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+# ADDITIVE POWERS-OF-TWO QUANTIZATION:AN EFFICIENT NON-UNIFORM DISCRETIZATION FORNEURAL NETWORKS
+
+Yuhang Li †∗, $\mathbf { X i n ~ D o n g ^ { \delta * } }$ , Wei Wang †   
+†National University of Singapore, §Harvard University   
+loafyuhang@gmail.com, xindong@g.harvard.edu, wangwei@comp.nus.edu.sg
+
+# ABSTRACT
+
+We propose Additive Powers-of-Two (APoT) quantization, an efficient nonuniform quantization scheme for the bell-shaped and long-tailed distribution of weights and activations in neural networks. By constraining all quantization levels as the sum of Powers-of-Two terms, APoT quantization enjoys high computational efficiency and a good match with the distribution of weights. A simple reparameterization of the clipping function is applied to generate a better-defined gradient for learning the clipping threshold. Moreover, weight normalization is presented to refine the distribution of weights to make the training more stable and consistent. Experimental results show that our proposed method outperforms state-of-the-art methods, and is even competitive with the full-precision models, demonstrating the effectiveness of our proposed APoT quantization. For example, our 4-bit quantized ResNet-50 on ImageNet achieves $7 6 . 6 \%$ top-1 accuracy without bells and whistles; meanwhile, our model is capable to decrease $22 \%$ computational cost compared with the uniformly quantized counterpart. 1
+
+# 1 INTRODUCTION
+
+Deep Neural Networks (DNNs) have made a significant improvement for various real-world applications. However, the huge memory and computational cost impede the mass deployment of DNNs, e.g., on resource-constrained devices. To reduce memory footprint and computational burden, several model compression methods such as quantization (Zhou et al., 2016), pruning (Han et al., 2015) and low-rank decomposition (Denil et al., 2013) have been widely explored.
+
+In this paper, we focus on the neural network quantization for efficient inference. Two operations are involved in the quantization process, namely clipping and projection. The clipping operation sets a full precision number to the range boundary if it is outside of the range; the projection operation maps each number (after clipping) into a predefined quantization level (a fixed number). We can see that both operations incur information loss. A good quantization method should resolve the two following questions/challenges, which correspond to two contradictions respectively.
+
+How to determine the optimal clipping threshold to balance clipping range and projection resolution? The resolution indicates the interval between two quantization levels; the smaller the interval, the higher the resolution. The first contradiction is that given a fixed number of bits to represent weights, the range and resolution are inversely proportional. For example, a larger range can clip fewer weights; however, the resolution becomes lower and thus damage the projection. Note that slipshod clipping of outliers can jeopardize the network a lot (Zhao et al., 2019) although they may only take $1 \%$ of all weights in one layer. Previous works have tried either pre-defined (Cai et al., 2017; Zhou et al., 2016) or trainable (Choi et al., 2018b) clipping thresholds, but how to find the optimal threshold during training automatically is still not resolved.
+
+How to design quantization levels with consideration for both the computational efficiency and the distribution of weights? Most of the existing quantization approaches (Cai et al., 2017; Gong et al., 2019) use uniform quantization although non-uniform quantization can usually achieve better accuracy (Zhu et al., 2016). The reason is that projection against uniform quantization levels are much more hardware-friendly (Zhou et al., 2016). However, empirical study (Han et al., 2015) has shown that weights in a layer of DNN follow a bell-shaped and long-tailed distribution instead of a uniform distribution (as shown in the right figure). In other words, a fair percentage of weights concentrate around the mean (peak area); and a few weights are of relatively high magnitude and out of the quantization range (called outliers). Such distribution also exists in activations (Miyashita et al., 2016). The second contradiction is: considering the bell-shaped distribution of weight, it is well-motivated to assign higher resolution (i.e. smaller quantization interval) around the mean; however, such non-uniform quantization levels will introduce high computational overhead. Powers-of-Two quantization levels (Miyashita et al., 2016; Zhou et al., 2017) are then proposed because of its cheap multiplication implemented by shift operations on hardware, and super high resolution around the mean. However, the vanilla powers-of-two quantization method only increases the resolution near the mean and ignores other regions at all when the bit-width is increased. Consequently, it assigns inordinate quantization levels for a tiny range around the mean. To this end, we propose additive Powers-of-Two (APoT) quantization to resolve these two contradictions, our contribution can be listed as follows:
+
+![](images/ac3ca25c85bb9d61445f20df0c9161b4ffa7e70781fbe8ea4ef4c47a284f6eb9.jpg)  
+Figure 1: Density of weights in ResNet-18
+
+1. We introduce the APoT quantization scheme for the weights and activations of DNNs. APoT is a non-uniform quantization scheme, in which the quantization levels is a sum of several PoT terms and can adapt well to the bell-shaped distribution of weights. APoT quantization enjoys an approximate $2 \times$ multiplication speed-up compared with uniform quantization on both generic and specific hardware. 2. We propose a Reparameterized Clipping Function (RCF) that can compute a more accurate gradient for the clipping threshold and thus facilitate the optimization of the clipping threshold. We also introduce weight normalization for neural network quantization. Normalized weights in the forward pass are more stable and consistent for clipping and projection. 3. Experimental results show that our proposed method outperforms state-of-the-art methods, and is even competitive with the full-precision implementation with higher computational efficiency. Specifically, our 4-bit quantized ResNet-50 on ImageNet achieve $7 6 . 6 \%$ Top-1 and $9 3 . 1 \%$ Top-5 accuracy. Compared with uniform quantization, our method can decrease $22 \%$ computational cost, demonstrating the proposed algorithm is hardware-friendly.
+
+# 2 METHODOLOGY
+
+# 2.1 PRELIMINARIES
+
+Suppose kernels in a convolutional layer are represented by a 4D tensor $\mathcal { W } \in \mathbb { R } ^ { C _ { o u t } \times C _ { i n } \times K \times K }$ , where $C _ { o u t }$ and $C _ { i n }$ are the number of output and input channels respectively, and $K$ is the kernel size. We denote the quantization of the weights as
+
+$$
+\begin{array} { r } { \hat { \mathcal { W } } = \Pi _ { \mathcal { Q } ( \alpha , b ) } \lfloor \mathcal { W } , \alpha \rceil , } \end{array}
+$$
+
+where $\alpha$ is the clipping threshold and the clipping function $[ \cdot , \alpha ]$ clips weights into $[ - \alpha , \alpha ]$ . After clipping, each element of $\mathcal { W }$ is projected by $\Pi ( \cdot )$ onto the quantization levels. We denote $\mathcal { Q } ( \alpha , b )$ for a set of quantization levels, where $b$ is the bit-width. For uniform quantization, the quantization levels are defined as
+
+$$
+\mathcal { Q } ^ { u } ( \alpha , b ) = \alpha \times \{ 0 , \frac { \pm 1 } { 2 ^ { b - 1 } - 1 } , \frac { \pm 2 } { 2 ^ { b - 1 } - 1 } , \frac { \pm 3 } { 2 ^ { b - 1 } - 1 } , \ldots , \pm 1 \} .
+$$
+
+For every floating-point number, uniform quantization maps it to a $b$ -bit fixed-point representation (quantization levels) in $\mathcal { Q } ^ { u } ( \alpha , b )$ . Note that $\alpha$ is stored separately as a full-precision floating-point number for each whole $\mathcal { W }$ . Convolution is done against the quantization levels first and the results are then multiplied by $\alpha$ . Arithmetical computation, e.g., convolution, can be implemented using low-precision fixed point operations on hardware, which are substantially cheaper than their floating-point contradictory (Goldberg, 1991). Nevertheless, uniform quantization does not match the distribution of weights (and activations), which is typically bell-shaped (Han et al., 2015). A straightforward solution is to assign more quantization levels (higher resolution) for the peak of the distribution and fewer levels (lower resolution) for the tails. However, it is difficult to implement the arithmetical operations for the non-uniform quantization levels efficiently.
+
+![](images/c73baa7a1cd6bb5846b071beac43506a8cd785aea78c61ec29270271ff274fee.jpg)  
+Figure 2: Quantization of unsigned data to 3-bit or 4-bit $( \alpha = 1 . 0 $ ) using three different quantization levels. APoT quantization has a more reasonable resolution assignment and it does not suffer from the rigid resolution.
+
+# 2.2 ADDITIVE POWERS-OF-TWO QUANTIZATION
+
+To solve the contradiction between non-uniform resolution and hardware efficiency, Powers-ofTwo (PoT) quantization (Miyashita et al., 2016; Zhou et al., 2017) is proposed by constraining quantization levels to be powers-of-two values or zero, i.e.,
+
+$$
+\begin{array} { r } { \underline { { Q } } ^ { p } ( \alpha , b ) = \alpha \times \{ 0 , \pm 2 ^ { - 2 ^ { b - 1 } + 1 } , \pm 2 ^ { - 2 ^ { b - 1 } + 2 } , . . . , \pm 2 ^ { - 1 } , \pm 1 \} . } \end{array}
+$$
+
+Apparently, as a non-uniform quantizer, PoT has a higher resolution for the value range with denser weights because of its exponential property. Furthermore, multiplication between a Powers-of-two number $2 ^ { x }$ and the other operand $r$ can be implemented by bit-wise shift instead of bulky digital multipliers, i.e.,
+
+$$
+2 ^ { x } r = { \left\{ \begin{array} { l l } { r } & { { \mathrm { ~ i f ~ } } x = 0 } \\ { r < < x } & { { \mathrm { ~ i f ~ } } x > 0 , } \\ { r > > x } & { { \mathrm { ~ i f ~ } } x < 0 } \end{array} \right. }
+$$
+
+where $> >$ denotes the right shift operation and is computationally cheap, which only takes 1 clock cycle in modern CPU architectures.
+
+However, we find that PoT quantization does not benefit from more bits. Assume $\alpha$ is 1, as shown in Equation (3), when we increase the bit-width from b to b + 1, the interval [−2−2b−1+1, 2−2b−1+1] will be split into $2 ^ { b - 1 } - 1$ sub-intervals, whereas all other intervals remain unchanged. In other words, by increasing the bit-width, the resolution will increase only for $[ - 2 ^ { - 2 ^ { b - 1 } + 1 } , 2 ^ { - 2 ^ { b - 1 } + 1 } ]$ . We refer this phenomenon as the rigid resolution of PoT quantization. Take $\mathcal { Q } ^ { p } ( 1 , 5 )$ as an example, the two smallest positive levels are $2 ^ { - 1 5 }$ and $2 ^ { - 1 4 }$ , which is excessively fine-grained. In contrast, the two largest levels are $2 ^ { - 1 }$ and $2 ^ { 0 }$ , whose interval is large enough to incur high projection error for weights between $[ 2 ^ { - 1 } , 2 ^ { 0 } ]$ , e.g., 0.75. The rigid resolution is demonstrated in Figure 2(b). When we change from from 3-bit to 4-bit, all new quantization levels concentrate around 0 and thus cannot increase the model’s expressiveness effectively.
+
+![](images/b463f44038b337fd8ebc5c453fd6d8c77e25551f39f417fe6b886196dc7ef5d3.jpg)  
+Figure 3: Hardware accelerator with different quantization schemes. When $k$ increase, weights usually has less PoT terms, thus accelerates the computation.
+
+To tackle the rigid resolution problem, we propose Additive Powers-of-Two (APoT) quantization. Without loss of generality, in this section, we only consider unsigned numbers for simplicity2. In APoT quantization, each level is the sum of n PoT terms as shown below,
+
+$$
+\mathcal { Q } ^ { a } ( \alpha , k n ) = \gamma \times \{ \sum _ { i = 0 } ^ { n - 1 } p _ { i } \} \mathrm { w h e r e } p _ { i } \in \{ 0 , \frac { 1 } { 2 ^ { i } } , \frac { 1 } { 2 ^ { i + n } } , . . . , \frac { 1 } { 2 ^ { i + ( 2 ^ { k } - 2 ) n } } \} ,
+$$
+
+where $\gamma$ is a scaling coefficient to make sure the maximum level in ${ \mathcal { Q } } ^ { a }$ is $\alpha , \ k$ is called the base bit-width, which is the bit-width for each additive term, and $n$ is the number of additive terms. When the bit-width $b$ and the base bit-width $k$ is set, $n$ can be calculated by $\begin{array} { r } { n = { \frac { b } { k } } } \end{array}$ . There are $2 ^ { k n } = 2 ^ { b }$ $b$ which provides a higher resolution for the non-uniform levels.
+
+We use $b \ = \ 4$ and $k \ = \ 2$ as an example to illustrate how APoT resolves the rigid resolution problem. For this example, we have $p _ { 0 } \in \{ 0 , 2 ^ { 0 } , 2 ^ { - 2 } , 2 ^ { - 4 } \}$ , $p _ { 1 } \in \{ 0 , 2 ^ { - 1 } , 2 ^ { - 3 } , 2 ^ { - 5 } \} ,$ $\gamma = 2 \alpha / 3$ , and ${ \mathcal { Q } } ^ { a } ( \alpha , k n ) = \{ \gamma \times ( p _ { 0 } + p _ { 1 } ) \}$ for all $( 2 ^ { b } = 1 6 )$ combinations of $p _ { 0 }$ and $p _ { 1 }$ . First, we can see the smallest positive quantization level in $\mathcal { Q } ^ { a } ( 1 , 4 )$ is $2 ^ { - 4 } / 3$ . Compared with the original PoT levels, APoT allocates quantization levels prudently for the central area. Second, APoT generates 3 new quantization levels between $2 ^ { 0 }$ and $\bar { 2 } ^ { - 1 }$ , to properly increase the resolution. In Figure 2, the second row compares the 3 quantization methods using 4 bits for range [0, 1]. APoT quantization has a reasonable distribution of quantization levels, with more levels in the peak area (near 0) and relatively higher resolution than the vanilla PoT quantization at the tail (near 1).
+
+Relation to other quantization schemes. On the one hand, the fixed-point number representations used in the uniform quantization is a special case of APoT. When $k ~ = ~ 1$ in Equation (5), the quantization levels is a sum of $b$ PoT terms or 0. In the fixed-point representations, each bit indicates one specific choice of the additive terms. On the other hand, when $k = b$ , there is only one PoT term and $\bar { \mathcal { Q } } ^ { a } ( \alpha , b )$ becomes $\mathcal { Q } ^ { p } ( \alpha , b )$ , i.e., PoT quantization. We can conclude that when $k$ decreases, APoT levels are decomposed into more PoT terms, and the distribution of levels becomes more uniform. Our experiments use $k = 2$ , which is an intermediate choice between the uniform case $k = 1 ,$ ) and the vanilla PoT case $k = b$ ).
+
+Computation. Multiplication for fixed-point numbers can be implemented by shifting the multiplicand (i.e., the activations) and adding the partial product. The $n$ in Equation (5) denotes the Since number of additive PoT terms in the multiplier (weights), and control the speed of computation. $\begin{array} { r } { n = \frac { b } { k } } \end{array}$ , either decreasing $b$ or increasing $k$ can accelerate the multiplication. Compared with uniform quantization $k = 1 ,$ ), our method $k = 2$ ) is approximately $2 \times$ faster in multiplication. As for the full precision $\alpha$ , it is a coefficient for all weights in a layer and can be multiplied only once after the multiply-accumulate operation is finished. Figure 3 shows the hardware accelerator, the weights buffer takes $k$ -bit as a PoT term and shift-adds the activations.
+
+Generalizing to $2 n + 1$ bits. When $k = 2$ , APoT quantization can only leverages $2 n$ -bit width for quantization. To deal with $2 n + 1$ -bit quantization, we choose to add $n + 1$ PoT terms, one of which only contains 2 levels. The formulation is given by
+
+$$
+\mathcal { Q } ^ { a } ( \alpha , 2 n + 1 ) = \gamma \times \ \{ \sum _ { i = 0 } ^ { n - 1 } p _ { i } + \tilde { p } \} \ \mathrm { w h e r e } \ p _ { i } \in \{ 0 , \frac { 1 } { 2 ^ { i } } , \frac { 1 } { 2 ^ { i + n } } , \frac { 1 } { 2 ^ { i + 2 n + 1 } } \} , \ \tilde { p } \in \{ 0 , \frac { 1 } { 2 ^ { i + 2 n } } \} .
+$$
+
+Take 3-bit APoT quantization as an example, every level is a sum of one $p _ { 0 }$ and one $\tilde { p }$ , where $p _ { 0 } \in \{ 0 , 2 ^ { - 1 } , 2 ^ { - 2 } , \bar { 2 } ^ { - 4 } \}$ and $\tilde { p } \in \{ 0 , 2 ^ { - 3 } \}$ . The forward function is plotted in Figure 2(c).
+
+# 2.3 REPARAMETERIZED CLIPPING FUNCTION
+
+Besides the projection operation, the clipping operation $[ \mathcal { W } , \alpha ]$ is also important for quantization. $\alpha$ is a threshold that determines the value range of weights in a quantized layer. Tuning the clipping threshold $\alpha$ is a key challenge because of the long-tail distribution of the weights. Particularly, if $\alpha$ is too large (e.g., the maximum absolute value of $\mathcal { W }$ ), $\mathcal { Q } ( \alpha , b )$ would have a wide range and then the projection will lead to large error as a result of insufficient resolution for the weights in the central area; if $\alpha$ is too small, more outliers will be clipped slipshodly. Considering the distribution of weights can be complex and differs across layers and training steps, a static clipping threshold $\alpha$ for all layers is not optimal.
+
+To jointly optimize the clipping threshold $\alpha$ and weights via SGD during training, Choi et al. (2018b) apply the Straight-Through Estimator (STE) (Bengio et al., 2013) to do the backward propagation for the projection operation. According to STE, the gradient to $\alpha$ is computed by $\begin{array} { r } { \frac { \partial \hat { \mathcal { W } } } { \partial \alpha } \approx \frac { \partial \lfloor \mathcal { W } , \alpha \rceil } { \partial \alpha } = } \end{array}$ $\mathrm { s i g n } ( { \mathscr W } )$ when $| \mathcal { W } | > \alpha$ otherwise 0, where the weights outside of the range cannot contribute to the gradients, which results in inaccurate gradient approximation. To provide a refined gradient for the clipping threshold, we design a Reparameterized Clipping Function (RCF) as
+
+$$
+\hat { \mathcal { W } } = \alpha \Pi _ { \mathcal { Q } ( 1 , b ) } \big [ \frac { \mathcal { W } } { \alpha } , 1 \big ] .
+$$
+
+Instead of directly clipping them to $[ - \alpha , \alpha ]$ , RCF outputs a constant clipping range and re-scales weights back after the projection, which is mathematically equivalent to Equation (1) during forward. In backpropagation, STE is adopted for the projection operation and the gradients of $\alpha$ are calculated by
+
+$$
+\frac { \partial \hat { \mathcal { W } } } { \partial \alpha } = \left\{ \begin{array} { l l } { \mathrm { s i g n } ( \mathcal { W } ) } & { \mathrm { i f ~ } | \mathcal { W } | > \alpha } \\ { \displaystyle \Pi _ { \mathcal { Q } ( 1 , b ) } \frac { \mathcal { W } } { \alpha } - \frac { \mathcal { W } } { \alpha } } & { \mathrm { i f ~ } | \mathcal { W } | \leq \alpha } \end{array} \right.
+$$
+
+The detail derivation of the gradients is shown in Appendix A. Compared with the normal clipping function, RCF provides more accurate gradient signals for the optimization because both weights inside $( | \mathcal { W } | \leq \alpha )$ and out of $( | \mathcal { W } | > \alpha )$ the range can contribute to the gradient for the clipping threshold. Particularly, the outliers are responsible for the clipping, and the weights in $[ - \alpha , \alpha ]$ are for projection. Therefore, the update of $\alpha$ considers both clipping and projection, and tries to find a balance between them. In experiments, we observe that the clipping threshold will become universally smaller when the bit-width is reduced to guarantee sufficient resolution, which further validates the efficaciousness of the gradient in RCF.
+
+# 2.4 WEIGHT NORMALIZATION
+
+In practice, we find that learning $\alpha$ for weights is quite arduous because the distribution of weights is pretty steep and changes frequently during training. As a result, jointly training the clipping threshold and weights parameters is hard to converge. Inspired by the crucial role of batch normalization (BN) (Ioffe $\&$ Szegedy, 2015) in activation quantization (Cai et al., 2017), we propose weight normalization (WN) to refine the distribution of weights with zero mean and unit variance,
+
+$$
+\tilde { \mathcal { W } } = \frac { \mathcal { W } - \mu } { \sigma + \epsilon } , \mathrm { w h e r e } \mu = \frac { 1 } { I } \sum _ { i = 1 } ^ { I } \mathcal { W } _ { i } , \sigma = \sqrt { \frac { 1 } { I } \sum _ { i = 1 } ^ { I } ( \mathcal { W } _ { i } - \mu ) ^ { 2 } } ,
+$$
+
+where $\epsilon$ is a small number (typically $1 0 ^ { - 5 }$ ) for numerical stability, and $I$ denotes the number of weights in one layer. Note that quantization of weights is applied right after this normalization.
+
+![](images/296afd41c25721ce18c2a3fba33b7a7c5e0585187412215ff4824d4f0da40e1f.jpg)  
+Figure 4: The evolution of clipping ratio of the first three layers in ResNet-20. (a) demonstrates clipping ratio is too sensitive to threshold to hurt its optimization without weights normalization. (b) shows that weights distribution after normalization is relatively more stable during training.
+
+# Algorithm 1 Forward and backward procedure for an APoT quantized convolutional layer
+
+Input: input activations $\mathcal { X } _ { i n }$ , the full precision weight tensor $\mathcal { W }$ , the clipping threshold for weights and activations $\alpha _ { \mathcal { W } }$ , $\alpha _ { \mathcal { X } }$ , the bit-width $b$ of quantized tensor.
+
+Output: the output activations $\mathcal { X } _ { o u t }$
+
+1: Normalize weights $\mathcal { W }$ to $\tilde { \mathcal W }$   
+2: Apply RCF and APoT quantization to the normalized weights $\begin{array} { r } { \hat { \mathcal { W } } = \alpha \nu \Pi _ { \mathcal { Q } ^ { a } ( 1 , b ) } \big \lfloor \frac { \tilde { \mathcal { W } } } { \alpha _ { \mathcal { W } } } , 1 \big \rceil } \end{array}$   
+3: Apply RCF and APoT quantization to the activations $\begin{array} { r } { \hat { \mathcal { X } _ { i n } } = \alpha _ { \mathcal { X } } \Pi _ { \mathcal { Q } ^ { a } ( 1 , b ) } \lfloor \frac { \mathcal { X } _ { i n } } { \alpha _ { \mathcal { X } } } , 1 \rceil } \end{array}$   
+4: Compute the output activations $\mathcal { X } _ { o u t } = C o n \nu ( \hat { \mathcal { W } } , \hat { \mathcal { X } _ { i n } } )$   
+5: Compute the loss $\mathcal { L }$ and the gradients $\frac { \partial \mathcal { L } } { \partial \mathcal { X } _ { o u t } }$ ,   
+6: Compute the gradients of convolution $\frac { \partial \mathcal { L } } { \partial \hat { \mathcal { X } } _ { i n } }$ $\textstyle \frac { \partial { \mathcal { L } } } { \partial { \hat { \mathcal { W } } } }$   
+7: Compute the gradients for clipping threshold $\frac { \partial \mathcal { L } } { \partial \alpha \ w }$ , $\frac { \partial \mathcal { L } } { \partial \alpha \boldsymbol { x } }$ based on Equation (8)   
+8: Compute the gradients to the full precision weights $\begin{array} { r } { \frac { \partial \mathcal { L } } { \partial \mathcal { W } } = \frac { \partial \mathcal { L } } { \partial \hat { \mathcal { W } } } \frac { \partial \hat { \mathcal { W } } } { \partial \tilde { \mathcal { W } } } \frac { \partial \tilde { \mathcal { W } } } { \partial \mathcal { W } } } \end{array}$   
+9: Update $\mathcal { W }$ , $\alpha _ { \mathcal { W } }$ , $\alpha _ { \mathcal { X } }$ with learning rate $\eta _ { \mathcal { W } } , \eta _ { \alpha _ { \mathcal { W } } } , \eta _ { \alpha _ { \mathcal { X } } }$
+
+Normalization is important to provide a relatively consistent and stable input distribution to both clipping and projection functions for smoother optimization of $\alpha$ over different layers and iterations during training. Besides, making the mean of weights to be zero can reap the benefits of the symmetric design of the quantization levels.
+
+Here, we conduct a case study of ResNet-20 on CIFAR10 to illustrate how normalization for weights can help quantization. For a certain layer (at a certain training step) in ResNet-18, We firstly fix the weights, and let $\alpha$ go from 0 to max $| \mathcal { W } |$ to plot the curve of the clipping ratio (i.e. the proportion of clipped weights). As shown in Figure 4a, the change of clipping ratio is much smoother after quantization. As a result, the optimization of $\alpha$ will be significantly smoother. In addition, normalization also makes the distribution of weights quite more consistent over training iterations. We fix the value of $\alpha$ , and visualize clipping ratio over training iterations in Figure 4b. After normalization, the same $\alpha$ will result in almost the same clipping ratio, which improves the consistency of optimization goal for $\alpha$ . More experimental analysis demonstrating the effectiveness of the normalization on weights can be found in Appendix B.
+
+# 2.5 TRAINING AND DEPLOYING
+
+We adopt APoT quantization for both weights and activations. Notwithstanding the effect in activations is not conspicuous, we adopt APoT quantization for consistency. During backpropagation, we use STE when computing the gradients of weights, i.e. $\begin{array} { r } { \frac { \partial \hat { \mathcal { W } } } { \partial \tilde { \mathcal { W } } } = 1 } \end{array}$ . The detailed training procedure is shown in Algorithm 1.
+
+To save memory cost during inference, we discard the full precision weights $\mathcal { W }$ and only store the quantized weights $\hat { \mathcal W }$ . Compared with other uniform quantization methods, APoT quantization is more efficient and effective during inference.
+
+# 3 RELATED WORKS
+
+Non-Uniform Quantization. Several methods are proposed for the non-uniform distribution of weights. LQ-Nets(Zhang et al., 2018) learns quantization levels based on the quantization error minimization (QEM) algorithm. Distillation (Polino et al., 2018) optimizes the quantization levels directly to minimize the task loss which reflects the behavior of their teacher network. These methods use finite floating-point numbers to quantize weights (and activations), bringing extra computation overhead compared with linear quantization. Logarithmic quantizers (Zhou et al., 2017; Miyashita et al., 2016) leverage powers-of-2 values to accelerate the computation by shift operations; however, they suffer from the rigid resolution problem.
+
+Jointly Training. Many works have explored to optimize the quantization parameters (e.g., $\alpha$ ) and the weights parameters simultaneously. Zhu et al. (2016) learns positive and negative scaling coefficients respectively. LQ-Nets jointly train these parameters to minimize the quantization error. QIL (Jung et al., 2019) introduces a learnable transformer to change the quantization intervals and optimize them based on the task loss. PACT (Choi et al., 2018b) parameterizes the clipping threshold in activations and optimize it through gradient descent. However, in PACT, the gradient of $\alpha$ is not accurate, which only includes the contribution from outliers and ignores the contribution from other weights.
+
+Weight Normalization. Previous works on weight normalization mainly focus on addressing the limitations of BatchNorm (Ioffe & Szegedy, 2015). Salimans & Kingma (2016); Hoffer et al. (2018) decouple direction from magnitude to accelerate the training procedure. Weight Standardization (Qiao et al., 2019) normalizes weights to zero mean and unit variance during the forward pass. However, there is limited literature that studies the normalization of weights for neural network quantization. (Zhu et al., 2016) uses feature-scaling to normalize weights by dividing the maximum absolute value. Weight Normalization based Quantization (Cai & Li, 2019) also uses this feature scaling and derive the gradient to eliminate the outliers in the weights tensor.
+
+# 4 EXPERIMENT
+
+In this section, we validate our proposed method on ImageNet-ILSVRC2012 (Russakovsky et al., 2015) and CIFAR10 (Krizhevsky et al., 2009). We also conduct ablation study for each component of our algorithm.
+
+# 4.1 EVALUATION ON IMAGENET
+
+We compare our methods with several strong baselines on ResNet architectures (He et al., 2016), including ABC-Net (Lin et al., 2017), DoReFa-Net (Zhou et al., 2016), PACT (Choi et al., 2018b), LQ-Net (Zhang et al., 2018), DSQ (Gong et al., 2019), QIL (Jung et al., 2019). Both weights and activations of the networks are quantized for comparisons. All the state-of-the-art methods use full precision (32 bits) for the first and the last layer, which incur more memory cost. In our implementation, we employ 8-bit quantization for them to balance the accuracy drop and the hardware overhead.
+
+For our proposed APoT quantization algorithm, four configurations of the bit-width, i.e., 2,3,4, and 5 ( $k = 2$ and $n = 1$ or 2 in Equation (5) and (6)) are tested, where one bit is used for the sign of the weights but not for activations. Note that for the 2-bit symmetric weight quantization method, $\mathcal { Q } ( \alpha , 2 )$ can only be $\{ \pm \alpha , 0 \}$ , therefore only RCF and WN are used in this setting. To obtain a reasonable initialization, we follow Lin et al. (2017); Jung et al. (2019) to initialize our model. Specifically, the 5-bit quantized model is initialized from the pre-trained full precision one3, while the 4-bit network is initialized from the trained 5-bit model. We compare the accuracy, memory cost, and the fixed point operations under different bit-width. To compare the operations with different bit-width, we use the bit-op computation scheme introduced in Zhou et al. (2016) where the multiplication between a $m$ -bit and a $l$ -bit uniform quantized number costs $m l$ binary operation. We define one FixOP as one operation between an 8-bit weight and an 8-bit activation which takes 64 binary operations if uniform quantization scheme is applied. In APoT scheme, the multiplication between a $m$ -bit activation and a $l = k n$ -bit weight only needs mn shift-adds operations, i.e., $\frac { n \times m } { 6 4 }$ FixOPs. More details of the implementation are in the Appendix C.2.
+
+Table 1: Comparison of accuracy performance as well as hardware performance of ResNets (He et al., 2016) on ImageNet with existing methods.   
