 3). But as you can see that even the time time taken for vertex embedding is similar or more than the total time taken by spectral methods for learning both hyperedge as well as vertex embedding, together. This indicates the robustness of the spectral methods, specially the e2v-hyp for different kinds of hypergraph data and edge densities (sparsity).
+
+# 5 RELATED WORKS
+
+Hypergraphs were studied rigorously by Berge (1976; 1984) as a generalization of graphs and directed hypergraphs have been introduced by Bretto (2013). Hypergraph were argued for the first time as a model to naturally capture higher-order relationships between sets of objects across variety of domains by Estrada & Rodriguez-Velazquez (2005). Hypergraphs have been used to model complex networks in different fields including biology (Klamt et al., 2009), databases (Fagin, 1983) and data mining (Han et al., 1997; Zhou et al., 2007). Within machine learning, algorithms guided by the structure of hypergraph were introduced by Zhou et al. (2006) and have found applications in a variety of domains (Tian et al., 2009; Gao et al., 2013; Li & Li, 2013; Sharma et al., 2015; 2017). Simplicial complex (Munkres, 1984) based view of hypergraph using hasse lattice (Skiena, 1990) within machine learning has recently been proposed by Sharma et al. (2017).
+
+Representation learning (RL) (Bengio et al., 2013) focuses on learning features (geometry) of the data (topology). Machine learning algorithms make use of these features for prediction. Traditionally these features are readily available within the data-set or are engineered manually. However, this is often a tedious and human labor-intensive process. RL addresses this issue by learning features automatically in a task-dependent supervised or a task-independent unsupervised manner.
+
+Node Representations in Graph: Traditionally unsupervised node embedding learning has been dong using latent models like matrix factorization (Ahmed et al., 2013) or by community detection (Tang & Liu, 2011) based techniques for networks (Roweis & Saul, 2000; Belkin & Niyogi, 2001; Tenenbaum et al., 2000; Cox & Cox, 2000). In each case there is a vector of features learned for a node, each of whose entries reflects node’s association with some latent dimension or a network community. More recently, there has been a revived interest in graph embedding in form of context oriented techniques (Perozzi et al., 2014; Tang et al., 2015b; Grover & Leskovec, 2016). These techniques are inspired by recent unsupervised RL methods in NLP (Mikolov et al., 2013b; Le & Mikolov, 2014) where word embeddings are learned that are similar to words in a given neighborhood or context. These techniques differ in the manner they generate this context as well as in the objective which they optimize. Also there are supervised algorithms learn embeddings which are optimal for the specific task at hand. This results in high accuracy but incurs significant computational cost for training. Recently, several supervised learning algorithms have been proposed for network analysis (Tian et al., 2014; Xiaoyi et al., 2013) and for text networks in a semi-supervised setting (Tang et al., 2015a). Finally, we refer readers to a very recent and comprehensive survey on graph embedding methods by Cai et al. (2017).
+
+Node Representations in Hypergraph: Learning embeddings for nodes within a hypergraph while incorporating the hypergraph topology using proxy graphs is introduced by Zhou et al. (2006). Using graph proxy destroys the hypredge-level joint information and thus, incur loss of information. Also, Agarwal et al. (2006) squarely criticize that such representations can be learned by constructing graphs, which are proxies for the hypergraph structure.
+
+Set Representations: RL for sets using neural networks has been proposed recently (Vinyals et al., 2016), where a memory network is used to compose features sequentially but in an order invariant manner. In their very recent paper, Rezatofighi et al. (2016) have tried to answer this set ordering issue by the use of random set theory. However, they do not consider embedding but focus on learning set-level probabilities. More importantly, both of these works, do not consider the hypergraph structure of overlapping sets which is the main focus of this paper.
+
+Tensors For comprehensive view of tensors, tensor decomposition as well as applications we refer to the survey by Kolda & Bader (2009). The connection between $k$ -way tensor and $\mathbf { k }$ -uniform hypergraph eigen values was established by Qi (2005). The use of $k$ -way symmetric tensor and their non-negative decomposition for uniform hypergraph partitioning was first introduced by Shashua et al. (2006). But they are again restricted to uniform hypergraphs.
+
+# 6 CONCLUSION
+
+In this paper we have proposed two methods to generate higher-order representations for both hyperedges (representing sets of nodes) and hypergraph nodes (that also take into account the hypergraph structure). First, is an auto-encoder based deep-learning method and second, is a tensor-based algebraic method. Both learning models are unique in the manner they leverage the existing structure present in network data as context. While introducing a new idea of a dual tensors corresponding to the hypergraph dual, we develop a novel approach of using factors from joint decomposition of $k$ -way tensors corresponding to $k$ -uniform sub-hypergraphs, as generic node & hyperedge representations. We show that that both methods perform comparably with several other baselines in terms of accuracy. We also observe that the proposed tensor based methods are more efficient and also have a natural hypergraph theoretic interpretation; unlike deep learning based black-box approach. We therefore, argue that we have proposed more general methods which are principally suited for hypergraphs (and therefore also for graphs) while maintaining accuracy and efficiency.
+
+# REFERENCES
+
+Pubmed data. https://www.ncbi.nlm.nih.gov/pubmed/.
+
+Sameer Agarwal, Kristin Branson, and Serge Belongie. Higher order learning with graphs. In Proceedings of the 23rd international conference on Machine learning, pp. 17–24. ACM, 2006.
+
+Amr Ahmed, Nino Shervashidze, Shravan Narayanamurthy, Vanja Josifovski, and Alexander J Smola. Distributed large-scale natural graph factorization. In Proceedings of the 22nd international conference on World Wide Web, pp. 37–48. ACM, 2013.
+
+Iftekhar Ahmed, Channing Brown, Andrew Pilny, Dora Cai, Yannick Atouba Ada, and Marshall Scott Poole. Identification of groups in online environments: The twist and turns of grouping groups. In Privacy, Security, Risk and Trust (PASSAT) and 2011 IEEE Third Inernational Conference on Social Computing (SocialCom), 2011 IEEE Third International Conference on, pp. 629–632. IEEE, 2011.
+
+Anthony F Bartholomay. Molecular set theory: A mathematical representation for chemical reaction mechanisms. Bulletin of Mathematical Biology, 22(3):285–307, 1960.
+
+Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps and spectral techniques for embedding and clustering. In NIPS, volume 14, pp. 585–591, 2001.
+
+Yoshua Bengio and Samy Bengio. Modeling high-dimensional discrete data with multi-layer neural networks. In Advances in Neural Information Processing Systems, pp. 400–406, 2000.
+
+Yoshua Bengio, Aaron Courville, and Pascal Vincent. Representation learning: A review and new perspectives. IEEE transactions on pattern analysis and machine intelligence, 35(8):1798–1828, 2013.
+
+Yoshua Bengio et al. Learning deep architectures for ai. Foundations and trends $\textsuperscript { \textregistered }$ in Machine Learning, 2(1):1–127, 2009.
+
+C. Berge. Graphs and hypergraphs, volume 6. Elsevier, 1976.
+
+Claude Berge. Hypergraphs: combinatorics of finite sets, volume 45. Elsevier, 1984.
+
+Alain Bretto. Hypergraph theory. An introduction. Mathematical Engineering. Cham: Springer, 2013.
+
+Samuel R Bulo and Marcello Pelillo. A game-theoretic approach to hypergraph clustering. In \` Advances in neural information processing systems, pp. 1571–1579, 2009.
+
+Hongyun Cai, Vincent W Zheng, and Kevin Chen-Chuan Chang. A comprehensive survey of graph embedding: Problems, techniques and applications. arXiv preprint arXiv:1709.07604, 2017.
+
+Evangelia Christakopoulou and George Karypis. Hoslim: higher-order sparse linear method for topn recommender systems. In Pacific-Asia Conference on Knowledge Discovery and Data Mining, pp. 38–49. Springer, 2014.
+
+Joshua Cooper and Aaron Dutle. Spectra of uniform hypergraphs. Linear Algebra and its Applications, 436(9):3268–3292, 2012.
+
+Trevor F Cox and Michael AA Cox. Multidimensional scaling. CRC press, 2000.
+
+Mukund Deshpande and George Karypis. Item-based top-n recommendation algorithms. ACM Transactions on Information Systems (TOIS), 22(1):143–177, 2004.
+
+Ernesto Estrada and Juan A Rodriguez-Velazquez. Complex networks as hypergraphs. arXiv preprint physics/0505137, 2005.
+
+R. Fagin. Degrees of acyclicity for hypergraphs and relational database schemes. Journal of the ACM (JACM), 30(3):514–550, 1983.
+
+Shenghua Gao, Ivor Wai-Hung Tsang, and Liang-Tien Chia. Laplacian sparse coding, hypergraph laplacian sparse coding, and applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(1):92–104, 2013.
+
+Aditya Grover and Jure Leskovec. node2vec: Scalable feature learning for networks. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, August 13-17, 2016, pp. 855–864, 2016. doi: 10.1145/2939672. 2939754.
+
+E.H. Han, G. Karypis, V. Kumar, and B. Mobasher. Clustering based on association rule hypergraphs. University of Minnesota, Department of Computer Science, 1997.
+
+Benjamin Heintz and Abhishek Chandra. Enabling scalable social group analytics via hypergraph analysis systems. In HotCloud, 2015.
+
+Benjamin Heintz, Shivangi Singh, Rankyung Hong, Guarav Khandelwal, Corey Tesdahl, and Abhishek Chandra. Mesh: A flexible distributed hypergraph processing system. 2017.
+
+Jin Huang, Rui Zhang, and Jeffrey Xu Yu. Scalable hypergraph learning and processing. In Data Mining (ICDM), 2015 IEEE International Conference on, pp. 775–780. IEEE, 2015.
+
+TaeHyun Hwang, Ze Tian, Rui Kuangy, and Jean-Pierre Kocher. Learning on weighted hypergraphs to integrate protein interactions and gene expressions for cancer outcome prediction. In Data Mining, 2008. ICDM’08. Eighth IEEE International Conference on, pp. 293–302. IEEE, 2008.
+
+S. Klamt, U.U. Haus, and F. Theis. Hypergraphs and cellular networks. PLoS computational biology, 5(5):e1000385, 2009.
+
+Tamara G Kolda and Brett W Bader. Tensor decompositions and applications. SIAM review, 51(3): 455–500, 2009.
+
+Quoc V Le and Tomas Mikolov. Distributed representations of sentences and documents. In ICML, volume 14, pp. 1188–1196, 2014.
+
+Lei Li and Tao Li. News recommendation via hypergraph learning: encapsulation of user behavior and news content. In Proceedings of the sixth ACM international conference on Web search and data mining, pp. 305–314. ACM, 2013.
+
+Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space. arXiv preprint arXiv:1301.3781, 2013a.
+
+Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg Corrado, and Jeffrey Dean. Distributed representations of words and phrases and their compositionality. In Proceedings of the 26th International Conference on Neural Information Processing Systems, NIPS’13, pp. 3111–3119, 2013b.
+
+James R Munkres. Elements of algebraic topology, volume 2. Addison-Wesley Menlo Park, 1984.
+
+Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 701–710. ACM, 2014.
+
+Liqun Qi. Eigenvalues of a real supersymmetric tensor. Journal of Symbolic Computation, 40(6): 1302–1324, 2005.
+
+Seyed Hamid Rezatofighi, Anton Milan, Ehsan Abbasnejad, Anthony Dick, Ian Reid, et al. Deepsetnet: Predicting sets with deep neural networks. arXiv preprint arXiv:1611.08998, 2016.
+
+Sam T Roweis and Lawrence K Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323–2326, 2000.
+
+Ankit Sharma, Rui Kuang, Jaideep Srivastava, Xiaodong Feng, and Kartik Singhal. Predicting small group accretion in social networks: A topology based incremental approach. In 2015 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining (ASONAM), pp. 408–415. IEEE, 2015.
+
+Ankit Sharma, Terrence J Moore, Ananthram Swami, and Jaideep Srivastava. Weighted simplicial complex: A novel approach for predicting small group evolution. In Pacific-Asia Conference on Knowledge Discovery and Data Mining, pp. 511–523. Springer, 2017.
+
+Amnon Shashua, Ron Zass, and Tamir Hazan. Multi-way clustering using super-symmetric nonnegative tensor factorization. Computer Vision–ECCV 2006, pp. 595–608, 2006.
+
+Steven Skiena. Hasse diagrams. Implementing Discrete Mathematics: Combinatorics and Graph Theory With Mathematica, pp. 163, 1990.
+
+Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher Manning, Andrew Ng, and Christopher Potts. Parsing With Compositional Vector Grammars. In EMNLP. 2013.
+
+Steven H Strogatz. Exploring complex networks. Nature, 410(6825):268–276, 2001.
+
+Karthik Subbian, Charu Aggarwal, and Jaideep Srivastava. Content-centric flow mining for influence analysis in social streams. In Proceedings of the 22nd ACM international conference on Conference on information & knowledge management, pp. 841–846. ACM, 2013.
+
+Jian Tang, Meng Qu, and Qiaozhu Mei. Pte: Predictive text embedding through large-scale heterogeneous text networks. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1165–1174. ACM, 2015a.
+
+Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. Line: Large-scale information network embedding. In Proceedings of the 24th International Conference on World Wide Web, pp. 1067–1077. ACM, 2015b.
+
+Lei Tang and Huan Liu. Leveraging social media networks for classification. Data Mining and Knowledge Discovery, 23(3):447–478, 2011.
+
+Joshua B Tenenbaum, Vin De Silva, and John C Langford. A global geometric framework for nonlinear dimensionality reduction. science, 290(5500):2319–2323, 2000.
+
+Fei Tian, Bin Gao, Qing Cui, Enhong Chen, and Tie-Yan Liu. Learning deep representations for graph clustering. In AAAI, pp. 1293–1299, 2014.
+
+Ze Tian, TaeHyun Hwang, and Rui Kuang. A hypergraph-based learning algorithm for classifying gene expression and arraycgh data with prior knowledge. Bioinformatics, 25(21):2831–2838, 2009.
+
+Pascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol. Extracting and composing robust features with denoising autoencoders. In Proceedings of the 25th international conference on Machine learning, pp. 1096–1103. ACM, 2008.
+
+Pascal Vincent, Hugo Larochelle, Isabelle Lajoie, Yoshua Bengio, and Pierre-Antoine Manzagol. Stacked denoising autoencoders: Learning useful representations in a deep network with a local denoising criterion. Journal of Machine Learning Research, 11(Dec):3371–3408, 2010.
+
+Oriol Vinyals, Samy Bengio, and Manjunath Kudlur. Order matters: Sequence to sequence for sets. In ICLR, 2016.
+
+Li Xiaoyi, Li Hui Du Nan, et al. A deep learning approach to link prediction in dynamic networks. In Proceedings of the 2013 SIAM International Conference on Data Mining. Philadelphia, PA, USA: SIAM, 2013.
+
+D. Zhou, J. Huang, and B. Scholkopf. Learning with hypergraphs: Clustering, classification, and embedding. Advances in Neural Information Processing Systems, 19:1601, 2007.
+
+Dengyong Zhou, Jiayuan Huang, and Bernhard Scholkopf. Learning with hypergraphs: Clustering, ¨ classification, and embedding. In Advances in neural information processing systems, pp. 1601– 1608, 2006.
+
+![](images/9f65e8e35f028b30bbbf3eeafa5cec3d720defbe7f139efbe3bbc6ceae666b33.jpg)  
+Figure 4: Distributed memory (DM) and Distributed bag of words (DBOW) versions of Sen2Vec for a sentence “I love soccer”.
+
+# A APPENDIX
+
+# A.1 HYPEREDGE2VEC USING SENTENCE EMBEDDINGS
+
+As mentioned before, the most commonly used model for studying complex interactions in networks is graphs, where each edge represents a dyadic interaction between nodes Strogatz (2001). Therefore, even if the original interaction is not dyadic like a set of researchers (3 or more) collaborating on a publication, we shall break down this joint interaction into dyadic interactions. Even though a large number of complex network data naturally occurs as hypergraphs Estrada & RodriguezVelazquez (2005), the popularity of “think like dyadic edges” and not like sets seems to us as a hindrance, specifically when the end aim is to model the joint distribution at the level of sets. In this paper, as it is obvious, we go by “think like set” paradigm, specifically, when the data is naturally occurring as network of sets (i.e., a hypergraph). Therefore, if we think of hyperedge as a set, a natural question arises is that are there any existing techniques that learn embeddings at set level? We, henceforth, explore the most popular discrete data on which representation learning has been applied is text data.