+
+<table><tr><td rowspan="3">METHOD</td><td rowspan="2">PRECISION (W/A)</td><td colspan="2">ACCURACY(%)</td><td rowspan="2">MODEL SIZE</td><td rowspan="3">FIXOPS</td><td rowspan="2">PRECISION</td><td colspan="2">ACCURACY(%)</td><td rowspan="3">MODEL SIZE</td><td rowspan="3">FIXOPS</td></tr><tr><td>ToP-1</td><td>TOP-5</td><td>(W/A)</td><td>TOP-1 TOP-5</td></tr><tr><td>FP.(RES18)</td><td>32/32</td><td>70.2</td><td>89.4</td><td>46.8MB</td><td>1.82G</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ABC-NETS</td><td>5/5</td><td>65.0</td><td>85.9</td><td>8.72 MB</td><td>781M</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DoREFA-NET</td><td>5/5</td><td>68.4</td><td>88.3</td><td>8.72MB</td><td>781M</td><td>4/4</td><td>68.1</td><td>88.1</td><td>7.39MB</td><td>542M</td></tr><tr><td>PACT</td><td>5/5</td><td>69.8</td><td>89.3</td><td>8.72MB</td><td>781M</td><td>4/4</td><td>69.2</td><td>89.0</td><td>7.39 MB</td><td>542M</td></tr><tr><td>LQ-NET</td><td></td><td></td><td></td><td></td><td></td><td>4/4</td><td>69.3</td><td>88.8</td><td>7.39MB</td><td>542M</td></tr><tr><td>DSQ</td><td></td><td></td><td></td><td></td><td></td><td>4/4</td><td>69.6</td><td></td><td>7.39 MB</td><td>542M</td></tr><tr><td>QIL</td><td>5/5</td><td>70.4</td><td>=</td><td>8.72MB</td><td>781M</td><td>4/4</td><td>70.1</td><td>=</td><td>7.39MB</td><td>542M</td></tr><tr><td>APoT (OURS)</td><td>5/5</td><td>70.9</td><td>89.7</td><td>7.22 MB</td><td>616M</td><td>4/4</td><td>70.7</td><td>89.6</td><td>5.89 MB</td><td>437M</td></tr><tr><td>ABC-NETS</td><td>3/3</td><td>61.0</td><td>83.2</td><td>6.06MB</td><td>357M</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DoREFA-NET</td><td>3/3</td><td>67.5</td><td>87.6</td><td>6.06MB</td><td>357M</td><td>2/2</td><td>62.6</td><td>84.6</td><td>4.73MB</td><td>225M</td></tr><tr><td>PACT</td><td>3/3</td><td>68.1</td><td>88.2</td><td>6.06MB</td><td>357M</td><td>2/2</td><td>64.4</td><td>85.6</td><td>4.73 MB</td><td>225M</td></tr><tr><td>LQ-NET</td><td>3/3</td><td>68.2</td><td>87.9</td><td>6.06MB</td><td>357M</td><td>2/2</td><td>64.9</td><td>85.9</td><td>4.73MB</td><td>225M</td></tr><tr><td>DSQ</td><td>3/3</td><td>68.7</td><td>-</td><td>6.06MB</td><td>357M</td><td>2/2</td><td>65.2</td><td>-</td><td>4.73MB</td><td>225M</td></tr><tr><td>QIL</td><td>3/3</td><td>69.2</td><td>-</td><td>6.06MB</td><td>357M</td><td>2/2</td><td>65.7</td><td>-</td><td>4.73MB</td><td>225M</td></tr><tr><td>PACT+SAWB</td><td></td><td></td><td></td><td></td><td></td><td>2/2</td><td>67.0</td><td></td><td>5.36MB</td><td>243M</td></tr><tr><td>APOT (OURS)</td><td>3/3</td><td>69.9</td><td>89.2</td><td>4.56MB</td><td>298M</td><td>2/2</td><td>67.3</td><td>87.5</td><td>3.23MB</td><td>198M</td></tr><tr><td>FP.(RES34)</td><td>32/32</td><td>73.7</td><td>91.3</td><td>83.2MB</td><td>3.68G</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ABC-NETS</td><td>5/5</td><td>68.4</td><td>88.2</td><td>14.8MB</td><td>1.50G</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DSQ</td><td></td><td></td><td></td><td></td><td></td><td>4/4</td><td>72.8</td><td></td><td>12.3MB</td><td>1.00G</td></tr><tr><td>QIL</td><td>5/5</td><td>73.7</td><td>-</td><td>14.8MB</td><td>1.50G</td><td>4/4</td><td>73.7</td><td>=</td><td>12.3MB</td><td>1.00G</td></tr><tr><td>APOT(OURS)</td><td>5/5</td><td>73.9</td><td>91.6</td><td>13.3 MB</td><td>1.15G</td><td>4/4</td><td>73.8</td><td>91.6</td><td>10.8 MB</td><td>784M</td></tr><tr><td>ABC-NETS</td><td>3/3</td><td>66.4</td><td>87.4</td><td>9.73MB</td><td>618M</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>LQ-NET</td><td>3/3</td><td>71.9</td><td>90.2</td><td>9.73MB</td><td>618M</td><td>2/2</td><td>69.8</td><td>89.1</td><td>7.20 MB</td><td>340M</td></tr><tr><td>DSQ</td><td>3/3</td><td>72.5</td><td>-</td><td>9.73MB</td><td>618M</td><td>2/2</td><td>70.0</td><td>1</td><td>7.20MB</td><td>340M</td></tr><tr><td>QIL</td><td>3/3</td><td>73.1</td><td>-</td><td>9.73MB</td><td>618M</td><td>2/2</td><td>70.6</td><td>-</td><td>7.20MB</td><td>340M</td></tr><tr><td>APOT(OURS)</td><td>3/3</td><td>73.4</td><td>91.1</td><td>8.23MB</td><td>493M</td><td>2/2</td><td>70.9</td><td>89.7</td><td>5.70MB</td><td>285M</td></tr><tr><td>FP.(RES50)</td><td>32/32</td><td>76.4</td><td>93.1</td><td>97.5MB</td><td>4.14G</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ABC-NETS</td><td>5/5</td><td>70.1</td><td>89.7</td><td>22.2MB</td><td>1.67G</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DOREFA-NET</td><td>5/5</td><td>71.4</td><td>93.3</td><td>22.2MB</td><td>1.67G</td><td>4/4</td><td>71.4</td><td>89.8</td><td>19.4MB</td><td>1.11G</td></tr><tr><td>LQ-NET</td><td></td><td></td><td></td><td></td><td></td><td>4/4</td><td>75.1</td><td>92.4</td><td>19.4MB</td><td>1.11G</td></tr><tr><td>PACT</td><td>5/5</td><td>76.7</td><td>93.3</td><td>22.2MB</td><td>1.67G</td><td>4/4</td><td>76.5</td><td>93.3</td><td>19.4 MB</td><td>1.11G</td></tr><tr><td>APOT (OURS)</td><td>5/5</td><td>76.7</td><td>93.3</td><td>16.3MB</td><td>1.28G</td><td>4/4</td><td>76.6</td><td>93.1</td><td>13.6MB</td><td>866M</td></tr><tr><td>DoREFA-NET</td><td>3/3</td><td>69.9</td><td>89.2</td><td>16.6 MB</td><td>680M</td><td>2/2</td><td>67.1</td><td>87.3</td><td>13.8MB</td><td>370M</td></tr><tr><td>PACT</td><td>3/3</td><td>75.3</td><td>92.6</td><td>16.6MB</td><td>680M</td><td>2/2</td><td>72.2</td><td>90.5</td><td>13.8MB</td><td>370M</td></tr><tr><td>LQ-NET PACT+SAWB</td><td>3/3</td><td>74.2</td><td>91.6</td><td>16.6MB</td><td>680M</td><td>2/2 2/2</td><td>71.5 74.2</td><td>90.3</td><td>13.8MB 23.7MB</td><td>370M 707M</td></table>
+
+Overall results are shown in Table 1. The results of DoReFa-Net are taken from Choi et al. (2018b), and the other results are quoted from the original papers. It can be observed that our 5-bit quantized network achieves even higher accuracy than the full precision baselines ( $0 . 7 \%$ Top-1 improvement on ResNet-18 and $0 . 2 \%$ Top-1 improvement on ResNet-34 and ResNet-50), which means quantization may serve the purpose of regularization. Along with the accuracy performance, our APoT quantization can achieve better hardware performance on model size and inference speed. For full precision models, the number in the column of FixOPs indicates FLOPs.
+
+4-bit and 3-bit quantized networks are also preserving (or approaching) the full-precision accuracy except for the 3-bit quantized ResNet-18 and ResNet-34, which only drops $0 . 5 \%$ and $0 . 3 \%$ accuracy respectively. When $b$ is further reduced to 2, our model still outperforms the baselines, which demonstrates the effectiveness of RCF and WN. Note that Choi et al. (2018a) use a full precision shortcut in the model, reaching higher accuracy on ResNet-50 however suffering from the hardware performance. In specific, the different precision between the main path and the residual path may result in greater latency in a pipelined implementation.
+
+# 4.2 EVALUATION ON CIFAR10
+
+We quantize ResNet-20 and ResNet-56 (He et al., 2016) on CIFAR10 for evaluation. We adopt progressive initialization and choose the quantization bit as 2, 3 and 4. More implementations can be found in the Appendix C.2.
+
+Table 2: Accuracy comparison of ResNet architectures on CIFAR10   
+
+<table><tr><td rowspan="2">MODELS METHODS</td><td rowspan="2"></td><td colspan="3">ACCURACY(%)</td></tr><tr><td>2 BITS</td><td>3 BITS</td><td>4 BITS</td></tr><tr><td rowspan="5">REsNET-20 (FP: 91.6)</td><td>DOREFA-NET (ZHOU ET AL.,2016)</td><td>88.2</td><td>89.9</td><td>90.5</td></tr><tr><td>PACT(CHOI ET AL., 2018B)</td><td>89.7</td><td>91.1</td><td>91.7</td></tr><tr><td>LQ-NET (ZHANG ET AL.,2018)</td><td>90.2</td><td>91.6</td><td>-</td></tr><tr><td>PACT+SAWB+FPSC(CHOI ET AL.,2018A)</td><td>90.5</td><td>-</td><td>=</td></tr><tr><td>APOT QUANTIZATION(OURS)</td><td>91.0</td><td>92.2</td><td>92.3</td></tr><tr><td rowspan="2">REsNET-56 (FP: 93.2)</td><td>PACT+SAWB+FPSC (CHOI ET AL., 2018A)</td><td>92.5</td><td>=</td><td>1</td></tr><tr><td>APOTQUANTIZATION (OURS)</td><td>92.9</td><td>93.9</td><td>94.0</td></tr></table>
+
+Table. 2 summarizes the accuracy of our APoT in comparison with baselines. For 3-bit and 4-bit models, APoT quantization has reached comparable results with the full precision baselines. It is worthwhile to note that all state-of-the-arts methods in the table use 4 levels to quantize weights into 2-bit. Our model only employs ternary weights for 2-bit representation and still outstrips existing quantization methods.
+
+# 4.3 ABLATION STUDY
+
+Table 3: Comparison of quantizer, weight normalization and RCF of ResNet-18 on ImageNet.   
+
+<table><tr><td>METHOD</td><td>PRECISION</td><td>WN</td><td>RCF</td><td>Acc.-1</td><td>RCF</td><td>Acc.-1</td><td>MODEL SIZE</td><td>FIxOPS</td></tr><tr><td>FULL PREC.</td><td>32/32</td><td>：</td><td>-</td><td>70.2</td><td>-</td><td>70.2</td><td>46.8 MB</td><td>1.82G</td></tr><tr><td>APoT</td><td>5/5</td><td></td><td></td><td>70.9</td><td>X</td><td>70.0</td><td>7.22 MB</td><td>616M</td></tr><tr><td>PoT</td><td>5/5</td><td></td><td></td><td>70.3</td><td>X</td><td>68.9</td><td>7.22 MB</td><td>582M</td></tr><tr><td>UNIFORM</td><td>5/5</td><td>&lt;&lt;&gt;</td><td></td><td>70.7</td><td>×</td><td>69.4</td><td>7.22 MB</td><td>781M</td></tr><tr><td>LLOYD</td><td>5/5</td><td>√</td><td>&lt;&lt;&lt;√</td><td>70.9</td><td>X</td><td>70.2</td><td>7.22 MB</td><td>1.81G</td></tr><tr><td>APoT</td><td>3/3</td><td></td><td></td><td>69.9</td><td>X</td><td>68.5</td><td>4.56 MB</td><td>298M</td></tr><tr><td>UNIFORM</td><td>3/3</td><td></td><td></td><td>69.4</td><td>×</td><td>67.8</td><td>4.56 MB</td><td>357M</td></tr><tr><td>LLOYD</td><td>3/3</td><td>/&lt;√</td><td>/&lt;&gt;</td><td>70.0</td><td>X</td><td>69.0</td><td>4.56 MB</td><td>1.81G</td></tr><tr><td>APoT</td><td>3/3</td><td></td><td>√</td><td>2.0</td><td>×</td><td>68.5</td><td>4.56 MB</td><td>198M</td></tr></table>
+
+The proposed algorithm consists of three techniques, APoT quantization levels to fit the bell-shaped distribution, RCF to learn the clipping threshold and WN to avoid the perturbation of the distribution of weights during training. In this section, we conduct an ablation study for these three techniques. We compare the APoT quantizer, the vanilla PoT quantizer, uniform quantizer and a non-uniform quantizer using Lloyd algorithm (Cai et al., 2017) to quantize the weights. And we either apply RCF to learn the optimal clipping range or do not clip any weights (i.e. $\alpha = \operatorname* { m a x } | \mathcal { W } | )$ . Weight Normalization is also adopted or discarded to justify the effectiveness of these techniques.
+
+Table 3 summarizes the results of ResNet-18 using different techniques. Quantizer using Lloyd achieves the highest accuracy, however, the irregular non-uniform quantized weights cannot utilize the fixed point arithmetic to accelerate the inference time. APoT quantization attends to the distribution of weights, which reaches the same accuracy in 5-bit and only decreases $0 . 2 \%$ accuracy in 3-bit quantization compared with Lloyd, and shares a better tradeoff between task performance and hardware performance. We also observe that the vanilla PoT quantization suffers from the rigid resolution, and has the lowest accuracy in the 5-bit model.
+
+Clipping range also matters in quantization, the comparison in Table 3 shows that a proper clipping range can help improve the robustness of the network. Especially when the network is quantized to 3-bit, the accuracy will drop significantly because of the quantization interval increases. Applying RCF to learn the optimal clipping range could improve at most $1 . 6 \%$ accuracy. As we mentioned before, normalization of weight is important to learn the clipping range, and the network diverges if RCF is applied without WN. We refer to the Appendix B for more details of weight normalization during training.
+
+# 5 CONCLUSION
+
+In this paper, we have introduced the additive powers-of-two (APoT) quantization algorithm for quantizing weights and activations in neural networks, which typically exhibit a bell-shaped and long-tailed distribution. Each quantization level of APoT is the sum of a set of powers-of-two terms, bringing roughly 2x speed-up in multiplication compared with uniform quantization. The distribution of the quantization levels matches that of the weights and activations better than existing quantization schemes. In addition, we propose to reparameterize the clipping function and normalize the weights to get a more stable and better-defined gradient for optimizing the clipping threshold. We reach state-of-the-art accuracy on ImageNet and CIFAR10 dataset compared to uniform or PoT quantization.
+
+# ACKNOWLEDGEMENT
+
+This work is supported by National University of Singapore FY2017 SUG Grant, and Singapore Ministry of Education Academic Research Fund Tier 3 under MOEs official grant number MOE2017-T3-1-007.
+
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+
+# APPENDICES
+
+# A GRADIENT DERIVATION
+
+In this section, we derive the gradient estimation of PACT (Choi et al., 2018b) along with our proposed Reparameterized Clipping Function and show the distinction of these two estimation.
+
+# A.1 PACT
+
+Equation (1) shows the forward of PACT. In backpropagation, PACT applies the Straight-Through Estimator for the projection operation. In particular, the STE assumes that
+
+$$
+{ \frac { \partial \Pi _ { \mathcal { Q } } X } { \partial X } } = 1 ,
+$$
+
+which means the variable before and after projection are treated the same in backpropagation. Therefore, the gradients of $\alpha$ in PACT is computed by:
+
+$$
+\frac { \partial \hat { W } } { \partial \alpha } = \frac { \partial \Pi _ { Q ( \alpha , b ) } \lfloor \mathcal { W } , \alpha \rfloor } { \partial \lfloor \mathcal { W } , \alpha \rfloor } \frac { \partial \lfloor \mathcal { W } , \alpha \rceil } { \partial \alpha } = \left\{ \begin{array} { l l } { \mathrm { s i g n } ( \mathcal { W } ) } & { \mathrm { i f } \left| \mathcal { W } \right| > \alpha } \\ { 0 } & { \mathrm { i f } \left| \mathcal { W } \right| \le \alpha } \end{array} , \right.
+$$
+
+where the first term is computed by STE and the second term is because the clip operation $[ \cdot , \alpha ]$ returns $\mathrm { s i g n } ( \cdot ) \alpha$ when $| \cdot | > \alpha$ . In this gradient estimation, the effect of $\alpha$ in the levels set $\mathcal { Q } ( \alpha , b )$ is ignored by the STE, leading to an inaccurate approximation.
+
+# A.2 RCF
+
+To avoid the elimination of STE, we reparameterize the clipping function so that the output clipping range before projection is settled and the range is re-scaled after the projection. We can define a general formation of RCF by
+
+$$
+\hat { \mathcal { W } } = \frac { \alpha } { c } \Pi _ { \mathcal { Q } ( c , b ) } \lfloor \frac { c } { \alpha } \mathcal { W } , c \rceil ,
+$$
+
+where $c > 0$ is a constant. This function clips the weights to $[ - c , c ]$ before projection and re-scaled to $[ - \alpha , \alpha ]$ after projection. Thus, the backpropagation is given by:
+
+$$
+\begin{array}{c} \begin{array} { l } { \displaystyle \frac { \partial \hat { \mathcal { W } } } { \partial \alpha } = \frac { \partial \alpha } { \partial \alpha } \times \frac 1 c \Pi _ { \mathcal { Q } ( c , b ) } \lfloor \frac c \alpha \mathcal { W } , c \rfloor + \displaystyle \frac { \partial \Pi _ { \mathcal { Q } ( c , b ) } \lfloor \frac c \alpha \mathcal { W } , c \rfloor } { \partial \lfloor \frac c \alpha \mathcal { W } , c \rfloor } \frac { \partial \lfloor \frac c \alpha \mathcal { W } , c \rfloor } { \partial \alpha } \times \frac \alpha c } \\ { = \displaystyle \left\{ \frac 1 c \times \mathrm { s i g n } ( \frac \alpha \mathcal { W } ) \times c + \frac \alpha c \times 0 \quad \right.} & { \mathrm { i f } | \mathcal { W } | > \alpha } \\ { \displaystyle \frac 1 c \Pi _ { \mathcal { Q } ( c , b ) } \frac { c } { \alpha } \mathcal { W } + \frac \alpha c \times ( - \frac { c } { \alpha ^ { 2 } } ) \mathcal { W } \quad \mathrm { i f } | \mathcal { W } | \le \alpha } \\ { = \displaystyle \left\{ \mathrm { s i g n } ( \frac \alpha \mathcal { W } ) \qquad \quad } & { \mathrm { i f } | \mathcal { W } | > \alpha \right. } \\ { \displaystyle \frac 1 c \Pi _ { \mathcal { Q } ( c , b ) } \frac { c } \alpha \mathcal { W } - \frac { 1 } { \alpha } \mathcal { W } \quad \mathrm { i f } | \mathcal { W } | \le \alpha } \end{array}  .  \end{array}
+$$
+
+Since the levels set $\mathcal { Q }$ is not parameterized by $\alpha$ , the gradients will flow to two parts in RCF: the re-scale coefficient and the scale in clipping function. The constant $c$ here do not impact the gradient estimation, therefore we choose 1 for simplicity in the implementation. Note that in uniform quantization scheme, this function is equivalent to the Learned Step Size Quantization (Esser et al., 2020), where the setp size is the same for all levels while RCF provides a more general formation for any levels set $\mathcal { Q }$ .
+
+# B HOW DOES NORMALIZATION HELP QUANTIZATION
+
+In this section, we show some experimental results to illustrate the effect of our weights normalization in quantization neural networks.
+
+![](images/d4f25614e44cf9c0581acb53bb10a413457c9bcdcdb956236f69dfb285cb7194.jpg)  
+Figure 5: When weights are normalized the distribution of weights are more stable. The dashed line shows the mean value of weights.
+
+# B.1 WEIGHTS DISTRIBUTION
+
+We visualize the density distribution of weights before normalization $\mathcal { W }$ and after normalization $\tilde { \mathcal W }$ during training to demonstrate its effectiveness.
+
+Figure 5a demonstrates the density distribution of the fifth layer of the 5-bit quantized ResNet-18, from which we can see that the density of the unnormalized weights could be extensive high $( > 8 )$ in the centered area. Such distribution indicates that even a tiny change of clipping threshold would bring a significant effect on clipping when $\alpha$ is small, as shown in Figure 4a. , which means a small learning rate for $\alpha$ is needed. However, if the learning rate is too small, the change of $\alpha$ cannot follow the change of weights distribution because weights are also updated according to Figure 5a. Thus it is unfavorable to train the clipping threshold for unnormalized weights, while Figure 5b shows that the normalized weights can have a stable distribution. Furthermore, the dashed line in the figure indicates $\mathcal { W }$ usually do not have zero mean, which may not utilize the symmetric design of quantization levels.
+
+# B.2 TRAINING BEHAVIOR
+
+The above experiments use normalization during training to compare the distribution of weights. In this section, we compare the training of quantization neural networks with and without normalization to investigate the real effect of WN. Here, we train a 3-bit quantized (full precision for activations) ResNet-20 from scratch, and compare the results under different learning rate for $\alpha$ . The results are shown in Table 4, from which we can find that if weights are normalized during training, the network can converge to descent performances and is robust to the learning rate of clipping threshold. However, if the weights are not normalized, the network would diverge if the learning rate for $\alpha$ is too high. Even if the learning rate is set to a lower value, the network does not outperform the normalized one. Based on the training behaviors, the learning rate for clipping threshold without WN in QNNs need a careful choice.
+
+Table 4: Accuracy comparison of 3-bit quantized ResNet-20 on CIFAR10.
+
+![](images/d2162a50a836907d469220f34e80810a93f268b39dcf0ad70382113424671738.jpg)  
+Figure 6: A summary of projection error and clipping error in different layers.
+
+# C EXPERIMENTAL DETAILS
+
+# C.1 REVISITING QUANTIZATION ERROR
+
+Typically, quantization error $( \Delta )$ is defined as the mean squared error between weights $\tilde { \mathcal W }$ and $\hat { \mathcal W }$ before and after quantization respectively, defined as $\Delta = \mathbb { E } [ \tilde { \mathcal { W } } - \hat { \mathcal { W } } ] ^ { 2 }$ . This quantization error is composed of two errors, the clipping error $\Delta _ { c l i p }$ produced by $\lfloor \cdot , \alpha \rceil$ and the projection error $\Delta _ { p r o j }$ produced by $\Pi _ { \mathfrak { Q } }$ . I.e.
+
+$$
+\Delta = \Delta _ { c l i p } + \Delta _ { p r o j } = \frac { 1 } { I } \sum _ { | \tilde { \mathcal { W } } _ { i } | > \alpha } \left( | \tilde { \mathcal { W } } _ { i } | - \alpha \right) ^ { 2 } + \frac { 1 } { I } \sum _ { | \tilde { \mathcal { W } } _ { i } | \leq \alpha } ( \tilde { \mathcal { W } } _ { i } - \hat { \mathcal { W } } _ { i } ) ^ { 2 } .
+$$
+
+Previous methods (Zhang et al., 2018; Cai et al., 2017) seek to minimize the quantization error to obtain the optimal clipping threshold (i.e. $\begin{array} { r } { \alpha = \arg \operatorname* { m i n } _ { \alpha } ( \Delta _ { c l i p } + \Delta _ { p r o j } ) ) } \end{array}$ , while RCF is directly optimized by the final training loss to balance projection error and clipping error. We compare the Quantization Error Minimization (QEM) method with our RCF on the quantized ResNet-18 model. Figure 6 gives an overview of the clipping error and projection error using RCF or QEM.
+
+For the 5-bit quantized model, RCF has a much higher quantization error. The projection error obtained by RCF is lower than QEM and QEM significantly reduces the clipping error. Therefore, we can infer that projection error has a higher priority in RCF. When quantizing to 3-bit, the clipping error in RCF still exceeds QEM except for the first quantized layer. This means RCF can identify whether the projection is more important than the clipping over different layers and bit-width. Generally, the insight behind is that simply minimizing the quantization error may not be the best choice and it is more direct to optimize threshold with respect to training loss.
+
+# C.2 IMPLEMENTATIONS DETAILS
+
+The ImageNet dataset consists of 1.2M training and 50K validation images. We use a standard data preprocess in the original paper (He et al., 2016). For training images, they are randomly cropped and resized to $2 2 4 \times 2 2 4$ . Validation images are center-cropped to the same size. We use the Pytorch official code 4 to construct ResNets, and they are initialized from the released pre-trained model. We use stochastic gradient descent (SGD) with the momentum of 0.9 to optimize both weight parameters and the clipping threshold simultaneously. Batch size is set to 1024 and the learning rate starts from 0.1 with a decay factor of 0.1 at epoch 30,60,80,100. The network is trained up to 120 epochs and weight decay is set to $1 0 ^ { - 4 }$ for 3-bit quantized models or higher and $2 \times 1 0 ^ { - 5 }$ for 2-bit model.
+
+The CIFAR10 dataset contains 50K training and 10K test images with $3 2 \times 3 2$ pixels. The ResNet architectures for CIFAR10 (He et al., 2016) contains a convolutional layer followed by 3 residual blocks and a final FC layer. We train full precision ResNet-20 and ResNet-56 firstly and use them as initialization for quantized models. All networks were trained for 200 epochs with a mini-batch size of 128. SGD with momentum of 0.9 was adopted to optimize the parameters. Learning rate started at 0.04 and was scaled by 0.1 at epoch 80,120. Weight decay was set to $1 0 ^ { - 4 }$ .
+
+For clipping threshold $\alpha$ , we set 8.0 for activations and 3.0 for weights as initial value when training a 5-bit quantized model. The learning rate of $\alpha$ is set to 0.01 and 0.03 for weights and activations, respectively. During practice, we found that the learning rate of $\alpha$ merely does not influence network performance. Different from PACT (Choi et al., 2018b), the update of $\alpha$ in our works already consider the projection error, so we do not require a relatively large L2-regularization. In practice, the network works fine when the weight decay for $\alpha$ is set to $\mathrm { \bar { 1 0 } } ^ { - 5 }$ and may increase to $1 0 ^ { - 4 }$ when bit-width is reduced.