+
+Recently, Le & Mikolov (2014) proposed two representation learning methods for sentences as shown in Figure 4: a) a distributed memory (DM) model, and b) a distributed bag of words (DBOW) model. In the DM model, every sentence in the dataset $G$ is represented by a $d$ dimensional vector in a shared lookup matrix $S \in \mathbb { R } ^ { | G | \times d }$ . Similarly, every word in the vocabulary $\Omega$ is represented by a $d$ dimensional vector in another shared lookup matrix $L \in \mathbb { R } ^ { | \Omega | \times d }$ . Given an input sentence $\mathbf { v } = \left( v _ { 1 } , v _ { 2 } \cdot \cdot \cdot v _ { m } \right)$ , the corresponding sentence vector from $S$ and the word vectors from $L$ are averaged to predict the next word in a context. More formally, let $\phi$ denote the mapping from sentence and word ids to their respective vectors in $S$ and $L$ , the DM model minimizes the following objective:
+
+$$
+\begin{array} { r c l } { { { \cal J } ( \phi ) } } & { { = } } & { { \displaystyle - \sum _ { t = k } ^ { m - k } \log P ( v _ { t } | { \bf v } ; v _ { t - k + 1 } , \cdots , v _ { t - 1 } ) } } \\ { { } } & { { = } } & { { \displaystyle - \sum _ { t = k } ^ { m - k } \log \frac { \exp ( \omega ( v _ { t } ) ^ { T } { \bf z } ) } { \sum _ { i } \exp ( \omega ( v _ { i } ) ^ { T } { \bf z } ) } } } \end{array}
+$$
+
+where $\mathbf { z }$ is the average of $\phi ( \mathbf { v } ) , \phi ( v _ { t - k + 1 } ) , \cdot \cdot \cdot , \phi ( v _ { t - 1 } )$ input vectors, and $\omega ( v _ { t } )$ is the output vector representation of word $\boldsymbol { v } _ { t } ~ \in ~ \Omega$ . The sentence vector $\phi ( \mathbf { v } )$ is shared across all (sliding window) contexts extracted from the same sentence, thus acts as a distributed memory.
+
+Instead of predicting the next word in the context, the DBOW model predicts the words in the context independently given the sentence id as input. More formally, DBOW minimizes the following objective:
+
+$$
+\begin{array} { r c l } { J ( \phi ) } & { = } & { \displaystyle - \sum _ { t = k } ^ { m - k } \sum _ { j = t - k + 1 } ^ { t } \log P ( v _ { j } | \mathbf { v } ) } \\ & { = } & { \displaystyle - \sum _ { t = k } ^ { m - k } \sum _ { j = t - k + 1 } ^ { t } \log \frac { \exp \left( \omega ( v _ { j } ) ^ { T } \phi ( \mathbf { v } ) \right) } { \sum _ { i } \exp \left( \omega ( v _ { i } ) ^ { T } \phi ( \mathbf { v } ) \right) } } \end{array}
+$$
+
+Both the methods take sentences as input and return embeddings for words as well as the sentence. To apply the sentence embedding models directly to hyperedges, as our first method we generate a proxy sentence for each hyperedge by leveraging the contextual information in the hyperedge.
+
+A proxy sentence $\mathbf { v } _ { i }$ is formed for each hyperedge $g _ { i } \in \textit { G }$ as a sequence made by concatenating all the permutations of the nodes (as words) in the hyperedge and further repeating this sequence as many time this hyperedge occurred. For example, for a three node hyperedge $\begin{array} { r c l } { { g _ { i } } } & { { = } } & { { \{ 1 , 4 , 7 \} } } \end{array}$ which has occurred two times we make the following sentence $\mathbf { v } _ { i }$ : {1,4,7,1,7,4,7,1,4,7,4,1,4,1,7,4,7,1,1,4,7,1,7,4,7,1,4,7,4,1,
+
+$\delta , 1 , 7 , 4 , 7 , 1 \}$ of length 12 (6 permutations times 2 occurrence). In this scheme we have permutations and repetitions. We take all the permutations hoping that it should fool the sequence based model to get a sequence independent embedding. On the other hand, repetition of the permutations acts similarly to observing the same sequence of words several times in the text corpus. Of course one can think of other alternative schemes, but the point we are trying to make is that naturally observed hyperedges captures important contextual information that can be leveraged to achieve better representations. We refer to the node-hyperedge embedding pairs resulting from DM, DBOW models as h2v-dm and $\mathbf { h } 2 \mathbf { v }$ -dbow, respectively.
+
+# A.2 HYPEREDGE2VEC USING NODE2VEC
+
+Grover & Leskovec (2016) propose a representation method for nodes in a graph called node2vec, which uses the skip-gram model Mikolov et al. (2013b) with the intuition that neighborhood nodes within a graph should have similar representations. The objective of the skip-gram model for graphs can be defined as:
+
+$$
+\begin{array} { r c l } { { J ( \phi ) } } & { { = } } & { { \displaystyle - \sum _ { v \in V } \log P ( N ( v ) | \phi ( v ) ) } } \\ { { } } & { { = } } & { { \displaystyle - \sum _ { v \in V } \sum _ { n _ { i } \in N ( v ) } \log P ( n _ { i } | \phi ( v ) ) } } \\ { { } } & { { = } } & { { \displaystyle - \sum _ { v \in V } \sum _ { n _ { i } \in N ( v ) } \log \frac { \exp \left( \omega ( n _ { i } ) ^ { T } \phi ( v ) \right) } { \sum _ { x \in V } \exp \left( \omega ( x ) ^ { T } \phi ( v ) \right) } } } \end{array}
+$$
+
+where as before, $\phi$ and $\omega$ denote the input and the output vector representations of the nodes. The neighboring nodes $N ( v )$ form the context for node $v$ . node2vec uses a biased random walk which adaptively combines breadth first search (BFS) and depth first search (DFS) to find the neighborhood of a node. The walk attempts to capture two properties of a graph often used for prediction tasks in networks: (i) homophily and (ii) structural equivalence. According to homophily, nodes in the same group or community should have similar representations. Structural equivalence suggests that nodes with similar structural roles (hub, bridge) should have similar representations. In a real-world network, nodes exhibit mixed properties.
+
+Our aim is to find both node and hyperedge level embeddings by taking into account the hypergraph structure. For the former, we leverage the adjacency matrix associated with a hypergraph Zhou et al. (2006), which is defined as:
+
+$$
+\mathbf { A _ { h y p } } = \mathbf { H } ^ { T } \mathbf { W _ { e } H } - \mathbf { D _ { v } }
+$$
+
+where ${ \bf W _ { e } }$ is a diagonal matrix containing the weights of each hyperedge and $\mathbf { D _ { v } }$ is a diagonal matrix containing the degree of each vertex. We take the weight of a hyperedge as its occurrence number, i.e. $\mathbf { W _ { e } } ( i , i ) = R ( g _ { i } ) , \forall \{ i \in \{ 1 , . . . , n \} \}$ . The adjacency matrix $\mathbf { \bar { \phi } } ( \mathbf { A _ { h y p } } \in \mathbb { R } ^ { | V | \times | V | } )$ associates a weight between a pair of nodes while taking into account the weights of all the hyperedges that encompass a pair of nodes. The weighted graph associated with $\mathbf { A _ { h y p } }$ in some sense serves as a proxy to the actual hypergraph structure. We can provide $\mathbf { A _ { h y p } }$ as input to node2vec and hypothesize that the random walk over this proxy hypergraph should allow the skip-gram model to learn more meaningful node level embeddings which can be combined to construct hyperedge level embeddings. We refer to these node level embeddings as N2V-hyp.
+
+We still have not met our second objective to learn hyperedge embeddings directly. We again wish to leverage node2vec for our purpose. However, node2vec only works for graphs by performing random walk over nodes. Therefore, we ask ourselves the question, that can we treat hyperedges as nodes? There can be other ways of doing so, but here we suggest two techniques. In the first technique, we simply consider the hypergraph dual Berge (1984), whose incidence matrix is $\mathbf { H } _ { \mathbf { d u a l } } = \bar { \mathbf { H } } ^ { T }$ . The adjacency matrix associated with the hypergraph dual is:
+
+$$
+\mathbf { A } _ { \mathbf { d u a l } } = \mathbf { H _ { d u a l } } ^ { T } \mathbf { W _ { v } } \mathbf { H _ { d u a l } } - \mathbf { D _ { e } } = \mathbf { H } \mathbf { W _ { v } } \mathbf { H } ^ { T } - \mathbf { D _ { e } }
+$$
+
+where $\mathbf { W _ { v } }$ is a diagonal matrix containing the weights of each node and $\mathbf { D _ { e } }$ is a diagonal matrix containing the degree of each hyperedge. We assume no weights on the nodes and take $\mathbf { W _ { v } } = \mathbf { I }$ . The matrix $\mathbf { A _ { d u a l } } ^ { } \mathbf { \bar { \Pi } } \mathbf { \in } \mathbb { R } ^ { | G | \times | G | }$ represents another hypergraph, but the roles of nodes and hyperedges have now switched. For example, in case of words and sentences, the hyperedges were sentences, but in the hypergraph dual the words become hyperedges and the nodes within a word’s hyperedge represent all the sentences in which the word has appeared. We again give $\mathbf { A } _ { \mathbf { d u a l } }$ (i.e., a graph proxy for the dual hypergraph) as input to node2vec, but this time we get output as the embeddings for the hyperedges associated with the nodes in dual. We refer to these hyperedge embeddings as h2v-dual.
+
+In the second technique, we consider the following adjacency matrix $\mathbf { A _ { i n v } } \in \mathbb { R } ^ { | G | \times | G | }$
+
+$$
+\mathbf { A } _ { \mathbf { i n v } } = \mathbf { H } \mathbf { H } ^ { T }
+$$
+
+associated with what we refer to as the inverted hypergraph. This inverted hypergraph is a graph (unlike the dual which is a hypergraph) and there is an edge between two nodes if the hyperedges corresponding to the nodes in the original hypergraph have nodes in common. Weight of this edge is the number of common nodes. We again give $\mathbf { A _ { i n v } }$ as input to node2vec to get embeddings for the hyperedges associated with the nodes in inverted hypergraph. We refer to these hyperedge embeddings as h2v-inv.
+
+# A.3 HYPEREDGE2VEC USING SPECTRAL EMBEDDINGS
+
+These set of methods extract embeddings as the eigenvectors associated with Laplacian matrices corresponding to the various adjacency matrices discussed in the previous section. We consider the following graph Laplacians:
+
+$$
+\mathbf { L _ { g r a p h } } = \mathbf { I } - \mathbf { D _ { v } } ^ { - 1 / 2 } \mathbf { A _ { g r a p h } } \mathbf { D _ { v } } ^ { - 1 / 2 }
+$$
+
+where $\mathbf { A } _ { \mathbf { g r a p h } } = \mathbf { H } ^ { T } \mathbf { W _ { e } H }$ is the weighted graph associated with the graph corresponding to adjacency matrix A,
+
+$$
+\begin{array} { r } { \begin{array} { c c l } { { \bf L } _ { \bf h y p } } & { = } & { { \bf I } - { \bf D } _ { \bf v } - ^ { - 1 / 2 } { \bf A } _ { \bf h y p } { \bf D } _ { \bf v } - ^ { - 1 / 2 } } \\ { { \bf L } _ { \bf i n v } } & { = } & { { \bf I } - { \bf D } _ { \bf e } - ^ { - 1 / 2 } { \bf A } _ { \bf i n v } { \bf D } _ { \bf e } - ^ { - 1 / 2 } } \\ { { \bf L } _ { \bf d u a l } } & { = } & { { \bf I } - { \bf D } _ { \bf e } - ^ { - 1 / 2 } { \bf A } _ { \bf d u a l } { \bf D } _ { \bf e } - ^ { - 1 / 2 } } \end{array} } \end{array}
+$$
+
+We get the $d$ eigenvectors associated with the smallest $d$ eigenvalues of the above graph Laplacians as the embeddings. We get vertex embeddings using Eq. 22 and hyperedge embedding using Eq. 24, and we refer to them together as e2v-hyp. Similarly, we get another pair of vertex embeddings (using Eq. 21) and hyperedge embedding (using Eq. 24). We refer to the later pair as $\mathbf { e } 2 \mathbf { v }$ .
+
+# B APPENDIX
+
+# B.1 TENSOR INITIALIZATION SCHEMES
+
+We can also have other initialization for hypergraph tensor, $\mathcal { A } _ { \mathbf { h } \mathbf { y } \mathbf { p } } ^ { \mathbf { k } } = ( a _ { p _ { 1 } , p _ { 2 } , . . , p _ { k } } ) \in \mathbb { R } ^ { [ k , n ] }$ , as follows:
+
+$$
+a _ { p _ { 1 } , p _ { 2 } , . . , p _ { k } } = \frac { R ( g _ { i } ) } { k ! }
+$$
+
+where $\left\{ v _ { p _ { 1 } } , v _ { p _ { 2 } } , . . . , v _ { p _ { k } } \right\} \in g _ { i }$ and $| g _ { i } | = k , \forall i \in \left. 1 , . . . , m \right.$ . This initialization normalizes the effect of occurrence counts of hyperedge $g _ { i }$ across all of its vertex permutations. It can be interpreted as each vertex permutation marginally contributes to the total occurrence count of the hyperedge. Now if a hyperedge is large intuitively they are less informative about a joint relationship between the vertices part of it. For example the members of a large social group are less likely to be related as compared to members of a small research collaboration who have published several papers together. To incorporate this large hyperedge issue, we normalize the occurrence count with the hyperedge cardinality, as follows:
+
+$$
+a _ { p _ { 1 } , p _ { 2 } , . . , p _ { k } } = \frac { R ( g _ { i } ) } { k ! \times k }
+$$
+
+In a similar manner we can also have different initialization schemes for dual tensor, $\mathbf { \mathcal { A } _ { d u a l } ^ { k } } =$ $\left( a _ { q _ { 1 } , q _ { 2 } , . . , q _ { k } } \right) \in \mathbb { R } ^ { [ k , m ] }$ , as follows:
+
+$$
+a _ { q _ { 1 } , q _ { 2 } , . . , q _ { k } } = \frac { 1 } { k ! }
+$$
+
+where $\left\{ g _ { q _ { 1 } } , g _ { q _ { 2 } } , . . . , g _ { q _ { k } } \right\} \ni v _ { j }$ and $d ( v _ { j } ) = k , \forall j \in \{ 1 , . . . , n \}$ . This initialization indicates that vertex $v _ { j }$ connects various permutations of the hyperedges $\left\{ g _ { q _ { 1 } } , g _ { q _ { 2 } } , . . . , g _ { q _ { k } } \right\}$ , each contributing fractionally. We can also have a weighted initialization like:
+
+$$
+a _ { q _ { 1 } , q _ { 2 } , . . , q _ { k } } = \frac { \sum _ { s = 1 } ^ { k } w _ { 1 } ( g _ { q _ { s } } ) } { k ! }
+$$
+
+If the weight $w _ { 1 } ( g _ { q _ { s } } ) = R ( g _ { q _ { s } } )$ , then the numerator $\textstyle \sum _ { s = 1 } ^ { k } w ( g _ { q _ { s } } )$ would be high if the vertex $v _ { j }$ participated in its neighboring hyperedges at several occurrences, therefore, is a strongly informative if we were to infer that the hyperedges $\left\{ g _ { q _ { 1 } } , g _ { q _ { 2 } } , . . . , g _ { q _ { k } } \right\}$ are related jointly to each other. For example, if an individual has participated a lot in a bunch of social groups then its reasonable to assume that these groups are related. Same argument goes for a common word in several sentence. However, if a word or a person is part of large sentences or big groups then this argument might not be that strong. Reason being, for example a student can be part of a big university and a big company where he works as a part time employee. This does not make the company and the university related. Another issue is that if a word or a person is very common or popular i.e. the vertex $v _ { j }$ has high degree $k ( = d ( v _ { j } ) )$ then its doesn’t mean the sentences or groups they are part of are related. Its just that they are common word or popular person that happen to be in a lot of sentence or groups. We can mitigate the above two issues by the following augmented initialization:
+
+$$
+a _ { q _ { 1 } , q _ { 2 } , . . , q _ { k } } = \frac { \sum _ { s = 1 } ^ { k } w _ { 2 } ( g _ { q _ { s } } ) } { k ! \times k }
+$$
+
+where $\begin{array} { r } { w _ { 2 } ( g _ { q _ { s } } ) = \frac { R ( g _ { q _ { s } } ) } { | g _ { q _ { s } } | } } \end{array}$ R(gqs )|g | is the occurrence counts normalized by the hyperedge size.