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@@ -0,0 +1,1745 @@
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+    {
+        "type": "text",
+        "text": "ADDITIVE POWERS-OF-TWO QUANTIZATION:AN EFFICIENT NON-UNIFORM DISCRETIZATION FORNEURAL NETWORKS",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Yuhang Li †∗, $\\mathbf { X i n ~ D o n g ^ { \\delta * } }$ , Wei Wang †   \n†National University of Singapore, §Harvard University   \nloafyuhang@gmail.com, xindong@g.harvard.edu, wangwei@comp.nus.edu.sg ",
+        "bbox": [
+            183,
+            194,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            275,
+            544,
+            290
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose Additive Powers-of-Two (APoT) quantization, an efficient nonuniform quantization scheme for the bell-shaped and long-tailed distribution of weights and activations in neural networks. By constraining all quantization levels as the sum of Powers-of-Two terms, APoT quantization enjoys high computational efficiency and a good match with the distribution of weights. A simple reparameterization of the clipping function is applied to generate a better-defined gradient for learning the clipping threshold. Moreover, weight normalization is presented to refine the distribution of weights to make the training more stable and consistent. Experimental results show that our proposed method outperforms state-of-the-art methods, and is even competitive with the full-precision models, demonstrating the effectiveness of our proposed APoT quantization. For example, our 4-bit quantized ResNet-50 on ImageNet achieves $7 6 . 6 \\%$ top-1 accuracy without bells and whistles; meanwhile, our model is capable to decrease $22 \\%$ computational cost compared with the uniformly quantized counterpart. 1 ",
+        "bbox": [
+            233,
+            311,
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+            506
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            545,
+            334,
+            560
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep Neural Networks (DNNs) have made a significant improvement for various real-world applications. However, the huge memory and computational cost impede the mass deployment of DNNs, e.g., on resource-constrained devices. To reduce memory footprint and computational burden, several model compression methods such as quantization (Zhou et al., 2016), pruning (Han et al., 2015) and low-rank decomposition (Denil et al., 2013) have been widely explored. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we focus on the neural network quantization for efficient inference. Two operations are involved in the quantization process, namely clipping and projection. The clipping operation sets a full precision number to the range boundary if it is outside of the range; the projection operation maps each number (after clipping) into a predefined quantization level (a fixed number). We can see that both operations incur information loss. A good quantization method should resolve the two following questions/challenges, which correspond to two contradictions respectively. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "How to determine the optimal clipping threshold to balance clipping range and projection resolution? The resolution indicates the interval between two quantization levels; the smaller the interval, the higher the resolution. The first contradiction is that given a fixed number of bits to represent weights, the range and resolution are inversely proportional. For example, a larger range can clip fewer weights; however, the resolution becomes lower and thus damage the projection. Note that slipshod clipping of outliers can jeopardize the network a lot (Zhao et al., 2019) although they may only take $1 \\%$ of all weights in one layer. Previous works have tried either pre-defined (Cai et al., 2017; Zhou et al., 2016) or trainable (Choi et al., 2018b) clipping thresholds, but how to find the optimal threshold during training automatically is still not resolved. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "How to design quantization levels with consideration for both the computational efficiency and the distribution of weights? Most of the existing quantization approaches (Cai et al., 2017; Gong et al., 2019) use uniform quantization although non-uniform quantization can usually achieve better accuracy (Zhu et al., 2016). The reason is that projection against uniform quantization levels are much more hardware-friendly (Zhou et al., 2016). However, empirical study (Han et al., 2015) has shown that weights in a layer of DNN follow a bell-shaped and long-tailed distribution instead of a uniform distribution (as shown in the right figure). In other words, a fair percentage of weights concentrate around the mean (peak area); and a few weights are of relatively high magnitude and out of the quantization range (called outliers). Such distribution also exists in activations (Miyashita et al., 2016). The second contradiction is: considering the bell-shaped distribution of weight, it is well-motivated to assign higher resolution (i.e. smaller quantization interval) around the mean; however, such non-uniform quantization levels will introduce high computational overhead. Powers-of-Two quantization levels (Miyashita et al., 2016; Zhou et al., 2017) are then proposed because of its cheap multiplication implemented by shift operations on hardware, and super high resolution around the mean. However, the vanilla powers-of-two quantization method only increases the resolution near the mean and ignores other regions at all when the bit-width is increased. Consequently, it assigns inordinate quantization levels for a tiny range around the mean. To this end, we propose additive Powers-of-Two (APoT) quantization to resolve these two contradictions, our contribution can be listed as follows: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "image",
+        "img_path": "images/ac3ca25c85bb9d61445f20df0c9161b4ffa7e70781fbe8ea4ef4c47a284f6eb9.jpg",
+        "image_caption": [
+            "Figure 1: Density of weights in ResNet-18 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            570,
+            165,
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+            310
+        ],
+        "page_idx": 1
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+    {
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+    {
+        "type": "text",
+        "text": "1. We introduce the APoT quantization scheme for the weights and activations of DNNs. APoT is a non-uniform quantization scheme, in which the quantization levels is a sum of several PoT terms and can adapt well to the bell-shaped distribution of weights. APoT quantization enjoys an approximate $2 \\times$ multiplication speed-up compared with uniform quantization on both generic and specific hardware. 2. We propose a Reparameterized Clipping Function (RCF) that can compute a more accurate gradient for the clipping threshold and thus facilitate the optimization of the clipping threshold. We also introduce weight normalization for neural network quantization. Normalized weights in the forward pass are more stable and consistent for clipping and projection. 3. Experimental results show that our proposed method outperforms state-of-the-art methods, and is even competitive with the full-precision implementation with higher computational efficiency. Specifically, our 4-bit quantized ResNet-50 on ImageNet achieve $7 6 . 6 \\%$ Top-1 and $9 3 . 1 \\%$ Top-5 accuracy. Compared with uniform quantization, our method can decrease $22 \\%$ computational cost, demonstrating the proposed algorithm is hardware-friendly. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 PRELIMINARIES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Suppose kernels in a convolutional layer are represented by a 4D tensor $\\mathcal { W } \\in \\mathbb { R } ^ { C _ { o u t } \\times C _ { i n } \\times K \\times K }$ , where $C _ { o u t }$ and $C _ { i n }$ are the number of output and input channels respectively, and $K$ is the kernel size. We denote the quantization of the weights as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b2652eafe223034310630dbd31c9a0b0b2cfb7e18f6a4a5a942bc12246010c96.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\mathcal { W } } = \\Pi _ { \\mathcal { Q } ( \\alpha , b ) } \\lfloor \\mathcal { W } , \\alpha \\rceil , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\alpha$ is the clipping threshold and the clipping function $[ \\cdot , \\alpha ]$ clips weights into $[ - \\alpha , \\alpha ]$ . After clipping, each element of $\\mathcal { W }$ is projected by $\\Pi ( \\cdot )$ onto the quantization levels. We denote $\\mathcal { Q } ( \\alpha , b )$ for a set of quantization levels, where $b$ is the bit-width. For uniform quantization, the quantization levels are defined as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/6d05b39e2cd93db294ae43e4f95b2739075a78802e27d65bca2534192bab5a99.jpg",
+        "text": "$$\n\\mathcal { Q } ^ { u } ( \\alpha , b ) = \\alpha \\times \\{ 0 , \\frac { \\pm 1 } { 2 ^ { b - 1 } - 1 } , \\frac { \\pm 2 } { 2 ^ { b - 1 } - 1 } , \\frac { \\pm 3 } { 2 ^ { b - 1 } - 1 } , \\ldots , \\pm 1 \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For every floating-point number, uniform quantization maps it to a $b$ -bit fixed-point representation (quantization levels) in $\\mathcal { Q } ^ { u } ( \\alpha , b )$ . Note that $\\alpha$ is stored separately as a full-precision floating-point number for each whole $\\mathcal { W }$ . Convolution is done against the quantization levels first and the results are then multiplied by $\\alpha$ . Arithmetical computation, e.g., convolution, can be implemented using low-precision fixed point operations on hardware, which are substantially cheaper than their floating-point contradictory (Goldberg, 1991). Nevertheless, uniform quantization does not match the distribution of weights (and activations), which is typically bell-shaped (Han et al., 2015). A straightforward solution is to assign more quantization levels (higher resolution) for the peak of the distribution and fewer levels (lower resolution) for the tails. However, it is difficult to implement the arithmetical operations for the non-uniform quantization levels efficiently. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/c73baa7a1cd6bb5846b071beac43506a8cd785aea78c61ec29270271ff274fee.jpg",
+        "image_caption": [
+            "Figure 2: Quantization of unsigned data to 3-bit or 4-bit $( \\alpha = 1 . 0 $ ) using three different quantization levels. APoT quantization has a more reasonable resolution assignment and it does not suffer from the rigid resolution. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.2 ADDITIVE POWERS-OF-TWO QUANTIZATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To solve the contradiction between non-uniform resolution and hardware efficiency, Powers-ofTwo (PoT) quantization (Miyashita et al., 2016; Zhou et al., 2017) is proposed by constraining quantization levels to be powers-of-two values or zero, i.e., ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/252fc28b1b8d0fea344f7e9480368f8f98f7b52dd5092ceb2ed3d65568c3c8e7.jpg",
+        "text": "$$\n\\begin{array} { r } { \\underline { { Q } } ^ { p } ( \\alpha , b ) = \\alpha \\times \\{ 0 , \\pm 2 ^ { - 2 ^ { b - 1 } + 1 } , \\pm 2 ^ { - 2 ^ { b - 1 } + 2 } , . . . , \\pm 2 ^ { - 1 } , \\pm 1 \\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Apparently, as a non-uniform quantizer, PoT has a higher resolution for the value range with denser weights because of its exponential property. Furthermore, multiplication between a Powers-of-two number $2 ^ { x }$ and the other operand $r$ can be implemented by bit-wise shift instead of bulky digital multipliers, i.e., ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/14d53c737b05cbe89ccdbec0f2a94d3a42c1a958a82a5393c9d33089f5d4736a.jpg",
+        "text": "$$\n2 ^ { x } r = { \\left\\{ \\begin{array} { l l } { r } & { { \\mathrm { ~ i f ~ } } x = 0 } \\\\ { r < < x } & { { \\mathrm { ~ i f ~ } } x > 0 , } \\\\ { r > > x } & { { \\mathrm { ~ i f ~ } } x < 0 } \\end{array} \\right. }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $> >$ denotes the right shift operation and is computationally cheap, which only takes 1 clock cycle in modern CPU architectures. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "However, we find that PoT quantization does not benefit from more bits. Assume $\\alpha$ is 1, as shown in Equation (3), when we increase the bit-width from b to b + 1, the interval [−2−2b−1+1, 2−2b−1+1] will be split into $2 ^ { b - 1 } - 1$ sub-intervals, whereas all other intervals remain unchanged. In other words, by increasing the bit-width, the resolution will increase only for $[ - 2 ^ { - 2 ^ { b - 1 } + 1 } , 2 ^ { - 2 ^ { b - 1 } + 1 } ]$ . We refer this phenomenon as the rigid resolution of PoT quantization. Take $\\mathcal { Q } ^ { p } ( 1 , 5 )$ as an example, the two smallest positive levels are $2 ^ { - 1 5 }$ and $2 ^ { - 1 4 }$ , which is excessively fine-grained. In contrast, the two largest levels are $2 ^ { - 1 }$ and $2 ^ { 0 }$ , whose interval is large enough to incur high projection error for weights between $[ 2 ^ { - 1 } , 2 ^ { 0 } ]$ , e.g., 0.75. The rigid resolution is demonstrated in Figure 2(b). When we change from from 3-bit to 4-bit, all new quantization levels concentrate around 0 and thus cannot increase the model’s expressiveness effectively. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/b463f44038b337fd8ebc5c453fd6d8c77e25551f39f417fe6b886196dc7ef5d3.jpg",
+        "image_caption": [
+            "Figure 3: Hardware accelerator with different quantization schemes. When $k$ increase, weights usually has less PoT terms, thus accelerates the computation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To tackle the rigid resolution problem, we propose Additive Powers-of-Two (APoT) quantization. Without loss of generality, in this section, we only consider unsigned numbers for simplicity2. In APoT quantization, each level is the sum of n PoT terms as shown below, ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/0fa4d056203e5ca001db674f57f7d88005f14db004aba127cf94cd179f2bd9fb.jpg",
+        "text": "$$\n\\mathcal { Q } ^ { a } ( \\alpha , k n ) = \\gamma \\times \\{ \\sum _ { i = 0 } ^ { n - 1 } p _ { i } \\} \\mathrm { w h e r e } p _ { i } \\in \\{ 0 , \\frac { 1 } { 2 ^ { i } } , \\frac { 1 } { 2 ^ { i + n } } , . . . , \\frac { 1 } { 2 ^ { i + ( 2 ^ { k } - 2 ) n } } \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\gamma$ is a scaling coefficient to make sure the maximum level in ${ \\mathcal { Q } } ^ { a }$ is $\\alpha , \\ k$ is called the base bit-width, which is the bit-width for each additive term, and $n$ is the number of additive terms. When the bit-width $b$ and the base bit-width $k$ is set, $n$ can be calculated by $\\begin{array} { r } { n = { \\frac { b } { k } } } \\end{array}$ . There are $2 ^ { k n } = 2 ^ { b }$ $b$ which provides a higher resolution for the non-uniform levels. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use $b \\ = \\ 4$ and $k \\ = \\ 2$ as an example to illustrate how APoT resolves the rigid resolution problem. For this example, we have $p _ { 0 } \\in \\{ 0 , 2 ^ { 0 } , 2 ^ { - 2 } , 2 ^ { - 4 } \\}$ , $p _ { 1 } \\in \\{ 0 , 2 ^ { - 1 } , 2 ^ { - 3 } , 2 ^ { - 5 } \\} ,$ $\\gamma = 2 \\alpha / 3$ , and ${ \\mathcal { Q } } ^ { a } ( \\alpha , k n ) = \\{ \\gamma \\times ( p _ { 0 } + p _ { 1 } ) \\}$ for all $( 2 ^ { b } = 1 6 )$ combinations of $p _ { 0 }$ and $p _ { 1 }$ . First, we can see the smallest positive quantization level in $\\mathcal { Q } ^ { a } ( 1 , 4 )$ is $2 ^ { - 4 } / 3$ . Compared with the original PoT levels, APoT allocates quantization levels prudently for the central area. Second, APoT generates 3 new quantization levels between $2 ^ { 0 }$ and $\\bar { 2 } ^ { - 1 }$ , to properly increase the resolution. In Figure 2, the second row compares the 3 quantization methods using 4 bits for range [0, 1]. APoT quantization has a reasonable distribution of quantization levels, with more levels in the peak area (near 0) and relatively higher resolution than the vanilla PoT quantization at the tail (near 1). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Relation to other quantization schemes. On the one hand, the fixed-point number representations used in the uniform quantization is a special case of APoT. When $k ~ = ~ 1$ in Equation (5), the quantization levels is a sum of $b$ PoT terms or 0. In the fixed-point representations, each bit indicates one specific choice of the additive terms. On the other hand, when $k = b$ , there is only one PoT term and $\\bar { \\mathcal { Q } } ^ { a } ( \\alpha , b )$ becomes $\\mathcal { Q } ^ { p } ( \\alpha , b )$ , i.e., PoT quantization. We can conclude that when $k$ decreases, APoT levels are decomposed into more PoT terms, and the distribution of levels becomes more uniform. Our experiments use $k = 2$ , which is an intermediate choice between the uniform case $k = 1 ,$ ) and the vanilla PoT case $k = b$ ). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Computation. Multiplication for fixed-point numbers can be implemented by shifting the multiplicand (i.e., the activations) and adding the partial product. The $n$ in Equation (5) denotes the Since number of additive PoT terms in the multiplier (weights), and control the speed of computation. $\\begin{array} { r } { n = \\frac { b } { k } } \\end{array}$ , either decreasing $b$ or increasing $k$ can accelerate the multiplication. Compared with uniform quantization $k = 1 ,$ ), our method $k = 2$ ) is approximately $2 \\times$ faster in multiplication. As for the full precision $\\alpha$ , it is a coefficient for all weights in a layer and can be multiplied only once after the multiply-accumulate operation is finished. Figure 3 shows the hardware accelerator, the weights buffer takes $k$ -bit as a PoT term and shift-adds the activations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Generalizing to $2 n + 1$ bits. When $k = 2$ , APoT quantization can only leverages $2 n$ -bit width for quantization. To deal with $2 n + 1$ -bit quantization, we choose to add $n + 1$ PoT terms, one of which only contains 2 levels. The formulation is given by ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1c4c5ff7d6f31c61b8d11b92d483882726f590cd6a112a00138fe5f31cff027.jpg",
+        "text": "$$\n\\mathcal { Q } ^ { a } ( \\alpha , 2 n + 1 ) = \\gamma \\times \\ \\{ \\sum _ { i = 0 } ^ { n - 1 } p _ { i } + \\tilde { p } \\} \\ \\mathrm { w h e r e } \\ p _ { i } \\in \\{ 0 , \\frac { 1 } { 2 ^ { i } } , \\frac { 1 } { 2 ^ { i + n } } , \\frac { 1 } { 2 ^ { i + 2 n + 1 } } \\} , \\ \\tilde { p } \\in \\{ 0 , \\frac { 1 } { 2 ^ { i + 2 n } } \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Take 3-bit APoT quantization as an example, every level is a sum of one $p _ { 0 }$ and one $\\tilde { p }$ , where $p _ { 0 } \\in \\{ 0 , 2 ^ { - 1 } , 2 ^ { - 2 } , \\bar { 2 } ^ { - 4 } \\}$ and $\\tilde { p } \\in \\{ 0 , 2 ^ { - 3 } \\}$ . The forward function is plotted in Figure 2(c). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.3 REPARAMETERIZED CLIPPING FUNCTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Besides the projection operation, the clipping operation $[ \\mathcal { W } , \\alpha ]$ is also important for quantization. $\\alpha$ is a threshold that determines the value range of weights in a quantized layer. Tuning the clipping threshold $\\alpha$ is a key challenge because of the long-tail distribution of the weights. Particularly, if $\\alpha$ is too large (e.g., the maximum absolute value of $\\mathcal { W }$ ), $\\mathcal { Q } ( \\alpha , b )$ would have a wide range and then the projection will lead to large error as a result of insufficient resolution for the weights in the central area; if $\\alpha$ is too small, more outliers will be clipped slipshodly. Considering the distribution of weights can be complex and differs across layers and training steps, a static clipping threshold $\\alpha$ for all layers is not optimal. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To jointly optimize the clipping threshold $\\alpha$ and weights via SGD during training, Choi et al. (2018b) apply the Straight-Through Estimator (STE) (Bengio et al., 2013) to do the backward propagation for the projection operation. According to STE, the gradient to $\\alpha$ is computed by $\\begin{array} { r } { \\frac { \\partial \\hat { \\mathcal { W } } } { \\partial \\alpha } \\approx \\frac { \\partial \\lfloor \\mathcal { W } , \\alpha \\rceil } { \\partial \\alpha } = } \\end{array}$ $\\mathrm { s i g n } ( { \\mathscr W } )$ when $| \\mathcal { W } | > \\alpha$ otherwise 0, where the weights outside of the range cannot contribute to the gradients, which results in inaccurate gradient approximation. To provide a refined gradient for the clipping threshold, we design a Reparameterized Clipping Function (RCF) as ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/8302bf6c0ffbebd49b59d3fbc1648a4f8097a35d00802f942b34b2c9a73f57e1.jpg",
+        "text": "$$\n\\hat { \\mathcal { W } } = \\alpha \\Pi _ { \\mathcal { Q } ( 1 , b ) } \\big [ \\frac { \\mathcal { W } } { \\alpha } , 1 \\big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Instead of directly clipping them to $[ - \\alpha , \\alpha ]$ , RCF outputs a constant clipping range and re-scales weights back after the projection, which is mathematically equivalent to Equation (1) during forward. In backpropagation, STE is adopted for the projection operation and the gradients of $\\alpha$ are calculated by ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/bb377852c69494e71e00922edffb3dc4ae9a39383406a81e12ddfb0557bbcd7e.jpg",
+        "text": "$$\n\\frac { \\partial \\hat { \\mathcal { W } } } { \\partial \\alpha } = \\left\\{ \\begin{array} { l l } { \\mathrm { s i g n } ( \\mathcal { W } ) } & { \\mathrm { i f ~ } | \\mathcal { W } | > \\alpha } \\\\ { \\displaystyle \\Pi _ { \\mathcal { Q } ( 1 , b ) } \\frac { \\mathcal { W } } { \\alpha } - \\frac { \\mathcal { W } } { \\alpha } } & { \\mathrm { i f ~ } | \\mathcal { W } | \\leq \\alpha } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The detail derivation of the gradients is shown in Appendix A. Compared with the normal clipping function, RCF provides more accurate gradient signals for the optimization because both weights inside $( | \\mathcal { W } | \\leq \\alpha )$ and out of $( | \\mathcal { W } | > \\alpha )$ the range can contribute to the gradient for the clipping threshold. Particularly, the outliers are responsible for the clipping, and the weights in $[ - \\alpha , \\alpha ]$ are for projection. Therefore, the update of $\\alpha$ considers both clipping and projection, and tries to find a balance between them. In experiments, we observe that the clipping threshold will become universally smaller when the bit-width is reduced to guarantee sufficient resolution, which further validates the efficaciousness of the gradient in RCF. ",
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+    {
+        "type": "text",
+        "text": "2.4 WEIGHT NORMALIZATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In practice, we find that learning $\\alpha$ for weights is quite arduous because the distribution of weights is pretty steep and changes frequently during training. As a result, jointly training the clipping threshold and weights parameters is hard to converge. Inspired by the crucial role of batch normalization (BN) (Ioffe $\\&$ Szegedy, 2015) in activation quantization (Cai et al., 2017), we propose weight normalization (WN) to refine the distribution of weights with zero mean and unit variance, ",
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+    {
+        "type": "equation",
+        "img_path": "images/58256393a051dd01e6f8db0f20d5dec97246338c89ba5bf0aced9c995621a500.jpg",
+        "text": "$$\n\\tilde { \\mathcal { W } } = \\frac { \\mathcal { W } - \\mu } { \\sigma + \\epsilon } , \\mathrm { w h e r e } \\mu = \\frac { 1 } { I } \\sum _ { i = 1 } ^ { I } \\mathcal { W } _ { i } , \\sigma = \\sqrt { \\frac { 1 } { I } \\sum _ { i = 1 } ^ { I } ( \\mathcal { W } _ { i } - \\mu ) ^ { 2 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon$ is a small number (typically $1 0 ^ { - 5 }$ ) for numerical stability, and $I$ denotes the number of weights in one layer. Note that quantization of weights is applied right after this normalization. ",
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+    {
+        "type": "image",
+        "img_path": "images/296afd41c25721ce18c2a3fba33b7a7c5e0585187412215ff4824d4f0da40e1f.jpg",
+        "image_caption": [
+            "Figure 4: The evolution of clipping ratio of the first three layers in ResNet-20. (a) demonstrates clipping ratio is too sensitive to threshold to hurt its optimization without weights normalization. (b) shows that weights distribution after normalization is relatively more stable during training. "
+        ],
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Forward and backward procedure for an APoT quantized convolutional layer ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Input: input activations $\\mathcal { X } _ { i n }$ , the full precision weight tensor $\\mathcal { W }$ , the clipping threshold for weights and activations $\\alpha _ { \\mathcal { W } }$ , $\\alpha _ { \\mathcal { X } }$ , the bit-width $b$ of quantized tensor. ",
+        "bbox": [
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+            328,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Output: the output activations $\\mathcal { X } _ { o u t }$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: Normalize weights $\\mathcal { W }$ to $\\tilde { \\mathcal W }$   \n2: Apply RCF and APoT quantization to the normalized weights $\\begin{array} { r } { \\hat { \\mathcal { W } } = \\alpha \\nu \\Pi _ { \\mathcal { Q } ^ { a } ( 1 , b ) } \\big \\lfloor \\frac { \\tilde { \\mathcal { W } } } { \\alpha _ { \\mathcal { W } } } , 1 \\big \\rceil } \\end{array}$   \n3: Apply RCF and APoT quantization to the activations $\\begin{array} { r } { \\hat { \\mathcal { X } _ { i n } } = \\alpha _ { \\mathcal { X } } \\Pi _ { \\mathcal { Q } ^ { a } ( 1 , b ) } \\lfloor \\frac { \\mathcal { X } _ { i n } } { \\alpha _ { \\mathcal { X } } } , 1 \\rceil } \\end{array}$   \n4: Compute the output activations $\\mathcal { X } _ { o u t } = C o n \\nu ( \\hat { \\mathcal { W } } , \\hat { \\mathcal { X } _ { i n } } )$   \n5: Compute the loss $\\mathcal { L }$ and the gradients $\\frac { \\partial \\mathcal { L } } { \\partial \\mathcal { X } _ { o u t } }$ ,   \n6: Compute the gradients of convolution $\\frac { \\partial \\mathcal { L } } { \\partial \\hat { \\mathcal { X } } _ { i n } }$ $\\textstyle \\frac { \\partial { \\mathcal { L } } } { \\partial { \\hat { \\mathcal { W } } } }$   \n7: Compute the gradients for clipping threshold $\\frac { \\partial \\mathcal { L } } { \\partial \\alpha \\ w }$ , $\\frac { \\partial \\mathcal { L } } { \\partial \\alpha \\boldsymbol { x } }$ based on Equation (8)   \n8: Compute the gradients to the full precision weights $\\begin{array} { r } { \\frac { \\partial \\mathcal { L } } { \\partial \\mathcal { W } } = \\frac { \\partial \\mathcal { L } } { \\partial \\hat { \\mathcal { W } } } \\frac { \\partial \\hat { \\mathcal { W } } } { \\partial \\tilde { \\mathcal { W } } } \\frac { \\partial \\tilde { \\mathcal { W } } } { \\partial \\mathcal { W } } } \\end{array}$   \n9: Update $\\mathcal { W }$ , $\\alpha _ { \\mathcal { W } }$ , $\\alpha _ { \\mathcal { X } }$ with learning rate $\\eta _ { \\mathcal { W } } , \\eta _ { \\alpha _ { \\mathcal { W } } } , \\eta _ { \\alpha _ { \\mathcal { X } } }$ ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Normalization is important to provide a relatively consistent and stable input distribution to both clipping and projection functions for smoother optimization of $\\alpha$ over different layers and iterations during training. Besides, making the mean of weights to be zero can reap the benefits of the symmetric design of the quantization levels. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, we conduct a case study of ResNet-20 on CIFAR10 to illustrate how normalization for weights can help quantization. For a certain layer (at a certain training step) in ResNet-18, We firstly fix the weights, and let $\\alpha$ go from 0 to max $| \\mathcal { W } |$ to plot the curve of the clipping ratio (i.e. the proportion of clipped weights). As shown in Figure 4a, the change of clipping ratio is much smoother after quantization. As a result, the optimization of $\\alpha$ will be significantly smoother. In addition, normalization also makes the distribution of weights quite more consistent over training iterations. We fix the value of $\\alpha$ , and visualize clipping ratio over training iterations in Figure 4b. After normalization, the same $\\alpha$ will result in almost the same clipping ratio, which improves the consistency of optimization goal for $\\alpha$ . More experimental analysis demonstrating the effectiveness of the normalization on weights can be found in Appendix B. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "2.5 TRAINING AND DEPLOYING ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We adopt APoT quantization for both weights and activations. Notwithstanding the effect in activations is not conspicuous, we adopt APoT quantization for consistency. During backpropagation, we use STE when computing the gradients of weights, i.e. $\\begin{array} { r } { \\frac { \\partial \\hat { \\mathcal { W } } } { \\partial \\tilde { \\mathcal { W } } } = 1 } \\end{array}$ . The detailed training procedure is shown in Algorithm 1. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To save memory cost during inference, we discard the full precision weights $\\mathcal { W }$ and only store the quantized weights $\\hat { \\mathcal W }$ . Compared with other uniform quantization methods, APoT quantization is more efficient and effective during inference. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3 RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Non-Uniform Quantization. Several methods are proposed for the non-uniform distribution of weights. LQ-Nets(Zhang et al., 2018) learns quantization levels based on the quantization error minimization (QEM) algorithm. Distillation (Polino et al., 2018) optimizes the quantization levels directly to minimize the task loss which reflects the behavior of their teacher network. These methods use finite floating-point numbers to quantize weights (and activations), bringing extra computation overhead compared with linear quantization. Logarithmic quantizers (Zhou et al., 2017; Miyashita et al., 2016) leverage powers-of-2 values to accelerate the computation by shift operations; however, they suffer from the rigid resolution problem. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Jointly Training. Many works have explored to optimize the quantization parameters (e.g., $\\alpha$ ) and the weights parameters simultaneously. Zhu et al. (2016) learns positive and negative scaling coefficients respectively. LQ-Nets jointly train these parameters to minimize the quantization error. QIL (Jung et al., 2019) introduces a learnable transformer to change the quantization intervals and optimize them based on the task loss. PACT (Choi et al., 2018b) parameterizes the clipping threshold in activations and optimize it through gradient descent. However, in PACT, the gradient of $\\alpha$ is not accurate, which only includes the contribution from outliers and ignores the contribution from other weights. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Weight Normalization. Previous works on weight normalization mainly focus on addressing the limitations of BatchNorm (Ioffe & Szegedy, 2015). Salimans & Kingma (2016); Hoffer et al. (2018) decouple direction from magnitude to accelerate the training procedure. Weight Standardization (Qiao et al., 2019) normalizes weights to zero mean and unit variance during the forward pass. However, there is limited literature that studies the normalization of weights for neural network quantization. (Zhu et al., 2016) uses feature-scaling to normalize weights by dividing the maximum absolute value. Weight Normalization based Quantization (Cai & Li, 2019) also uses this feature scaling and derive the gradient to eliminate the outliers in the weights tensor. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we validate our proposed method on ImageNet-ILSVRC2012 (Russakovsky et al., 2015) and CIFAR10 (Krizhevsky et al., 2009). We also conduct ablation study for each component of our algorithm. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 EVALUATION ON IMAGENET ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We compare our methods with several strong baselines on ResNet architectures (He et al., 2016), including ABC-Net (Lin et al., 2017), DoReFa-Net (Zhou et al., 2016), PACT (Choi et al., 2018b), LQ-Net (Zhang et al., 2018), DSQ (Gong et al., 2019), QIL (Jung et al., 2019). Both weights and activations of the networks are quantized for comparisons. All the state-of-the-art methods use full precision (32 bits) for the first and the last layer, which incur more memory cost. In our implementation, we employ 8-bit quantization for them to balance the accuracy drop and the hardware overhead. ",
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+    {
+        "type": "text",
+        "text": "For our proposed APoT quantization algorithm, four configurations of the bit-width, i.e., 2,3,4, and 5 ( $k = 2$ and $n = 1$ or 2 in Equation (5) and (6)) are tested, where one bit is used for the sign of the weights but not for activations. Note that for the 2-bit symmetric weight quantization method, $\\mathcal { Q } ( \\alpha , 2 )$ can only be $\\{ \\pm \\alpha , 0 \\}$ , therefore only RCF and WN are used in this setting. To obtain a reasonable initialization, we follow Lin et al. (2017); Jung et al. (2019) to initialize our model. Specifically, the 5-bit quantized model is initialized from the pre-trained full precision one3, while the 4-bit network is initialized from the trained 5-bit model. We compare the accuracy, memory cost, and the fixed point operations under different bit-width. To compare the operations with different bit-width, we use the bit-op computation scheme introduced in Zhou et al. (2016) where the multiplication between a $m$ -bit and a $l$ -bit uniform quantized number costs $m l$ binary operation. We define one FixOP as one operation between an 8-bit weight and an 8-bit activation which takes 64 binary operations if uniform quantization scheme is applied. In APoT scheme, the multiplication between a $m$ -bit activation and a $l = k n$ -bit weight only needs mn shift-adds operations, i.e., $\\frac { n \\times m } { 6 4 }$ FixOPs. More details of the implementation are in the Appendix C.2. ",
+        "bbox": [
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+            722,
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/2d216214d930e58604a4dd12a6a819a0de3c528086e0d88d81901c87b07cde8c.jpg",
+        "table_caption": [
+            "Table 1: Comparison of accuracy performance as well as hardware performance of ResNets (He et al., 2016) on ImageNet with existing methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">METHOD</td><td rowspan=\"2\">PRECISION (W/A)</td><td colspan=\"2\">ACCURACY(%)</td><td rowspan=\"2\">MODEL SIZE</td><td rowspan=\"3\">FIXOPS</td><td rowspan=\"2\">PRECISION</td><td colspan=\"2\">ACCURACY(%)</td><td rowspan=\"3\">MODEL SIZE</td><td rowspan=\"3\">FIXOPS</td></tr><tr><td>ToP-1</td><td>TOP-5</td><td>(W/A)</td><td>TOP-1 TOP-5</td></tr><tr><td>FP.(RES18)</td><td>32/32</td><td>70.2</td><td>89.4</td><td>46.8MB</td><td>1.82G</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ABC-NETS</td><td>5/5</td><td>65.0</td><td>85.9</td><td>8.72 MB</td><td>781M</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DoREFA-NET</td><td>5/5</td><td>68.4</td><td>88.3</td><td>8.72MB</td><td>781M</td><td>4/4</td><td>68.1</td><td>88.1</td><td>7.39MB</td><td>542M</td></tr><tr><td>PACT</td><td>5/5</td><td>69.8</td><td>89.3</td><td>8.72MB</td><td>781M</td><td>4/4</td><td>69.2</td><td>89.0</td><td>7.39 MB</td><td>542M</td></tr><tr><td>LQ-NET</td><td></td><td></td><td></td><td></td><td></td><td>4/4</td><td>69.3</td><td>88.8</td><td>7.39MB</td><td>542M</td></tr><tr><td>DSQ</td><td></td><td></td><td></td><td></td><td></td><td>4/4</td><td>69.6</td><td></td><td>7.39 MB</td><td>542M</td></tr><tr><td>QIL</td><td>5/5</td><td>70.4</td><td>=</td><td>8.72MB</td><td>781M</td><td>4/4</td><td>70.1</td><td>=</td><td>7.39MB</td><td>542M</td></tr><tr><td>APoT (OURS)</td><td>5/5</td><td>70.9</td><td>89.7</td><td>7.22 MB</td><td>616M</td><td>4/4</td><td>70.7</td><td>89.6</td><td>5.89 MB</td><td>437M</td></tr><tr><td>ABC-NETS</td><td>3/3</td><td>61.0</td><td>83.2</td><td>6.06MB</td><td>357M</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DoREFA-NET</td><td>3/3</td><td>67.5</td><td>87.6</td><td>6.06MB</td><td>357M</td><td>2/2</td><td>62.6</td><td>84.6</td><td>4.73MB</td><td>225M</td></tr><tr><td>PACT</td><td>3/3</td><td>68.1</td><td>88.2</td><td>6.06MB</td><td>357M</td><td>2/2</td><td>64.4</td><td>85.6</td><td>4.73 MB</td><td>225M</td></tr><tr><td>LQ-NET</td><td>3/3</td><td>68.2</td><td>87.9</td><td>6.06MB</td><td>357M</td><td>2/2</td><td>64.9</td><td>85.9</td><td>4.73MB</td><td>225M</td></tr><tr><td>DSQ</td><td>3/3</td><td>68.7</td><td>-</td><td>6.06MB</td><td>357M</td><td>2/2</td><td>65.2</td><td>-</td><td>4.73MB</td><td>225M</td></tr><tr><td>QIL</td><td>3/3</td><td>69.2</td><td>-</td><td>6.06MB</td><td>357M</td><td>2/2</td><td>65.7</td><td>-</td><td>4.73MB</td><td>225M</td></tr><tr><td>PACT+SAWB</td><td></td><td></td><td></td><td></td><td></td><td>2/2</td><td>67.0</td><td></td><td>5.36MB</td><td>243M</td></tr><tr><td>APOT (OURS)</td><td>3/3</td><td>69.9</td><td>89.2</td><td>4.56MB</td><td>298M</td><td>2/2</td><td>67.3</td><td>87.5</td><td>3.23MB</td><td>198M</td></tr><tr><td>FP.(RES34)</td><td>32/32</td><td>73.7</td><td>91.3</td><td>83.2MB</td><td>3.68G</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ABC-NETS</td><td>5/5</td><td>68.4</td><td>88.2</td><td>14.8MB</td><td>1.50G</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DSQ</td><td></td><td></td><td></td><td></td><td></td><td>4/4</td><td>72.8</td><td></td><td>12.3MB</td><td>1.00G</td></tr><tr><td>QIL</td><td>5/5</td><td>73.7</td><td>-</td><td>14.8MB</td><td>1.50G</td><td>4/4</td><td>73.7</td><td>=</td><td>12.3MB</td><td>1.00G</td></tr><tr><td>APOT(OURS)</td><td>5/5</td><td>73.9</td><td>91.6</td><td>13.3 MB</td><td>1.15G</td><td>4/4</td><td>73.8</td><td>91.6</td><td>10.8 MB</td><td>784M</td></tr><tr><td>ABC-NETS</td><td>3/3</td><td>66.4</td><td>87.4</td><td>9.73MB</td><td>618M</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>LQ-NET</td><td>3/3</td><td>71.9</td><td>90.2</td><td>9.73MB</td><td>618M</td><td>2/2</td><td>69.8</td><td>89.1</td><td>7.20 MB</td><td>340M</td></tr><tr><td>DSQ</td><td>3/3</td><td>72.5</td><td>-</td><td>9.73MB</td><td>618M</td><td>2/2</td><td>70.0</td><td>1</td><td>7.20MB</td><td>340M</td></tr><tr><td>QIL</td><td>3/3</td><td>73.1</td><td>-</td><td>9.73MB</td><td>618M</td><td>2/2</td><td>70.6</td><td>-</td><td>7.20MB</td><td>340M</td></tr><tr><td>APOT(OURS)</td><td>3/3</td><td>73.4</td><td>91.1</td><td>8.23MB</td><td>493M</td><td>2/2</td><td>70.9</td><td>89.7</td><td>5.70MB</td><td>285M</td></tr><tr><td>FP.(RES50)</td><td>32/32</td><td>76.4</td><td>93.1</td><td>97.5MB</td><td>4.14G</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ABC-NETS</td><td>5/5</td><td>70.1</td><td>89.7</td><td>22.2MB</td><td>1.67G</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>DOREFA-NET</td><td>5/5</td><td>71.4</td><td>93.3</td><td>22.2MB</td><td>1.67G</td><td>4/4</td><td>71.4</td><td>89.8</td><td>19.4MB</td><td>1.11G</td></tr><tr><td>LQ-NET</td><td></td><td></td><td></td><td></td><td></td><td>4/4</td><td>75.1</td><td>92.4</td><td>19.4MB</td><td>1.11G</td></tr><tr><td>PACT</td><td>5/5</td><td>76.7</td><td>93.3</td><td>22.2MB</td><td>1.67G</td><td>4/4</td><td>76.5</td><td>93.3</td><td>19.4 MB</td><td>1.11G</td></tr><tr><td>APOT (OURS)</td><td>5/5</td><td>76.7</td><td>93.3</td><td>16.3MB</td><td>1.28G</td><td>4/4</td><td>76.6</td><td>93.1</td><td>13.6MB</td><td>866M</td></tr><tr><td>DoREFA-NET</td><td>3/3</td><td>69.9</td><td>89.2</td><td>16.6 MB</td><td>680M</td><td>2/2</td><td>67.1</td><td>87.3</td><td>13.8MB</td><td>370M</td></tr><tr><td>PACT</td><td>3/3</td><td>75.3</td><td>92.6</td><td>16.6MB</td><td>680M</td><td>2/2</td><td>72.2</td><td>90.5</td><td>13.8MB</td><td>370M</td></tr><tr><td>LQ-NET PACT+SAWB</td><td>3/3</td><td>74.2</td><td>91.6</td><td>16.6MB</td><td>680M</td><td>2/2 2/2</td><td>71.5 74.2</td><td>90.3</td><td>13.8MB 23.7MB</td><td>370M 707M</td></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Overall results are shown in Table 1. The results of DoReFa-Net are taken from Choi et al. (2018b), and the other results are quoted from the original papers. It can be observed that our 5-bit quantized network achieves even higher accuracy than the full precision baselines ( $0 . 7 \\%$ Top-1 improvement on ResNet-18 and $0 . 2 \\%$ Top-1 improvement on ResNet-34 and ResNet-50), which means quantization may serve the purpose of regularization. Along with the accuracy performance, our APoT quantization can achieve better hardware performance on model size and inference speed. For full precision models, the number in the column of FixOPs indicates FLOPs. ",
+        "bbox": [
+            174,
+            632,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4-bit and 3-bit quantized networks are also preserving (or approaching) the full-precision accuracy except for the 3-bit quantized ResNet-18 and ResNet-34, which only drops $0 . 5 \\%$ and $0 . 3 \\%$ accuracy respectively. When $b$ is further reduced to 2, our model still outperforms the baselines, which demonstrates the effectiveness of RCF and WN. Note that Choi et al. (2018a) use a full precision shortcut in the model, reaching higher accuracy on ResNet-50 however suffering from the hardware performance. In specific, the different precision between the main path and the residual path may result in greater latency in a pipelined implementation. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 EVALUATION ON CIFAR10 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
+            403,
+            868
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We quantize ResNet-20 and ResNet-56 (He et al., 2016) on CIFAR10 for evaluation. We adopt progressive initialization and choose the quantization bit as 2, 3 and 4. More implementations can be found in the Appendix C.2. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b19c2b3dbe21aa2b70246f29b1aa95525250fbae8b3cdd2dfd6a2f735853ce0.jpg",
+        "table_caption": [
+            "Table 2: Accuracy comparison of ResNet architectures on CIFAR10 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">MODELS METHODS</td><td rowspan=\"2\"></td><td colspan=\"3\">ACCURACY(%)</td></tr><tr><td>2 BITS</td><td>3 BITS</td><td>4 BITS</td></tr><tr><td rowspan=\"5\">REsNET-20 (FP: 91.6)</td><td>DOREFA-NET (ZHOU ET AL.,2016)</td><td>88.2</td><td>89.9</td><td>90.5</td></tr><tr><td>PACT(CHOI ET AL., 2018B)</td><td>89.7</td><td>91.1</td><td>91.7</td></tr><tr><td>LQ-NET (ZHANG ET AL.,2018)</td><td>90.2</td><td>91.6</td><td>-</td></tr><tr><td>PACT+SAWB+FPSC(CHOI ET AL.,2018A)</td><td>90.5</td><td>-</td><td>=</td></tr><tr><td>APOT QUANTIZATION(OURS)</td><td>91.0</td><td>92.2</td><td>92.3</td></tr><tr><td rowspan=\"2\">REsNET-56 (FP: 93.2)</td><td>PACT+SAWB+FPSC (CHOI ET AL., 2018A)</td><td>92.5</td><td>=</td><td>1</td></tr><tr><td>APOTQUANTIZATION (OURS)</td><td>92.9</td><td>93.9</td><td>94.0</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table. 2 summarizes the accuracy of our APoT in comparison with baselines. For 3-bit and 4-bit models, APoT quantization has reached comparable results with the full precision baselines. It is worthwhile to note that all state-of-the-arts methods in the table use 4 levels to quantize weights into 2-bit. Our model only employs ternary weights for 2-bit representation and still outstrips existing quantization methods. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/dade630c22cf7c7821dc33c52458476cc3b56d6fa1e49de901599460bc4a242f.jpg",
+        "table_caption": [
+            "Table 3: Comparison of quantizer, weight normalization and RCF of ResNet-18 on ImageNet. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>METHOD</td><td>PRECISION</td><td>WN</td><td>RCF</td><td>Acc.-1</td><td>RCF</td><td>Acc.-1</td><td>MODEL SIZE</td><td>FIxOPS</td></tr><tr><td>FULL PREC.</td><td>32/32</td><td>：</td><td>-</td><td>70.2</td><td>-</td><td>70.2</td><td>46.8 MB</td><td>1.82G</td></tr><tr><td>APoT</td><td>5/5</td><td></td><td></td><td>70.9</td><td>X</td><td>70.0</td><td>7.22 MB</td><td>616M</td></tr><tr><td>PoT</td><td>5/5</td><td></td><td></td><td>70.3</td><td>X</td><td>68.9</td><td>7.22 MB</td><td>582M</td></tr><tr><td>UNIFORM</td><td>5/5</td><td>&lt;&lt;&gt;</td><td></td><td>70.7</td><td>×</td><td>69.4</td><td>7.22 MB</td><td>781M</td></tr><tr><td>LLOYD</td><td>5/5</td><td>√</td><td>&lt;&lt;&lt;√</td><td>70.9</td><td>X</td><td>70.2</td><td>7.22 MB</td><td>1.81G</td></tr><tr><td>APoT</td><td>3/3</td><td></td><td></td><td>69.9</td><td>X</td><td>68.5</td><td>4.56 MB</td><td>298M</td></tr><tr><td>UNIFORM</td><td>3/3</td><td></td><td></td><td>69.4</td><td>×</td><td>67.8</td><td>4.56 MB</td><td>357M</td></tr><tr><td>LLOYD</td><td>3/3</td><td>/&lt;√</td><td>/&lt;&gt;</td><td>70.0</td><td>X</td><td>69.0</td><td>4.56 MB</td><td>1.81G</td></tr><tr><td>APoT</td><td>3/3</td><td></td><td>√</td><td>2.0</td><td>×</td><td>68.5</td><td>4.56 MB</td><td>198M</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The proposed algorithm consists of three techniques, APoT quantization levels to fit the bell-shaped distribution, RCF to learn the clipping threshold and WN to avoid the perturbation of the distribution of weights during training. In this section, we conduct an ablation study for these three techniques. We compare the APoT quantizer, the vanilla PoT quantizer, uniform quantizer and a non-uniform quantizer using Lloyd algorithm (Cai et al., 2017) to quantize the weights. And we either apply RCF to learn the optimal clipping range or do not clip any weights (i.e. $\\alpha = \\operatorname* { m a x } | \\mathcal { W } | )$ . Weight Normalization is also adopted or discarded to justify the effectiveness of these techniques. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 3 summarizes the results of ResNet-18 using different techniques. Quantizer using Lloyd achieves the highest accuracy, however, the irregular non-uniform quantized weights cannot utilize the fixed point arithmetic to accelerate the inference time. APoT quantization attends to the distribution of weights, which reaches the same accuracy in 5-bit and only decreases $0 . 2 \\%$ accuracy in 3-bit quantization compared with Lloyd, and shares a better tradeoff between task performance and hardware performance. We also observe that the vanilla PoT quantization suffers from the rigid resolution, and has the lowest accuracy in the 5-bit model. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Clipping range also matters in quantization, the comparison in Table 3 shows that a proper clipping range can help improve the robustness of the network. Especially when the network is quantized to 3-bit, the accuracy will drop significantly because of the quantization interval increases. Applying RCF to learn the optimal clipping range could improve at most $1 . 6 \\%$ accuracy. As we mentioned before, normalization of weight is important to learn the clipping range, and the network diverges if RCF is applied without WN. We refer to the Appendix B for more details of weight normalization during training. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have introduced the additive powers-of-two (APoT) quantization algorithm for quantizing weights and activations in neural networks, which typically exhibit a bell-shaped and long-tailed distribution. Each quantization level of APoT is the sum of a set of powers-of-two terms, bringing roughly 2x speed-up in multiplication compared with uniform quantization. The distribution of the quantization levels matches that of the weights and activations better than existing quantization schemes. In addition, we propose to reparameterize the clipping function and normalize the weights to get a more stable and better-defined gradient for optimizing the clipping threshold. We reach state-of-the-art accuracy on ImageNet and CIFAR10 dataset compared to uniform or PoT quantization. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            282,
+            356,
+            296
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work is supported by National University of Singapore FY2017 SUG Grant, and Singapore Ministry of Education Academic Research Fund Tier 3 under MOEs official grant number MOE2017-T3-1-007. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            377,
+            285,
+            392
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+            176,
+            479,
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+            523
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "bbox": [
+            173,
+            534,
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+        ],
+        "page_idx": 9
+    },
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+    {
+        "type": "text",
+        "text": "APPENDICES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            101,
+            303,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A GRADIENT DERIVATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            136,
+            411,
+            154
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we derive the gradient estimation of PACT (Choi et al., 2018b) along with our proposed Reparameterized Clipping Function and show the distinction of these two estimation. ",
+        "bbox": [
+            173,
+            167,
+            823,
+            198
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 PACT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            214,
+            261,
+            229
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Equation (1) shows the forward of PACT. In backpropagation, PACT applies the Straight-Through Estimator for the projection operation. In particular, the STE assumes that ",
+        "bbox": [
+            173,
+            242,
+            823,
+            271
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0848ecee7d2d5af86b6128d427fc1e6a68ec0e9c5b18e16c6cf8cb1d9befe48a.jpg",
+        "text": "$$\n{ \\frac { \\partial \\Pi _ { \\mathcal { Q } } X } { \\partial X } } = 1 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            454,
+            277,
+            544,
+            309
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "which means the variable before and after projection are treated the same in backpropagation. Therefore, the gradients of $\\alpha$ in PACT is computed by: ",
+        "bbox": [
+            171,
+            315,
+            823,
+            344
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f863b7d2f7b324cb69b97f60cfec562baa0b7531f26e0dba242d87723fe14a8c.jpg",
+        "text": "$$\n\\frac { \\partial \\hat { W } } { \\partial \\alpha } = \\frac { \\partial \\Pi _ { Q ( \\alpha , b ) } \\lfloor \\mathcal { W } , \\alpha \\rfloor } { \\partial \\lfloor \\mathcal { W } , \\alpha \\rfloor } \\frac { \\partial \\lfloor \\mathcal { W } , \\alpha \\rceil } { \\partial \\alpha } = \\left\\{ \\begin{array} { l l } { \\mathrm { s i g n } ( \\mathcal { W } ) } & { \\mathrm { i f } \\left| \\mathcal { W } \\right| > \\alpha } \\\\ { 0 } & { \\mathrm { i f } \\left| \\mathcal { W } \\right| \\le \\alpha } \\end{array} , \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            352,
+            710,
+            388
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where the first term is computed by STE and the second term is because the clip operation $[ \\cdot , \\alpha ]$ returns $\\mathrm { s i g n } ( \\cdot ) \\alpha$ when $| \\cdot | > \\alpha$ . In this gradient estimation, the effect of $\\alpha$ in the levels set $\\mathcal { Q } ( \\alpha , b )$ is ignored by the STE, leading to an inaccurate approximation. ",
+        "bbox": [
+            174,
+            397,
+            823,
+            440
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 RCF ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            457,
+            253,
+            472
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To avoid the elimination of STE, we reparameterize the clipping function so that the output clipping range before projection is settled and the range is re-scaled after the projection. We can define a general formation of RCF by ",
+        "bbox": [
+            174,
+            483,
+            825,
+            526
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9bbec8041ce4d628b59a4e8f942f89dc62e76428ac26373dabc453817a745c72.jpg",
+        "text": "$$\n\\hat { \\mathcal { W } } = \\frac { \\alpha } { c } \\Pi _ { \\mathcal { Q } ( c , b ) } \\lfloor \\frac { c } { \\alpha } \\mathcal { W } , c \\rceil ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            526,
+            583,
+            554
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $c > 0$ is a constant. This function clips the weights to $[ - c , c ]$ before projection and re-scaled to $[ - \\alpha , \\alpha ]$ after projection. Thus, the backpropagation is given by: ",
+        "bbox": [
+            174,
+            558,
+            825,
+            587
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/22f74219b8851ab8c61f79c921ff8a53ad25157efd108f4c9cd194e191557324.jpg",
+        "text": "$$\n\\begin{array}{c} \\begin{array} { l } { \\displaystyle \\frac { \\partial \\hat { \\mathcal { W } } } { \\partial \\alpha } = \\frac { \\partial \\alpha } { \\partial \\alpha } \\times \\frac 1 c \\Pi _ { \\mathcal { Q } ( c , b ) } \\lfloor \\frac c \\alpha \\mathcal { W } , c \\rfloor + \\displaystyle \\frac { \\partial \\Pi _ { \\mathcal { Q } ( c , b ) } \\lfloor \\frac c \\alpha \\mathcal { W } , c \\rfloor } { \\partial \\lfloor \\frac c \\alpha \\mathcal { W } , c \\rfloor } \\frac { \\partial \\lfloor \\frac c \\alpha \\mathcal { W } , c \\rfloor } { \\partial \\alpha } \\times \\frac \\alpha c } \\\\ { = \\displaystyle \\left\\{ \\frac 1 c \\times \\mathrm { s i g n } ( \\frac \\alpha \\mathcal { W } ) \\times c + \\frac \\alpha c \\times 0 \\quad \\right.} & { \\mathrm { i f } | \\mathcal { W } | > \\alpha } \\\\ { \\displaystyle \\frac 1 c \\Pi _ { \\mathcal { Q } ( c , b ) } \\frac { c } { \\alpha } \\mathcal { W } + \\frac \\alpha c \\times ( - \\frac { c } { \\alpha ^ { 2 } } ) \\mathcal { W } \\quad \\mathrm { i f } | \\mathcal { W } | \\le \\alpha } \\\\ { = \\displaystyle \\left\\{ \\mathrm { s i g n } ( \\frac \\alpha \\mathcal { W } ) \\qquad \\quad } & { \\mathrm { i f } | \\mathcal { W } | > \\alpha \\right. } \\\\ { \\displaystyle \\frac 1 c \\Pi _ { \\mathcal { Q } ( c , b ) } \\frac { c } \\alpha \\mathcal { W } - \\frac { 1 } { \\alpha } \\mathcal { W } \\quad \\mathrm { i f } | \\mathcal { W } | \\le \\alpha } \\end{array}  .  \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            594,
+            733,
+            753
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Since the levels set $\\mathcal { Q }$ is not parameterized by $\\alpha$ , the gradients will flow to two parts in RCF: the re-scale coefficient and the scale in clipping function. The constant $c$ here do not impact the gradient estimation, therefore we choose 1 for simplicity in the implementation. Note that in uniform quantization scheme, this function is equivalent to the Learned Step Size Quantization (Esser et al., 2020), where the setp size is the same for all levels while RCF provides a more general formation for any levels set $\\mathcal { Q }$ . ",
+        "bbox": [
+            173,
+            757,
+            825,
+            842
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B HOW DOES NORMALIZATION HELP QUANTIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            862,
+            640,
+            880
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this section, we show some experimental results to illustrate the effect of our weights normalization in quantization neural networks. ",
+        "bbox": [
+            174,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/d4f25614e44cf9c0581acb53bb10a413457c9bcdcdb956236f69dfb285cb7194.jpg",
+        "image_caption": [
+            "Figure 5: When weights are normalized the distribution of weights are more stable. The dashed line shows the mean value of weights. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            103,
+            803,
+            421
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.1 WEIGHTS DISTRIBUTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            503,
+            392,
+            518
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We visualize the density distribution of weights before normalization $\\mathcal { W }$ and after normalization $\\tilde { \\mathcal W }$ during training to demonstrate its effectiveness. ",
+        "bbox": [
+            174,
+            537,
+            823,
+            566
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 5a demonstrates the density distribution of the fifth layer of the 5-bit quantized ResNet-18, from which we can see that the density of the unnormalized weights could be extensive high $( > 8 )$ in the centered area. Such distribution indicates that even a tiny change of clipping threshold would bring a significant effect on clipping when $\\alpha$ is small, as shown in Figure 4a. , which means a small learning rate for $\\alpha$ is needed. However, if the learning rate is too small, the change of $\\alpha$ cannot follow the change of weights distribution because weights are also updated according to Figure 5a. Thus it is unfavorable to train the clipping threshold for unnormalized weights, while Figure 5b shows that the normalized weights can have a stable distribution. Furthermore, the dashed line in the figure indicates $\\mathcal { W }$ usually do not have zero mean, which may not utilize the symmetric design of quantization levels. ",
+        "bbox": [
+            174,
+            573,
+            825,
+            713
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.2 TRAINING BEHAVIOR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            750,
+            367,
+            765
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The above experiments use normalization during training to compare the distribution of weights. In this section, we compare the training of quantization neural networks with and without normalization to investigate the real effect of WN. Here, we train a 3-bit quantized (full precision for activations) ResNet-20 from scratch, and compare the results under different learning rate for $\\alpha$ . The results are shown in Table 4, from which we can find that if weights are normalized during training, the network can converge to descent performances and is robust to the learning rate of clipping threshold. However, if the weights are not normalized, the network would diverge if the learning rate for $\\alpha$ is too high. Even if the learning rate is set to a lower value, the network does not outperform the normalized one. Based on the training behaviors, the learning rate for clipping threshold without WN in QNNs need a careful choice. ",
+        "bbox": [
+            174,
+            784,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "",
+        "table_caption": [
+            "Table 4: Accuracy comparison of 3-bit quantized ResNet-20 on CIFAR10. "
+        ],
+        "table_footnote": [],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/d2162a50a836907d469220f34e80810a93f268b39dcf0ad70382113424671738.jpg",
+        "image_caption": [
+            "Figure 6: A summary of projection error and clipping error in different layers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            125,
+            818,
+            310
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            357,
+            419,
+            373
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.1 REVISITING QUANTIZATION ERROR ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            387,
+            467,
+            404
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Typically, quantization error $( \\Delta )$ is defined as the mean squared error between weights $\\tilde { \\mathcal W }$ and $\\hat { \\mathcal W }$ before and after quantization respectively, defined as $\\Delta = \\mathbb { E } [ \\tilde { \\mathcal { W } } - \\hat { \\mathcal { W } } ] ^ { 2 }$ . This quantization error is composed of two errors, the clipping error $\\Delta _ { c l i p }$ produced by $\\lfloor \\cdot , \\alpha \\rceil$ and the projection error $\\Delta _ { p r o j }$ produced by $\\Pi _ { \\mathfrak { Q } }$ . I.e. ",
+        "bbox": [
+            173,
+            415,
+            825,
+            474
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/50505ac11fca163c2df95b959196aaefeefec744553504c937eb59a994017e3e.jpg",
+        "text": "$$\n\\Delta = \\Delta _ { c l i p } + \\Delta _ { p r o j } = \\frac { 1 } { I } \\sum _ { | \\tilde { \\mathcal { W } } _ { i } | > \\alpha } \\left( | \\tilde { \\mathcal { W } } _ { i } | - \\alpha \\right) ^ { 2 } + \\frac { 1 } { I } \\sum _ { | \\tilde { \\mathcal { W } } _ { i } | \\leq \\alpha } ( \\tilde { \\mathcal { W } } _ { i } - \\hat { \\mathcal { W } } _ { i } ) ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            478,
+            736,
+            520
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Previous methods (Zhang et al., 2018; Cai et al., 2017) seek to minimize the quantization error to obtain the optimal clipping threshold (i.e. $\\begin{array} { r } { \\alpha = \\arg \\operatorname* { m i n } _ { \\alpha } ( \\Delta _ { c l i p } + \\Delta _ { p r o j } ) ) } \\end{array}$ , while RCF is directly optimized by the final training loss to balance projection error and clipping error. We compare the Quantization Error Minimization (QEM) method with our RCF on the quantized ResNet-18 model. Figure 6 gives an overview of the clipping error and projection error using RCF or QEM. ",
+        "bbox": [
+            173,
+            523,
+            825,
+            594
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For the 5-bit quantized model, RCF has a much higher quantization error. The projection error obtained by RCF is lower than QEM and QEM significantly reduces the clipping error. Therefore, we can infer that projection error has a higher priority in RCF. When quantizing to 3-bit, the clipping error in RCF still exceeds QEM except for the first quantized layer. This means RCF can identify whether the projection is more important than the clipping over different layers and bit-width. Generally, the insight behind is that simply minimizing the quantization error may not be the best choice and it is more direct to optimize threshold with respect to training loss. ",
+        "bbox": [
+            173,
+            601,
+            825,
+            699
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.2 IMPLEMENTATIONS DETAILS ",
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+        "text": "The ImageNet dataset consists of 1.2M training and 50K validation images. We use a standard data preprocess in the original paper (He et al., 2016). For training images, they are randomly cropped and resized to $2 2 4 \\times 2 2 4$ . Validation images are center-cropped to the same size. We use the Pytorch official code 4 to construct ResNets, and they are initialized from the released pre-trained model. We use stochastic gradient descent (SGD) with the momentum of 0.9 to optimize both weight parameters and the clipping threshold simultaneously. Batch size is set to 1024 and the learning rate starts from 0.1 with a decay factor of 0.1 at epoch 30,60,80,100. The network is trained up to 120 epochs and weight decay is set to $1 0 ^ { - 4 }$ for 3-bit quantized models or higher and $2 \\times 1 0 ^ { - 5 }$ for 2-bit model. ",
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+        "type": "text",
+        "text": "The CIFAR10 dataset contains 50K training and 10K test images with $3 2 \\times 3 2$ pixels. The ResNet architectures for CIFAR10 (He et al., 2016) contains a convolutional layer followed by 3 residual blocks and a final FC layer. We train full precision ResNet-20 and ResNet-56 firstly and use them as initialization for quantized models. All networks were trained for 200 epochs with a mini-batch size of 128. SGD with momentum of 0.9 was adopted to optimize the parameters. Learning rate started at 0.04 and was scaled by 0.1 at epoch 80,120. Weight decay was set to $1 0 ^ { - 4 }$ . ",
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+    {
+        "type": "text",
+        "text": "For clipping threshold $\\alpha$ , we set 8.0 for activations and 3.0 for weights as initial value when training a 5-bit quantized model. The learning rate of $\\alpha$ is set to 0.01 and 0.03 for weights and activations, respectively. During practice, we found that the learning rate of $\\alpha$ merely does not influence network performance. Different from PACT (Choi et al., 2018b), the update of $\\alpha$ in our works already consider the projection error, so we do not require a relatively large L2-regularization. In practice, the network works fine when the weight decay for $\\alpha$ is set to $\\mathrm { \\bar { 1 0 } } ^ { - 5 }$ and may increase to $1 0 ^ { - 4 }$ when bit-width is reduced. ",
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+]

parse/train/H1gX8C4YPr/H1gX8C4YPr.md

@@ -0,0 +1,375 @@
+# DD-PPO: LEARNING NEAR-PERFECT POINTGOAL NAVIGATORS FROM 2.5 BILLION FRAMES
+
+Erik Wijmans1,2∗Abhishek Kadian2 Ari Morcos2 Stefan Lee1,3 Irfan Essa1   
+Devi Parikh1,2 Manolis Savva2,4 Dhruv Batra1,2   
+1Georgia Institute of Technology 2Facebook AI Research   
+3Oregon State University 4Simon Fraser University
+
+# ABSTRACT
+
+We present Decentralized Distributed Proximal Policy Optimization (DD-PPO), a method for distributed reinforcement learning in resource-intensive simulated environments. DD-PPO is distributed (uses multiple machines), decentralized (lacks a centralized server), and synchronous (no computation is ever ‘stale’), making it conceptually simple and easy to implement. In our experiments on training virtual robots to navigate in Habitat-Sim (Savva et al., 2019), DD-PPO exhibits near-linear scaling – achieving a speedup of $1 0 7 \mathrm { x }$ on 128 GPUs over a serial implementation. We leverage this scaling to train an agent for 2.5 Billion steps of experience (the equivalent of 80 years of human experience) – over 6 months of GPU-time training in under 3 days of wall-clock time with 64 GPUs.
+
+This massive-scale training not only sets the state of art on Habitat Autonomous Navigation Challenge 2019, but essentially ‘solves’ the task – near-perfect autonomous navigation in an unseen environment without access to a map, directly from an RGB-D camera and a $\mathrm { G P S 4 C }$ ompass sensor. Fortuitously, error vs computation exhibits a power-law-like distribution; thus, $90 \%$ of peak performance is obtained relatively early (at 100 million steps) and relatively cheaply (under 1 day with 8 GPUs). Finally, we show that the scene understanding and navigation policies learned can be transferred to other navigation tasks – the analog of ‘ImageNet pre-training $^ +$ task-specific fine-tuning’ for embodied AI. Our model outperforms ImageNet pre-trained CNNs on these transfer tasks and can serve as a universal resource (all models and code are publicly available).
+
+Code: https://github.com/facebookresearch/habitat-api Video: https://www.youtube.com/watch?v=5PBp V5i1v4
+
+# 1 INTRODUCTION
+
+Recent advances in deep reinforcement learning (RL) have given rise to systems that can outperform human experts at variety of games (Silver et al., 2017; Tian et al., 2019; OpenAI, 2018). These advances, even more-so than those from supervised learning, rely on significant numbers of training samples, making them impractical without large-scale, distributed parallelization. Thus, scaling RL via multi-node distribution is of importance to AI – that is the focus of this work.
+
+Several works have proposed systems for distributed RL (Heess et al., 2017; Liang et al., 2018a; Tian et al., 2019; Silver et al., 2016; OpenAI, 2018; Espeholt et al., 2018). These works utilize two core components: 1) workers that collect experience (‘rollout workers’), and 2) a parameter server that optimizes the model. The rollout workers are then distributed across, potentially, thousands of CPUs1. However, synchronizing thousands of workers introduces significant overhead (the parameter server must wait for the slowest worker, which can be costly as the number of workers grows). To combat this, they wait for only a few rollout workers, and then asynchronously optimize the model.
+
+However, this paradigm – of a single parameter server and thousands of (typically CPU) workers – appears to be fundamentally incompatible with the needs of modern computer vision and robotics communities. Over the last few years, a large number of works have proposed training virtual robots (or ‘embodied agents’) in rich 3D simulators before transferring the learned skills to reality (Beattie et al., 2016; Chaplot et al., 2017; Das et al., 2018; Gordon et al., 2018; Anderson et al., 2018b; Wijmans et al., 2019; Savva et al., 2019). Unlike Gym or Atari, 3D simulators require GPU acceleration, and, consequently, the number of workers is greatly limited ( $2 ^ { 5 }$ to 8 vs. $2 ^ { 1 2 }$ to 15). The desired agents operate from high dimensional inputs (pixels) and, consequentially, use deep networks (ResNet50) that strain the parameter server. Thus, there is a need to develop a new distributed architecture.
+
+![](images/21bf2fd54b08beb184a14d21af7c8f027f4f2c1da0c98afce402df4bf44a191a.jpg)  
+Figure 1: Left: In PointGoal Navigation, an agent must navigate from a random starting location (blue) to a target location (red) specified relative to the agent $\mathbf { \bar { \Sigma } } ^ { 6 6 } \mathbf { G o } 5 \mathbf { m }$ north, $1 0 \mathrm { m }$ east of you”) in a previously unseen environment without access to a map. Right: Performance (SPL; higher is better) of an agent equipped with RGB-D and $\mathrm { G P S 4 C }$ ompass sensors on the Habitat Challenge 2019 (Savva et al., 2019) train & val sets. Using DD-PPO, we train agents for over 180 days of GPU-time in under 3 days of wall-clock time with 64 GPUs, achieving state-of-art results and ‘solving’ the task.
+
+Contributions. We propose a simple, synchronous, distributed RL method that scales well. We call this method Decentralized Distributed Proximal Policy Optimization (DD-PPO) as it is decentralized (has no parameter server), distributed (runs across many different machines), and we use it to scale Proximal Policy Optimization (Schulman et al., 2017).