+
+B.2 INCIDENT HYPEREDGE PARTITIONING SCHEMES FOR HIGH DEGREE VERTICES
+
+Here we briefly discuss various partitioning schemes capturing different intuitive objectives. As mentioned in the Section 4.4 we wish to limit ourselves to vertices of degree $d ( v ) \leq \alpha$ . One straight forward method is to simply ignore the vertices with $d ( v ) > \alpha$ . However, this would result in loss of the information. In the sense that each vertex connects a set of incident hyperedges and this connection implies that these set of hyperedges are related to each other and that too in a joint fashion with this vertex acting as a common bridge. Therefore, we don’t wish to ignore vertices with $d ( v ) > \alpha$ but rather partition the $d ( v )$ incident hyperedges into sets that have size $\leq \alpha$ . Here we enlist a few possible ways:
+
+1. Linear Partitioning In this we order the hyperedges in a certain order and then equally partition them into $\textstyle \left[ { \frac { d ( v ) } { \alpha } } \right]$ number of $\alpha$ size partitions (if $\alpha$ is a divisor of $d ( v )$ ), else there shall be one more partition of size $\{ d ( v ) - ( \left\lfloor { \frac { d ( v ) } { \alpha } } \right\rfloor ) \}$ . There can be various ways to order:
+
+(a) Order by hyperedge cardinality $( | g _ { i } | )$   
+(b) Order by hyperedge occurrence count $( R ( g _ { i } ) )$ )   
+(c) Order by hyperedge occurrence count normalized by cardinality $( \frac { R ( g _ { i } ) } { | g _ { i } | } )$
+
+(d) Segregate the critical hyperedges for vertex $v$ (i.e. the set $\{ g _ { i } | g _ { i } \in G , v \in g _ { i } , d ( x ) >$ $\alpha , \forall x \in g _ { i } \}$ ) followed by the remaining hyperedges.
+
+2. Subset based partitioning In this we take subsets of the $d ( v )$ hyperedges incident on vertex $v$ . This can be done in couple of different ways:
+
+(a) Take all the $\textstyle { \binom { d ( v ) } { \alpha } }$ subsets of $d ( v )$ hyperedges and initialize the various permutation for each subset in the tensor Aαdual. (b) Take all the $\binom { d ( v ) } { \alpha } , \binom { d ( v ) } { \alpha - 1 } , . . . , \binom { d ( v ) } { 2 }$ subsets of $d ( v )$ hyperedges and initialize the various permutation for each subset in the tensors $\mathcal { A } _ { \bf d u a l } ^ { \alpha } , \mathcal { A } _ { \bf d u a l } ^ { \alpha - 1 } , . . . , \mathcal { A } _ { \bf d u a l } ^ { 2 }$ , respectively.
+
+3. Sampling subsets In this we do not exhaustively use the entire set of $d ( v )$ hyperedges but rather sample subsets of size $\alpha$ from the $d ( v )$ hyperedges. This sampling is performed in proportion to:
+
+(a) Hyperedge cardinality $( | g _ { i } | )$   
+(b) Hyperedge occurrence count $( R ( g _ { i } ) )$ )   
+(c) Hyperedge occurrence count normalized by cardinality $( \frac { R ( g _ { i } ) } { | g _ { i } | } )$
+
+Further, we can also perform the above sampling of subsets from only among the critical hyperedges, or from only among the non-critical hyperedges or in a mixed manner.

parse/train/ryl8-3AcFX/ryl8-3AcFX.md

@@ -0,0 +1,274 @@
+# ENVIRONMENT PROBING INTERACTION POLICIES
+
+Wenxuan Zhou1, Lerrel Pinto1, Abhinav Gupta1,2 1The Robotics Institute, Carnegie Mellon University 2Facebook AI Research
+
+# ABSTRACT
+
+A key challenge in reinforcement learning (RL) is environment generalization: a policy trained to solve a task in one environment often fails to solve the same task in a slightly different test environment. A common approach to improve interenvironment transfer is to learn policies that are invariant to the distribution of testing environments. However, we argue that instead of being invariant, the policy should identify the specific nuances of an environment and exploit them to achieve better performance. In this work, we propose the “Environment-Probing” Interaction (EPI) policy, a policy that probes a new environment to extract an implicit understanding of that environment’s behavior. Once this environmentspecific information is obtained, it is used as an additional input to a task-specific policy that can now perform environment-conditioned actions to solve a task. To learn these EPI-policies, we present a reward function based on transition predictability. Specifically, a higher reward is given if the trajectory generated by the EPI-policy can be used to better predict transitions. We experimentally show that EPI-conditioned task-specific policies significantly outperform commonly used policy generalization methods on novel testing environments.
+
+# 1 INTRODUCTION
+
+Over the last few years deep reinforcement learning (RL) has shown tremendous progress. From beating the world’s best at Go (Silver et al., 2016) to generating agile and dexterous motor policies (Schulman et al., 2015; Lillicrap et al., 2015; Peng et al., 2018), deep RL has been used to solve a variety of sequential decision-making tasks. However, one key issue that deep RL faces is transfer: a policy trained in one environment is extremely dependent on the specific parameters of that environment (Rajeswaran et al., 2017) and fails to generalize to a new environment. Because of this, training agents that can generalize to novel environments has become an active area of research (Rajeswaran et al., 2017; Pinto et al., 2017b; Yu et al., 2017; Peng et al., 2017).
+
+The most common approach for environment generalization is to train a single policy over a variety of different environments (Rajeswaran et al., 2017; Peng et al., 2017), thus allowing the policy network to learn invariances to the underlying environmental factors. However, a monolithic policy that works on multiple environments often ends up being conservative since it is unable to exploit the dynamics of a given environment. We argue that instead of learning a policy that is invariant to environment dynamics we need a policy that adapts to its environment. A popular approach for this is to first perform explicit system identification of the environment, and then use the estimated parameters to generate an environment specific policy (Yu et al., 2017). However, there are three reasons why this approach might be infeasible: (a) it is based on the structural assumption that only a known set of parameters matter; (b) estimating those parameters from few observations is an extremely challenging problem; (c) these approaches require access to environment parameters during training time, which might be impractical. So is it possible to learn an environment dependent policy without estimating every environment parameter?
+
+When humans are tasked to perform in a new environment, we do not explicitly know what parameters affect performance (Braun et al., 2009). Instead, we probe the environment to gain an intuitive understanding of its behavior. The purpose of these initial interactions is not to complete the task immediately, but to extract information about the environment. This process facilitates learning in that environment. Inspired by this, we present a framework in which the agent first performs “environment-probing” interactions that extract information from an environment, then leverages this information to achieve the goal with a task-specific policy. This setup entails three challenges: 1) How should an agent interact with an environment? 2) How can information be extracted from such interactions? 3) How can the extracted information be used to learn a better policy?
+
+One way to interact is by executing random actions in the environment. However, this may not expose the nuances of an environment. Another way is to use samples from the task-specific policy to infer the environment (Yu et al., 2017). However, a good task-specific policy may not be optimal for understanding an environment. Instead, we separately optimize an “Environment-Probing” Interaction policy (EPI-policy) to extract environment information. To assign rewards to the EPIpolicy, our key insight is that the agent’s ability to predict transitions in an environment can be used as a metric of its understanding of that environment. If an “environment-probing” trajectory can help in predicting transitions, then it should be a good trajectory. More specifically, we use the “environment-probing” trajectory to generate an environment embedding vector. This compact embedding vector is passed to a prediction model that provides higher rewards to the trajectories that are useful in estimating environment-dependent transitions.
+
+Once we have learned the EPI-policy, we can use the generated environment embedding as an input to the task-specific policy. Since this task-specific policy is conditioned on the information from the environment, it can now produce actions dependent on the nuances of the specific environment.
+
+We evaluate the effectiveness of using EPI on Hopper and Striker from OpenAI Gym (Brockman et al., 2016) with randomized mass, damping and friction. In both of these environments, the performance of our policies is better than several commonly used techniques and is even comparable to an oracle policy that has full access to the environment information. These results, along with a detailed ablation analysis, demonstrates that implicitly encoding the environment using EPI-policies improves task performance in unseen testing environments.
+
+# 2 RELATED WORK
+
+Developing generalizable machine learning models is a longstanding challenge. One approach to transfer models across different input distributions is domain adaptation. Duan et al. (2012) along with several others (Hoffman et al., 2014; Kulis et al., 2011; Long et al., 2015) domain-adapt visual models. With the onset of deep learning, several works have investigated finetuning (retraining) pretrained networks in different target domains (Yosinski et al., 2014; Li et al., 2016; Tzeng et al., 2014). In reinforcement learning, Gupta et al. (2017) learns different encoders for different environments that share a common feature space to enable transfer. Improving the generalizability of models can also be formulated as a meta-learning problem. Finn et al. (2017) and Yu et al. (2018) finetune the models in new environments. We argue that agents should have the capacity to generalize without updating their policy. Duan et al. (2016b) and Wang et al. (2016) optimize the policy over multiple episodes and use RNN to implicitly model the underlying changing dynamics in the hidden states. Hausman et al. (2018) purposes to learn an embedding in a supervised manner by performing variational inference on task ids. These methods are based on model-free meta-RL, while Sæmundsson et al. (2018) and Clavera et al. (2018) investigate in model-based meta-RL. Sæmundsson et al. (2018) infers latent variables using Gaussian Process, while we use neural networks to create an embedding and experimented with more complex high-dimensional continuous control problems. Concurrent work Clavera et al. (2018) proposes to train a dynamics model that adapts online using either gradient based method (Finn et al., 2017) or recurrent based method (Duan et al., 2016b). The most important difference is that we have a separate policy to probe the environment instead of using the trajectories from the execution of the task. In our method, training prediction models is somewhat related to learning dynamics model in the model-based RL. However, the prediction model in our method does not to be very accurate since we are only taking the difference.
+
+Another stream of research that is relevant to our work is simulation-to-real transfer. Here the goal is to train a policy in a physics simulator and then transfer that policy to a real robot. The challenge here is to bridge the “Reality Gap” between the real world and the simulator. Rusu et al. (2016) attempt to do this via architectural novelty for visual domain adaptation. Sadeghi & Levine (2016); Tobin et al. (2017); Pinto et al. (2017a) and James et al. (2017) perform domain randomization on the inputs during training to learn a transferable policy. These methods target the mismatch of observation space. However, we target the mismatch in the dynamics of the environment. Peng et al. (2017) proposes domain randomization of the environment parameters and learning an LSTM policy to implicitly learn the environment. We compare our method to this approach and show that a separate process that extracts the nuances of an environment is vital for better generalization. Rajeswaran et al. (2017) and Pinto et al. (2017b) also target generalization to environments, however they do not incorporate environment information and hence are only able to learn a conservative policy.
+
+The traditional approach of system identification (Milanese et al., 2013; Gevers et al., 2006; Bongard & Lipson, 2005; Armstrong, 1987) aims to explicitly estimate the varying environment parameters. Yu et al. (2017) propose the UP-OSI framework that can perform online system identification and takes the estimated environment parameters as additional input to the policy. This works well when the number of parameters to estimate is few. However, explicit identification of every environment parameter is a extremely challenging. Denil et al. (2016) trains an policy to gather information about physical properties through interactions in intuitive physics to answer questions, while we are trying do dealt with a more complex continuous control setting.
+
+A recent approach by Pathak et al. (2017; 2018) uses errors in prediction models as an intrinsic reward (Chentanez et al., 2005) to improve exploration in RL. The strategy is to take actions to reach the states that maximize the error in the learned prediction model. In contrast, our method selects actions that reduce the prediction error on the heldout prediction dataset of task policy trajectories.
+
+# 3 BACKGROUND: REINFORCEMENT LEARNING
+
+We use the framework of reinforcement learning to learn the interaction policy and the final task target policy. This section contains brief preliminaries for reinforcement learning. Note that a more comprehensive exposition can be found in Kaelbling et al. (1996) and Sutton $\&$ Barto (1998).
+
+We model our problem as a continuous space Markov Decision Process represented as the tuple $( S , \mathcal { A } , \mathcal { P } , r , \gamma , \bar { \mathbb { S } } )$ , where $s$ is a set of continuous states of the agent, $\mathcal { A }$ is a set of continuous actions the agent can take at a given state, $\mathcal { P } : \mathcal { S } \times \mathcal { A } \times \mathcal { S }  \mathbb { R }$ is the transition probability function, $r : S \times \mathcal { A }  \mathbb { R }$ is the reward function, $\gamma$ is the discount factor, and $\mathbb { S }$ is the initial state distribution.
+
+The goal for an agent is to maximize the expected cumulative discounted reward $\scriptstyle \sum _ { t = 0 } ^ { T - 1 } \gamma ^ { t } r \left( s _ { t } , a _ { t } \right)$ by performing actions according to its policy $\pi _ { \theta } : { \mathcal { S } }  A$ . Here, denotes the parameters for the policy $\pi$ which takes action $a _ { t }$ given state $s _ { t }$ at timestep $t$ . There are various techniques to optimize $\pi _ { \theta }$ . The one we use in this paper is a policy gradient approach, TRPO (Schulman et al., 2015).
+
+# 4 APPROACH
+
+Our objective is to learn a generalizable policy that is trained on a set of training environments, but can succeed across unseen testing environments. To this end, instead of a single policy, we have two separate policies: (a) an Environment-Probing Interaction policy (EPI-policy) that efficiently interacts with an environment to collect environment-specific information, and (b) a task-specific policy that uses this additional environment information (estimated via EPI-policy) to achieve high rewards for a given task. In this section, we present our methodology for learning these two policies.
+
+# 4.1 LEARNING ENVIRONMENT-PROBING INTERACTIONS (EPI)
+
+To learn the EPI-policy, we introduce a reward formulation based on the prediction models (Fig. 1). The key insight here is that a good EPI-policy generates trajectories in an environment from which predicting transitions in that environment is easier. To quantify this notion of how easy or difficult it is to predict transitions, we use the error transition prediction models.
+
+# 4.1.1 TRANSITION PREDICTION MODELS
+
+To train our prediction models, a dataset of transition data $\left( { { s _ { t } } , { a _ { t } } , { s _ { t + 1 } } } \right)$ is collected in the training environments using a pre-trained task policy (Sec. 4.1.3). This data is split into a training set and a validation set. With the training set, we train two prediction models: (a) a vanilla prediction model $f ( s _ { t } , a _ { t } )$ that predicts the next state $s _ { t + 1 }$ given the current state $s _ { t }$ and action $a _ { t }$ , and (b) an EPI-conditioned prediction model $f _ { e p i } ( s _ { t } , a _ { t } ; \psi ( \tau _ { e p i } ) )$ , which takes the embedded EPI-trajectory $\psi ( \tau _ { e p i } )$ as an additional input. Here $\tau _ { e p i }$ is a trajectory in the environment which contains a small fixed number of observations and actions generated by $\pi _ { e p i }$ . $\psi ( \tau _ { e p i } )$ is the low dimensional embedding of the trajectory $\tau _ { e p i }$ . We can now quantify the reward for $\pi _ { e p i }$ as the difference between their performance. The intuition behind this is that if an interaction trajectory generated from $\pi _ { e p i }$ is informative, $f _ { e p i }$ will be able to use it to give smaller prediction error than $f$ .