+
+In DD-PPO, each worker alternates between collecting experience in a resource-intensive and GPU accelerated simulated environment and optimizing the model. This distribution is synchronous – there is an explicit communication stage where workers synchronize their updates to the model (the gradients). To avoid delays due to stragglers, we propose a preemption threshold where the experience collection of stragglers is forced to end early once a pre-specified percentage of the other workers finish collecting experience. All workers then begin optimizing the model.
+
+We characterize the scaling of DD-PPO by the steps of experience per second with N workers relative to 1 worker. We consider two different workloads, 1) simulation time is roughly equivalent for all environments, and 2) simulation time can vary dramatically due to large differences in environment complexity. Under both workloads, we find that DD-PPO scales near-linearly. While we only examined our method with PPO, other on-policy RL algorithms can easily be used and we believe the method is general enough to be adapted to off -policy RL algorithms.
+
+We leverage these large-scale engineering contributions to answer a key scientific question arising in embodied navigation. Mishkin et al. (2019) benchmarked classical (mapping $^ +$ planning) and learning-based methods for agents with RGB-D and $\mathrm { G P S + } \mathrm { c }$ ompass sensors on PointGoal Navigation (Anderson et al., 2018a) (PointGoalNav), see Fig. 1, and showed that classical methods outperform learning-based. However, they trained for ‘only’ 5 million steps of experience. Savva et al. (2019) then scaled this training to 75 million steps and found that this trend reverses – learningbased outperforms classical, even in unseen environments! However, even with an order of magnitude more experience (75M vs 5M), they found that learning had not yet saturated. This begs the question – what are the fundamental limits of learnability in PointGoalNav? Is this task entirely learnable? We answer this question affirmatively via an ‘existence proof’.
+
+Utilizing DD-PPO, we find that agents continue to improve for a long time (Fig. 1) – not only setting the state of art in Habitat Autonomous Navigation Challenge 2019 (Savva et al., 2019), but essentially ‘solving’ PointGoalNav (for agents with GPS $^ +$ Compass). Specifically, these agents 1) almost always reach the goal (failing on 1/1000 val episodes on average), and 2) reach it nearly as efficiently as possible – nearly matching (within $3 \%$ of) the performance of a shortest-path oracle! It is worth stressing how uncompromising that comparison is – in a new environment, an agent navigating without a map traverses a path nearly matching the shortest path on the map. This means there is no scope for mistakes of any kind – no wrong turn at a crossroad, no back-tracking from a dead-end, no exploration or deviation of any kind from the shortest-path. Our hypothesis is that the model learns to exploit the statistical regularities in the floor-plans of indoor environments (apartments, offices) in our datasets. The more challenging task of navigating purely from an RGB camera without GPS $+$ Compass demonstrates progress but remains an open frontier.
+
+![](images/332df56e623c0f624a0eebf02fae9bb2ccb30196ae3b468e2b75d7c3a37378f0.jpg)  
+Figure 2: Comparison of asynchronous distribution (left) and synchronous distribution via distributed data parallelism (right) for RL. Left: rollout workers collect experience and asynchronously send it to the parameter-server. Right: a worker alternates between collecting experience, synchronizing gradients, and optimization. We find this highly effective in resource-intensive environments.
+
+Finally, we show that the scene understanding and navigation policies learned on PointGoalNav can be transferred to other tasks (Flee and Explore (Gordon et al., 2019)) – the analog of ‘ImageNet pre-training $^ +$ task-specific fine-tuning’ for Embodied AI. Our models are able to rapidly learn these new tasks (outperforming ImageNet pre-trained CNNs) and can be utilized as near-perfect neural PointGoal controllers, a universal resource for other high-level navigation tasks (Anderson et al., 2018b; Das et al., 2018). We make code and trained models publicly available.
+
+# 2 PRELIMINARIES: RL AND PPO
+
+Reinforcement learning (RL) is concerned with decision making in Markov decision processes. In a partially observable MDP (POMDP), the agent receives an observation that does not fully specify the state $\left( { { s _ { t } } } \right)$ of the environment, $o _ { t }$ (e.g. an egocentric RGB image), takes an action $a _ { t }$ , and is given a reward $r _ { t }$ . The objective is to maximize cumulative reward over an episode, Formally, let $\tau$ be a sequence of $\left( o _ { t } , a _ { t } , r _ { t } \right)$ where $a _ { t } \sim \pi ( \cdot \mid o _ { t } )$ , and $s _ { t + 1 } \sim \mathcal { T } ( s _ { t } , a _ { t } )$ . For a discount factor $\gamma$ , which balances the trade-off between exploration and exploitation, the optimal policy, $\pi ^ { * }$ , is specified by
+
+$$
+\pi ^ { * } = \underset { \pi } { \operatorname { a r g m a x } } \mathbb { E } _ { \tau \sim \pi } \left[ R _ { T } \right] , \quad \mathrm { w h e r e } , R _ { T } = \sum _ { t = 1 } ^ { T } \gamma ^ { t - 1 } r _ { t } .
+$$
+
+One technique to find $\pi ^ { * }$ is Proximal Policy Optimization (PPO) (Schulman et al., 2017), an on-policy algorithm in the policy-gradient family. Given a $\theta$ -parameterized policy $\pi _ { \theta }$ and a set of trajectories collected with it (commonly referred to as a ‘rollout’), PPO updates $\pi _ { \theta }$ as follows. Let $\widehat { A } _ { t } ^ { \mathrm { ~ \scriptsize ~ - ~ } }$ , be the estimate of the advantage, where $\begin{array} { r } { R _ { t } = \sum _ { i = t } ^ { T } \gamma ^ { i - t } r _ { i } } \end{array}$ , and $\hat { V } _ { t }$ is the expected value of $R _ { t }$ , and $\begin{array} { r } { r _ { t } ( \theta ) = \frac { \pi _ { \theta } \left( a _ { t } | o _ { t } \right) } { \pi _ { \theta _ { t } } \left( a _ { t } | o _ { t } \right) } } \end{array}$ be the ratio of the probability of the action $a _ { t }$ under the current policy and the policy used to collect the rollout. The parameters are then updated by maximizing
+
+$$
+\mathcal { I } ^ { P P O } ( \theta ) = E _ { t } \Bigg [ \operatorname* { m i n } \Big ( \underbrace { r _ { t } ( \theta ) \hat { A } _ { t } } _ { \mathrm { i m p o r t a n c e - w e i g h t e d a d v a n t a g e } } , \quad \underbrace { \mathrm { c l i p } ( r _ { t } ( \theta ) , 1 - \epsilon , 1 + \epsilon ) \hat { A } _ { t } } _ { \mathrm { p r o x i m i t y c l i p p i n g ~ t e r m } } \Big ) \Bigg ]
+$$
+
+This clipped objective keeps this ratio within $\epsilon$ and functions as a trust-region optimization method;   
+allowing for the multiple gradient updates using the rollout, thereby improving sample efficiency.
+
+![](images/bf1241387fc742f2e852ff22264473db8b27bf1f5d3a2c3dd7491d9496d81e71.jpg)  
+Figure 3: Our agent for PointGoalNav. At very time-step, the agent receives an egocentric Depth or RGB (shown here) observation, utilizes its $\mathrm { G P S + C }$ ompass sensor to update the target position to be relative to its current position, and outputs the next action and an estimate of the value function.
+
+# 3 DECENTRALIZED DISTRIBUTED PROXIMAL POLICY OPTIMIZATION
+
+In reinforcement learning, the dominant paradigm for distribution is asynchronous (see Fig. 2). Asynchronous distribution is notoriously difficult – even minor errors can result in opaque crashes – and the parameter server and rollout workers necessitate separate programs.
+
+In supervised learning, however, synchronous distributed training via data parallelism (Hillis & Steele Jr, 1986) dominates. As a general abstraction, this method implements the following: at step $k$ , worker $n$ has a copy of the parameters, $\theta _ { n } ^ { k }$ , calculates the gradient, $\partial \theta _ { n } ^ { k }$ , and updates $\theta$ via
+
+$$
+\theta _ { n } ^ { k + 1 } = \mathsf { P a r a m l p o d a t e } \Big ( \theta _ { n } ^ { k } , \mathsf { A l l R e d u c e } \big ( \partial \theta _ { 1 } ^ { k } , \ldots , \partial \theta _ { N } ^ { k } \big ) \Big ) = \mathsf { P a r a m l p o d a t e } \Big ( \theta _ { n } ^ { k } , \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \partial \theta _ { i } ^ { k } \Big ) ,
+$$
+
+where ParamUpdate is any first-order optimization technique (e.g. gradient descent) and AllReduce performs a reduction (e.g. mean) over all copies of a variable and returns the result to all workers. Distributed DataParallel scales very well (near-linear scaling up to 32,000 GPUs (Kurth et al., 2018)), and is reasonably simple to implement (all workers synchronously running identical code).
+
+We adapt this to on-policy RL as follows: At step $k$ , a worker $n$ has a copy of the parameters $\theta _ { n } ^ { k }$ ; it gathers experience (rollout) using $\pi _ { \theta _ { n } ^ { k } }$ , calculates the parameter-gradients $\nabla _ { \theta }$ via any policy-gradient method (e.g. PPO), synchronizes these gradients with other workers, and updates the model:
+
+$$
+\theta _ { n } ^ { k + 1 } = \mathsf { P a r a m l y n d a t e } \left( \theta _ { n } ^ { k } , \mathsf { A l l R e d u c e } \left( \nabla _ { \theta } \mathcal { I } ^ { P P O } ( \theta _ { 1 } ^ { k } ) , \ldots , \nabla _ { \theta } \mathcal { I } ^ { P P O } ( \theta _ { N } ^ { k } ) \right) \right) .
+$$
+
+A key challenge to using this method in RL is variability in experience collection run-time. In supervised learning, all gradient computations take approximately the same time. In RL, some resourceintensive environments can take significantly longer to simulate. This introduces significant synchronization overhead as every worker must wait for the slowest to finish collecting experience. To combat this, we introduce a preemption threshold where the rollout collection stage of these stragglers is preempted (forced to end early) once some percentage, $p \%$ , (we find $6 0 \%$ to work well) of the other workers are finished collecting their rollout; thereby dramatically improving scaling. We weigh all worker’s contributions to the loss equally and limit the minimum number of steps before preemption to one-fourth the maximum to ensure all environments contribute to learning.
+
+While we only examined our method with PPO, other on-policy RL algorithms can easily be used and we believe the method can be adapted to off -policy RL algorithms. Off-policy RL algorithms also alternate between experience collection and optimization, but differ in how experience is collected/used and the parameter update rule. Our adaptations simply add synchronization to the optimization stage and a preemption to the experience collection stage.
+
+Implementation. We leverage PyTorch’s (Paszke et al., 2017) DistributedDataParallel to synchronize gradients, and TCPStore – a simple distributed key-value storage – to track how many workers have finished collecting experience. See Apx. E for a detailed description with code.
+
+# 4 EXPERIMENTAL SETUP: POINTGOAL NAVIGATION, AGENTS, SIMULATOR
+
+PointGoal Navigation (PointGoalNav). An agent is initialized at a random starting position and orientation in a new environment and asked to navigate to target coordinates specified relative to the agents position; no map is available and the agent must navigate using only its sensors – in our case RGB-D (or RGB) and $\mathrm { G P S } { + } 0$ Compass (providing current position and orientation relative to start).
+
+The evaluation criteria for an episode is as follows (Anderson et al., 2018a): Let $S$ indicate ‘success’ (did the agent stop within 0.2 meters of the target?), $l$ be the length of the shortest path between start and target, and $p$ be the length of the agent’s path, then Success weighted by (normalized inverse) Path Length $\begin{array} { r } { \mathrm { S P L } = S \frac { \iota } { \operatorname* { m a x } ( l , p ) } } \end{array}$ . It is worth stressing that SPL is a highly punitive metric – to achieve $\mathrm { S P L } = 1$ , the agent (navigating without the map) must match the performance of the shortest-path oracle that has access to the map! There is no scope for any mistake – no wrong turn at a crossroad, no back-tracking from a dead-end, no exploration or deviation from the shortest path. In general, this may not even be possible in a new environment (certainly not if an adversary designs the map).
+
+Agent. As in Savva et al. (2019), the agent has 4 actions, stop, which indicates the agent has reached the goal, move forward $( 0 . 2 5 \mathrm { m } )$ , turn left $( 1 0 ^ { \circ } )$ , and turn right $( 1 0 ^ { \circ } )$ . It receives $2 5 6 \times 2 5 6$ sized images and uses the GPS $^ +$ Compass to compute target coordinates relative to its current state. The RGB-D agent is limited to only Depth as Savva et al. (2019) found this to perform best.
+
+Our agent architecture (Fig. 3) has two main components – a visual encoder and a policy network.
+
+The visual encoder is based on either ResNet (He et al., 2016) or SE (Hu et al., 2018)-ResNeXt (Xie et al., 2017) with the number of output channels at every layer reduced by half. We use a first layer of 2x2-AvgPool to reduce resolution (essentially performing low-pass filtering $^ +$ down-sampling) – we find this to have no impact on performance while allowing faster training. From our initial experiments, we found it necessary to replace every BatchNorm layer (Ioffe & Szegedy, 2015) with GroupNorm (Wu & He, 2018) to account for highly correlated inputs seen in on-policy RL.
+
+The policy is parameterized by a 2-layer LSTM with a 512-dimensional hidden state. It takes three inputs: the previous action, the target relative to the current state, and the output of the visual encoder. The LSTM’s output is used to produce a softmax distribution over the action space and an estimate of the value function. See Appendix C for full details.
+
+Training. We use PPO with Generalized Advantage Estimation (Schulman et al., 2015). We set the discount factor $\gamma$ to 0.99 and the GAE parameter $\tau$ to 0.95. Each worker collects (up to) 128 frames of experience from 4 agents running in parallel (all in different environments) and then performs 2 epochs of PPO with 2 mini-batches per epoch. We use Adam (Kingma & Ba, 2014) with a learning rate of $2 . 5 \times 1 0 ^ { - 4 }$ . Unlike popular implementations of PPO, we do not normalize advantages as we find this leads to instabilities. We use DD-PPO to train with 64 workers on 64 GPUs.
+
+The agent receives terminal reward $r _ { T } = 2 . 5 \mathrm { S P L }$ , and shaped reward $r _ { t } ( a _ { t } , s _ { t } ) = - \Delta _ { \mathrm { g e o } _ { - } \mathrm { d i s t } } - 0 . 0 1$ , where $\Delta _ { \mathrm { g e o \_ d i s t } }$ is the change in geodesic distance to the goal by performing action $a _ { t }$ in state $s _ { t }$ .
+
+Simulator+Datasets. Our experiments are conducted using Habitat, a 3D simulation platform for embodied AI research (Savva et al., 2019). Habitat is a modular framework with a highly performant and stable simulator, making it an ideal framework for simulating billions of steps of experience.
+
+We experiment with several different sources of data. First, we utilize the training data released as part of the Habitat Challenge 2019, consisting of 72 scenes from the Gibson dataset (Xia et al., 2018). We then augment this with all 90 scenes in the Matterport3D dataset (Chang et al., 2017) to create a larger training set (note that Matterport3D meshes tend to be larger and of better quality).2 Furthermore, Savva et al. (2019) curated the Gibson dataset by rating every mesh reconstruction on a quality scale of 0 to 5 and then filtered all splits such that each only contains scenes with a rating of 4 or above (Gibson- $^ { . 4 + }$ ), leaving all scenes with a lower rating previously unexplored. We examine training on the 332 scenes from the original train split with a rating of 2 or above (Gibson- $^ { 2 + }$ ).
+
+# 5 BENCHMARKING: HOW DOES DD-PPO SCALE?
+
+In this section, we examine how DD-PPO scales under two different workload regimes – homogeneous (every environment takes approximately the same amount of time to simulate) and heterogeneous (different environments can take orders of magnitude more/less time to simulate). We examine the number of steps of experience per second with N workers relative to 1 worker. We compare different values of the preemption threshold $p \%$ . We benchmark training our ResNet50 PointGoalNav agent with Depth on a cluster with Nvidia V100 GPUs and NCCL2.4.7 with Infiniband interconnect.
+
+![](images/435c1b7eeb301c7b83385d03704adc322e277f00ae6437de8b0ef2d9ac70b409.jpg)  
+Figure 4: Scaling performance (in steps of experience per second relative to 1 GPU) of DD-PPO for various preemption threshold, $p \%$ , values. Shading represents a $9 5 \%$ confidence interval.
+
+Homogeneous. To create a homogeneous workload, we train on scenes from the Gibson dataset, which require very similar times to simulate agent steps. As shown in Fig. 4 (left), DD-PPO exhibits near-linear scaling (linear $=$ ideal) for preemption thresholds larger than $50 \%$ , achieving a $1 9 6 \mathrm { x }$ speed up with 256 GPUs relative to 1 GPU and an $7 . 3 \mathrm { x }$ speed up with 8 GPUs relative to 1.
+
+Heterogeneous. To create a heterogeneous workload, we train on scenes from both Gibson and Matterport3D. Unlike Gibson, MP3D scenes vary significantly in complexity and time to simulate – the largest contains 8GB of data while the smallest is only 135MB. DD-PPO scales poorly at a preemption threshold of $100 \%$ (no preemption) due to the substantial straggler effect (one rollout taking substantially longer than the others); see Fig. 4 (right). However, with a preemption threshold of $80 \%$ or $60 \%$ , we achieve near-identical scaling to the homogeneous workload! We found no degradation in performance of models trained with any of these values for the preemption threshold despite learning in large scenes occurring at a lower frequency.
+
+# 6 MASTERING POINTGOAL NAVIGATION WITH GPS $^ +$ COMPASS
+
+In this section, we answer the following questions: 1) What are the fundamental limits of learnability in PointGoalNav navigation? 2) Do more training scenes improve performance? 3) Do better visual encoders improve performance? 4) Is PointGoalNav ‘solvable’ when navigating from RGB instead of Depth? 5) What are the open/unsolved problems – specifically, how does navigation without GPS+Compass perform? 6) Can agents trained for PointGoalNav be transferred to new tasks?
+
+Agents continue to improve for a long time. Using DD-PPO, we train agents for 2.5 Billion steps of experience with 64 Tesla V100 GPUs in 2.75 days – 180 GPU-days of training, the equivalent of 80 years of human experience (assuming 1 human second per step). As a comparison, Savva et al. (2019) reached 75 million steps (an order of magnitude more than prior work) in 2.5 days using 2 GPUs – at that rate, it would take them over a month (wall-clock time) to achieve the scale of our study. Fig. 1 shows the performance of an agent with RGB-D and $\mathrm { G P S + C }$ ompass sensors, utilizing an SE-ResNeXt50 visual encoder, trained on Gibson- $^ { 2 + }$ – it does not saturate before 1 billion steps3, suggesting that previous studies were incomplete by 1-2 orders of magnitude. Fortuitously, error vs computation exhibits a power-law-like distribution; $90 \%$ of peak performance is obtained relatively early (100M steps) and relatively cheaply $( \mathrm { i n } 0 . 1$ day with 64 GPUs and in 1 day with ${ 8 \mathrm { G P U s } ^ { 4 } }$ ). Also noteworthy in Fig. 1 is the strong generalization (train to val) and corresponding lack of overfitting.
+
+Increasing training data helps. Tab. 1 presents results with different training datasets and visual encoders for agent with RGB-D and GPS $^ +$ Compass. Our most basic setting (ResNet50, Gibson- $^ { . 4 + }$ training) already achieves SPL of 0.922 (val), 0.917 (test), which nearly misses (by 0.003) the top of the leaderboard for the Habitat Challenge $2 0 1 9 \ \mathsf { R G B - D \ t r a c k }$ . Next, we increase the size of the training data by adding in all Matterport3D scenes and see an improvement of ${ \sim } 0 . 0 3$ SPL – to 0.956 (val), 0.941 (test). Next, we compare training on Gibson- $^ { . 4 + }$ and Gibson- $^ { 2 + }$ . Recall that Gibson-{2, $3 \}$ corresponds to poorly reconstructed scenes (see Fig. 11). A priori, it is unclear whether the net effect of this addition would be positive or negative; adding them provides diverse experience to the agent, however, it is poor quality data. We find a potentially counter-intuitive result – adding poor 3D reconstructions to the train set improves performance on good reconstructions in val/test by ${ \sim } 0 . 0 3$ SPL – from 0.922 (val), 0.917 (test) to 0.956 (val), 0.944 (test). Our conjecture is that training on poor (Gibson-{2,3}) and good $^ { ( 4 + ) }$ reconstructions leads to robustness in representations learned.
+
+Table 1: Performance (higher is better) of different architectures for agents with RGB-D and $\mathrm { G P S } { + }$ Compass sensors on the Habitat Challenge 2019 (Savva et al., 2019) validation and test-std splits (checkpoint selected on val). 10 samples taken for each episode on val. Gibson- $\cdot 4 + ( 2 + )$ refers to the subset of Gibson train scenes (Xia et al., 2018) with a quality rating of 4 (2) or higher. See Tab. 2 for results of the best DD-PPO agent for Blind, RGB, and RGB-D and other baselines.   
+
+<table><tr><td rowspan="2">Training Dataset</td><td rowspan="2">Agent Visual Encoder</td><td colspan="2">Validation</td><td colspan="2">Test Standard</td></tr><tr><td>SPL</td><td>Success</td><td>SPL</td><td>Success</td></tr><tr><td>Gibson-4+</td><td>ResNet50</td><td>0.922 ± 0.004</td><td>0.967 ±0.003</td><td>0.917</td><td>0.970</td></tr><tr><td>Gibson-4+ and MP3D</td><td>ResNet50</td><td>0.956 ± 0.002</td><td>0.996 ± 0.002</td><td>0.941</td><td>0.996</td></tr><tr><td>Gibson-2+</td><td>ResNet50</td><td>0.956± 0.003</td><td>0.994± 0.002</td><td>0.944</td><td>0.982</td></tr><tr><td></td><td>SE-ResNeXt50</td><td>0.959 ± 0.002</td><td>0.999 ± 0.001</td><td>0.943</td><td>0.988</td></tr><tr><td></td><td>SE-ResNeXt101+1024-d LSTM</td><td>0.969 ±0.002</td><td>0.997 ± 0.001</td><td>0.948</td><td>0.980</td></tr></table>
+
+Better visual encoders and more parameters help. Using a better visual encoder, SE (Hu et al., 2018)-ResNeXt50 (Xie et al., 2017) instead of ResNet50, improves performance by 0.003 SPL (Tab. 1). Adding capacity to the visual encoder (SE-ResNeXt101 vs SE-ResNeXt50) and navigation policy (1024-d vs 512-d LSTM) further improves performance by 0.010 SPL.
+
+PointGoalNav ‘solved’ with RGB-D and GPS $^ +$ Compass. Our best agent – SE-ResNeXt101 + 1024-d LSTM trained on Gibson- $^ { 2 + }$ – achieves SPL of 0.969 (val), 0.948 (test), which not only sets the state of art on the Habitat Challenge 2019 RGB-D track but is also within $3 { - } 5 \%$ of the shortest-path oracle6. Given the challenges with achieving near-perfect SPL in new environments, it is important to dig deeper. Fig. 13 shows (a) distribution of episode lengths in val and (b) SPL vs episode length. We see that while the dataset is dominated by short episodes $( 2 \mathrm { - } 1 2 \mathrm { m } )$ , the performance of the agent is remarkably stable over long distances and average SPL is not necessarily inflated. Our hypothesis is the agent has learned to exploit the structural regularities in layouts of real indoor environments. One (admittedly imperfect) way to test this is by training a Blind agent with only a $\mathrm { G P S + }$ Compass sensor. Fig. 13 shows that this agent is able to handle short-range navigation (which primarily involve turning to face the target and walking straight) but performs very poorly on longer trajectories – SPL of 0.3 (Blind) vs 0.95 (RGB-D) at $2 0 { - } 2 5 \mathrm { m }$ navigation. Thus, structural regularities, in part, explain performance for short-range navigation. For long-range navigation, the RGB-D agent is extracting overwhelming signal from its Depth sensor. We repeat this analysis on two additional navigation datasets proposed by Chaplot et al. (2019) – longer episodes and ‘harder’ episodes (more navigation around obstacles) – and find similar trends (Fig. 14). This discussion continues in Apx. A.
+
+Performance with RGB is also improved. So far we studied RGB-D as this performed best in Savva et al. (2019). We now study RGB (with SE-ResNeXt50 encoder). We found it crucial to train on Gibson- $^ { 2 + }$ and all of Matterport3D, ensuring diversity in both layouts (Gibson- $^ { 2 + }$ ) and appearance (Matterport3D), and to channel-wise normalize RGB (subtract by mean and divide by standard deviation) as our networks lack BatchNorm. Performance improves dramatically from 0.57 (val), 0.47 (test) SPL in Savva et al. (2019) to near-perfect success 0.991 (val), 0.977 (test) and high SPL 0.929 (val), 0.920 (test). While SPL is considerably lower than the Depth agent, (0.929 vs 0.959), interestingly, the RGB agent still reaches the goal a similar percentage of the time $( 9 9 . 1 \%$ vs $9 9 . 9 \%$ ). This agent achieves state-of-art on the Habitat Challenge 2019 RGB track (rank 2 entry has 0.89 SPL).5
+
+No GPS $^ +$ Compass remains unsolved. Finally, we examine if we also achieve better performance on the significantly more challenging task of navigation from RGB without GPS+Compass. At 100 million steps (an amount equivalent to Savva et al. (2019)), the agent achieves 0 SPL. By training to 2.5 billion steps, we make some progress and achieve 0.15 SPL. While this is a substantial improvement, the task continues to remain an open frontier for research in embodied AI.
+
+Transfer Learning. We examine transferring our agents to the following tasks (Gordon et al., 2019)
+
+![](images/c9cfb838a867db299130f5b389dfb94f653429bcdff0be5f38adc41a18592352.jpg)  
+Figure 5: Performance (higher is better) on Flee (left) and Exploration (right) under five settings.
+
+– Flee The agent maximizes its geodesic distance from its starting location. Let $s _ { t }$ be the agent’s position at time $t$ , and $M a x ( s _ { 0 } )$ denote the maximum distance over all reachable points, then the agent maximizes $D _ { T } = G e o ( s _ { T } , s _ { 0 } ) / M a x ( s _ { 0 } )$ . The reward is $r _ { t } = 5 ( D _ { t } - D _ { t - 1 } )$ .
+
+Exploration The agent maximizes the number of locations (specified by 1m cubes) visited. Let Visitedt denote the number of location visited at time $t$ , then the agent maximizes $| { \mathrm { V i s i t e d } } _ { T } |$ The reward is $r _ { t } = 0 . 2 5 ( | \mathrm { V i s i t e d } _ { t } | - | \mathrm { V i s i t e d } _ { t - 1 } | )$ .
+
+We use a PointGoalNav-trained agent with RGB and ${ \mathrm { G P S } } { + } { \mathsf { C } }$ ompass, remove the GPS+Compass, and transfer to these tasks under five different settings:
+
+Scratch. All parameters (visual encoder $^ +$ policy) are trained from scratch for each new task. Improvements over this baseline demonstrate benefits of transfer learning.   
+ImageNetEncoder-ScratchPolicy. The visual encoder is initialized with ImageNet pre-trained weights and frozen; the navigation policy is trained from scratch.   
+PointGoalNavEncoder-ScratchPolicy. The visual encoder is initialized from PointGoalNav and frozen; the navigation policy is trained from scratch.   
+PointGoalNavEncoder-FinetunePolicy. Both visual encoder and policy parameters are initialized from PointGoalNav (critic layers are reinitialized). Encoder is frozen, policy is fine-tuned.7 ∇ Neural Controller We treat our agent as a differentiable neural controller, a closed-loop lowlevel controller than can navigate to a specified coordinate. We utilize this controller in a new task by training a light-weight high-level planner that predicts a goal-coordinate (at each time-step) for the controller to navigate to. Since the controller is fully differentiable, we can backprop through it. We freeze the controller, train the planner+controller system with PPO for the new task. The planner is a 2-layer LSTM and shares the (frozen) visual encoder with the controller.
+
+Fig. 5 shows performance vs. experience results (higher is better). Nearly all methods outperform learning from scratch, establishing the value of transfer learning. PointGoalNav pre-trained visual encoders dramatically outperforms ImageNet pre-trained ones, indicating that the agent has learned generally useful scene understanding. For both tasks, fine-tuning an existing policy allows it to rapidly learn the new task, indicating that the agent has learned general navigation skills. ∇Neural Controller outperforms PointGoalNavEncoder-ScratchPolicy on Flee and is competitive on Exploration, indicating that the agent can indeed be ‘controlled’ or directed to target locations by a planner. Overall, these results demonstrate that our trained model is useful for more than just PointGoalNav.
+
+# 7 RELATED WORK
+
+Visual Navigation. Visual navigation in indoor environments has been the subject of many recent works (Gupta et al., 2017; Das et al., 2018; Anderson et al., 2018b; Savva et al., 2019; Mishkin et al., 2019). Our primary contribution is DD-PPO, thus we discuss other distributed works.
+
+In the general case, computation in reinforcement learning (RL) in simulators can be broken down into 4 roles: 1) Simulation: Takes actions performed by the agent as input, simulates the new state, returns observations, reward, etc. 2) Inference: Takes observations as input and utilizes the agent policy to return actions, value estimate, etc. 3) Learner: Takes rollouts as input and computes gradients to update the policy’s parameters. 4) Parameter server/master: Holds the source of truth for the policy’s parameters and coordinates workers.
+
+Synchronous RL. Synchronous RL systems utilize a single processes to perform all four roles; this design is found in RL libraries like OpenAI Baselines (Dhariwal et al., 2017) and PytorchRL (Kostrikov, 2018). This method is limited to a single nodes worth of GPUs.
+
+Synchronous Distributed RL. The works most closely related to DD-PPO also propose to scale synchronous RL by replicating this simulation/inference/learner process across multiple GPUs and then synchronize gradients with AllReduce. Stooke & Abbeel (2018) experiment with Atari and find it not effective however. We hypothesize that this is due to a subtle difference – this distribution design relies on a single worker collecting experience from multiple environments, stepping through them in lock step. This introduces significant synchronization and communication costs as every step in the rollout must be synchronized across as many as 64 processes (possible because each environment is resource-light, e.g. Atari). For instance, taking 1 step in 8 parallel pong environments takes approximately the same wall-clock time as 1 pong environment, but it takes 10 times longer to take 64 steps in lock-step; thus gains from parallelization are washed out due to the lock-step synchronization. In contrast, we study resource-intensive environments, where only 2 or 4 environments per worker is possible, and find this technique to be effective. Liang et al. (2018b) mirror our findings (this distribution method can be effective for resource intensive simulation) in GPUaccelerated physics simulation, specifically MuJoCo (Todorov et al., 2012) with NVIDIA Flex. In contrast to our work, they examine scaling up to only 32 GPUs and only for homogeneous workloads. In contrast to both, we propose an adaption to mitigate the straggler effect – preempting the experience collection (rollout) of stragglers and then beginning optimization. This improves scaling for homogeneous workloads and dramatically improves scaling for heterogeneous workloads.
+
+Asynchronous Distributed RL. Existing public frameworks for asynchronous distributed reinforcement learning (Heess et al., 2017; Liang et al., 2018a; Espeholt et al., 2018) use a single (CPU-only) process to perform the simulation and inference roles (and then replicate this process to scale). A separate process asynchronously performs the learner and parameter server roles (note its not clear how to use more than one these processes as it holds the source of truth for the parameters). Adapting these methods to the resource-intensive environments studied in this work (e.g. Habtiat (Savva et al., 2019)) encounters the following issues: 1) Limiting the inference/simulation processes to CPU-only is untenable (deep networks and need for GPU-accelerated simulation). While the inference/simulation processes could be moved to the GPU, this would be ineffective for the following: GPUs operate most efficiently with large batch sizes (each inference/simulation process would have a batch size of 1), CUDA runtime requires ${ \sim } 6 0 0 \mathrm { M B }$ of GPU memory per process, and only one CUDA kernel (function that runs on the GPU) can executed by the GPU at a time. These issue contribute and lead to low GPU utilization. In contrast, DD-PPO utilizes a single process per GPU and batches observations from multiple environments for inference. 2) The single process learner/- parameter server is limited to a single node’s worth of GPUs. While this not a limitation for small networks and low dimensional inputs, our agents take high dimensional inputs (e.g. a Depth sensor) and utilize large neural networks (ResNet50), thereby requiring considerable computation to compute gradients. In contrast, DD-PPO has no parameter server and every GPU computes gradients, supporting even very large networks (SE-ResNeXt101).
+
+Straggler Effect Mitigation. In supervised learning, the straggler effect is commonly caused by heterogeneous hardware or hardware failures. Chen et al. (2016) propose a pool of $b$ “back-up” workers (there are $N + b$ workers total) and perform the parameter update once $N$ workers finish. In comparison, their method a) requires a parameter server, and b) discards all work done by the stragglers. Chen et al. (2018) propose to dynamically adjust the batch size of each worker such that all workers perform their forward and backward pass in the same amount of time. Our method aims to reduce variance in experience collection times. DD-PPO dynamically adjusts a worker’s batch size as a necessary side-effect of preempting experience collection in on-policy RL.
+
+Distributed Synchronous SGD. Data parallelism is a common paradigm in high performance computing (Hillis & Steele Jr, 1986). In this paradigm, parallelism is achieved by workers performing the same work on different data. This paradigm can be naturally adapted to supervised deep learning (Chen et al., 2016). Works have used this to achieve state-of-the-art results in tasks ranging from computer vision (Goyal et al., 2017; He et al., 2017) to natural language processing (Peters et al., 2018; Devlin et al., 2018; Ott et al., 2019). Furthermore, multiple deep learning frameworks provide simple-to-use wrappers supporting this parallelism model (Paszke et al., 2017; Abadi et al., 2015; Sergeev & Balso, 2018). We adapt this framework to reinforcement learning.
+
+# 8 ACKNOWLEDGEMENTS
+
+The Georgia Tech effort was supported in part by NSF, AFRL, DARPA, ONR YIPs, ARO PECASE. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the U.S. Government, or any sponsor.
+
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+
+![](images/81c837610c64d0e34826f47afbe2b738e6e256363d65e3ed0343b307afa6c0ac.jpg)  
+Figure 6: Example episodes broken down by geodesic distance between agent’s spawn location and target (on rows) vs SPL achieved by the agent (on cols). Gray represents navigable regions on the map while white is non-navigable. The agent begins at the blue square and navigates to the red square. The green line shows the shortest path on the map (or oracle navigation). The blue line shows the agent’s trajectory. The color of the agent’s trajectory changes changes from dark to light over time. Navigation dataset from the longer validation episodes proposed in Chaplot et al. (2019).
+
+# A ADDITIONAL ANALYSIS AND DISCUSSION
+
+In this section, we continue the analysis of our agent and examine differences in its behavior from a classical, hand-designed agent – the map-and-plan baseline agent proposed in Gupta et al. (2017).
+
+Intricacies of SPL. Given an agent that always reaches the goal ${ \approx } 1 0 0 \%$ success), SPL can be seen as measuring the efficiency of an agent vs. an oracle $- i . e .$ . an SPL of 0.95 means the agent is $5 \%$ less efficient than an oracle. Given the challenges of near-perfect autonomous navigation without a map in novel environments we outlined, being $5 \%$ less efficient than an oracle seems near-impossible. However, this comparison/view is potentially miss-leading. Percentage errors are potentially miss-leading for long paths. Over a 10 meter episode, the agent can deviate from the oracle path by up-to a meter and still be within $10 \%$ . As a consequence, significant qualitative errors can result in an insignificant quantitative error (see Fig. 6).
+
+Error recovery. Given the near-perfect performance of our agent (on average), we explicitly examine if it is able to recover from its own navigation errors. Fig. 6 column 3 shows several examples of error recovery, including several well executed backtracks (video: https://www.youtube.com/watch?v=a8AugVLSJ50), indicating that the agent is effective at recovering from its own navigation errors. Next, we look at the statistics of non-perfect ${ \mathrm { \ S P L { < } } } 0 . 9 9 $ ) episodes on the longer validation episodes proposed in Chaplot et al. (2019). Non-perfect episodes make up the majority of episodes $54 \%$ , see Fig. 7) with an average SPL of 0.85 $9 9 . 0 \%$ success) – compared to 0.92 SPL $9 9 . 5 \%$ success) over all episodes. Thus there are many episodes where the agent makes significant deviation from the shortest path and reaches the goal (a $15 \%$ deviation on long trajectories $( > 1 0 \mathrm { m } )$ is significant).
+
+![](images/63d58cd2df8d1f4b8cef78bfd5f321c6b966a92873e1041b58c5adc14208a446.jpg)  
+Figure 7: Histogram of SPL for non-perfect (S $\mathrm { ; P L } { < } 0 . 9 9 $ ) episodes.
+
+When does the agent fail? Column 2 in Fig. 6 shows that the agent performs poorly when the ratio of the geodesic distance to goal and euclidean distance to goal. However, the agent is able to eventually overcome this failure mode and reach the goal in most cases.
+
+Row 1 column 1 in Fig. 6 shows that the agent fails or performs poorly when it needs to go slightly up/down stairs. The data-set generation process used in Savva et al. (2019) only guarantees a start and goal pair won’t be on different floors, but there remains a possibility that the agent will need to traverse the stairs slightly. However, these situations are rare, and, in general, the stairs should be avoided. Furthermore, the GPS sensor provides location in 2D, not 3D.
+
+The remaining failure cases of column 1 in Fig. 6 show that a singular location in one environment acts as a sink for the agent (once it enters this location, it is almost never able to leave it). At this location, there is a large hole in the mesh (an entire wall is missing). Utilizing visual encoders that explicitly handle missing values may allow the agent to overcome this failure mode.
+
+Differences from a classical agent. We compare the behavior of our agent with the classical mapand-plan baseline agent proposed in Gupta et al. (2017). This agent achieves 0.92 val (0.89 test) SPL with 0.976 success.8 By comparing and contrasting qualitative behaviors, we can determine what behaviors learning-based methods enable. We make the following observation.
+
+The learned agent is able to recover from unexpected collisions without hurting SPL. The mapand-plan baseline agent incorporates a specific collision recovery behavior where, after repeated collisions, the agent turns around and backs up $1 . 2 5 \mathrm { m }$ . This behavior brings the obstacle into view, maps it, and then allows the agent to create a plan to avoid it. In contrast, our agent is able to navigate around unseen obstacles without such a large impact on SPL. Determining the set of action sequences and heuristics necessary to do this is what learning enables.
+
+# B RELATED WORK CONTINUED
+
+Straggler Effect Mitigation. In supervised learning, the straggler effect is commonly caused by heterogeneous hardware or hardware failures. Chen et al. (2016) propose a pool of $b$ “back-up” workers (there are $N + b$ workers total) and perform the parameter update once $N$ workers finish. In comparison, their method a) requires a parameter server, and b) discards all work done by the stragglers. Chen et al. (2018) propose to dynamically adjust the batch size of each worker such that all workers perform their forward and backward pass in the same amount of time. Our method aims to reduce variance in experience collection times. DD-PPO dynamically adjusts a worker’s batch size as a necessary side-effect of preempting experience collection in on-policy RL.
+
+Distributed Synchronous SGD. Data parallelism is a common paradigm in high performance computing (Hillis & Steele Jr, 1986). In this paradigm, parallelism is achieved by workers performing the same work on different data. This paradigm can be naturally adapted to supervised deep learning (Chen et al., 2016). Works have used this to achieve state-of-the-art results in tasks ranging from computer vision (Goyal et al., 2017; He et al., 2017) to natural language processing (Peters et al., 2018; Devlin et al., 2018; Ott et al., 2019). Furthermore, multiple deep learning frameworks provide simple-to-use wrappers supporting this parallelism model (Paszke et al., 2017; Abadi et al., 2015; Sergeev & Balso, 2018). We adapt this framework to reinforcement learning.
+
+# C AGENT DESIGN
+
+In this section, we outline the exact agent design we use. We break the agent into three components: a visual encoder, a goal encoder, and a navigation policy.
+
+Visual Encoder. Out visual encoder uses one of three different backbones, ResNet50 (He et al., 2016), Squeeze-Excite(SE) (Hu et al., 2018)-ResNeXt50 (Xie et al., 2017), and SE-ResNeXt101. For all backbones, we reduce the number of output channels at each layer by half. We also add a 2x2-AvgPool before each backbone so that the effective resolution is 128x128. Given these modifications, each backbone produces a 1024x4x4 feature map. We then convert this to a 128x4x4 feature map with a 3x3-Conv.
+
+We replace every BatchNorm layer with GroupNorm (Wu & He, 2018) to account for the highly correlated trajectories seen in on-policy RL and massively distributed training.
+
+![](images/b144c3f28a4d7d1f698e1ae00b22e8bf5d00c1424e34b32573287b7200215b45.jpg)  
+Figure 8: Scaling of DD-PPO under homogeneous and heterogeneous workloads for various different values of the percentage of rollouts that are fully completed by optimizing the model. Shading represents a bootstrapped $9 5 \%$ confidence interval.
+
+Goal encoder. Habitat (Savva et al., 2019) provides the vector pointing to the goal in ego-centric polar coordinates. We convert this to magnitude and a unit vector, i.e. [d, $\theta ]$ to [d, $\cos ( \theta )$ , $\sin ( \theta ) ]$ , to account for the discontinuity at the $x$ -axis in polar coordinates. We pass the goal vector to a fully connected layer, resulting in a 32-dimensional representation.
+
+Navigation Policy. Our navigation policy takes the $6 4 \times 4 \times 4$ feature map from the visual encoder, flattens it, and then converts the 2048-d vector to the same size as the hidden size via a fully-connected layer. It then concatenates this vector with output of the goal encoder, and a 32-dimensional embedding of the previous action taken (or the start-token in the case of the first action) and then passes this to a 2-layer LSTM with either a 512-dimensional or 1024-dimensional hidden dimension. The output of the LSTM is used as input to a fully connected layer, resulting in a soft-max distribution of the action space and an estimate of the value function.
+
+# D ADDITIONAL SCALING DETAILS
+
+We use the following procedure for benchmarking the throughput of our proposed DD-PPO: Each optimizer selects 4 scenes at random and then performs the process of collecting experience and optimizing the model based on that experience 10 times. We calculate throughput as the total number of steps of experience collected over the last 5 rollout/optimizing steps divided by the amount of time taken. We repeat this procedure over 10 different random seeds (we use the same random seeds for all variations of number of GPUs and sync-fraction values).
+
+# E DD-PPO IMPLEMENTATION
+
+Utilizing Distributed Data Parallel in supervised learning is straightforward as frameworks such as PyTorch (Paszke et al., 2017) provide a simple wrapper. The recommended way to use these wrappers is to first write training code that runs on a single GPU and then enable distributed training via the wrapper. We follow a similar approach. Given an implementation of
+
+Table 2: Performance (higher is better) of various sensors and agent methods on the Habitat Challenge 2019 (Savva et al., 2019) validation and test splits (checkpoint selected on val). Random, Forward-only, and Goal-follower taken from Savva et al. (2019). Best visual encoder reported for DD-PPO.   
+
+<table><tr><td rowspan="2">Perception</td><td rowspan="2">Method</td><td colspan="2">Validation</td><td colspan="2">Test Standard</td></tr><tr><td>SPL</td><td>Success</td><td>SPL</td><td>Success</td></tr><tr><td rowspan="4">Blind</td><td>Random</td><td>0.02</td><td>0.03</td><td>0.02</td><td>1</td></tr><tr><td>Forward-only</td><td>0.00</td><td>0.00</td><td>0.00</td><td>1</td></tr><tr><td>Goal-follower</td><td>0.23</td><td>0.23</td><td>0.23</td><td>1</td></tr><tr><td>DD-PPO (RL)</td><td>0.729 ± 0.005</td><td>0.973 ± 0.003</td><td>0.676</td><td>0.947</td></tr><tr><td>RGB</td><td>DD-PPO (RL)</td><td>0.929 ± 0.003</td><td>0.991 ± 0.002</td><td>0.920</td><td>0.977</td></tr><tr><td>RGB-D (Depth)</td><td>DD-PPO (RL)</td><td>0.969 ± 0.002</td><td>0.997 ± 0.001</td><td>0.948</td><td>0.980</td></tr></table>
+
+PPO that runs on one GPU we create a decentralized distributed variant by adding gradient synchronization, leveraging highly performant code written for this purpose in popular deep-learning frameworks, e.g. tf.distribute.MirroredStrategy in TensorFlow (Abadi et al., 2015) and torch.nn.parallel.DistributedDataParallel in PyTorch. Note that care must be taken to synchronize any training or rollout statistics between workers – in most cases these can also be synchronized via AllReduce.
+
+We track how many workers have finished the experience collection stage with a distributed keyvalue storage – we use PyTorch’s torch.distributed.TCPStore, however almost any distributed key-value storage would be sufficient.
+
+See Fig. 9 for an example implementation which adds 1) gradient synchronization via torch.nn.parallel.DistributedDataParallel, and 2) preempts stragglers by tracking the number of workers have finished the experience collection stage with a torch.distributed.TCPStore.
+
+See Fig. 10 for a visual depiction of DD-PPO.
+
+# F TRANSFER EXPERIMENTS ADDITIONAL DETAILS
+
+For the transfer learning experiments, we utilize the same PPO hyper-parameters as the PointGoalNav experiments. We use DD-PPO to train with 8 workers on 8 GPUs. We train our agents on Gibson- $^ { . 4 + }$ and evaluate on the Habitat Challenge 2019 Validation scene and starting locations (the goal location is simply discarded).
+
+The ImageNet encoder is trained using the same hyper-parameters and training procedure as Xie et al. (2017) with no data-augmentation.
+
+# G NEURAL CONTROLLER ADDITIONAL DETAILS
+
+The planner for neural controller used in Sec. 6 shares the same architecture as our agent’s policy, but utilizes a 512-d hidden state. It takes as input the previous action of the controller (or the start token), and the output of the visual encoder (which is shared with the controller). The output of the LSTM is then used to produced an estimate of the value function and a 3-dimensional vector specifying the PointGoal in magnitude and unit direction vector format. The magnitude competent is passed through an ELU activation and offset by 0.75. Each component of the unit direction vector is passed through a tanh activation – note that we do not re-normalize this vector have a length of 1 as we find doing so both unnecessary and harder to optimize.
+
+![](images/09aedb209ebc7a0632dd4fb2613d9e9c2ffd4f40b36fe80a2ff6738e63925f03.jpg)  
+Figure 9: Implementation of DD-PPO using PyTorch (Paszke et al., 2017) v1.1 and the NCCL backend. We use SLURM to populate the world rank, world size, and local rank fields.
+
+![](images/548a7b1636c4069fa10ecafdb847ecaa144309bb2924a4be4a9b9235dc06bf67.jpg)  
+Figure 10: Illustration of DD-PPO. Processes collecting experience in environments that are more costly to simulate (stragglers) have their experience collection stage preempted such that other processes do not have to wait for them. Note that we implement the monitor with a simple key-value storage and have processes preempt themselves. Note that the order of processes is irrelevant and done solely for aesthetic purposes.
+
+![](images/feecc15c6e42afc065761488dc668b9410f131d8a32b24553b8830bc13dee11f.jpg)  
+2: big holes or significant texture issues, but good reconstruction
+
+![](images/238990f692e58257508b5a837c9a392c6cfb81286f98ed9ee1f5420f912f4545.jpg)  
+3: small holes, some texture issues, good reconstruction
+
+![](images/cf91fd17d0f210f1d58b53676e9e03f2fd37699470a52f36eddb58fe8baeb331.jpg)
+
+![](images/a5e2e679c00c83f03d488f0cb714e28c3a397dd6e78d46506d2ff9990e6aeca6.jpg)
+
+4: no holes, some texture issues, good reconstruction
+
+Figure 11: Examples of Gibson meshes for a given quality rating from Savva et al. (2019)
+
+![](images/11d28acfcfc73db15805e66b4444d205e36f47717fcf3ee6a14865fb35e1e59b.jpg)  
+Figure 12: Training and validation performance (in SPL; higher is better) of different architectures for Depth agents with $\mathrm { G P S } { + }$ Compass on the Habitat Challenge 2019 (Savva et al., 2019). Gibson (Xia et al., 2018)- $^ { . 4 + }$ refers to the subset of Gibson train scenes with a quality rating of 4 or better. Gibson- $^ { . 4 + }$ and MP3D refers to training on both Gibson- $^ { . 4 + }$ and all of Matterport3D. Gibson$^ { 2 + }$ refers to training on the subset of Gibson train scenes with a quality rating of 2 or better.
+
+![](images/caa878e17343d85a028e79410d5c25d85bad02b79e6abee9f14f813eb0e258da.jpg)  
+Figure 13: Performance vs. Geodesic Distance from start to goal for Blind, RGB, and RGB-D (using Depth only) models trained with DD-PPO on the Habitat Challenge 2019 (Savva et al., 2019) validation split. Bars at the bottom represent the fraction of episodes within each geodesic distance bin.
+
+![](images/99f5a5fc9997891ca3846c5e4879e1ea272013650165d78b65e0d27cdeac44c6.jpg)  
+Figure 14: Performance vs. Geodesic Distance from start to goal for Blind, RGB, and RGB-D (using Depth only) models trained with DD-PPO on the longer and harder validation episodes proposed in Chaplot et al. (2019). Bars at the bottom represent the fraction of episodes within each geodesic distance bin.

parse/train/OmtmcPkkhT/OmtmcPkkhT.md

@@ -0,0 +1,258 @@
+# MULTIPLICATIVE FILTER NETWORKS
+
+Rizal Fathony∗   
+Bosch Center for Artificial Intelligence Pittsburgh, PA   
+rizal.fathony@us.bosch.com
+
+Anit Kumar Sahu∗† Amazon Alexa AI Seattle, WA anit.sahu@gmail.com
+
+Devin Willmott∗   
+Bosch Center for Artificial Intelligence Pittsburgh, PA   
+devin.willmott@us.bosch.com   
+J. Zico Kolter   
+Bosch Center for Artificial Intelligence   
+Carnegie Mellon University   
+Pittsburgh, PA   
+zkolter@cs.cmu.edu
+
+# ABSTRACT
+
+Although deep networks are typically used to approximate functions over high dimensional inputs, recent work has increased interest in neural networks as function approximators for low-dimensional-but-complex functions, such as representing images as a function of pixel coordinates, solving differential equations, or representing signed distance functions or neural radiance fields. Key to these recent successes has been the use of new elements such as sinusoidal nonlinearities or Fourier features in positional encodings, which vastly outperform simple ReLU networks. In this paper, we propose and empirically demonstrate that an arguably simpler class of function approximators can work just as well for such problems: multiplicative filter networks. In these networks, we avoid traditional compositional depth altogether, and simply multiply together (linear functions of) sinusoidal or Gabor wavelet functions applied to the input. This representation has the notable advantage that the entire function can simply be viewed as a linear function approximator over an exponential number of Fourier or Gabor basis functions, respectively. Despite this simplicity, when compared to recent approaches that use Fourier features with ReLU networks or sinusoidal activation networks, we show that these multiplicative filter networks largely outperform or match the performance of these approaches on the domains highlighted in these past works.
+
+# 1 INTRODUCTION
+
+Neural networks are most commonly used to approximate functions over high-dimensional input spaces, such as functions that operate on images or long text sequences. However, there has been a recent growing interest in neural networks used to approximate low-dimensional-but-complex functions: for example, one could represent a continuous image as a function $f : \mathbb { R } ^ { 2 } \to \mathbb { R } ^ { 3 }$ where the input to this function specifies $( x , y )$ coordinates of a location in the image, and the output specifies the RGB value of the pixel at that location. However, two recent papers in particular have argued that specific architectural changes are required to make (fully-connected) deep networks suitable to this task: Sitzmann et al. (2020) employ sinusoidal activation functions within a multi-layer networks (called the SIREN architecture); and Tancik et al. (2020) propose random Fourier features input to a traditional ReLU-based network. Both papers show that the resulting networks can approximate these low-dimensional functions much better than simple feedforward ReLU networks, and achieve striking results in representing fairly complex functions (e.g. 3D signed distance fields or neural radiance fields) with a high degree of fidelity. However, the precise benefit of sinusoidal bases or a first layer of Fourier features seems difficult to characterize, and it remains unclear why such representations work well for these tasks.
+
+In this paper, however, we argue and empirically demonstrate that an arguably simpler class of functions can work as well or better than these previously-proposed networks on this task. Specifically, we propose an architecture we call the multiplicative filter network (MFN). Unlike a traditional multi-layer network that achieves representation power through compositional depth, the MFN instead simply repeatedly applies nonlinear filters (such as a sinusoid or a Gabor wavelet function) to the network’s input, then multiplies together linear functions of these features. The notable advantage of this representation that, owing to the multiplicative properties of Fourier and Gabor filters, the entire function is ultimately just a linear function of (an exponential number of) these Fourier or Gabor features of the input. Indeed, we can express the exact linear form of these MFNs, which can make their analysis considerably simpler than that for deep networks, where compositions of nonlinear activation’s make the entire function difficult to characterize.
+
+In this work, we show that despite this simplicity, the proposed networks often perform as well or better than the previously proposed SIREN or Fourier feature networks. Specifically, we compare our approach on networks with comparable numbers of parameters to the exact benchmarks proposed in the SIREN and Fourier features papers. We show that MFNs achieve better performance deltas when increasing the depth or width of the networks. Despite this, we do emphasize that SIREN networks, in particular, appear to retain some notable advantages over MFNs, such as a bias towards smoother regions in the represented function and its gradients. However, especially given the fact that MFNs ultimately just correspond to a linear Fourier or Wavelet representation of a lowdimensional function, we believe they should be considered a standard benchmark for future work on such problems, to indicate where the compositional depth of typical deep networks can propose a substantial benefit.
+
+# 2 BACKGROUND AND RELATED WORK
+
+Our approach is related to many previous works in Fourier and Wavelet transforms, random Fourier features, and implicit neural representations. We explore the connection among the areas below.
+
+Fourier and Wavelet transforms. Transforming time or space domain signals to frequency domain using transforms such as Fourier and Wavelet transforms have been at the heart of many developments in image processing, signal processing, and computer vision. In particular, the Fourier transform (Bracewell & Bracewell, 1986; Vetterli et al., 2014) and its various forms have found usage in myriad applications, such as spectroscopy, quantum mechanics, signal processing. Wavelet transforms, which in particular aid in multi-scale analysis, have been found to be particularly useful in data compression, JPEG2000 (Rabbani, 2002) being one example.
+
+Random Fourier features. A seminal work by Rahimi & Recht (2008) demonstrates the power of Fourier transform in machine learning applications. They show that simply projecting the original dataset into random Fourier bases vastly improves the expressiveness of models as it approximates kernel computations. Many subsequent works apply the Fourier features and variations (Rahimi & Recht, 2009; Le et al., 2013; Yu et al., 2016) to improve machine learning algorithm performance in many domain areas, including classification (Sun et al., 2018; Rawat et al., 2019), regression (Avron et al., 2017; Brault et al., 2016), clustering (Chitta et al., 2012; Liu et al., 2019), online learning (Lin et al., 2014; Hu et al., 2015), and deep learning (Xue et al., 2019; Mehrkanoon & Suykens, 2018; Rick Chang et al., 2016; Mairal et al., 2014; Jacot et al., 2018; Tancik et al., 2020).
+
+Implicit neural representations. A recent line of work in representing signals as a continuous function parameterized by neural network (instead of using the traditional discrete representation) is gaining popularity. This strategy has been used to represent different objects such as images (Nguyen et al., 2015; Stanley, 2007), shapes (Park et al., 2019; Genova et al., 2019; Chen & Zhang, 2019; Chabra et al., 2020), scenes (Mildenhall et al., 2020; Sitzmann et al., 2019; Jiang et al., 2020; Niemeyer et al., 2020), and textures (Oechsle et al., 2019; Henzler et al., 2020). In most of these applications, the standard neural networks architecture with multi-layer perceptrons and ReLU activation function is often used. Recently, motivated by the success of Fourier transform in machine learning, a few papers have suggested architectural changes that integrate periodic nonlinearities into the network. Mildenhall et al. (2020); Zhong et al. (2020); Tancik et al. (2020) proposed the use of sinusoidal mapping of the input features (Rahimi & Recht, 2008) that uses positional encoding and Gaussian random distribution in the mapping. Others (Klocek et al., 2019; Sitzmann et al., 2020) have proposed the use of sinusoidal activation function within a multi-layer perceptron architecture.
+
+Both of these strategies are demonstrated to vastly improve the results on many object representation tasks.
+
+# 3 MULTIPLICATIVE FILTER NETWORKS
+
+A traditional $k$ -layer deep network $f : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ is typically defined by a recurrence such as:
+
+$$
+\begin{array} { r } { z ^ { ( 1 ) } = x \qquad } \\ { z ^ { ( i + 1 ) } = \sigma \left( W ^ { ( i ) } z ^ { ( i ) } + b ^ { ( i ) } \right) , i = 1 , \dots , k - 1 \qquad } \\ { f ( x ) = W ^ { ( k ) } z ^ { ( k ) } + b ^ { ( k ) } \qquad } \end{array}
+$$
+
+where $\sigma$ denotes a nonlinearity applied elementwise, $W ^ { ( i ) } \in \mathbb { R } ^ { d _ { i + 1 } \times d _ { i } }$ and $b ^ { ( i ) } \in \mathbb { R } ^ { d _ { i + 1 } }$ denote the weight and bias of the ith layer, and $z ^ { ( i ) } \in \mathbb { R } ^ { d _ { i } }$ denotes the hidden unit at layer $i$ . We refer to these networks as compositional depth networks, because each nonlinearity is applied compositionally to outputs of the previous nonlinearity in order to achieve its representational complexity.
+
+The SIREN or Fourier feature networks of (Sitzmann et al., 2020) and Tancik et al. (2020) respectively can be viewed as simple specializations of this structure. In a SIREN network, one uses the sinusoid $\sigma ( x ) = \sin ( x )$ as the nonlinearity, plus proper initialization of the weights and scaling of the input. In a Fourier features network, one replaces the input layer with
+
+$$
+z ^ { ( 1 ) } = \left[ \begin{array} { l } { \sin ( \Omega x + \phi ) } \\ { \cos ( \Omega x + \phi ) } \end{array} \right]
+$$
+
+where $\Omega \in \mathbb { R } ^ { \frac { d _ { 1 } } { 2 } \times n }$ is matrix of random $\mathcal { N } ( 0 , \tau ^ { 2 } )$ variables $\mathit { \Pi } _ { \tau }$ being a hyperparameter of the method), but with the typical ReLU nonlinearities $\sigma ( x ) = \mathrm { R e L U } ( x )$ .
+
+Our proposed multiplicative filter network, in contrast, uses a different recursion that never results in composition of nonlinear functions. Specifically, an MFN is defined via the following recursion
+
+$$
+\begin{array} { r l } & { z ^ { ( 1 ) } = g \left( x ; \theta ^ { ( 1 ) } \right) } \\ & { z ^ { ( i + 1 ) } = \left( W ^ { ( i ) } z ^ { ( i ) } + b ^ { ( i ) } \right) \circ g \left( x ; \theta ^ { ( i + 1 ) } \right) , i = 1 , \dots , k - 1 } \\ & { f ( x ) = W ^ { ( k ) } z ^ { ( k ) } + b ^ { ( k ) } } \end{array}
+$$
+
+where $\circ$ denotes elementwise multiplication, $W ^ { ( i ) } , \boldsymbol { b } ^ { ( i ) } ,$ $z ^ { ( i ) }$ are all defined as above, but where $g : \mathbb { R } ^ { n }  \mathbb { R } ^ { d _ { i } }$ is parameterized by parameters $\theta ^ { ( i ) }$ (the size of $\theta ^ { ( i ) }$ can vary to implicitly define the output dimensions $d _ { i }$ ) and denotes a nonlinear filter applied to the input directly. Of immediate importance here is that in such a network, we never apply a nonlinearity to the output of a previous nonlinearity. All the nonlinearity of the network occurs within the $g$ functions; layers $z ^ { ( i ) }$ , after passing through a linear function, are simply multiplied by new filters of the input. This results in a considerably different type of function that is currently employed by most multi-layer networks, and indeed it is largely only by convention that we refer to such a function as a “network” at all.
+
+We now present two instantiations of the MFN, using sinusoids or a Gabor wavelet as the filter $g$ ; we call these two networks the FOURIERNET and GABORNET respectively. As we show, the crucial property of a function $f$ represented by a FOURIERNET or GABORNET is that the entire function $f$ can also be written as a linear combination of sinusoids and Gabor wavelets of the input respectively (albeit an exponentially large number of such features, but of course also with a highly reduced space of allowable coefficients on this exponential number of terms, since there are only a polynomial number of parameters that define the MFN). Thus, we would claim that the MFN really looks more like a (rich) Fourier or Wavelet representation of the underlying signal, just one that happens to have a similar parameterization as deep networks (and which can be tuned by typical gradient descent methods).
+
+# 3.1 MULTIPLICATIVE FOURIER NETWORKS
+
+As our first instantiation of the MFN, we consider using a simple sinusoidal filter
+
+$$
+g ( x ; \theta ^ { ( i ) } ) = \sin ( \omega ^ { ( i ) } x + \phi ^ { ( i ) } )
+$$
+
+with parameters $\theta ^ { ( i ) } = \{ \omega ^ { ( i ) } \in \mathbb { R } ^ { d _ { i } \times n } , \phi ^ { ( i ) } \in \mathbb { R } ^ { d _ { i } } \}$ . We term such a network the FOURIERNET, as the sinusoidal activation (with arbitrary phase shifts, to represent sine or cosine functions equally) corresponds naturally to a Fourier random feature representation of the entire function.
+
+An immediate and compelling feature of the FOURIERNET, compared to networks based upon composition, is that its output can be directly viewed as a linear function of (an exponential number of) Fourier bases, with a low-rank set of coefficients determined by the parameters of the network. This is conveyed by the following theorem.