+
+![](images/454d3d5c07e46c47836db203be42a162039ca2b89ed604e4b93d3f54b9439450.jpg)  
+Figure 1: We illustrate the architecture for learning the EPI-Policy. Trajectories $\tau _ { e p i }$ generated from the EPI policy are passed through an embedding network to obtain the environment embeddings $\psi$ . The difference in performance between a simple prediction model $f$ and the embedding-conditioned prediction model $f _ { e p i }$ is used to train the EPI-policy.
+
+To train the prediction models, we use a mean squared error loss, i.e the loss for $f$ is:
+
+$$
+\mathcal { L } _ { p r e d } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \left. f ( s _ { t } , a _ { t } ) - s _ { t + 1 } \right. _ { 2 } ^ { 2 }
+$$
+
+Here $s _ { t + 1 }$ is the target next state and $\left\| \cdot \right\| _ { 2 }$ is the L2 norm. Similarly, the loss for $f _ { e p i }$ is:
+
+$$
+\mathcal { L } _ { e p i . p r e d } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \left\| { f ( s _ { t } , a _ { t } ; \psi ( \tau _ { e p i } ) ) - s _ { t + 1 } } \right\| _ { 2 } ^ { 2 }
+$$
+
+During training, gradients are back-propagated to train both the prediction model $f _ { e p i }$ and the embedding network $\psi$ . The embedding network $\psi$ will be then directly used for the task-specific policy.
+
+# 4.1.2 EPI REWARD FORMULATION
+
+The reward for the EPI-policy is computed as the difference in performance between the simple prediction model $f$ and the EPI dependent prediction model $f _ { e p i }$ over the validation transition dataset. For a given training environment, the trajectory $\tau _ { e p i }$ generated by EPI-policy $\pi _ { e p i }$ is fed into $f _ { e p i }$ to calculate the prediction loss $\mathcal { L } _ { e p i - p r e d }$ . The simple prediction loss $\mathcal { L } _ { p r e d }$ is calculated using $f$ . This leads us to the prediction reward function:
+
+$$
+R _ { p } ( \pi _ { e p i } ) = \mathcal { L } _ { p r e d } - \mathbb { E } _ { \tau _ { e p i } \sim \pi _ { e p i } } [ \mathcal { L } _ { e p i . p r e d } ( \tau _ { e p i } ) ]
+$$
+
+Note that this formulation is different from the ”curiosity” reward proposed by Pathak et al. (2017) which encourages the policy to visit unexplored regions of the state space. Our reward function is to encourage the policy to extract information to improve prediction, rather than explore.
+
+# 4.1.3 ADDITIONAL TRAINING DETAILS
+
+Collecting transition dataset: To collect the transition dataset, we first train a task policy over a fixed environment. This policy is then executed with an $\epsilon$ -greedy strategy over the training environments to generate the transition data. To make this dataset more environment-dependent, we collect more data by setting different environments to the same state and executing the same action, following the Vine method (Schulman et al., 2015). Different environments would cause different observations in the next timestep, thus forcing the prediction model to use environment information given by the interaction. The dataset is kept fixed throughout the updates of the EPI-policy.
+
+Regularization via Separation: An additional regularization can be added to the loss function of the EPI-conditioned prediction model $f _ { e p i }$ . We implement a separation loss back-propagated from the outputs of the embedding network $\dot { \psi } ( \tau _ { e p i } )$ . The idea is to additionally force the embeddings from different environments to be apart from each other when the interaction trajectories are not yet informative enough for prediction. To calculate this loss, we assume the knowledge of which trajectory is from the same environment. We then fit a multivariate Gaussian distribution over the embeddings in an environment $\mathcal { E } _ { i }$ and encourage the mean vector $\mu _ { i }$ of one environment to be at least a fixed distance away from every other environment $\mu _ { j \neq i }$ . In addition, we encourage the diagonal elements of the covariance matrix to be reasonably small.
+
+Interleaved training: Since the EPI-policy and the prediction model depend on each other, they are optimized in an alternating fashion. The prediction model is first trained using a randomly initialized EPI-policy. Then the EPI-policy is optimized using TRPO for a few iterations with rewards from the learned prediction model. After that, the prediction model is updated again using a batch of trajectories generated from the EPI-policy. This cycle of alternating optimization is repeated until the EPI-policy converges. Details on the exact parameters used can be found in Appendix C.
+
+# 4.2 LEARNING TASK-SPECIFIC POLICIES
+
+After the EPI-policy and the embedding network are optimized, they are fixed and used for training the task-specific policy. For each episode, the agent first follows the EPI-policy to generate a trajectory $\tau _ { I }$ . The trajectory is then converted to an environment embedding $\psi ( \tau _ { I } )$ using the trained embedding network $\psi$ from the prediction model. After the interaction stage, the agent starts an episode to achieve its goal. This environment embedding $\psi ( \tau _ { I } )$ is now a part of the observation for each time step of the task-specific policy $\pi _ { t a s k } ( s _ { t } , \psi ( \tau _ { I } ) )$ . The task-specific policy is then optimized using TRPO. During testing, the agent follows the same procedure: it first executes the EPI-policy to get the environment embedding, followed by the task-specific policy to achieve the goal.
+
+# 5 RESULTS
+
+We now present experimental results to demonstrate how the EPI-policy helps the agent understand the environment, and allows the task-specific policy to generalize better to unseen test environments. Code is available at https:/ g hu com/Wenxuan-Zhou/EPI.
+
+Environments: To evaluate the EPI-policy, we need appropriate simulation environments that are sensitive to environment parameters. For this, we use the Striker and the Hopper MuJoCo (Todorov et al., 2012) environments from OpenAI Gym (Brockman et al., 2016). Both of the agents are torque-controlled. For Striker, the goal is to strike an object on the table with a robot arm to make the object reach at a target position when it stops. The mass and damping of the object are varied (2 parameters). The goal of Hopper is to move as fast as possible in the forward direction without falling. Here, we vary the mass of four body parts, the damping of three joints and the friction coefficient of the ground (8 parameters). During training, each parameter is randomized uniformly among five values from its range at the beginning of an episode. During testing, the parameters are sampled from a set of unseen environment parameters. Details can be found in Appendix A and Appendix B.
+
+Baselines: To show that the EPI-policy improves generalization, we compare our results with three categories of baselines. Training details for the baselines can be found in Appendix D. First, we show the importance of extra environment information:
+
+• Simple Policy: a MLP (Multi-Layer Perceptron) policy $a _ { t } = \pi ( s _ { t } )$ trained only in one fixed environment of fixed parameters. This is how RL tasks are normally formulated. Invariant Policy (inv-pol): a MLP policy $\boldsymbol { a } _ { t } = \pi ( \boldsymbol { s } _ { t } )$ trained across the randomized environments similar to Rajeswaran et al. (2017). This policy will be more robust than Simple Policy since it encounters various environments during training. However, it is only able to output the same action from the same input in different environments. Oracle Policy: a MLP policy takes explicit environment parameters as additional input: $a _ { t } = \pi ( s _ { t } , \rho )$ , where $\rho$ is a vector of the environment parameters such as mass, friction and damping. This baseline has full information of the environment, thus this is meant to put our results in context with the best possible task-specific policy.
+
+![](images/e79e124b60d044cab8e51548fd8fd661c41ff4d077667c39bbbd1a8ba161936c.jpg)  
+Figure 2: Striker Environment: (a) Illustration of the environment (b) Training curves of the EPIpolicy comparing to baselines. The x-axis shows training iterations for TRPO. The y-axis shows the average reward of an episode.(c) Two-dimensional environment embedding of the EPI-policy after training. Embeddings from the same environment have the same color.
+
+![](images/b697477b9cb79ead3917daf560ff441c1c390142121c5691f6596dab62dd9294.jpg)  
+Figure 3: Hopper Environment: (a) Illustration of the environment (b) Training curves of the EPIpolicy comparing to baselines. The $\mathbf { X }$ -axis shows training iterations. The y-axis shows the average reward of an episode. (c) t-SNE of the 8-dimensional Environment embedding of the EPI-policy after training. To visualize, the color is given by (R,G,B) where R:average of mass, G:friction, B:average of damping.
+
+In the second category of baselines, we want to demonstrate the necessity of learning a separate interaction policy:
+
+• Random Interaction Policy: takes 10 timesteps of random interactions as additional input. It covers the possibility that random actions are good enough to capture environment nuances. • History Policy: takes 10 timesteps of most recent states and actions as additional input. (a) (b) (c)Mnih et al. (2015) adopts similar idea where it stacks 3 to 5 most recent frames as input. It covers the possibility that the trajectories from the task-specific policy are informative. • Recurrent Policy (lstm-pol): using a LSTM layer instead of MLP. This baseline is similar to Peng et al. (2017). LSTM is supposed to encode the past trajectories, so it serves the same idea as the History Policy but from a different approach. • System Identified Policy: This baseline is implemented based on Yu et al. (2017). It first trains an Oracle Policy. Then it performs explicit system identification that estimates environment parameters using a history of states and actions from the Oracle Policy. Finally, the estimated value are fed into the trained Oracle Policy during test time.
+
+Finally, to show that the prediction reward is an appropriate proxy for the EPI-policy update, we implemented the following baseline that directly updates the EPI-policy:
+
+• Direct Reward Policy: this baseline evaluates the interactions by directly taking the final reward of the following task-specific policy trajectory instead of the prediction reward.
+
+Table 1: Comparison of testing results. Hopper is evaluated by the accumulated reward in one episode. Higher reward implies better performance. Striker is evaluated by the final distance between the object and the goal. A smaller distance implies a better performance.   
+
+<table><tr><td colspan="2">METHOD</td><td>Hopper:Reward (↑)</td><td>Striker: Final Distance (↓)</td></tr><tr><td rowspan="7">Baselines</td><td>Simple Policy</td><td>414± 313</td><td>1.660± 2.010</td></tr><tr><td>Invariant Policy</td><td>1025± 49</td><td>0.297± 0.068</td></tr><tr><td>Random Interaction Policy</td><td>1101± 27</td><td>0.410 ± 0.047</td></tr><tr><td>History Policy</td><td>1143 ± 156</td><td>0.259 ± 0.038</td></tr><tr><td>Recurrent Policy</td><td>917 ± 180</td><td>0.418 ± 0.051</td></tr><tr><td>System Id Policy</td><td>1033 ± 81</td><td>1.113 ± 0.106</td></tr><tr><td>Direct Reward</td><td>1057 ± 310</td><td>0.458 ± 0.004</td></tr><tr><td>Ours</td><td>EPI + Task-specific Policy</td><td>1303±173</td><td>0.162 ± 0.015</td></tr><tr><td rowspan="3">Ablations</td><td>No Vine Data</td><td>1214± 138</td><td>0.293± 0.018</td></tr><tr><td>No Regularization</td><td>1203 ± 397</td><td>0.308 ± 0.019</td></tr><tr><td>No Vine and No Regularization Oracle Policy</td><td>1237± 78</td><td>0.324 ± 0.057</td></tr><tr><td>Oracle</td><td></td><td>1474 ± 205</td><td>0.133± 0.034</td></tr></table>
+
+# 5.1 PERFORMANCE OF TASK POLICY
+
+The training curves for the EPI Policy and three representative baselines are presented in Fig. 2(b) and Fig. 3(b). The final performance of the best Oracle Policy across random seeds is plotted as a dotted line as an upper bound. Average curves are plotted for other policies with standard deviation across seeds. We observe that for both the Hopper and the Striker environments, the EPI Policy achieves better performance than the baselines and is comparable to the performance of the Oracle Policy. To further validate our results, we test our final learned task polices $\pi _ { t a s k }$ on heldout test environments and present these results in Table. 1. For Hopper, our method achieves an average episode reward of $1 3 0 3 \pm 1 7 3$ , at least $1 4 . 0 \%$ better than all the baselines. For Striker, our method achieves a final distance of $0 . 1 6 2 \pm 0 . 0 1 5 ( m )$ to the goal, at least $3 7 . 5 \%$ more accurate than all the baselines.
+
+The baselines under-perform due to multiple reasons. Without any information of the environment, Simple Policy and Invariant Policy are not able to adapt to the different testing environments. The Random Interaction Policy, the History Policy and the Recurrent Policy, are all able to use environment information and are hence better than the invariant Policy. However, the EPI Policy can learn a lot more efficiently since it has a separate policy that explicitly probes the environment.
+
+Even though a System Identified Policy has extra access to true environment parameters during training time, its performance will be limited when the number of changing parameters increases. To further verify this, we trained System Identified Policy for Hopper when only 2 variables are changing. It achieves a low mean squared error of $1 0 ^ { - 5 }$ for environment parameter estimation. However, when 8 parameters of the Hopper are changing, the mean squared error is around $1 0 ^ { - 2 }$ . Furthermore, this baseline still relies on the assumption that the task-specific policy will capture environment nuances just like the History Policy. This may limit the performance in Striker even when only 2 parameters are changing.
+
+Finally, the Direct Reward Policy learns slowly and is hence worse than using EPI-policies. We attribute this to the added difficulty in credit assignment, i.e. a high reward for the task-specific policy is not necessarily caused by a good interaction, but the interaction policy would still receive the high reward.
+
+# 5.2 ENVIRONMENT EMBEDDING
+
+To understand what the EPI-policy learns, we visualize the embeddings given by interactions. We run interactions over the randomized environments and plot the output embedding in Fig. 2(c) for the Hopper and in Fig. 3(c) for the Striker. The embedding shows a correspondence to environment parameters with similar environments being closer to each other. This means that the agent learns to disentangle state transition differences induced by different environment parameters. This also demonstrates that it has learned to generate distinguishable trajectories in differing environments.
+
+![](images/84b82f134a4ce9d29b16e0d278118163c84dbd0c43cdb209fe29834e61b4f3d0.jpg)  
+Figure 4: Generalizability of the EPI-policy on Hopper. The grey area represents the training range while the rest represents testing environments.
+
+# 5.3 ANALYSIS ON GENERALIZABILITY
+
+To evaluate the generalizability of the EPI-policy, we further analyze the performance of Hopper when only damping, friction or mass is changing in Fig. 4. During the analysis of one parameter, the other parameters are set to default values. Grey area represents training range, while the rest of the values is only seen in testing range. From the figure we can see that our policy outperforms the invariant policy in almost every setting.
+
+# 5.4 ABLATIVE EXPERIMENTS
+
+Effect of Vine-Method for Initial Dataset: Vine method allows the transition prediction model to not overfit on samples based on their state. Training without the Vine method, leads to dropping performance by about 89 reward points on Hopper and $0 . 1 3 \mathrm { m }$ accuracy on Striker. This is still better than all baselines for Hopper. This shows that the vine method is important for certain environments like Striker, but it is not necessary.
+
+Effect of Regularization: The only thing the additional regularization loss does is trying to cluster embeddings from the same environment. It does not serve for the EPI-policy updates as prediction loss does. However, even without separation loss, EPI is significantly better than the Invariant policy baseline by 178 points on Hopper. Similar to Vine method, separation loss is necessary for Striker but not for Hopper to beat baselines. In addition, we did experiments without either Vine-Method or Regularization. Hopper performs 66 points worse than following the full method, while still beating all the baselines. Striker performs $0 . 1 6 2 \mathrm { m }$ worse than the full method. Although it still beats 5 of the 7 baselines, this is a significant loss in performance and highlights the importance of using both the Vine data and regularization.
+
+Experimental Setup: Our method probes the environment using the EPI-policy followed by solving the task itself. During task-solving, the agent starts from the same initial states as baselines (using reset). While in most cases, this might be a feasible assumption, initial environment probing might not be allowed and resets might not be available in some scenarios. Therefore, we also performed experiments when the EPI-policy probes the environment for the first 10 steps and then the taskpolicy takes over from where the EPI-policy left. This essentially leads to a larger range of initial states for the task-specific policy than all the baselines. Even in these strict conditions, EPI-policy is quite useful. For Hopper, we achieve a reward of $1 1 9 3 \pm 3 0 4$ , outperforming all the baselines as shown in Table. 1. Even in the case of Striker we perform on par with invariant policy and outperform all other baselines.
+
+# 6 CONCLUSION
+
+We have presented an approach to extract environment behavior information by probing it using an EPI-policy. To learn the EPI-policy we use a reward function which prefers trajectories that improve environment transition predictability. We show that such a reward objective enables the agent to perform meaningful interactions and extract implicit information (environment embedding) from the environment. Given unseen test environments on Hopper and Striker, we show that our embedding conditioned task policies improve generalization to unseen test environments.
+
+# REFERENCES
+
+B. Armstrong. On finding ’exciting’ trajectories for identification experiments involving systems with non-linear dynamics. In Proceedings. 1987 IEEE International Conference on Robotics and Automation, volume 4, pp. 1131–1139, March 1987. doi: 10.1109/ROBOT.1987.1087968.
+
+Josh C Bongard and Hod Lipson. Nonlinear system identification using coevolution of models and tests. IEEE Transactions on Evolutionary Computation, 9(4):361–384, 2005.
+
+Daniel A Braun, Ad Aertsen, Daniel M Wolpert, and Carsten Mehring. Learning optimal adaptation strategies in unpredictable motor tasks. Journal of Neuroscience, 29(20):6472–6478, 2009.
+
+Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym. arXiv preprint arXiv:1606.01540, 2016.
+
+Nuttapong Chentanez, Andrew G Barto, and Satinder P Singh. Intrinsically motivated reinforcement learning. In Advances in neural information processing systems, pp. 1281–1288, 2005.
+
+Ignasi Clavera, Anusha Nagabandi, Ronald S Fearing, Pieter Abbeel, Sergey Levine, and Chelsea Finn. Learning to adapt: Meta-learning for model-based control. arXiv preprint arXiv:1803.11347, 2018.
+
+Misha Denil, Pulkit Agrawal, Tejas D Kulkarni, Tom Erez, Peter Battaglia, and Nando de Freitas. Learning to perform physics experiments via deep reinforcement learning. arXiv preprint arXiv:1611.01843, 2016.
+
+Lixin Duan, Dong Xu, and Ivor Tsang. Learning with augmented features for heterogeneous domain adaptation. arXiv preprint arXiv:1206.4660, 2012.
+
+Yan Duan, Xi Chen, Rein Houthooft, John Schulman, and Pieter Abbeel. Benchmarking deep reinforcement learning for continuous control. In International Conference on Machine Learning, pp. 1329–1338, 2016a.
+
+Yan Duan, John Schulman, Xi Chen, Peter L Bartlett, Ilya Sutskever, and Pieter Abbeel. Rl2 : Fast reinforcement learning via slow reinforcement learning. arXiv preprint arXiv:1611.02779, 2016b.
+
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. arXiv preprint arXiv:1703.03400, 2017.
+
+Michel Gevers et al. System identification without lennart ljung: what would have been different? Forever Ljung in System Identification, Studentlitteratur AB, Norrtalje, 2006.
+
+Abhishek Gupta, Coline Devin, YuXuan Liu, Pieter Abbeel, and Sergey Levine. Learning invariant feature spaces to transfer skills with reinforcement learning. arXiv preprint arXiv:1703.02949, 2017.
+
+Karol Hausman, Jost Tobias Springenberg, Ziyu Wang, Nicolas Heess, and Martin Riedmiller. Learning an embedding space for transferable robot skills. 2018.
+
+Judy Hoffman, Sergio Guadarrama, Eric S Tzeng, Ronghang Hu, Jeff Donahue, Ross Girshick, Trevor Darrell, and Kate Saenko. Lsda: Large scale detection through adaptation. In Advances in Neural Information Processing Systems, pp. 3536–3544, 2014.
+
+Stephen James, Andrew J Davison, and Edward Johns. Transferring end-to-end visuomotor control from simulation to real world for a multi-stage task. arXiv preprint arXiv:1707.02267, 2017.
+
+Leslie Pack Kaelbling, Michael L Littman, and Andrew W Moore. Reinforcement learning: A survey. Journal of artificial intelligence research, 1996.
+
+Diederik 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 NIPS 2012.
+
+Brian Kulis, Kate Saenko, and Trevor Darrell. What you saw is not what you get: Domain adaptation using asymmetric kernel transforms. In Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, pp. 1785–1792. IEEE, 2011.
+
+Yanghao Li, Naiyan Wang, Jianping Shi, Jiaying Liu, and Xiaodi Hou. Revisiting batch normalization for practical domain adaptation. arXiv preprint arXiv:1603.04779, 2016.
+
+Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. arXiv preprint arXiv:1509.02971, 2015.
+
+Mingsheng Long, Yue Cao, Jianmin Wang, and Michael I Jordan. Learning transferable features with deep adaptation networks. arXiv preprint arXiv:1502.02791, 2015.
+
+Mario Milanese, John Norton, Hel´ ene Piet-Lahanier, and \` Eric Walter. ´ Bounding approaches to system identification. Springer Science & Business Media, 2013.
+
+Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529, 2015.
+
+Deepak Pathak, Pulkit Agrawal, Alexei A Efros, and Trevor Darrell. Curiosity-driven exploration by self-supervised prediction. In International Conference on Machine Learning (ICML), volume 2017, 2017.
+
+Deepak Pathak, Parsa Mahmoudieh, Guanghao Luo, Pulkit Agrawal, Dian Chen, Yide Shentu, Evan Shelhamer, Jitendra Malik, Alexei A Efros, and Trevor Darrell. Zero-shot visual imitation. arXiv preprint arXiv:1804.08606, 2018.
+
+Xue Bin Peng, Marcin Andrychowicz, Wojciech Zaremba, and Pieter Abbeel. Sim-to-real transfer of robotic control with dynamics randomization. arXiv preprint arXiv:1710.06537, 2017.
+
+Xue Bin Peng, Pieter Abbeel, Sergey Levine, and Michiel van de Panne. Deepmimic: Example-guided deep reinforcement learning of physics-based character skills. arXiv preprint arXiv:1804.02717, 2018.
+
+Lerrel Pinto, Marcin Andrychowicz, Peter Welinder, Wojciech Zaremba, and Pieter Abbeel. Asymmetric actor critic for image-based robot learning. arXiv preprint arXiv:1710.06542, 2017a.
+
+Lerrel Pinto, James Davidson, Rahul Sukthankar, and Abhinav Gupta. Robust adversarial reinforcement learning. ICML, 2017b.
+
+Aravind Rajeswaran, Sarvjeet Ghotra, Sergey Levine, and Balaraman Ravindran. Epopt: Learning robust neural network policies using model ensembles. ICLR, 2017.
+
+Andrei A Rusu, Matej Vecerik, Thomas Rothorl, Nicolas Heess, Razvan Pascanu, and Raia Hadsell. ¨ Sim-to-real robot learning from pixels with progressive nets. arXiv preprint arXiv:1610.04286, 2016.
+
+Fereshteh Sadeghi and Sergey Levine. (cad)2rl: Real single-image flight without a single real image. arXiv preprint arXiv:1611.04201, 2016.
+
+Steindor Sæmundsson, Katja Hofmann, and Marc Peter Deisenroth. Meta reinforcement learning ´ with latent variable gaussian processes. arXiv preprint arXiv:1803.07551, 2018.
+
+John Schulman, Sergey Levine, Pieter Abbeel, Michael Jordan, and Philipp Moritz. Trust region policy optimization. In International Conference on Machine Learning, pp. 1889–1897, 2015.
+
+David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. nature, 529(7587):484–489, 2016.
+
+Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction, volume 1. MIT press Cambridge, 1998.
+
+Josh Tobin, Rachel Fong, Alex Ray, Jonas Schneider, Wojciech Zaremba, and Pieter Abbeel. Domain randomization for transferring deep neural networks from simulation to the real world. IROS, 2017.
+
+Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International Conference on, pp. 5026– 5033. IEEE, 2012.
+
+Eric Tzeng, Judy Hoffman, Ning Zhang, Kate Saenko, and Trevor Darrell. Deep domain confusion: Maximizing for domain invariance. arXiv preprint arXiv:1412.3474, 2014.
+
+Jane X Wang, Zeb Kurth-Nelson, Dhruva Tirumala, Hubert Soyer, Joel Z Leibo, Remi Munos, Charles Blundell, Dharshan Kumaran, and Matt Botvinick. Learning to reinforcement learn. arXiv preprint arXiv:1611.05763, 2016.
+
+Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? In Advances in neural information processing systems, pp. 3320–3328, 2014.
+
+Tianhe Yu, Chelsea Finn, Annie Xie, Sudeep Dasari, Tianhao Zhang, Pieter Abbeel, and Sergey Levine. One-shot imitation from observing humans via domain-adaptive meta-learning. arXiv preprint arXiv:1802.01557, 2018.
+
+Wenhao Yu, Jie Tan, C Karen Liu, and Greg Turk. Preparing for the unknown: Learning a universal policy with online system identification. arXiv preprint arXiv:1702.02453, 2017.
+
+# APPENDIX A ENVIRONMENT DESCRIPTIONS
+
+We used Hopper and Striker environments from OpenAI Gym (Brockman et al., 2016). We will describe the details of the environments in this section.
+
+Hopper: Hopper consists of four body parts and three joints. It has an 3-dimensional action space including motor commands for all the joints. The observation space is 11-dimensional, inluding joint angles, joint velocities, y,z positions of the torso and x,y,z velocities of the torso. The initial positions and velocities are randomized over $\pm 0 . 0 0 5$ . The reward function is $r ( s , a ) = v _ { x } + 1 - 0 . 0 0 1 \| a \| ^ { 2 }$ . The first term is to encourage the agent to move forward. The second term is a reward for being alive. The third term is to disencourage unnecessary movements. The criteria for being alive is that the height of the torso should be larger than 0.7 and the angle of the torso should be smaller than 0.2 with respect to the horizontal plane. It has a maximum length of 1000 timesteps for each episode. If it falls down and breaks the alive condition, the episode will end immediately.
+
+Striker: Striker environment has a 7-dof robot arm and a table. There is a ball and a goal location marker on the table. The action space is 7-dimensional, including the motor commands for each joints on the robot. The 23-dimensional observation space consists of the joint angles, joint velocities, the end-effector (hand) position, the object position and the goal position. The initial conditions randomize the velocity of all the joints over $\pm 0 . 1$ . The reward is given by $r ( s , a ) = - 3 \| s _ { o b j e c t } - s _ { g o a l } \| - 0 . 1 \| a \| ^ { 2 } - 0 . 5 \| s _ { o b j e c t } - s _ { h a n d } \| ^ { 2 }$ . The first term is to encourage the object to reach the goal and stay at the goal. The second term is to disencourage unnecessary movements. The third term is to encourage the hand to hit the object. After the end-effector hits the object, $s h a n d$ will be replaced by the hitting location of the end-effector. It has a fixed length of 200 timesteps for each episode.
+
+# APPENDIX B RANDOMIZED ENVIRONMENT PARAMETERS
+
+Every environment parameter is randomized among 5 evenly spread discrete values. Striker has 2 varying parameters which create 25 possible environment. Hopper is varying 8 parameters, so there are $\mathrm { \bar { 5 } ^ { \bar { 8 } } }$ possible environments. In order to calculate separation loss for Hopper, we sample 100 environments from all $5 ^ { 8 }$ possible environment to train the EPI-policy. Otherwise it will not be able to cluster the embeddings from the same environment. Note that when training task policy with interactions, the environment is still randomized from all possible combinations.
+
+During testing, all environment parameters are sampled from a set of discrete values that the policy has never seen in any stage of training. For example, if mass is sampled from $\{ 1 . 0 , 2 . 0 , 3 . 0 , 4 . 0 \bar { , } 5 . 0 \}$ , it is sampled from $\{ 1 . 5 , \bar { 2 } . 5 , 3 . 5 , 4 . 5 , 5 . 5 \}$ during testing.
+
+# APPENDIX C TRAINING DETAILS
+
+An EPI-trajectory contains 10 steps of observations and actions for both Hopper and Striker. The embedding network $\psi$ that inputs an EPI-trajectory $\tau _ { e p i }$ and outputs environment embedding $\varepsilon$ has two fully connected layers with 32 neurons each and followed by ReLU (Krizhevsky et al.). The prediction models $f _ { p r e d }$ and $f _ { e p i . p r e d }$ has four fully connected layers with 128 neurons each. The prediction model is optimized by Adam (Kingma & Ba, 2014). Both the EPI-policy $\pi _ { e p i }$ and the final goal policy $\pi _ { g }$ are optimized using TRPO (Schulman et al., 2015) with rllab implementation (Duan et al., 2016a). The prediction model is trained from scratch every 50 policy updates. Training from scratch is to avoid overfitting which will lead to unintended increasing reward for the EPI-policy. The EPI-policy is trained for $2 0 0 { \sim } 4 0 0$ iterations in total with a batch size of 10000 timesteps. The task policy will then use the trained EPI-policy and the embedding network to update for 1000 iterations with a batch size of 100000 timesteps.
+
+# APPENDIX D TRAINING DETAILS OF BASELINES
+
+All MLP baselines have the same network architecture as the task policy: two hidden layers with 32 hidden units each with ReLU activations. They are trained with the same batch size, optimizer, number of iterations and other hyperparameters as the task policy. We made sure that all training curves saturate before the final iteration.
+
+• Simple Policy: a MLP policy trained only in one fixed environment of fixed parameters. All the other baselines are trained across the randomized environments.   
+• Invariant Policy (inv-pol): a default MLP policy $a _ { t } = \pi ( s _ { t } )$ .   
+• Oracle Policy: a MLP policy takes explicit environment parameters as additional input: $a _ { t } = \pi ( s _ { t } , \rho )$ , where $\rho$ is a vector of the environment parameters such as mass, friction and damping.   
+Random Interaction Policy: a MLP policy that takes 10 timesteps of random interactions as additional input. $a _ { t } = \pi ( s _ { t } , s _ { i 0 } , a _ { i 0 } , s _ { i 1 } , a _ { i 1 } . . . s _ { i 9 } , a _ { i 9 } )$ , where $s _ { i n }$ and $a _ { i n }$ are from a random policy.   
+• History Policy: a MLP policy that takes 10 timesteps of most recent states and actions as additional input. Mnih et al. (2015) adopts similar idea where it stacks 3 to 5 most recent frames as input. $a _ { t } = \pi ( s _ { t } , s _ { t - 1 } , a _ { t - 1 } , s _ { t - 2 } , a _ { t - 2 } . . . s _ { t - 1 0 } , a _ { t - 1 0 } )$ . When $t < 1 0$ , $s _ { t - j }$ are filled by zeros for $j > t$ .   
+• Recurrent Policy (lstm-pol): The Recurrent baseline is using an LSTM with 32 hidden units from rllab. $a _ { t } = \pi _ { L S T M } ( s _ { t } )$ .   
+• System Identified Policy: This baseline is implemented based on Yu et al. (2017). We first trains an Oracle Policy $a _ { t } ~ = ~ \pi ( s _ { t } , \rho )$ as we described above, which is called the Universal Policy (UP) in their paper. To train Online Sytem Identification (OSI) model, we collect trajectories on randomized environments following the Oracle Policy. With the trajectories, we trained the OSI model, which is a regression model that inputs a history of states and actions and outputs $\rho$ . The network architecture strictly follows the paper. Then the estimated $\rho$ are fed into the trained Oracle Policy $a _ { t } = \pi ( s _ { t } , \rho )$ to collect more trajectories for training OSI again. We followed the original paper and trained OSI for 5 times.   
+Direct Reward Policy: takes the first 10 steps of actions as additional input. In this way, these steps are taking the future reward of the whole trajectory. $\begin{array} { r l } { a _ { t } } & { { } = } \end{array}$ $\pi ( s _ { t } , s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } . . . s _ { 9 } , a _ { 9 } )$ . When $t < 1 0$ , $s _ { j }$ are filled by zeros for $j > t$ .