+
+Theorem 1. The output of a Fourier Network is given by a linear combination of sinusoidal bases,
+
+$$
+f _ { j } ( x ) = \sum _ { t = 1 } ^ { T } \bar { \alpha } _ { t } \sin ( \bar { \omega } _ { t } x + \bar { \phi } _ { t } ) + \bar { b } ,
+$$
+
+for some coefficients $\bar { \alpha } _ { 1 : T }$ , frequencies $\bar { \omega } _ { 1 : T }$ , phase offsets $\bar { \phi } _ { 1 : T }$ , and bias term $\bar { b }$
+
+In other words, the FOURIERNET represents its final function as a linear combination of traditional Fourier bases, just as do “classical” random Fourier features, for instance. The key element of the proof (given in the appendix) is the fact that for two Fourier filters, with parameters $\omega , \phi$ and $\tau , \psi$ respectively, their elementwise product can be transformed to a sum of the same type of filters
+
+$$
+\sin ( \omega x + \phi ) \circ \sin ( \tau x + \psi ) = { \frac { 1 } { 2 } } \cos \left( ( \omega - \tau ) x + \phi - \psi \right) - { \frac { 1 } { 2 } } \cos \left( ( \omega + \tau ) x + \phi + \psi \right)
+$$
+
+(note that the cosine can be expressed as a sine with a separate phase offset). Moreover, an inspection of the proof also lets us compute the exact coefficients of the linear expansion, as a function of the network parameters. This is shown in the following corollary.
+
+Corollary 1. Let $i _ { 1 } , i _ { 2 } , \ldots , i _ { k - 1 }$ range over all $\textstyle \prod _ { j = 1 } ^ { k - 1 } d _ { j }$ possible indices of each hidden unit of each layer of an MFN, and let $s _ { 2 } , \ldots , s _ { k } \in \{ - 1 , + 1 \}$ range over all $2 ^ { k - 1 }$ possible binary signs; then the expansion of $z _ { i _ { k } } ^ { ( k ) }$ from (2) is given by all the terms
+
+$$
+\begin{array} { l } { \bar { \alpha } = \left\{ \displaystyle \frac { 1 } { 2 ^ { k - 1 } } W _ { i _ { k } , i _ { k - 1 } } ^ { ( k - 1 ) } \cdot \cdot \cdot W _ { i _ { 3 } , i _ { 2 } } ^ { ( 2 ) } W _ { i _ { 2 } , i _ { 1 } } ^ { ( 1 ) } \right\} } \\ { \bar { \omega } = \left\{ s _ { k } \omega _ { i _ { k } } ^ { ( k ) } + \ldots + s _ { 2 } \omega _ { i _ { 2 } } ^ { ( 2 ) } + \omega _ { i _ { 1 } } ^ { ( 1 ) } \right\} } \\ { \bar { \phi } = \left\{ s _ { k } \phi _ { i _ { k } } ^ { ( k ) } + \ldots + s _ { 2 } \phi _ { i _ { 2 } } ^ { ( 2 ) } + \phi _ { i _ { 1 } } ^ { ( 1 ) } + \displaystyle \frac { \pi } { 2 } \sum _ { i = 2 } ^ { k } s _ { k } \right\} . } \end{array}
+$$
+
+with a similar form for terms that begin at the $i > 1$ layer, multiplied by the corresponding $b _ { i _ { j } } ^ { ( j ) }$ term.
+
+This corollary follows simply by inspection of the proof in the appendix, noting that each additional multiplicative layer creates both a positive and negative combination of frequencies in the sinusoid terms, and a multiplication of the corresponding entries of $W$ . In other words, the multiplicative “depth” of the FOURIERNET allows it to represent an exponential number of sinusoidal functions, but with the constraint that the actual number of coefficients on these features is given by a “lowrank” tensor consisting mainly of the coefficients in the $W$ matrices. This expansion also suggests a method for initializing parameters specific to this network in a manner that scales appropriately with the network size. Specifically, however $W ^ { ( i ) } \mathbf { s }$ are initialized (typically random uniform or Gaussian, though here with an additional scaling factor that depends on the relative scale of the input), one should divide these terms by $\sqrt { k }$ , to ensure that the variance of the final frequency $\omega _ { t }$ is independent of the number of layers.
+
+# 3.2 MULTIPLICATIVE GABOR NETWORKS
+
+A well-known deficiency of the pure Fourier bases is that they have global support, and thus may have difficulty representing more local features. A common alternative to these bases is the use of Gabor filter to capture both a frequency and spatial locality component. Specifically, we consider a Gabor filter of the form
+
+$$
+g _ { j } ( x ; \theta ^ { ( i ) } ) = \exp \left( - \frac { \gamma _ { j } ^ { ( i ) } } { 2 } \left\| x - \mu _ { j } ^ { ( i ) } \right\| _ { 2 } ^ { 2 } \right) \sin \left( \omega _ { j } ^ { ( i ) } x + \phi _ { j } ^ { ( i ) } \right)
+$$
+
+![](images/235304ef33435a84be48f88e9c2d75cda774c4b8f31dc7e4bfc65f9c5f8fd31b.jpg)  
+Figure 1: Left: Performance of various models on an image representation task (top row) and three frames from a video representation task (remaining rows). Leftmost column shows ground truth. Right: PSNR of each model in the image reconstruction task for the first 1000 training iterations.
+
+with parameters $\theta ^ { ( i ) } \ = \ \left\{ \gamma _ { 1 : d _ { i } } ^ { ( i ) } \in \mathbb { R } , \mu _ { 1 : d _ { i } } ^ { ( i ) } \in \mathbb { R } ^ { n } , \omega _ { 1 : d _ { i } } ^ { ( i ) } \in \mathbb { R } ^ { n } , \phi _ { 1 : d _ { i } } ^ { ( i ) } \in \mathbb { R } \right\}$ (here $\mu _ { j } ^ { ( i ) }$ denotes the mean of the $j$ th Gabor filter and $\gamma _ { j } ^ { ( i ) }$ denotes the scale term), and where for simplicity we specify the functional form of each $j = 1 \ldots , d _ { i }$ coordinates of the function $g : \mathbb { R } ^ { n }  \mathbb { R } ^ { d _ { i } }$ . We call the MFN using this filter the GaborNet.
+
+As with the FourierNet, a compelling feature of the Gabor network is that the final function $f$ can be represented as a linear combination of Gabor filters. This is captured by the following theorem:
+
+Theorem 2. The output of a Gabor Network is given by a linear combination of Gabor bases,
+
+$$
+f _ { j } ( x ) = \sum _ { t = 1 } ^ { T } \bar { \alpha } _ { t } \exp \left( - \frac { 1 } { 2 } \bar { \gamma } _ { t } \| x - \bar { \mu } _ { t } \| ^ { 2 } \right) \sin ( \bar { \omega } _ { t } x + \bar { \phi } _ { t } ) + \bar { b } ,
+$$
+
+for coefficients $\bar { \alpha } _ { 1 : T }$ , scales $\bar { \gamma } _ { 1 : T }$ , means $\bar { \mu } _ { 1 : T ; }$ , frequencies $\bar { \omega } _ { 1 : T }$ , phase offsets $\bar { \phi } _ { 1 : T }$ , and bias term $\bar { b }$
+
+The proof is given in the appendix, but the basic procedure is the same as the above: using the fact that just like Fourier filters, the product of Gabor filters is also a linear combination of (a different set of) Gabor filters. Likewise, we can also compute the explicit form of the coefficients and for this linear basis expansion, with the explicit form again given in the appendix. One relevant point, though, is how we choose initializations for the $\gamma$ and $\mu$ parameters. Since $\gamma$ effectively acts as an inverse covariance term of a Gaussian, a ${ \mathrm { G a m m a } } ( \alpha , \beta )$ random variable (the conjugate prior of the Gaussian inverse covariance), is a reasonable choice for this parameter. And since the $\bar { \gamma }$ functions in the final linear expansion end up being a sum of the individual random $\gamma ^ { ( i ) }$ terms at each layer, we scale each layer’s $\alpha$ term by $1 / k$ to effectively control this parameter at the final layer. We also simply choose each $\mu ^ { ( i ) }$ to be uniformly distributed over the range of the allowable input space $x$ .
+
+# 4 EXPERIMENTAL RESULTS
+
+We test MFNs on a broad range of representation tasks, showing that the relative simplicity of MFNs improves upon the performance of existing neural representation methods. Our set of experiments draws from those presented in Sitzmann et al. (2020) alongside SIREN (image representation, shape representation, and differential equation experiments) and in Tancik et al. (2020) alongside Fourier feature networks with Gaussian random features, which we call FF Gaussian (image generalization and 3D inverse rendering experiments). In each case, we compare against the set of models tested in the original experiment (generally, either SIREN or FF Gaussian, along with a basic ReLU MLP). A PyTorch implementation of MFN is available at https: //github.com/boschresearch/multiplicative-filter-networks, and full details on hyperparameters and training specifications are available in the appendix.
+
+![](images/c01ca2434e5a670f74ba1b510eb04b01d7fc87e132f9988f555fea2f0dbf540c.jpg)  
+Figure 2: Image generalization samples from the Natural and Text datasets.
+
+# 4.1 IMAGE REPRESENTATION & GENERALIZATION
+
+We first examine the ability of several networks architectures in the task of image representation as described in Section 1, where we fit the network to a function $f : \mathbb { R } ^ { 2 } \to \mathbb { R } ^ { c }$ using a dataset where input coordinates $( x , y )$ corresponding to the output pixel value at those coordinates (with $c = 1$ or 3 for grayscale or RGB images, respectively). To demonstrate this, we construct such a dataset from a $2 5 6 \times 2 5 6$ pixel grayscale image and fit various models using a simple mean squared error (MSE) loss, including SIREN, GABORNET, FOURIERNET, and ReLU MLPs with and without positional encoding (PE). Visual results and a plot of PSNR early in training are shown in (Figure 1). In particular, both FOURIERNET and GABORNET show quicker initial convergence than other architectures. PSNRs after training (Table 1) show that SIREN eventually outperforms FOURIERNET, while GABORNET remains the best model throughout the training. Indeed, after only 1000 training iterations, GABORNET performs reconstruction better than all other models trained for 10 times longer.
+
+Table 1: PSNR of each model’s reconstruction in image and video representation tasks after 10,000 training iterations. For video representation, mean $\pm$ standard deviation over all frames is reported.   
+
+<table><tr><td>Method</td><td colspan="2">PSNR (in dB) Image Video</td></tr><tr><td>FFBasic</td><td>20.13</td><td>24.09 ±1.03</td></tr><tr><td>FFPositional</td><td>40.09</td><td>27.90 ± 0.99</td></tr><tr><td>SIREN</td><td>56.54</td><td>30.58 ± 0.93</td></tr><tr><td>FOURIERNET</td><td>43.32</td><td>27.93 ± 0.91</td></tr><tr><td>GABORNET</td><td>73.98</td><td>29.83 ± 0.71</td></tr></table>
+
+We can broaden the task above to represent video by appending a third dimension to the input: the output corresponding to the input $( x , y , t )$ is the pixel value at $( x , y )$ at frame $t$ . We aim to represent a 300 frame color video with $5 1 2 \times 5 1 2$ resolution in this manner, testing all of the architectures used in the previous experiment. As shown in Figure 1, SIREN and GABORNET are most capable of reproducing fine details of the original video, such as whiskers and and eye color. This is reflected in the PSNR of these reconstructions (Table 1); SIREN performs best with a PSNR over $3 0 \mathrm { d B }$ , while the GABORNET reconstruction comes within 1 dB of SIREN and exhibits much lower variation across frames than any other model.
+
+In addition to representing images, we demonstrate that the MFNs are able to generalize the representation to unseen pixels. We train the networks using only $2 5 \%$ of the image pixels (every other pixel in the width and height dimensions) and evaluate using the complete images. We compare the results of our methods with the Fourier feature networks on two datasets (natural and text images) presented in Tancik et al. (2020). The peak signal-to-noise ratio (PSNR) metric is used to evaluate the performance. As we can see from Table 2, both FOURIERNET and GABOR
+
+Table 2: Image generalization results (mean $\pm$ standard deviation of PSNR).   
+
+<table><tr><td>Method</td><td>Natural</td><td>Text</td></tr><tr><td>FF Basic</td><td>21.61 ± 2.62</td><td>20.50± 2.13</td></tr><tr><td>FF Positional</td><td>25.13 ± 4.01</td><td>26.49 ± 3.11</td></tr><tr><td>FF Gaussian</td><td>25.57 ± 4.18</td><td>30.46 ± 1.97</td></tr><tr><td>FOURIERNET</td><td>26.03 ± 2.77</td><td>31.02 ± 2.04</td></tr><tr><td>GABORNET</td><td>26.18 ± 2.95</td><td>31.19 ± 2.00</td></tr></table>
+
+NET outperform all versions of the Fourier feature networks that uses basic, positional encoding, and random Gaussian features. Some examples of the generated images are presented in Figure
+
+![](images/afd1188c01f127ebf76a845eb95c19c023e2ebf71bc071e7e0b0d2192512d9a1.jpg)  
+Figure 3: Poisson Image Reconstruction: In the left and right figures, an image on the left is reconstructed using gradients and Laplacians respectively; the top row depicts the reconstructed images, while the bottom row indicates the fitted gradients and the fitted Laplacians.
+
+2, with additional images in Appendix B.2. Visually, the MFNs’ generated images also have better quality over the baselines, particularly in Text datasets. Some parts of the text are missing in the baselines images (highlighted with red rectangles in Figure 2), whereas the MFNs completely generate all parts of the text in the images.
+
+# 4.2 DIFFERENTIAL EQUATIONS
+
+In this section, we aim to solve boundary value problems which are supervised by different forms of gradient information from the functional at hand. We first focus on the Poisson equation, where we demonstrate image reconstruction in two settings where the supervision for the model is brought about by gradients and Laplacians respectively. It is worth noting that the model is never presented with real function values. We then focus on two 2nd order differential equations, namely, the Helmholtz equation and the wave equation, where we solve for the wave field, where the network is supervised by a known source function.
+
+We demonstrate image reconstruction using gradients and compare the performance of FOURIERNET and GABORNET with SIREN and ReLU MLP for the Poisson equation. We use the same loss function as in (Sitzmann et al., 2020) in (5). Figures 3a and 3b show that the image is reconstructed successfully when the networks are trained while being supervised by gradients and Laplacians respectively, while ReLU fails spectacularly. Table 3 depicts the losses of each method after 10000 iterations, where it can be seen that GABORNET beats other baselines in terms of performance.
+
+The Helmholtz and wave equations are related to the physical modeling of diffusion and waves and are closely related to a Fourier transform. Hence, we focus our attention on describing the Helmholtz equation. We aim to solve for the wave field and compare the performance of FOURIERNET and GABORNET with SIREN and ReLU MLP. To accommodate for complex-valued solutions, the network is configured to output two values which can be interpreted as the real and imaginary parts. We use the same loss function as used in (Sitzmann et al., 2020) (see Section 4.3 of (Sitzmann et al., 2020) for details). Figure 4 shows the magnitude and the phase of the reconstructed wave front for a single Gaussian source placed at the center of a medium with uniform wave propagation velocity. Table 3 depicts the losses of each method after 50000 iterations, and shows that GABORNET beats FOURIERNET and SIREN in terms of performance, while, as previously shown in (Sitzmann et al., 2020), ReLU MLP fails miserably. Details pertaining to the network and training are relegated to Appendix B.3.
+
+![](images/c062514c30b6bc4c47c8c4562d915d23494ee00f6ecae08bd017078eea666466.jpg)  
+Figure 5: Shape representations from fitting signed distance functions (a). 2D rendered photographs from view synthesis experiments (b).
+
+# 4.3 SHAPE REPRESENTATION VIA SIGNED DISTANCE FIELDS
+
+Recent work (Park et al., 2019) has explored the problem of 3D shape representation with neural architectures, often with surprisingly effective results. This is done by training on raw geometric data: given an oriented point cloud, we seek to learn a function $f : \mathbb { R } ^ { 3 }  \mathbb { R }$ that takes points as input such that the zero level set $\{ x \mid f ( x ) = 0 \}$ of the network accurately represents the surfaces of the shape. Effective training objectives for this task has been explored in considerable depth; we use the training loss presented by Park et al. (2019) and used in Sitzmann et al. (2020), which includes terms involving the
+
+![](images/a5ab3ebc60c65f0b4561c4260f876ee0e4e7b98a41a7ee7956dae03448454241.jpg)  
+Figure 4: Solving Helmholtz equation for a single point source placed at the center of a medium with uniform wave propagation velocity. The top row presents the magnitude, while the bottom row presents the phase.
+
+network’s output (penalties to encourage SDF values near and away from 0 for surface and offsurface points, respectively) as well as its gradients (a term encouraging the gradient to match the surface points’ normals, and a gradient norm penalty throughout the entire 3D space). A complete description of the loss function is left to the appendix.
+
+Figure 5a shows the results of our shape representation task on SIREN, a standard ReLU MLP, and both variants of MFN. As can be seen, the ReLU network’s fails to represent some features of the scene entirely, such as doorways, picture frames, and pillows. Both MFN architectures far outperform this baseline, and are able to reconstruct the room and objects within it to a recognizable degree. However, likely owing to its strong ability to produce smooth outputs and gradients, SIREN is largely able to avoid the visual artifacts on flat surfaces like walls that remain in reconstructions by FOURIERNET and, to a lesser extent, GABORNET.
+
+# 4.4 3D INVERSE RENDERING FOR VIEW SYNTHESIS
+
+In this view synthesis task, we aim to reconstruct 3D representation from the observed 2D photographs. Using the reconstructed 3D representation, we then render 2D images from new viewpoints. We use the “simplified Neural Radiance Fields (NeRF)” task on Lego dataset presented in Tancik et al. (2020). The networks are trained to predict the color (in RGB format) and the volume density at a given 3D location of the viewpoint. Volumetric rendering is then used to re-render the 2D image photograph at the viewpoint. The training loss is computed as the mean squared error between the rendered photograph and the actual 2D image observations. The results in Table 4 show that both MFNs perform competitively in the task. GABORNET has a slight advantage over the baselines and FOURIERNET in terms of the overall PSNR metric. It also arguably produces a slightly better image rendering quality as shown in Figure 5b and other images in Appendix B.5.
+
+# 5 CONCLUSION
+
+We have introduced multiplicative filter networks (MFNs), a class of neural representation architectures that forego the usual compositional notion of network depth in favor of a similarly expressive multiplicative operation. They also admit a natural signal processing interpretation, as in the two instantiations of MFNs, FOURIERNET and GABORNET, which are proven to be exactly equivalent to a linear combination of sinusoidal or Gabor wavelet bases, respectively. In experiments, we show that, despite their simplicity relative to other deep architectures designed for implicit representation, MFNs stand up to or surpass the previous state of the art on a battery of representation tasks.
+
+Table 4: View synthesis results (mean $\pm$ st.d. of PSNR).   
+
+<table><tr><td>Method</td><td>PSNR</td></tr><tr><td>FF Basic</td><td>23.37 ± 0.96</td></tr><tr><td>FF Positional</td><td>25.76 ± 0.79</td></tr><tr><td>FF Gaussian</td><td>25.76 ± 0.92</td></tr><tr><td>FOURIERNET</td><td>25.20 ± 0.72</td></tr><tr><td>GABORNET</td><td>25.81 ± 0.76</td></tr></table>
+
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+        "text": "MULTIPLICATIVE FILTER NETWORKS ",
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+    },
+    {
+        "type": "text",
+        "text": "Rizal Fathony∗   \nBosch Center for Artificial Intelligence Pittsburgh, PA   \nrizal.fathony@us.bosch.com ",
+        "bbox": [
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+    {
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+        "text": "Anit Kumar Sahu∗† Amazon Alexa AI Seattle, WA anit.sahu@gmail.com ",
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+    {
+        "type": "text",
+        "text": "Devin Willmott∗   \nBosch Center for Artificial Intelligence Pittsburgh, PA   \ndevin.willmott@us.bosch.com   \nJ. Zico Kolter   \nBosch Center for Artificial Intelligence   \nCarnegie Mellon University   \nPittsburgh, PA   \nzkolter@cs.cmu.edu ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "Although deep networks are typically used to approximate functions over high dimensional inputs, recent work has increased interest in neural networks as function approximators for low-dimensional-but-complex functions, such as representing images as a function of pixel coordinates, solving differential equations, or representing signed distance functions or neural radiance fields. Key to these recent successes has been the use of new elements such as sinusoidal nonlinearities or Fourier features in positional encodings, which vastly outperform simple ReLU networks. In this paper, we propose and empirically demonstrate that an arguably simpler class of function approximators can work just as well for such problems: multiplicative filter networks. In these networks, we avoid traditional compositional depth altogether, and simply multiply together (linear functions of) sinusoidal or Gabor wavelet functions applied to the input. This representation has the notable advantage that the entire function can simply be viewed as a linear function approximator over an exponential number of Fourier or Gabor basis functions, respectively. Despite this simplicity, when compared to recent approaches that use Fourier features with ReLU networks or sinusoidal activation networks, we show that these multiplicative filter networks largely outperform or match the performance of these approaches on the domains highlighted in these past works. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Neural networks are most commonly used to approximate functions over high-dimensional input spaces, such as functions that operate on images or long text sequences. However, there has been a recent growing interest in neural networks used to approximate low-dimensional-but-complex functions: for example, one could represent a continuous image as a function $f : \\mathbb { R } ^ { 2 } \\to \\mathbb { R } ^ { 3 }$ where the input to this function specifies $( x , y )$ coordinates of a location in the image, and the output specifies the RGB value of the pixel at that location. However, two recent papers in particular have argued that specific architectural changes are required to make (fully-connected) deep networks suitable to this task: Sitzmann et al. (2020) employ sinusoidal activation functions within a multi-layer networks (called the SIREN architecture); and Tancik et al. (2020) propose random Fourier features input to a traditional ReLU-based network. Both papers show that the resulting networks can approximate these low-dimensional functions much better than simple feedforward ReLU networks, and achieve striking results in representing fairly complex functions (e.g. 3D signed distance fields or neural radiance fields) with a high degree of fidelity. However, the precise benefit of sinusoidal bases or a first layer of Fourier features seems difficult to characterize, and it remains unclear why such representations work well for these tasks. ",
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+        "text": "In this paper, however, we argue and empirically demonstrate that an arguably simpler class of functions can work as well or better than these previously-proposed networks on this task. Specifically, we propose an architecture we call the multiplicative filter network (MFN). Unlike a traditional multi-layer network that achieves representation power through compositional depth, the MFN instead simply repeatedly applies nonlinear filters (such as a sinusoid or a Gabor wavelet function) to the network’s input, then multiplies together linear functions of these features. The notable advantage of this representation that, owing to the multiplicative properties of Fourier and Gabor filters, the entire function is ultimately just a linear function of (an exponential number of) these Fourier or Gabor features of the input. Indeed, we can express the exact linear form of these MFNs, which can make their analysis considerably simpler than that for deep networks, where compositions of nonlinear activation’s make the entire function difficult to characterize. ",
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+    {
+        "type": "text",
+        "text": "In this work, we show that despite this simplicity, the proposed networks often perform as well or better than the previously proposed SIREN or Fourier feature networks. Specifically, we compare our approach on networks with comparable numbers of parameters to the exact benchmarks proposed in the SIREN and Fourier features papers. We show that MFNs achieve better performance deltas when increasing the depth or width of the networks. Despite this, we do emphasize that SIREN networks, in particular, appear to retain some notable advantages over MFNs, such as a bias towards smoother regions in the represented function and its gradients. However, especially given the fact that MFNs ultimately just correspond to a linear Fourier or Wavelet representation of a lowdimensional function, we believe they should be considered a standard benchmark for future work on such problems, to indicate where the compositional depth of typical deep networks can propose a substantial benefit. ",
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND AND RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our approach is related to many previous works in Fourier and Wavelet transforms, random Fourier features, and implicit neural representations. We explore the connection among the areas below. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Fourier and Wavelet transforms. Transforming time or space domain signals to frequency domain using transforms such as Fourier and Wavelet transforms have been at the heart of many developments in image processing, signal processing, and computer vision. In particular, the Fourier transform (Bracewell & Bracewell, 1986; Vetterli et al., 2014) and its various forms have found usage in myriad applications, such as spectroscopy, quantum mechanics, signal processing. Wavelet transforms, which in particular aid in multi-scale analysis, have been found to be particularly useful in data compression, JPEG2000 (Rabbani, 2002) being one example. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Random Fourier features. A seminal work by Rahimi & Recht (2008) demonstrates the power of Fourier transform in machine learning applications. They show that simply projecting the original dataset into random Fourier bases vastly improves the expressiveness of models as it approximates kernel computations. Many subsequent works apply the Fourier features and variations (Rahimi & Recht, 2009; Le et al., 2013; Yu et al., 2016) to improve machine learning algorithm performance in many domain areas, including classification (Sun et al., 2018; Rawat et al., 2019), regression (Avron et al., 2017; Brault et al., 2016), clustering (Chitta et al., 2012; Liu et al., 2019), online learning (Lin et al., 2014; Hu et al., 2015), and deep learning (Xue et al., 2019; Mehrkanoon & Suykens, 2018; Rick Chang et al., 2016; Mairal et al., 2014; Jacot et al., 2018; Tancik et al., 2020). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Implicit neural representations. A recent line of work in representing signals as a continuous function parameterized by neural network (instead of using the traditional discrete representation) is gaining popularity. This strategy has been used to represent different objects such as images (Nguyen et al., 2015; Stanley, 2007), shapes (Park et al., 2019; Genova et al., 2019; Chen & Zhang, 2019; Chabra et al., 2020), scenes (Mildenhall et al., 2020; Sitzmann et al., 2019; Jiang et al., 2020; Niemeyer et al., 2020), and textures (Oechsle et al., 2019; Henzler et al., 2020). In most of these applications, the standard neural networks architecture with multi-layer perceptrons and ReLU activation function is often used. Recently, motivated by the success of Fourier transform in machine learning, a few papers have suggested architectural changes that integrate periodic nonlinearities into the network. Mildenhall et al. (2020); Zhong et al. (2020); Tancik et al. (2020) proposed the use of sinusoidal mapping of the input features (Rahimi & Recht, 2008) that uses positional encoding and Gaussian random distribution in the mapping. Others (Klocek et al., 2019; Sitzmann et al., 2020) have proposed the use of sinusoidal activation function within a multi-layer perceptron architecture. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Both of these strategies are demonstrated to vastly improve the results on many object representation tasks. ",
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+    },
+    {
+        "type": "text",
+        "text": "3 MULTIPLICATIVE FILTER NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A traditional $k$ -layer deep network $f : \\mathbb { R } ^ { n }  \\mathbb { R } ^ { m }$ is typically defined by a recurrence such as: ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/e1400faac2929343c3f67560feb80a6f30bc075b7a5d0c85232bc5a5c888ac41.jpg",
+        "text": "$$\n\\begin{array} { r } { z ^ { ( 1 ) } = x \\qquad } \\\\ { z ^ { ( i + 1 ) } = \\sigma \\left( W ^ { ( i ) } z ^ { ( i ) } + b ^ { ( i ) } \\right) , i = 1 , \\dots , k - 1 \\qquad } \\\\ { f ( x ) = W ^ { ( k ) } z ^ { ( k ) } + b ^ { ( k ) } \\qquad } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ denotes a nonlinearity applied elementwise, $W ^ { ( i ) } \\in \\mathbb { R } ^ { d _ { i + 1 } \\times d _ { i } }$ and $b ^ { ( i ) } \\in \\mathbb { R } ^ { d _ { i + 1 } }$ denote the weight and bias of the ith layer, and $z ^ { ( i ) } \\in \\mathbb { R } ^ { d _ { i } }$ denotes the hidden unit at layer $i$ . We refer to these networks as compositional depth networks, because each nonlinearity is applied compositionally to outputs of the previous nonlinearity in order to achieve its representational complexity. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The SIREN or Fourier feature networks of (Sitzmann et al., 2020) and Tancik et al. (2020) respectively can be viewed as simple specializations of this structure. In a SIREN network, one uses the sinusoid $\\sigma ( x ) = \\sin ( x )$ as the nonlinearity, plus proper initialization of the weights and scaling of the input. In a Fourier features network, one replaces the input layer with ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/c345822878416ce679333d6387f21268b7af516e4355dc71f217939edf1acf75.jpg",
+        "text": "$$\nz ^ { ( 1 ) } = \\left[ \\begin{array} { l } { \\sin ( \\Omega x + \\phi ) } \\\\ { \\cos ( \\Omega x + \\phi ) } \\end{array} \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\Omega \\in \\mathbb { R } ^ { \\frac { d _ { 1 } } { 2 } \\times n }$ is matrix of random $\\mathcal { N } ( 0 , \\tau ^ { 2 } )$ variables $\\mathit { \\Pi } _ { \\tau }$ being a hyperparameter of the method), but with the typical ReLU nonlinearities $\\sigma ( x ) = \\mathrm { R e L U } ( x )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our proposed multiplicative filter network, in contrast, uses a different recursion that never results in composition of nonlinear functions. Specifically, an MFN is defined via the following recursion ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/39dd151aeceae6c202291aa952b860e4a86436f83f21cc1064a5f1d8780d2bfb.jpg",
+        "text": "$$\n\\begin{array} { r l } & { z ^ { ( 1 ) } = g \\left( x ; \\theta ^ { ( 1 ) } \\right) } \\\\ & { z ^ { ( i + 1 ) } = \\left( W ^ { ( i ) } z ^ { ( i ) } + b ^ { ( i ) } \\right) \\circ g \\left( x ; \\theta ^ { ( i + 1 ) } \\right) , i = 1 , \\dots , k - 1 } \\\\ & { f ( x ) = W ^ { ( k ) } z ^ { ( k ) } + b ^ { ( k ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\circ$ denotes elementwise multiplication, $W ^ { ( i ) } , \\boldsymbol { b } ^ { ( i ) } ,$ $z ^ { ( i ) }$ are all defined as above, but where $g : \\mathbb { R } ^ { n }  \\mathbb { R } ^ { d _ { i } }$ is parameterized by parameters $\\theta ^ { ( i ) }$ (the size of $\\theta ^ { ( i ) }$ can vary to implicitly define the output dimensions $d _ { i }$ ) and denotes a nonlinear filter applied to the input directly. Of immediate importance here is that in such a network, we never apply a nonlinearity to the output of a previous nonlinearity. All the nonlinearity of the network occurs within the $g$ functions; layers $z ^ { ( i ) }$ , after passing through a linear function, are simply multiplied by new filters of the input. This results in a considerably different type of function that is currently employed by most multi-layer networks, and indeed it is largely only by convention that we refer to such a function as a “network” at all. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now present two instantiations of the MFN, using sinusoids or a Gabor wavelet as the filter $g$ ; we call these two networks the FOURIERNET and GABORNET respectively. As we show, the crucial property of a function $f$ represented by a FOURIERNET or GABORNET is that the entire function $f$ can also be written as a linear combination of sinusoids and Gabor wavelets of the input respectively (albeit an exponentially large number of such features, but of course also with a highly reduced space of allowable coefficients on this exponential number of terms, since there are only a polynomial number of parameters that define the MFN). Thus, we would claim that the MFN really looks more like a (rich) Fourier or Wavelet representation of the underlying signal, just one that happens to have a similar parameterization as deep networks (and which can be tuned by typical gradient descent methods). ",
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+    {
+        "type": "text",
+        "text": "3.1 MULTIPLICATIVE FOURIER NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As our first instantiation of the MFN, we consider using a simple sinusoidal filter ",
+        "bbox": [
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+    {
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+        "img_path": "images/5a431c4a500bae452567ca87edd012951290731e14b3e766d0769d58c6628113.jpg",