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+        "text": "ENVIRONMENT PROBING INTERACTION POLICIES ",
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+        "text": "Wenxuan Zhou1, Lerrel Pinto1, Abhinav Gupta1,2 1The Robotics Institute, Carnegie Mellon University 2Facebook AI Research ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "A key challenge in reinforcement learning (RL) is environment generalization: a policy trained to solve a task in one environment often fails to solve the same task in a slightly different test environment. A common approach to improve interenvironment transfer is to learn policies that are invariant to the distribution of testing environments. However, we argue that instead of being invariant, the policy should identify the specific nuances of an environment and exploit them to achieve better performance. In this work, we propose the “Environment-Probing” Interaction (EPI) policy, a policy that probes a new environment to extract an implicit understanding of that environment’s behavior. Once this environmentspecific information is obtained, it is used as an additional input to a task-specific policy that can now perform environment-conditioned actions to solve a task. To learn these EPI-policies, we present a reward function based on transition predictability. Specifically, a higher reward is given if the trajectory generated by the EPI-policy can be used to better predict transitions. We experimentally show that EPI-conditioned task-specific policies significantly outperform commonly used policy generalization methods on novel testing environments. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
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+        "text": "Over the last few years deep reinforcement learning (RL) has shown tremendous progress. From beating the world’s best at Go (Silver et al., 2016) to generating agile and dexterous motor policies (Schulman et al., 2015; Lillicrap et al., 2015; Peng et al., 2018), deep RL has been used to solve a variety of sequential decision-making tasks. However, one key issue that deep RL faces is transfer: a policy trained in one environment is extremely dependent on the specific parameters of that environment (Rajeswaran et al., 2017) and fails to generalize to a new environment. Because of this, training agents that can generalize to novel environments has become an active area of research (Rajeswaran et al., 2017; Pinto et al., 2017b; Yu et al., 2017; Peng et al., 2017). ",
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+        "text": "The most common approach for environment generalization is to train a single policy over a variety of different environments (Rajeswaran et al., 2017; Peng et al., 2017), thus allowing the policy network to learn invariances to the underlying environmental factors. However, a monolithic policy that works on multiple environments often ends up being conservative since it is unable to exploit the dynamics of a given environment. We argue that instead of learning a policy that is invariant to environment dynamics we need a policy that adapts to its environment. A popular approach for this is to first perform explicit system identification of the environment, and then use the estimated parameters to generate an environment specific policy (Yu et al., 2017). However, there are three reasons why this approach might be infeasible: (a) it is based on the structural assumption that only a known set of parameters matter; (b) estimating those parameters from few observations is an extremely challenging problem; (c) these approaches require access to environment parameters during training time, which might be impractical. So is it possible to learn an environment dependent policy without estimating every environment parameter? ",
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+        "text": "When humans are tasked to perform in a new environment, we do not explicitly know what parameters affect performance (Braun et al., 2009). Instead, we probe the environment to gain an intuitive understanding of its behavior. The purpose of these initial interactions is not to complete the task immediately, but to extract information about the environment. This process facilitates learning in that environment. Inspired by this, we present a framework in which the agent first performs “environment-probing” interactions that extract information from an environment, then leverages this information to achieve the goal with a task-specific policy. This setup entails three challenges: 1) How should an agent interact with an environment? 2) How can information be extracted from such interactions? 3) How can the extracted information be used to learn a better policy? ",
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+        "text": "One way to interact is by executing random actions in the environment. However, this may not expose the nuances of an environment. Another way is to use samples from the task-specific policy to infer the environment (Yu et al., 2017). However, a good task-specific policy may not be optimal for understanding an environment. Instead, we separately optimize an “Environment-Probing” Interaction policy (EPI-policy) to extract environment information. To assign rewards to the EPIpolicy, our key insight is that the agent’s ability to predict transitions in an environment can be used as a metric of its understanding of that environment. If an “environment-probing” trajectory can help in predicting transitions, then it should be a good trajectory. More specifically, we use the “environment-probing” trajectory to generate an environment embedding vector. This compact embedding vector is passed to a prediction model that provides higher rewards to the trajectories that are useful in estimating environment-dependent transitions. ",
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+    {
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+        "text": "Once we have learned the EPI-policy, we can use the generated environment embedding as an input to the task-specific policy. Since this task-specific policy is conditioned on the information from the environment, it can now produce actions dependent on the nuances of the specific environment. ",
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+        "text": "We evaluate the effectiveness of using EPI on Hopper and Striker from OpenAI Gym (Brockman et al., 2016) with randomized mass, damping and friction. In both of these environments, the performance of our policies is better than several commonly used techniques and is even comparable to an oracle policy that has full access to the environment information. These results, along with a detailed ablation analysis, demonstrates that implicitly encoding the environment using EPI-policies improves task performance in unseen testing environments. ",
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+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            467,
+            343,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Developing generalizable machine learning models is a longstanding challenge. One approach to transfer models across different input distributions is domain adaptation. Duan et al. (2012) along with several others (Hoffman et al., 2014; Kulis et al., 2011; Long et al., 2015) domain-adapt visual models. With the onset of deep learning, several works have investigated finetuning (retraining) pretrained networks in different target domains (Yosinski et al., 2014; Li et al., 2016; Tzeng et al., 2014). In reinforcement learning, Gupta et al. (2017) learns different encoders for different environments that share a common feature space to enable transfer. Improving the generalizability of models can also be formulated as a meta-learning problem. Finn et al. (2017) and Yu et al. (2018) finetune the models in new environments. We argue that agents should have the capacity to generalize without updating their policy. Duan et al. (2016b) and Wang et al. (2016) optimize the policy over multiple episodes and use RNN to implicitly model the underlying changing dynamics in the hidden states. Hausman et al. (2018) purposes to learn an embedding in a supervised manner by performing variational inference on task ids. These methods are based on model-free meta-RL, while Sæmundsson et al. (2018) and Clavera et al. (2018) investigate in model-based meta-RL. Sæmundsson et al. (2018) infers latent variables using Gaussian Process, while we use neural networks to create an embedding and experimented with more complex high-dimensional continuous control problems. Concurrent work Clavera et al. (2018) proposes to train a dynamics model that adapts online using either gradient based method (Finn et al., 2017) or recurrent based method (Duan et al., 2016b). The most important difference is that we have a separate policy to probe the environment instead of using the trajectories from the execution of the task. In our method, training prediction models is somewhat related to learning dynamics model in the model-based RL. However, the prediction model in our method does not to be very accurate since we are only taking the difference. ",
+        "bbox": [
+            174,
+            500,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another stream of research that is relevant to our work is simulation-to-real transfer. Here the goal is to train a policy in a physics simulator and then transfer that policy to a real robot. The challenge here is to bridge the “Reality Gap” between the real world and the simulator. Rusu et al. (2016) attempt to do this via architectural novelty for visual domain adaptation. Sadeghi & Levine (2016); Tobin et al. (2017); Pinto et al. (2017a) and James et al. (2017) perform domain randomization on the inputs during training to learn a transferable policy. These methods target the mismatch of observation space. However, we target the mismatch in the dynamics of the environment. Peng et al. (2017) proposes domain randomization of the environment parameters and learning an LSTM policy to implicitly learn the environment. We compare our method to this approach and show that a separate process that extracts the nuances of an environment is vital for better generalization. Rajeswaran et al. (2017) and Pinto et al. (2017b) also target generalization to environments, however they do not incorporate environment information and hence are only able to learn a conservative policy. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The traditional approach of system identification (Milanese et al., 2013; Gevers et al., 2006; Bongard & Lipson, 2005; Armstrong, 1987) aims to explicitly estimate the varying environment parameters. Yu et al. (2017) propose the UP-OSI framework that can perform online system identification and takes the estimated environment parameters as additional input to the policy. This works well when the number of parameters to estimate is few. However, explicit identification of every environment parameter is a extremely challenging. Denil et al. (2016) trains an policy to gather information about physical properties through interactions in intuitive physics to answer questions, while we are trying do dealt with a more complex continuous control setting. ",
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+        "type": "text",
+        "text": "A recent approach by Pathak et al. (2017; 2018) uses errors in prediction models as an intrinsic reward (Chentanez et al., 2005) to improve exploration in RL. The strategy is to take actions to reach the states that maximize the error in the learned prediction model. In contrast, our method selects actions that reduce the prediction error on the heldout prediction dataset of task policy trajectories. ",
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+    {
+        "type": "text",
+        "text": "3 BACKGROUND: REINFORCEMENT LEARNING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We use the framework of reinforcement learning to learn the interaction policy and the final task target policy. This section contains brief preliminaries for reinforcement learning. Note that a more comprehensive exposition can be found in Kaelbling et al. (1996) and Sutton $\\&$ Barto (1998). ",
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+        "type": "text",
+        "text": "We model our problem as a continuous space Markov Decision Process represented as the tuple $( S , \\mathcal { A } , \\mathcal { P } , r , \\gamma , \\bar { \\mathbb { S } } )$ , where $s$ is a set of continuous states of the agent, $\\mathcal { A }$ is a set of continuous actions the agent can take at a given state, $\\mathcal { P } : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S }  \\mathbb { R }$ is the transition probability function, $r : S \\times \\mathcal { A }  \\mathbb { R }$ is the reward function, $\\gamma$ is the discount factor, and $\\mathbb { S }$ is the initial state distribution. ",
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+        "type": "text",
+        "text": "The goal for an agent is to maximize the expected cumulative discounted reward $\\scriptstyle \\sum _ { t = 0 } ^ { T - 1 } \\gamma ^ { t } r \\left( s _ { t } , a _ { t } \\right)$ by performing actions according to its policy $\\pi _ { \\theta } : { \\mathcal { S } }  A$ . Here, denotes the parameters for the policy $\\pi$ which takes action $a _ { t }$ given state $s _ { t }$ at timestep $t$ . There are various techniques to optimize $\\pi _ { \\theta }$ . The one we use in this paper is a policy gradient approach, TRPO (Schulman et al., 2015). ",
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+        "type": "text",
+        "text": "4 APPROACH",
+        "text_level": 1,
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+        "text": "Our objective is to learn a generalizable policy that is trained on a set of training environments, but can succeed across unseen testing environments. To this end, instead of a single policy, we have two separate policies: (a) an Environment-Probing Interaction policy (EPI-policy) that efficiently interacts with an environment to collect environment-specific information, and (b) a task-specific policy that uses this additional environment information (estimated via EPI-policy) to achieve high rewards for a given task. In this section, we present our methodology for learning these two policies. ",
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+        "type": "text",
+        "text": "4.1 LEARNING ENVIRONMENT-PROBING INTERACTIONS (EPI) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To learn the EPI-policy, we introduce a reward formulation based on the prediction models (Fig. 1). The key insight here is that a good EPI-policy generates trajectories in an environment from which predicting transitions in that environment is easier. To quantify this notion of how easy or difficult it is to predict transitions, we use the error transition prediction models. ",
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+    {
+        "type": "text",
+        "text": "4.1.1 TRANSITION PREDICTION MODELS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To train our prediction models, a dataset of transition data $\\left( { { s _ { t } } , { a _ { t } } , { s _ { t + 1 } } } \\right)$ is collected in the training environments using a pre-trained task policy (Sec. 4.1.3). This data is split into a training set and a validation set. With the training set, we train two prediction models: (a) a vanilla prediction model $f ( s _ { t } , a _ { t } )$ that predicts the next state $s _ { t + 1 }$ given the current state $s _ { t }$ and action $a _ { t }$ , and (b) an EPI-conditioned prediction model $f _ { e p i } ( s _ { t } , a _ { t } ; \\psi ( \\tau _ { e p i } ) )$ , which takes the embedded EPI-trajectory $\\psi ( \\tau _ { e p i } )$ as an additional input. Here $\\tau _ { e p i }$ is a trajectory in the environment which contains a small fixed number of observations and actions generated by $\\pi _ { e p i }$ . $\\psi ( \\tau _ { e p i } )$ is the low dimensional embedding of the trajectory $\\tau _ { e p i }$ . We can now quantify the reward for $\\pi _ { e p i }$ as the difference between their performance. The intuition behind this is that if an interaction trajectory generated from $\\pi _ { e p i }$ is informative, $f _ { e p i }$ will be able to use it to give smaller prediction error than $f$ . ",
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+        "img_path": "images/454d3d5c07e46c47836db203be42a162039ca2b89ed604e4b93d3f54b9439450.jpg",
+        "image_caption": [
+            "Figure 1: We illustrate the architecture for learning the EPI-Policy. Trajectories $\\tau _ { e p i }$ generated from the EPI policy are passed through an embedding network to obtain the environment embeddings $\\psi$ . The difference in performance between a simple prediction model $f$ and the embedding-conditioned prediction model $f _ { e p i }$ is used to train the EPI-policy. "
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+        "type": "text",
+        "text": "To train the prediction models, we use a mean squared error loss, i.e the loss for $f$ is: ",
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+        "type": "equation",
+        "img_path": "images/35b7e6a24f0c5bed7b34598c68f8edc46e80acf25e85cb864608de6a62da3468.jpg",
+        "text": "$$\n\\mathcal { L } _ { p r e d } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\left. f ( s _ { t } , a _ { t } ) - s _ { t + 1 } \\right. _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Here $s _ { t + 1 }$ is the target next state and $\\left\\| \\cdot \\right\\| _ { 2 }$ is the L2 norm. Similarly, the loss for $f _ { e p i }$ is: ",
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+    {
+        "type": "equation",
+        "img_path": "images/fe5bbe6d4f99b0bec4e85cfef403473e032c312e72e9fe563e65f89c8a69f52a.jpg",
+        "text": "$$\n\\mathcal { L } _ { e p i . p r e d } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\left\\| { f ( s _ { t } , a _ { t } ; \\psi ( \\tau _ { e p i } ) ) - s _ { t + 1 } } \\right\\| _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "During training, gradients are back-propagated to train both the prediction model $f _ { e p i }$ and the embedding network $\\psi$ . The embedding network $\\psi$ will be then directly used for the task-specific policy. ",
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+    {
+        "type": "text",
+        "text": "4.1.2 EPI REWARD FORMULATION ",
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+        "type": "text",
+        "text": "The reward for the EPI-policy is computed as the difference in performance between the simple prediction model $f$ and the EPI dependent prediction model $f _ { e p i }$ over the validation transition dataset. For a given training environment, the trajectory $\\tau _ { e p i }$ generated by EPI-policy $\\pi _ { e p i }$ is fed into $f _ { e p i }$ to calculate the prediction loss $\\mathcal { L } _ { e p i - p r e d }$ . The simple prediction loss $\\mathcal { L } _ { p r e d }$ is calculated using $f$ . This leads us to the prediction reward function: ",
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+        "text": "$$\nR _ { p } ( \\pi _ { e p i } ) = \\mathcal { L } _ { p r e d } - \\mathbb { E } _ { \\tau _ { e p i } \\sim \\pi _ { e p i } } [ \\mathcal { L } _ { e p i . p r e d } ( \\tau _ { e p i } ) ]\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Note that this formulation is different from the ”curiosity” reward proposed by Pathak et al. (2017) which encourages the policy to visit unexplored regions of the state space. Our reward function is to encourage the policy to extract information to improve prediction, rather than explore. ",
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+    {
+        "type": "text",
+        "text": "4.1.3 ADDITIONAL TRAINING DETAILS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Collecting transition dataset: To collect the transition dataset, we first train a task policy over a fixed environment. This policy is then executed with an $\\epsilon$ -greedy strategy over the training environments to generate the transition data. To make this dataset more environment-dependent, we collect more data by setting different environments to the same state and executing the same action, following the Vine method (Schulman et al., 2015). Different environments would cause different observations in the next timestep, thus forcing the prediction model to use environment information given by the interaction. The dataset is kept fixed throughout the updates of the EPI-policy. ",
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+    {
+        "type": "text",
+        "text": "Regularization via Separation: An additional regularization can be added to the loss function of the EPI-conditioned prediction model $f _ { e p i }$ . We implement a separation loss back-propagated from the outputs of the embedding network $\\dot { \\psi } ( \\tau _ { e p i } )$ . The idea is to additionally force the embeddings from different environments to be apart from each other when the interaction trajectories are not yet informative enough for prediction. To calculate this loss, we assume the knowledge of which trajectory is from the same environment. We then fit a multivariate Gaussian distribution over the embeddings in an environment $\\mathcal { E } _ { i }$ and encourage the mean vector $\\mu _ { i }$ of one environment to be at least a fixed distance away from every other environment $\\mu _ { j \\neq i }$ . In addition, we encourage the diagonal elements of the covariance matrix to be reasonably small. ",
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+    {
+        "type": "text",
+        "text": "Interleaved training: Since the EPI-policy and the prediction model depend on each other, they are optimized in an alternating fashion. The prediction model is first trained using a randomly initialized EPI-policy. Then the EPI-policy is optimized using TRPO for a few iterations with rewards from the learned prediction model. After that, the prediction model is updated again using a batch of trajectories generated from the EPI-policy. This cycle of alternating optimization is repeated until the EPI-policy converges. Details on the exact parameters used can be found in Appendix C. ",
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+    {
+        "type": "text",
+        "text": "4.2 LEARNING TASK-SPECIFIC POLICIES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "After the EPI-policy and the embedding network are optimized, they are fixed and used for training the task-specific policy. For each episode, the agent first follows the EPI-policy to generate a trajectory $\\tau _ { I }$ . The trajectory is then converted to an environment embedding $\\psi ( \\tau _ { I } )$ using the trained embedding network $\\psi$ from the prediction model. After the interaction stage, the agent starts an episode to achieve its goal. This environment embedding $\\psi ( \\tau _ { I } )$ is now a part of the observation for each time step of the task-specific policy $\\pi _ { t a s k } ( s _ { t } , \\psi ( \\tau _ { I } ) )$ . The task-specific policy is then optimized using TRPO. During testing, the agent follows the same procedure: it first executes the EPI-policy to get the environment embedding, followed by the task-specific policy to achieve the goal. ",
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+        "type": "text",
+        "text": "5 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We now present experimental results to demonstrate how the EPI-policy helps the agent understand the environment, and allows the task-specific policy to generalize better to unseen test environments. Code is available at https:/ g hu com/Wenxuan-Zhou/EPI. ",
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+        "text": "Environments: To evaluate the EPI-policy, we need appropriate simulation environments that are sensitive to environment parameters. For this, we use the Striker and the Hopper MuJoCo (Todorov et al., 2012) environments from OpenAI Gym (Brockman et al., 2016). Both of the agents are torque-controlled. For Striker, the goal is to strike an object on the table with a robot arm to make the object reach at a target position when it stops. The mass and damping of the object are varied (2 parameters). The goal of Hopper is to move as fast as possible in the forward direction without falling. Here, we vary the mass of four body parts, the damping of three joints and the friction coefficient of the ground (8 parameters). During training, each parameter is randomized uniformly among five values from its range at the beginning of an episode. During testing, the parameters are sampled from a set of unseen environment parameters. Details can be found in Appendix A and Appendix B. ",
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+        "type": "text",
+        "text": "Baselines: To show that the EPI-policy improves generalization, we compare our results with three categories of baselines. Training details for the baselines can be found in Appendix D. First, we show the importance of extra environment information: ",
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+        "text": "• Simple Policy: a MLP (Multi-Layer Perceptron) policy $a _ { t } = \\pi ( s _ { t } )$ trained only in one fixed environment of fixed parameters. This is how RL tasks are normally formulated. Invariant Policy (inv-pol): a MLP policy $\\boldsymbol { a } _ { t } = \\pi ( \\boldsymbol { s } _ { t } )$ trained across the randomized environments similar to Rajeswaran et al. (2017). This policy will be more robust than Simple Policy since it encounters various environments during training. However, it is only able to output the same action from the same input in different environments. Oracle Policy: a MLP policy takes explicit environment parameters as additional input: $a _ { t } = \\pi ( s _ { t } , \\rho )$ , where $\\rho$ is a vector of the environment parameters such as mass, friction and damping. This baseline has full information of the environment, thus this is meant to put our results in context with the best possible task-specific policy. ",
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+        "img_path": "images/e79e124b60d044cab8e51548fd8fd661c41ff4d077667c39bbbd1a8ba161936c.jpg",
+        "image_caption": [
+            "Figure 2: Striker Environment: (a) Illustration of the environment (b) Training curves of the EPIpolicy comparing to baselines. The x-axis shows training iterations for TRPO. The y-axis shows the average reward of an episode.(c) Two-dimensional environment embedding of the EPI-policy after training. Embeddings from the same environment have the same color. "
+        ],
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+        "page_idx": 5
+    },
+    {
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+        "img_path": "images/b697477b9cb79ead3917daf560ff441c1c390142121c5691f6596dab62dd9294.jpg",
+        "image_caption": [
+            "Figure 3: Hopper Environment: (a) Illustration of the environment (b) Training curves of the EPIpolicy comparing to baselines. The $\\mathbf { X }$ -axis shows training iterations. The y-axis shows the average reward of an episode. (c) t-SNE of the 8-dimensional Environment embedding of the EPI-policy after training. To visualize, the color is given by (R,G,B) where R:average of mass, G:friction, B:average of damping. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "In the second category of baselines, we want to demonstrate the necessity of learning a separate interaction policy: ",
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+    {
+        "type": "text",
+        "text": "• Random Interaction Policy: takes 10 timesteps of random interactions as additional input. It covers the possibility that random actions are good enough to capture environment nuances. • History Policy: takes 10 timesteps of most recent states and actions as additional input. (a) (b) (c)Mnih et al. (2015) adopts similar idea where it stacks 3 to 5 most recent frames as input. It covers the possibility that the trajectories from the task-specific policy are informative. • Recurrent Policy (lstm-pol): using a LSTM layer instead of MLP. This baseline is similar to Peng et al. (2017). LSTM is supposed to encode the past trajectories, so it serves the same idea as the History Policy but from a different approach. • System Identified Policy: This baseline is implemented based on Yu et al. (2017). It first trains an Oracle Policy. Then it performs explicit system identification that estimates environment parameters using a history of states and actions from the Oracle Policy. Finally, the estimated value are fed into the trained Oracle Policy during test time. ",
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+    {
+        "type": "text",
+        "text": "Finally, to show that the prediction reward is an appropriate proxy for the EPI-policy update, we implemented the following baseline that directly updates the EPI-policy: ",
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+    {
+        "type": "text",
+        "text": "• Direct Reward Policy: this baseline evaluates the interactions by directly taking the final reward of the following task-specific policy trajectory instead of the prediction reward. ",
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+    {
+        "type": "table",
+        "img_path": "images/0c2af4f20bff019aa7b4451d8c8c17d18c418539da9d6fdb1731801104ee13f4.jpg",
+        "table_caption": [
+            "Table 1: Comparison of testing results. Hopper is evaluated by the accumulated reward in one episode. Higher reward implies better performance. Striker is evaluated by the final distance between the object and the goal. A smaller distance implies a better performance. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">METHOD</td><td>Hopper:Reward (↑)</td><td>Striker: Final Distance (↓)</td></tr><tr><td rowspan=\"7\">Baselines</td><td>Simple Policy</td><td>414± 313</td><td>1.660± 2.010</td></tr><tr><td>Invariant Policy</td><td>1025± 49</td><td>0.297± 0.068</td></tr><tr><td>Random Interaction Policy</td><td>1101± 27</td><td>0.410 ± 0.047</td></tr><tr><td>History Policy</td><td>1143 ± 156</td><td>0.259 ± 0.038</td></tr><tr><td>Recurrent Policy</td><td>917 ± 180</td><td>0.418 ± 0.051</td></tr><tr><td>System Id Policy</td><td>1033 ± 81</td><td>1.113 ± 0.106</td></tr><tr><td>Direct Reward</td><td>1057 ± 310</td><td>0.458 ± 0.004</td></tr><tr><td>Ours</td><td>EPI + Task-specific Policy</td><td>1303±173</td><td>0.162 ± 0.015</td></tr><tr><td rowspan=\"3\">Ablations</td><td>No Vine Data</td><td>1214± 138</td><td>0.293± 0.018</td></tr><tr><td>No Regularization</td><td>1203 ± 397</td><td>0.308 ± 0.019</td></tr><tr><td>No Vine and No Regularization Oracle Policy</td><td>1237± 78</td><td>0.324 ± 0.057</td></tr><tr><td>Oracle</td><td></td><td>1474 ± 205</td><td>0.133± 0.034</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.1 PERFORMANCE OF TASK POLICY ",
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+    {
+        "type": "text",
+        "text": "The training curves for the EPI Policy and three representative baselines are presented in Fig. 2(b) and Fig. 3(b). The final performance of the best Oracle Policy across random seeds is plotted as a dotted line as an upper bound. Average curves are plotted for other policies with standard deviation across seeds. We observe that for both the Hopper and the Striker environments, the EPI Policy achieves better performance than the baselines and is comparable to the performance of the Oracle Policy. To further validate our results, we test our final learned task polices $\\pi _ { t a s k }$ on heldout test environments and present these results in Table. 1. For Hopper, our method achieves an average episode reward of $1 3 0 3 \\pm 1 7 3$ , at least $1 4 . 0 \\%$ better than all the baselines. For Striker, our method achieves a final distance of $0 . 1 6 2 \\pm 0 . 0 1 5 ( m )$ to the goal, at least $3 7 . 5 \\%$ more accurate than all the baselines. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The baselines under-perform due to multiple reasons. Without any information of the environment, Simple Policy and Invariant Policy are not able to adapt to the different testing environments. The Random Interaction Policy, the History Policy and the Recurrent Policy, are all able to use environment information and are hence better than the invariant Policy. However, the EPI Policy can learn a lot more efficiently since it has a separate policy that explicitly probes the environment. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Even though a System Identified Policy has extra access to true environment parameters during training time, its performance will be limited when the number of changing parameters increases. To further verify this, we trained System Identified Policy for Hopper when only 2 variables are changing. It achieves a low mean squared error of $1 0 ^ { - 5 }$ for environment parameter estimation. However, when 8 parameters of the Hopper are changing, the mean squared error is around $1 0 ^ { - 2 }$ . Furthermore, this baseline still relies on the assumption that the task-specific policy will capture environment nuances just like the History Policy. This may limit the performance in Striker even when only 2 parameters are changing. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, the Direct Reward Policy learns slowly and is hence worse than using EPI-policies. We attribute this to the added difficulty in credit assignment, i.e. a high reward for the task-specific policy is not necessarily caused by a good interaction, but the interaction policy would still receive the high reward. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 ENVIRONMENT EMBEDDING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            811,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To understand what the EPI-policy learns, we visualize the embeddings given by interactions. We run interactions over the randomized environments and plot the output embedding in Fig. 2(c) for the Hopper and in Fig. 3(c) for the Striker. The embedding shows a correspondence to environment parameters with similar environments being closer to each other. This means that the agent learns to disentangle state transition differences induced by different environment parameters. This also demonstrates that it has learned to generate distinguishable trajectories in differing environments. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/84b82f134a4ce9d29b16e0d278118163c84dbd0c43cdb209fe29834e61b4f3d0.jpg",
+        "image_caption": [
+            "Figure 4: Generalizability of the EPI-policy on Hopper. The grey area represents the training range while the rest represents testing environments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 ANALYSIS ON GENERALIZABILITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            296,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the generalizability of the EPI-policy, we further analyze the performance of Hopper when only damping, friction or mass is changing in Fig. 4. During the analysis of one parameter, the other parameters are set to default values. Grey area represents training range, while the rest of the values is only seen in testing range. From the figure we can see that our policy outperforms the invariant policy in almost every setting. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 ABLATIVE EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            411,
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+            425
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Effect of Vine-Method for Initial Dataset: Vine method allows the transition prediction model to not overfit on samples based on their state. Training without the Vine method, leads to dropping performance by about 89 reward points on Hopper and $0 . 1 3 \\mathrm { m }$ accuracy on Striker. This is still better than all baselines for Hopper. This shows that the vine method is important for certain environments like Striker, but it is not necessary. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Effect of Regularization: The only thing the additional regularization loss does is trying to cluster embeddings from the same environment. It does not serve for the EPI-policy updates as prediction loss does. However, even without separation loss, EPI is significantly better than the Invariant policy baseline by 178 points on Hopper. Similar to Vine method, separation loss is necessary for Striker but not for Hopper to beat baselines. In addition, we did experiments without either Vine-Method or Regularization. Hopper performs 66 points worse than following the full method, while still beating all the baselines. Striker performs $0 . 1 6 2 \\mathrm { m }$ worse than the full method. Although it still beats 5 of the 7 baselines, this is a significant loss in performance and highlights the importance of using both the Vine data and regularization. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Experimental Setup: Our method probes the environment using the EPI-policy followed by solving the task itself. During task-solving, the agent starts from the same initial states as baselines (using reset). While in most cases, this might be a feasible assumption, initial environment probing might not be allowed and resets might not be available in some scenarios. Therefore, we also performed experiments when the EPI-policy probes the environment for the first 10 steps and then the taskpolicy takes over from where the EPI-policy left. This essentially leads to a larger range of initial states for the task-specific policy than all the baselines. Even in these strict conditions, EPI-policy is quite useful. For Hopper, we achieve a reward of $1 1 9 3 \\pm 3 0 4$ , outperforming all the baselines as shown in Table. 1. Even in the case of Striker we perform on par with invariant policy and outperform all other baselines. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            808,
+            318,
+            823
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We have presented an approach to extract environment behavior information by probing it using an EPI-policy. To learn the EPI-policy we use a reward function which prefers trajectories that improve environment transition predictability. We show that such a reward objective enables the agent to perform meaningful interactions and extract implicit information (environment embedding) from the environment. Given unseen test environments on Hopper and Striker, we show that our embedding conditioned task policies improve generalization to unseen test environments. ",
+        "bbox": [
+            174,
+            840,
+            825,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            285,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "B. Armstrong. On finding ’exciting’ trajectories for identification experiments involving systems with non-linear dynamics. In Proceedings. 1987 IEEE International Conference on Robotics and Automation, volume 4, pp. 1131–1139, March 1987. doi: 10.1109/ROBOT.1987.1087968. ",
+        "bbox": [
+            176,
+            126,
+            821,
+            169
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Josh C Bongard and Hod Lipson. Nonlinear system identification using coevolution of models and tests. IEEE Transactions on Evolutionary Computation, 9(4):361–384, 2005. ",
+        "bbox": [
+            171,
+            176,
+            821,
+            205
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Daniel A Braun, Ad Aertsen, Daniel M Wolpert, and Carsten Mehring. Learning optimal adaptation strategies in unpredictable motor tasks. Journal of Neuroscience, 29(20):6472–6478, 2009. ",
+        "bbox": [
+            173,
+            213,
+            823,
+            243
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym. arXiv preprint arXiv:1606.01540, 2016. ",
+        "bbox": [
+            174,
+            251,
+            823,
+            281
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Nuttapong Chentanez, Andrew G Barto, and Satinder P Singh. Intrinsically motivated reinforcement learning. In Advances in neural information processing systems, pp. 1281–1288, 2005. ",
+        "bbox": [
+            173,
+            287,
+            823,
+            319
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ignasi Clavera, Anusha Nagabandi, Ronald S Fearing, Pieter Abbeel, Sergey Levine, and Chelsea Finn. Learning to adapt: Meta-learning for model-based control. arXiv preprint arXiv:1803.11347, 2018. ",
+        "bbox": [
+            173,
+            325,
+            823,
+            369
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Misha Denil, Pulkit Agrawal, Tejas D Kulkarni, Tom Erez, Peter Battaglia, and Nando de Freitas. Learning to perform physics experiments via deep reinforcement learning. arXiv preprint arXiv:1611.01843, 2016. ",
+        "bbox": [
+            173,
+            377,
+            823,
+            420
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Lixin Duan, Dong Xu, and Ivor Tsang. Learning with augmented features for heterogeneous domain adaptation. arXiv preprint arXiv:1206.4660, 2012. ",
+        "bbox": [
+            171,
+            429,
+            823,
+            458
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yan Duan, Xi Chen, Rein Houthooft, John Schulman, and Pieter Abbeel. Benchmarking deep reinforcement learning for continuous control. In International Conference on Machine Learning, pp. 1329–1338, 2016a. ",
+        "bbox": [
+            174,
+            465,
+            823,
+            508
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yan Duan, John Schulman, Xi Chen, Peter L Bartlett, Ilya Sutskever, and Pieter Abbeel. Rl2 : Fast reinforcement learning via slow reinforcement learning. arXiv preprint arXiv:1611.02779, 2016b. ",
+        "bbox": [
+            176,
+            517,
+            825,
+            559
+        ],
+        "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": [
+            169,
+            568,
+            823,
+            598
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Michel Gevers et al. System identification without lennart ljung: what would have been different? Forever Ljung in System Identification, Studentlitteratur AB, Norrtalje, 2006. ",
+        "bbox": [
+            171,
+            606,
+            823,
+            635
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Abhishek Gupta, Coline Devin, YuXuan Liu, Pieter Abbeel, and Sergey Levine. Learning invariant feature spaces to transfer skills with reinforcement learning. arXiv preprint arXiv:1703.02949, 2017. ",
+        "bbox": [
+            174,
+            642,
+            823,
+            685
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Karol Hausman, Jost Tobias Springenberg, Ziyu Wang, Nicolas Heess, and Martin Riedmiller. Learning an embedding space for transferable robot skills. 2018. ",
+        "bbox": [
+            174,
+            694,
+            821,
+            723
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Judy Hoffman, Sergio Guadarrama, Eric S Tzeng, Ronghang Hu, Jeff Donahue, Ross Girshick, Trevor Darrell, and Kate Saenko. Lsda: Large scale detection through adaptation. In Advances in Neural Information Processing Systems, pp. 3536–3544, 2014. ",
+        "bbox": [
+            178,
+            731,
+            821,
+            775
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Stephen James, Andrew J Davison, and Edward Johns. Transferring end-to-end visuomotor control from simulation to real world for a multi-stage task. arXiv preprint arXiv:1707.02267, 2017. ",
+        "bbox": [
+            174,
+            782,
+            821,
+            813
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Leslie Pack Kaelbling, Michael L Littman, and Andrew W Moore. Reinforcement learning: A survey. Journal of artificial intelligence research, 1996. ",
+        "bbox": [
+            173,
+            820,
+            821,
+            849
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. ",
+        "bbox": [
+            174,
+            858,
+            821,
+            887
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In NIPS 2012. ",
+        "bbox": [
+            174,
+            895,
+            820,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Brian Kulis, Kate Saenko, and Trevor Darrell. What you saw is not what you get: Domain adaptation using asymmetric kernel transforms. In Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, pp. 1785–1792. IEEE, 2011. ",
+        "bbox": [
+            176,
+            103,
+            821,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yanghao Li, Naiyan Wang, Jianping Shi, Jiaying Liu, and Xiaodi Hou. Revisiting batch normalization for practical domain adaptation. arXiv preprint arXiv:1603.04779, 2016. ",
+        "bbox": [
+            171,
+            155,
+            823,
+            184
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. arXiv preprint arXiv:1509.02971, 2015. ",
+        "bbox": [
+            174,
+            193,
+            821,
+            236
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mingsheng Long, Yue Cao, Jianmin Wang, and Michael I Jordan. Learning transferable features with deep adaptation networks. arXiv preprint arXiv:1502.02791, 2015. ",
+        "bbox": [
+            173,
+            243,
+            823,
+            273
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mario Milanese, John Norton, Hel´ ene Piet-Lahanier, and \\` Eric Walter. ´ Bounding approaches to system identification. Springer Science & Business Media, 2013. ",
+        "bbox": [
+            173,
+            282,
+            821,
+            313
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529, 2015. ",
+        "bbox": [
+            176,
+            320,
+            823,
+            364
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Deepak Pathak, Pulkit Agrawal, Alexei A Efros, and Trevor Darrell. Curiosity-driven exploration by self-supervised prediction. In International Conference on Machine Learning (ICML), volume 2017, 2017. ",
+        "bbox": [
+            173,
+            372,
+            823,
+            415
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Deepak Pathak, Parsa Mahmoudieh, Guanghao Luo, Pulkit Agrawal, Dian Chen, Yide Shentu, Evan Shelhamer, Jitendra Malik, Alexei A Efros, and Trevor Darrell. Zero-shot visual imitation. arXiv preprint arXiv:1804.08606, 2018. ",
+        "bbox": [
+            174,
+            424,
+            821,
+            467
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xue Bin Peng, Marcin Andrychowicz, Wojciech Zaremba, and Pieter Abbeel. Sim-to-real transfer of robotic control with dynamics randomization. arXiv preprint arXiv:1710.06537, 2017. ",
+        "bbox": [
+            173,
+            476,
+            823,
+            505
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xue Bin Peng, Pieter Abbeel, Sergey Levine, and Michiel van de Panne. Deepmimic: Example-guided deep reinforcement learning of physics-based character skills. arXiv preprint arXiv:1804.02717, 2018. ",
+        "bbox": [
+            174,
+            513,
+            821,
+            556
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Lerrel Pinto, Marcin Andrychowicz, Peter Welinder, Wojciech Zaremba, and Pieter Abbeel. Asymmetric actor critic for image-based robot learning. arXiv preprint arXiv:1710.06542, 2017a. ",
+        "bbox": [
+            171,
+            565,
+            823,
+            594
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Lerrel Pinto, James Davidson, Rahul Sukthankar, and Abhinav Gupta. Robust adversarial reinforcement learning. ICML, 2017b. ",
+        "bbox": [
+            171,
+            603,
+            821,
+            632
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Aravind Rajeswaran, Sarvjeet Ghotra, Sergey Levine, and Balaraman Ravindran. Epopt: Learning robust neural network policies using model ensembles. ICLR, 2017. ",
+        "bbox": [
+            171,
+            640,
+            823,
+            670
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Andrei A Rusu, Matej Vecerik, Thomas Rothorl, Nicolas Heess, Razvan Pascanu, and Raia Hadsell. ¨ Sim-to-real robot learning from pixels with progressive nets. arXiv preprint arXiv:1610.04286, 2016. ",
+        "bbox": [
+            173,
+            678,
+            823,
+            722
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Fereshteh Sadeghi and Sergey Levine. (cad)2rl: Real single-image flight without a single real image. arXiv preprint arXiv:1611.04201, 2016. ",
+        "bbox": [
+            169,
+            729,
+            823,
+            760
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Steindor Sæmundsson, Katja Hofmann, and Marc Peter Deisenroth. Meta reinforcement learning ´ with latent variable gaussian processes. arXiv preprint arXiv:1803.07551, 2018. ",
+        "bbox": [
+            171,
+            767,
+            823,
+            797
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "John Schulman, Sergey Levine, Pieter Abbeel, Michael Jordan, and Philipp Moritz. Trust region policy optimization. In International Conference on Machine Learning, pp. 1889–1897, 2015. ",
+        "bbox": [
+            173,
+            805,
+            823,
+            835
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. nature, 529(7587):484–489, 2016. ",
+        "bbox": [
+            176,
+            843,
+            821,
+            886
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction, volume 1. MIT press Cambridge, 1998. ",
+        "bbox": [
+            173,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Josh Tobin, Rachel Fong, Alex Ray, Jonas Schneider, Wojciech Zaremba, and Pieter Abbeel. Domain randomization for transferring deep neural networks from simulation to the real world. IROS, 2017. ",
+        "bbox": [
+            174,
+            103,
+            821,
+            145
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International Conference on, pp. 5026– 5033. IEEE, 2012. ",
+        "bbox": [
+            173,
+            155,
+            821,
+            196
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Eric Tzeng, Judy Hoffman, Ning Zhang, Kate Saenko, and Trevor Darrell. Deep domain confusion: Maximizing for domain invariance. arXiv preprint arXiv:1412.3474, 2014. ",
+        "bbox": [
+            169,
+            205,
+            823,
+            234
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jane X Wang, Zeb Kurth-Nelson, Dhruva Tirumala, Hubert Soyer, Joel Z Leibo, Remi Munos, Charles Blundell, Dharshan Kumaran, and Matt Botvinick. Learning to reinforcement learn. arXiv preprint arXiv:1611.05763, 2016. ",
+        "bbox": [
+            173,
+            243,
+            821,
+            286
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? In Advances in neural information processing systems, pp. 3320–3328, 2014. ",
+        "bbox": [
+            171,
+            295,
+            821,
+            325
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Tianhe Yu, Chelsea Finn, Annie Xie, Sudeep Dasari, Tianhao Zhang, Pieter Abbeel, and Sergey Levine. One-shot imitation from observing humans via domain-adaptive meta-learning. arXiv preprint arXiv:1802.01557, 2018. ",
+        "bbox": [
+            174,
+            333,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Wenhao Yu, Jie Tan, C Karen Liu, and Greg Turk. Preparing for the unknown: Learning a universal policy with online system identification. arXiv preprint arXiv:1702.02453, 2017. ",
+        "bbox": [
+            173,
+            385,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX A ENVIRONMENT DESCRIPTIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We used Hopper and Striker environments from OpenAI Gym (Brockman et al., 2016). We will describe the details of the environments in this section. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hopper: Hopper consists of four body parts and three joints. It has an 3-dimensional action space including motor commands for all the joints. The observation space is 11-dimensional, inluding joint angles, joint velocities, y,z positions of the torso and x,y,z velocities of the torso. The initial positions and velocities are randomized over $\\pm 0 . 0 0 5$ . The reward function is $r ( s , a ) = v _ { x } + 1 - 0 . 0 0 1 \\| a \\| ^ { 2 }$ . The first term is to encourage the agent to move forward. The second term is a reward for being alive. The third term is to disencourage unnecessary movements. The criteria for being alive is that the height of the torso should be larger than 0.7 and the angle of the torso should be smaller than 0.2 with respect to the horizontal plane. It has a maximum length of 1000 timesteps for each episode. If it falls down and breaks the alive condition, the episode will end immediately. ",
+        "bbox": [
+            173,
+            170,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Striker: Striker environment has a 7-dof robot arm and a table. There is a ball and a goal location marker on the table. The action space is 7-dimensional, including the motor commands for each joints on the robot. The 23-dimensional observation space consists of the joint angles, joint velocities, the end-effector (hand) position, the object position and the goal position. The initial conditions randomize the velocity of all the joints over $\\pm 0 . 1$ . The reward is given by $r ( s , a ) = - 3 \\| s _ { o b j e c t } - s _ { g o a l } \\| - 0 . 1 \\| a \\| ^ { 2 } - 0 . 5 \\| s _ { o b j e c t } - s _ { h a n d } \\| ^ { 2 }$ . The first term is to encourage the object to reach the goal and stay at the goal. The second term is to disencourage unnecessary movements. The third term is to encourage the hand to hit the object. After the end-effector hits the object, $s h a n d$ will be replaced by the hitting location of the end-effector. It has a fixed length of 200 timesteps for each episode. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX B RANDOMIZED ENVIRONMENT PARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            467,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Every environment parameter is randomized among 5 evenly spread discrete values. Striker has 2 varying parameters which create 25 possible environment. Hopper is varying 8 parameters, so there are $\\mathrm { \\bar { 5 } ^ { \\bar { 8 } } }$ possible environments. In order to calculate separation loss for Hopper, we sample 100 environments from all $5 ^ { 8 }$ possible environment to train the EPI-policy. Otherwise it will not be able to cluster the embeddings from the same environment. Note that when training task policy with interactions, the environment is still randomized from all possible combinations. ",
+        "bbox": [
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+            500,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "During testing, all environment parameters are sampled from a set of discrete values that the policy has never seen in any stage of training. For example, if mass is sampled from $\\{ 1 . 0 , 2 . 0 , 3 . 0 , 4 . 0 \\bar { , } 5 . 0 \\}$ , it is sampled from $\\{ 1 . 5 , \\bar { 2 } . 5 , 3 . 5 , 4 . 5 , 5 . 5 \\}$ during testing. ",
+        "bbox": [
+            176,
+            590,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX C TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            655,
+            465,
+            671
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "An EPI-trajectory contains 10 steps of observations and actions for both Hopper and Striker. The embedding network $\\psi$ that inputs an EPI-trajectory $\\tau _ { e p i }$ and outputs environment embedding $\\varepsilon$ has two fully connected layers with 32 neurons each and followed by ReLU (Krizhevsky et al.). The prediction models $f _ { p r e d }$ and $f _ { e p i . p r e d }$ has four fully connected layers with 128 neurons each. The prediction model is optimized by Adam (Kingma & Ba, 2014). Both the EPI-policy $\\pi _ { e p i }$ and the final goal policy $\\pi _ { g }$ are optimized using TRPO (Schulman et al., 2015) with rllab implementation (Duan et al., 2016a). The prediction model is trained from scratch every 50 policy updates. Training from scratch is to avoid overfitting which will lead to unintended increasing reward for the EPI-policy. The EPI-policy is trained for $2 0 0 { \\sim } 4 0 0$ iterations in total with a batch size of 10000 timesteps. The task policy will then use the trained EPI-policy and the embedding network to update for 1000 iterations with a batch size of 100000 timesteps. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX D TRAINING DETAILS OF BASELINES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            862,
+            593,
+            878
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "All MLP baselines have the same network architecture as the task policy: two hidden layers with 32 hidden units each with ReLU activations. They are trained with the same batch size, optimizer, number of iterations and other hyperparameters as the task policy. We made sure that all training curves saturate before the final iteration. ",
+        "bbox": [
+            174,
+            895,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Simple Policy: a MLP policy trained only in one fixed environment of fixed parameters. All the other baselines are trained across the randomized environments.   \n• Invariant Policy (inv-pol): a default MLP policy $a _ { t } = \\pi ( s _ { t } )$ .   \n• Oracle Policy: a MLP policy takes explicit environment parameters as additional input: $a _ { t } = \\pi ( s _ { t } , \\rho )$ , where $\\rho$ is a vector of the environment parameters such as mass, friction and damping.   \nRandom Interaction Policy: a MLP policy that takes 10 timesteps of random interactions as additional input. $a _ { t } = \\pi ( s _ { t } , s _ { i 0 } , a _ { i 0 } , s _ { i 1 } , a _ { i 1 } . . . s _ { i 9 } , a _ { i 9 } )$ , where $s _ { i n }$ and $a _ { i n }$ are from a random policy.   \n• History Policy: a MLP policy that takes 10 timesteps of most recent states and actions as additional input. Mnih et al. (2015) adopts similar idea where it stacks 3 to 5 most recent frames as input. $a _ { t } = \\pi ( s _ { t } , s _ { t - 1 } , a _ { t - 1 } , s _ { t - 2 } , a _ { t - 2 } . . . s _ { t - 1 0 } , a _ { t - 1 0 } )$ . When $t < 1 0$ , $s _ { t - j }$ are filled by zeros for $j > t$ .   \n• Recurrent Policy (lstm-pol): The Recurrent baseline is using an LSTM with 32 hidden units from rllab. $a _ { t } = \\pi _ { L S T M } ( s _ { t } )$ .   \n• System Identified Policy: This baseline is implemented based on Yu et al. (2017). We first trains an Oracle Policy $a _ { t } ~ = ~ \\pi ( s _ { t } , \\rho )$ as we described above, which is called the Universal Policy (UP) in their paper. To train Online Sytem Identification (OSI) model, we collect trajectories on randomized environments following the Oracle Policy. With the trajectories, we trained the OSI model, which is a regression model that inputs a history of states and actions and outputs $\\rho$ . The network architecture strictly follows the paper. Then the estimated $\\rho$ are fed into the trained Oracle Policy $a _ { t } = \\pi ( s _ { t } , \\rho )$ to collect more trajectories for training OSI again. We followed the original paper and trained OSI for 5 times.   \nDirect Reward Policy: takes the first 10 steps of actions as additional input. In this way, these steps are taking the future reward of the whole trajectory. $\\begin{array} { r l } { a _ { t } } & { { } = } \\end{array}$ $\\pi ( s _ { t } , s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } . . . s _ { 9 } , a _ { 9 } )$ . When $t < 1 0$ , $s _ { j }$ are filled by zeros for $j > t$ . ",
+        "bbox": [
+            214,
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+        ],
+        "page_idx": 12
+    }
+]