+        "text": "$$\ng ( x ; \\theta ^ { ( i ) } ) = \\sin ( \\omega ^ { ( i ) } x + \\phi ^ { ( i ) } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "with parameters $\\theta ^ { ( i ) } = \\{ \\omega ^ { ( i ) } \\in \\mathbb { R } ^ { d _ { i } \\times n } , \\phi ^ { ( i ) } \\in \\mathbb { R } ^ { d _ { i } } \\}$ . We term such a network the FOURIERNET, as the sinusoidal activation (with arbitrary phase shifts, to represent sine or cosine functions equally) corresponds naturally to a Fourier random feature representation of the entire function. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "An immediate and compelling feature of the FOURIERNET, compared to networks based upon composition, is that its output can be directly viewed as a linear function of (an exponential number of) Fourier bases, with a low-rank set of coefficients determined by the parameters of the network. This is conveyed by the following theorem. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Theorem 1. The output of a Fourier Network is given by a linear combination of sinusoidal bases, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c5f759caad666848513ac573e806cd71c9bc2288bcfd02f2c869076319a5eaf0.jpg",
+        "text": "$$\nf _ { j } ( x ) = \\sum _ { t = 1 } ^ { T } \\bar { \\alpha } _ { t } \\sin ( \\bar { \\omega } _ { t } x + \\bar { \\phi } _ { t } ) + \\bar { b } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for some coefficients $\\bar { \\alpha } _ { 1 : T }$ , frequencies $\\bar { \\omega } _ { 1 : T }$ , phase offsets $\\bar { \\phi } _ { 1 : T }$ , and bias term $\\bar { b }$ ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In other words, the FOURIERNET represents its final function as a linear combination of traditional Fourier bases, just as do “classical” random Fourier features, for instance. The key element of the proof (given in the appendix) is the fact that for two Fourier filters, with parameters $\\omega , \\phi$ and $\\tau , \\psi$ respectively, their elementwise product can be transformed to a sum of the same type of filters ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/3f23d45d7e8cc74cc821fe7220ee36ae3720ce1b9f9a4ce39fe75816a00cdb80.jpg",
+        "text": "$$\n\\sin ( \\omega x + \\phi ) \\circ \\sin ( \\tau x + \\psi ) = { \\frac { 1 } { 2 } } \\cos \\left( ( \\omega - \\tau ) x + \\phi - \\psi \\right) - { \\frac { 1 } { 2 } } \\cos \\left( ( \\omega + \\tau ) x + \\phi + \\psi \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(note that the cosine can be expressed as a sine with a separate phase offset). Moreover, an inspection of the proof also lets us compute the exact coefficients of the linear expansion, as a function of the network parameters. This is shown in the following corollary. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Corollary 1. Let $i _ { 1 } , i _ { 2 } , \\ldots , i _ { k - 1 }$ range over all $\\textstyle \\prod _ { j = 1 } ^ { k - 1 } d _ { j }$ possible indices of each hidden unit of each layer of an MFN, and let $s _ { 2 } , \\ldots , s _ { k } \\in \\{ - 1 , + 1 \\}$ range over all $2 ^ { k - 1 }$ possible binary signs; then the expansion of $z _ { i _ { k } } ^ { ( k ) }$ from (2) is given by all the terms ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0071fcb5fea3bd1b4ecde688ba4efe1d119724e9334872de217195e8c6d3b9f0.jpg",
+        "text": "$$\n\\begin{array} { l } { \\bar { \\alpha } = \\left\\{ \\displaystyle \\frac { 1 } { 2 ^ { k - 1 } } W _ { i _ { k } , i _ { k - 1 } } ^ { ( k - 1 ) } \\cdot \\cdot \\cdot W _ { i _ { 3 } , i _ { 2 } } ^ { ( 2 ) } W _ { i _ { 2 } , i _ { 1 } } ^ { ( 1 ) } \\right\\} } \\\\ { \\bar { \\omega } = \\left\\{ s _ { k } \\omega _ { i _ { k } } ^ { ( k ) } + \\ldots + s _ { 2 } \\omega _ { i _ { 2 } } ^ { ( 2 ) } + \\omega _ { i _ { 1 } } ^ { ( 1 ) } \\right\\} } \\\\ { \\bar { \\phi } = \\left\\{ s _ { k } \\phi _ { i _ { k } } ^ { ( k ) } + \\ldots + s _ { 2 } \\phi _ { i _ { 2 } } ^ { ( 2 ) } + \\phi _ { i _ { 1 } } ^ { ( 1 ) } + \\displaystyle \\frac { \\pi } { 2 } \\sum _ { i = 2 } ^ { k } s _ { k } \\right\\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with a similar form for terms that begin at the $i > 1$ layer, multiplied by the corresponding $b _ { i _ { j } } ^ { ( j ) }$ term. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This corollary follows simply by inspection of the proof in the appendix, noting that each additional multiplicative layer creates both a positive and negative combination of frequencies in the sinusoid terms, and a multiplication of the corresponding entries of $W$ . In other words, the multiplicative “depth” of the FOURIERNET allows it to represent an exponential number of sinusoidal functions, but with the constraint that the actual number of coefficients on these features is given by a “lowrank” tensor consisting mainly of the coefficients in the $W$ matrices. This expansion also suggests a method for initializing parameters specific to this network in a manner that scales appropriately with the network size. Specifically, however $W ^ { ( i ) } \\mathbf { s }$ are initialized (typically random uniform or Gaussian, though here with an additional scaling factor that depends on the relative scale of the input), one should divide these terms by $\\sqrt { k }$ , to ensure that the variance of the final frequency $\\omega _ { t }$ is independent of the number of layers. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "3.2 MULTIPLICATIVE GABOR NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A well-known deficiency of the pure Fourier bases is that they have global support, and thus may have difficulty representing more local features. A common alternative to these bases is the use of Gabor filter to capture both a frequency and spatial locality component. Specifically, we consider a Gabor filter of the form ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1130d3ddd0e72e6caf4d58ca79c205c6c3f50ea58087d0c43ebda2e1e01d1d4.jpg",
+        "text": "$$\ng _ { j } ( x ; \\theta ^ { ( i ) } ) = \\exp \\left( - \\frac { \\gamma _ { j } ^ { ( i ) } } { 2 } \\left\\| x - \\mu _ { j } ^ { ( i ) } \\right\\| _ { 2 } ^ { 2 } \\right) \\sin \\left( \\omega _ { j } ^ { ( i ) } x + \\phi _ { j } ^ { ( i ) } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "image",
+        "img_path": "images/235304ef33435a84be48f88e9c2d75cda774c4b8f31dc7e4bfc65f9c5f8fd31b.jpg",
+        "image_caption": [
+            "Figure 1: Left: Performance of various models on an image representation task (top row) and three frames from a video representation task (remaining rows). Leftmost column shows ground truth. Right: PSNR of each model in the image reconstruction task for the first 1000 training iterations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "with parameters $\\theta ^ { ( i ) } \\ = \\ \\left\\{ \\gamma _ { 1 : d _ { i } } ^ { ( i ) } \\in \\mathbb { R } , \\mu _ { 1 : d _ { i } } ^ { ( i ) } \\in \\mathbb { R } ^ { n } , \\omega _ { 1 : d _ { i } } ^ { ( i ) } \\in \\mathbb { R } ^ { n } , \\phi _ { 1 : d _ { i } } ^ { ( i ) } \\in \\mathbb { R } \\right\\}$ (here $\\mu _ { j } ^ { ( i ) }$ denotes the mean of the $j$ th Gabor filter and $\\gamma _ { j } ^ { ( i ) }$ denotes the scale term), and where for simplicity we specify the functional form of each $j = 1 \\ldots , d _ { i }$ coordinates of the function $g : \\mathbb { R } ^ { n }  \\mathbb { R } ^ { d _ { i } }$ . We call the MFN using this filter the GaborNet. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As with the FourierNet, a compelling feature of the Gabor network is that the final function $f$ can be represented as a linear combination of Gabor filters. This is captured by the following theorem: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. The output of a Gabor Network is given by a linear combination of Gabor bases, ",
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/dfb90d1f414706c16a71d93c5380b4c7382976c6e168ee2c83e414264137ac64.jpg",
+        "text": "$$\nf _ { j } ( x ) = \\sum _ { t = 1 } ^ { T } \\bar { \\alpha } _ { t } \\exp \\left( - \\frac { 1 } { 2 } \\bar { \\gamma } _ { t } \\| x - \\bar { \\mu } _ { t } \\| ^ { 2 } \\right) \\sin ( \\bar { \\omega } _ { t } x + \\bar { \\phi } _ { t } ) + \\bar { b } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "for coefficients $\\bar { \\alpha } _ { 1 : T }$ , scales $\\bar { \\gamma } _ { 1 : T }$ , means $\\bar { \\mu } _ { 1 : T ; }$ , frequencies $\\bar { \\omega } _ { 1 : T }$ , phase offsets $\\bar { \\phi } _ { 1 : T }$ , and bias term $\\bar { b }$ ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The proof is given in the appendix, but the basic procedure is the same as the above: using the fact that just like Fourier filters, the product of Gabor filters is also a linear combination of (a different set of) Gabor filters. Likewise, we can also compute the explicit form of the coefficients and for this linear basis expansion, with the explicit form again given in the appendix. One relevant point, though, is how we choose initializations for the $\\gamma$ and $\\mu$ parameters. Since $\\gamma$ effectively acts as an inverse covariance term of a Gaussian, a ${ \\mathrm { G a m m a } } ( \\alpha , \\beta )$ random variable (the conjugate prior of the Gaussian inverse covariance), is a reasonable choice for this parameter. And since the $\\bar { \\gamma }$ functions in the final linear expansion end up being a sum of the individual random $\\gamma ^ { ( i ) }$ terms at each layer, we scale each layer’s $\\alpha$ term by $1 / k$ to effectively control this parameter at the final layer. We also simply choose each $\\mu ^ { ( i ) }$ to be uniformly distributed over the range of the allowable input space $x$ . ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We test MFNs on a broad range of representation tasks, showing that the relative simplicity of MFNs improves upon the performance of existing neural representation methods. Our set of experiments draws from those presented in Sitzmann et al. (2020) alongside SIREN (image representation, shape representation, and differential equation experiments) and in Tancik et al. (2020) alongside Fourier feature networks with Gaussian random features, which we call FF Gaussian (image generalization and 3D inverse rendering experiments). In each case, we compare against the set of models tested in the original experiment (generally, either SIREN or FF Gaussian, along with a basic ReLU MLP). A PyTorch implementation of MFN is available at https: //github.com/boschresearch/multiplicative-filter-networks, and full details on hyperparameters and training specifications are available in the appendix. ",
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+        "img_path": "images/c01ca2434e5a670f74ba1b510eb04b01d7fc87e132f9988f555fea2f0dbf540c.jpg",
+        "image_caption": [
+            "Figure 2: Image generalization samples from the Natural and Text datasets. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.1 IMAGE REPRESENTATION & GENERALIZATION ",
+        "text_level": 1,
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+        "text": "We first examine the ability of several networks architectures in the task of image representation as described in Section 1, where we fit the network to a function $f : \\mathbb { R } ^ { 2 } \\to \\mathbb { R } ^ { c }$ using a dataset where input coordinates $( x , y )$ corresponding to the output pixel value at those coordinates (with $c = 1$ or 3 for grayscale or RGB images, respectively). To demonstrate this, we construct such a dataset from a $2 5 6 \\times 2 5 6$ pixel grayscale image and fit various models using a simple mean squared error (MSE) loss, including SIREN, GABORNET, FOURIERNET, and ReLU MLPs with and without positional encoding (PE). Visual results and a plot of PSNR early in training are shown in (Figure 1). In particular, both FOURIERNET and GABORNET show quicker initial convergence than other architectures. PSNRs after training (Table 1) show that SIREN eventually outperforms FOURIERNET, while GABORNET remains the best model throughout the training. Indeed, after only 1000 training iterations, GABORNET performs reconstruction better than all other models trained for 10 times longer. ",
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+        "table_caption": [
+            "Table 1: PSNR of each model’s reconstruction in image and video representation tasks after 10,000 training iterations. For video representation, mean $\\pm$ standard deviation over all frames is reported. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"2\">PSNR (in dB) Image Video</td></tr><tr><td>FFBasic</td><td>20.13</td><td>24.09 ±1.03</td></tr><tr><td>FFPositional</td><td>40.09</td><td>27.90 ± 0.99</td></tr><tr><td>SIREN</td><td>56.54</td><td>30.58 ± 0.93</td></tr><tr><td>FOURIERNET</td><td>43.32</td><td>27.93 ± 0.91</td></tr><tr><td>GABORNET</td><td>73.98</td><td>29.83 ± 0.71</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We can broaden the task above to represent video by appending a third dimension to the input: the output corresponding to the input $( x , y , t )$ is the pixel value at $( x , y )$ at frame $t$ . We aim to represent a 300 frame color video with $5 1 2 \\times 5 1 2$ resolution in this manner, testing all of the architectures used in the previous experiment. As shown in Figure 1, SIREN and GABORNET are most capable of reproducing fine details of the original video, such as whiskers and and eye color. This is reflected in the PSNR of these reconstructions (Table 1); SIREN performs best with a PSNR over $3 0 \\mathrm { d B }$ , while the GABORNET reconstruction comes within 1 dB of SIREN and exhibits much lower variation across frames than any other model. ",
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+    {
+        "type": "text",
+        "text": "In addition to representing images, we demonstrate that the MFNs are able to generalize the representation to unseen pixels. We train the networks using only $2 5 \\%$ of the image pixels (every other pixel in the width and height dimensions) and evaluate using the complete images. We compare the results of our methods with the Fourier feature networks on two datasets (natural and text images) presented in Tancik et al. (2020). The peak signal-to-noise ratio (PSNR) metric is used to evaluate the performance. As we can see from Table 2, both FOURIERNET and GABOR",
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+        "table_caption": [
+            "Table 2: Image generalization results (mean $\\pm$ standard deviation of PSNR). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Natural</td><td>Text</td></tr><tr><td>FF Basic</td><td>21.61 ± 2.62</td><td>20.50± 2.13</td></tr><tr><td>FF Positional</td><td>25.13 ± 4.01</td><td>26.49 ± 3.11</td></tr><tr><td>FF Gaussian</td><td>25.57 ± 4.18</td><td>30.46 ± 1.97</td></tr><tr><td>FOURIERNET</td><td>26.03 ± 2.77</td><td>31.02 ± 2.04</td></tr><tr><td>GABORNET</td><td>26.18 ± 2.95</td><td>31.19 ± 2.00</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "NET outperform all versions of the Fourier feature networks that uses basic, positional encoding, and random Gaussian features. Some examples of the generated images are presented in Figure ",
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+    {
+        "type": "image",
+        "img_path": "images/afd1188c01f127ebf76a845eb95c19c023e2ebf71bc071e7e0b0d2192512d9a1.jpg",
+        "image_caption": [
+            "Figure 3: Poisson Image Reconstruction: In the left and right figures, an image on the left is reconstructed using gradients and Laplacians respectively; the top row depicts the reconstructed images, while the bottom row indicates the fitted gradients and the fitted Laplacians. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "2, with additional images in Appendix B.2. Visually, the MFNs’ generated images also have better quality over the baselines, particularly in Text datasets. Some parts of the text are missing in the baselines images (highlighted with red rectangles in Figure 2), whereas the MFNs completely generate all parts of the text in the images. ",
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+    {
+        "type": "text",
+        "text": "4.2 DIFFERENTIAL EQUATIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we aim to solve boundary value problems which are supervised by different forms of gradient information from the functional at hand. We first focus on the Poisson equation, where we demonstrate image reconstruction in two settings where the supervision for the model is brought about by gradients and Laplacians respectively. It is worth noting that the model is never presented with real function values. We then focus on two 2nd order differential equations, namely, the Helmholtz equation and the wave equation, where we solve for the wave field, where the network is supervised by a known source function. ",
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+    {
+        "type": "text",
+        "text": "We demonstrate image reconstruction using gradients and compare the performance of FOURIERNET and GABORNET with SIREN and ReLU MLP for the Poisson equation. We use the same loss function as in (Sitzmann et al., 2020) in (5). Figures 3a and 3b show that the image is reconstructed successfully when the networks are trained while being supervised by gradients and Laplacians respectively, while ReLU fails spectacularly. Table 3 depicts the losses of each method after 10000 iterations, where it can be seen that GABORNET beats other baselines in terms of performance. ",
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+    {
+        "type": "text",
+        "text": "The Helmholtz and wave equations are related to the physical modeling of diffusion and waves and are closely related to a Fourier transform. Hence, we focus our attention on describing the Helmholtz equation. We aim to solve for the wave field and compare the performance of FOURIERNET and GABORNET with SIREN and ReLU MLP. To accommodate for complex-valued solutions, the network is configured to output two values which can be interpreted as the real and imaginary parts. We use the same loss function as used in (Sitzmann et al., 2020) (see Section 4.3 of (Sitzmann et al., 2020) for details). Figure 4 shows the magnitude and the phase of the reconstructed wave front for a single Gaussian source placed at the center of a medium with uniform wave propagation velocity. Table 3 depicts the losses of each method after 50000 iterations, and shows that GABORNET beats FOURIERNET and SIREN in terms of performance, while, as previously shown in (Sitzmann et al., 2020), ReLU MLP fails miserably. Details pertaining to the network and training are relegated to Appendix B.3. ",
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+        "page_idx": 6
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+    {
+        "type": "image",
+        "img_path": "images/c062514c30b6bc4c47c8c4562d915d23494ee00f6ecae08bd017078eea666466.jpg",
+        "image_caption": [
+            "Figure 5: Shape representations from fitting signed distance functions (a). 2D rendered photographs from view synthesis experiments (b). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3 SHAPE REPRESENTATION VIA SIGNED DISTANCE FIELDS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Recent work (Park et al., 2019) has explored the problem of 3D shape representation with neural architectures, often with surprisingly effective results. This is done by training on raw geometric data: given an oriented point cloud, we seek to learn a function $f : \\mathbb { R } ^ { 3 }  \\mathbb { R }$ that takes points as input such that the zero level set $\\{ x \\mid f ( x ) = 0 \\}$ of the network accurately represents the surfaces of the shape. Effective training objectives for this task has been explored in considerable depth; we use the training loss presented by Park et al. (2019) and used in Sitzmann et al. (2020), which includes terms involving the ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/a5ab3ebc60c65f0b4561c4260f876ee0e4e7b98a41a7ee7956dae03448454241.jpg",
+        "image_caption": [
+            "Figure 4: Solving Helmholtz equation for a single point source placed at the center of a medium with uniform wave propagation velocity. The top row presents the magnitude, while the bottom row presents the phase. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "network’s output (penalties to encourage SDF values near and away from 0 for surface and offsurface points, respectively) as well as its gradients (a term encouraging the gradient to match the surface points’ normals, and a gradient norm penalty throughout the entire 3D space). A complete description of the loss function is left to the appendix. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 5a shows the results of our shape representation task on SIREN, a standard ReLU MLP, and both variants of MFN. As can be seen, the ReLU network’s fails to represent some features of the scene entirely, such as doorways, picture frames, and pillows. Both MFN architectures far outperform this baseline, and are able to reconstruct the room and objects within it to a recognizable degree. However, likely owing to its strong ability to produce smooth outputs and gradients, SIREN is largely able to avoid the visual artifacts on flat surfaces like walls that remain in reconstructions by FOURIERNET and, to a lesser extent, GABORNET. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 3D INVERSE RENDERING FOR VIEW SYNTHESIS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this view synthesis task, we aim to reconstruct 3D representation from the observed 2D photographs. Using the reconstructed 3D representation, we then render 2D images from new viewpoints. We use the “simplified Neural Radiance Fields (NeRF)” task on Lego dataset presented in Tancik et al. (2020). The networks are trained to predict the color (in RGB format) and the volume density at a given 3D location of the viewpoint. Volumetric rendering is then used to re-render the 2D image photograph at the viewpoint. The training loss is computed as the mean squared error between the rendered photograph and the actual 2D image observations. The results in Table 4 show that both MFNs perform competitively in the task. GABORNET has a slight advantage over the baselines and FOURIERNET in terms of the overall PSNR metric. It also arguably produces a slightly better image rendering quality as shown in Figure 5b and other images in Appendix B.5. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have introduced multiplicative filter networks (MFNs), a class of neural representation architectures that forego the usual compositional notion of network depth in favor of a similarly expressive multiplicative operation. They also admit a natural signal processing interpretation, as in the two instantiations of MFNs, FOURIERNET and GABORNET, which are proven to be exactly equivalent to a linear combination of sinusoidal or Gabor wavelet bases, respectively. In experiments, we show that, despite their simplicity relative to other deep architectures designed for implicit representation, MFNs stand up to or surpass the previous state of the art on a battery of representation tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/61b8b60fd88e6db39fedfdf68fecb1533e24755108a577d58fdab6a371755321.jpg",
+        "table_caption": [
+            "Table 4: View synthesis results (mean $\\pm$ st.d. of PSNR). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>PSNR</td></tr><tr><td>FF Basic</td><td>23.37 ± 0.96</td></tr><tr><td>FF Positional</td><td>25.76 ± 0.79</td></tr><tr><td>FF Gaussian</td><td>25.76 ± 0.92</td></tr><tr><td>FOURIERNET</td><td>25.20 ± 0.72</td></tr><tr><td>GABORNET</td><td>25.81 ± 0.76</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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parse/train/rk9eAFcxg/rk9eAFcxg.md

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+# VARIATIONAL RECURRENT ADVERSARIALDEEP DOMAIN ADAPTATION
+
+Sanjay Purushotham\*, Wilka Carvalho\*, Tanachat Nilanon, Yan Liu
+
+Department of Computer Science   
+University of Southern California   
+Los Angeles, CA 90089, USA   
+{spurusho,wcarvalh,nilanon,yanliu.cs}@usc.edu
+
+# ABSTRACT
+
+We study the problem of learning domain invariant representations for time series data while transferring the complex temporal latent dependencies between domains. Our model termed as Variational Recurrent Adversarial Deep Domain Adaptation (VRADA) is built atop a variational recurrent neural network (VRNN) and trains adversarially to capture complex temporal relationships that are domain-invariant. This is (as far as we know) the first to capture and transfer temporal latent dependencies of multivariate time-series data. Through experiments on real-world multivariate healthcare time-series datasets, we empirically demonstrate that learning temporal dependencies helps our model’s ability to create domain-invariant representations, allowing our model to outperform current state-of-the-art deep domain adaptation approaches.
+
+# 1 INTRODUCTION
+
+Many real-world applications require effective machine learning algorithms that can learn invariant representations across related time-series datasets. For example, precision medicine for patients of various age groups, mobile application recommendation for users based on locations, and so on. In these examples, while the domains (i.e. age group and location) may vary, there exist common predictive patterns that can aid in inferring knowledge from one domain to another. More often than not, some domains have a significantly larger number of observations than others (e.g., respiratory failure in adults vs. children). Therefore effective domain adaption of time-series data is in great demand.
+
+The general approach to tackling domain adaptation has been explored under many facets which include reducing the domain discrepancy between the source and target domains(Ben-David et al. (2007)), instance re-weighting (Jiang & Zhai (2007)), subspace alignment (Fernando et al. (2013)), and deep learning (Tzeng et al. (2015); Ganin & Lempitsky (2014)). Many of these approaches work very well for non-sequential data but are not suitable for multivariate time-series data as they do not usually capture the temporal dependencies present in the data. For sequential data, earlier work has successfully used dynamic Bayesian Networks(Huang & Yates (2009)) and Recurrent Neural Networks (Socher et al. (2011)) to learn latent feature representations which were domaininvariant. Unfortunately, these works were not flexible enough to model non-linear dynamics or did not explicitly capture and transfer the complex latent dependencies needed to perform domain adaptation of time-series data.
+
+In this paper, we address this problem with a model that learns temporal latent dependencies (i.e. dependencies between the latent variables across timesteps) that can be transferred across domains that experience different distributions in their features. We draw inspiration from the Variational Recurrent Neural Network (Chung et al. (2016)) and use variational methods to produce a latent representation that captures underlying temporal latent dependencies. Motivated by the theory of domain adaptation (Ben-David et al. (2010)), we perform adversarial training on this representation
+
+![](images/5994455d34a7df0125ff448c1f0d78bf48aa6fe58d9be5e243ebb4424444504a.jpg)  
+Figure 1: A Story of Temporal Dependency and Domain Invariance
+
+t-SNE projections for the latent representations of DNN, R-DANN, and our VRADA model. We show adaption   
+from Adult-AHRF to Child-AHRF data. Source data is represented with red circles and target data with blue   
+circles. From left to right, one can see that domain adaptation results in mixing the source and target domain data distributions. We can also see a story of how encoding more temporal dependency into the latent   
+representation induces more domain-invariant representations. As models capture more underlying factors of   
+variation, post domain adaptation representations gradually smoothen and become evenly dispersed, indicating that temporal dependency acts synergestically with domain adaptation.
+
+similarly to the Domain Adversarial Neural Network (DANN) (Ganin et al. (2016)) to make the representations invariant across domains. We call our model the Variational Recurrent Adversarial Deep Domain Adaptation (VRADA) model. As far as we know, this is the first model capable of accomplishing unsupervised domain adaptation while transferring temporal latent dependencies for complex multivariate time-series data. Figure 1 shows an example of the domain invariant representations learned by different deep learning models including our VRADA model. From this figure, we can see that our model (VRADA) shows better mixing of the domain distributions than the competing models indicating that it learns better domain invariant representations.
+
+In order to prove the efficacy of our model, we perform domain adaptation using real-world healthcare time-series data. We choose healthcare data for two primary reasons. (1) Currently, a standard protocol in healthcare is to build, evaluate, and deploy machine learning models for particular datasets that may perform poorly on unseen datasets with different distributions. For example, models built around patient data from particular age groups perform poorly on other age groups because the features used to train the models have different distributions across the groups (Alemayehu & Warner (2004); Lao et al. (2004); Seshamani & Gray (2004)). Knowledge learned from one group is not transferrable to the other group. Domain adaptation seems like a natural solution to this problem as knowledge needs to be transferred across domains which share features that exhibit different distributions. (2) Healthcare data has multiple attributes recorded per patient visit, and it is longitudinal and episodic in nature. Thus, healthcare data is a suitable platform on which to study a model which seeks to capture complex temporal representations and transfer this knowledge across domains.
+
+The rest of the paper is structured as follows. In the following section, we briefly discuss the current state-of-the-art deep domain adaptation approaches. Afterwards, we present our model mathematically, detailing how it simultaneously learns to capture temporal latent dependencies and create domain-invariant representations. In Section 4, we compare and contrast the performance of proposed approach with other approaches on two real-world health care datasets, and provide analysis on our domain-invariant representations.
+
+# 2 RELATED WORK
+
+Domain adaptation is a specific instance of transfer learning in which the feature spaces are shared but their marginal distributions are different. A good survey on the two has been done in several previous works (Pan & Yang (2009); Jiang (2008); Patel et al. (2015)). Domain adaptation has been thoroughly studied in computer vision(Saenko et al. (2010); Gong et al. (2012); Fernando et al. (2013)) and natural language processing (NLP) (Blitzer (2007); Foster et al. (2010)) applications. Recently, the deep learning paradigm has become popular in domain adaptation (Chen et al. (2012); Tzeng et al. (2015); Yang & Eisenstein; Long & Wang (2015)) due to its ability to learn rich, flexible, non-linear domain-invariant representations. Here, we briefly discuss two deep domain adaptation approaches which are closely related to our proposed model. Domain Adversarial Neural Networks (DANN)
+
+![](images/cdec1408957e6500c20e8bea104c23b625844077c8fbf34a2b1ef94f9c4d6fa5.jpg)  
+Figure 2: Block diagram of VRADA. Blue lines show the inference process, $q _ { \theta _ { e } } \left( z _ { t } | \boldsymbol { x } _ { \le t } , \boldsymbol { z } _ { < t } \right)$ . Brown lines show the generation process, $p _ { \theta _ { g } } ( x _ { t } | \boldsymbol { z } _ { \le t } , \boldsymbol { x } _ { < t } )$ . Red lines show the recurrence process where $h _ { t }$ is informed by $h _ { t - 1 }$ , which is informed by $z _ { t - 1 }$ and $x _ { t - 1 }$ . Black lines indicate classification.
+
+(Ganin et al. (2016)) is a deep domain adaptation model which uses two core components to create domain-invariant representations, a feature extractor that produces the data’s latent representation, and an adversarial domain labeler that attempts to classify that data’s domain to help the feature extractor produce latent representations which are domain-invariant. In Louizos et al. (2015), the authors propose Variational Fair AutoEncoder, which uses Variational Autoencoding architecture (Kingma & Welling (2013)) to learn latent representations where most of the information about certain known factors of variation are purged from the representation while still retaining as much information about the data as possible. While, these deep learning approaches learn domain-invariant representations, they fail to capture and transfer the underlying complex temporal latent relationships from one domain to another as they use convolutional or feed forward neural networks which we claim are not suitable for multivariate time-series data.
+
+Other works such as Huang & Yates (2009); Xiao & Guo (2013) have used distributed representations for domain adaptation in NLP sequence labeling tasks. However, they either induce hidden states as latent features using dynamic Bayesian networks (DBNs) or learn generalizable distributed representations of words using Recurrent Neural Networks (RNN) (Socher et al. (2011)) to enable domain adaptation. These works either model the highly non-linear dynamics, as one can with RNN, or capture the complex latent dependencies present in sequential data, as one can with DBNs, but not both. To overcome the challenges of DBNs and RNNs, Variational Recurrent Neural Network (VRNN)( Chung et al. (2016)) was proposed recently to capture the complex relationship between the underlying hidden factors of variation and the output variables at different time-steps. The VRNN uses Variational Autoencoders (VAEs)( Kingma & Welling (2013); Goodfellow et al. (2016)) at each time-step to learn a complex relationship between the latent hidden factors across time-steps. Like the VAE, its latent variable is parametric. Combined, these things make it well-suited for multimodal sequential data such as multivariate time-series. In the following section, we discuss our approach, Variational Adversarial Deep Domain Adaptation (VRADA), which uses a VRNN to model and transfer complex domain-invariant temporal latent relationships for unsupervised domain adaptation of multivariate time-series.
+
+# 3 VARIATIONAL RECURRENT ADVERSARIAL DEEP DOMAIN ADAPTATION
+
+In this section, we present our Variational Recurrent Adversarial Deep Domain Adaptation (VRADA) model for the purpose of capturing and transferring temporal latent dependencies across domains via domain-invariant representations. First, we introduce the notations used in this paper and then discuss our VRADA model in detail.
+
+# 3.1 NOTATIONS
+
+Let us denote a multivariate variable-length time series with $N$ data samples as $\{ \mathbf { x ^ { i } } = ( x _ { t } ^ { i } ) _ { t = 1 } ^ { T ^ { i } } \} _ { i = 1 } ^ { N }$ where . (Note: in our experiments, for all data samples , but for generality we maintain $T ^ { i }$ ). We denote $\{ \mathbf { x } _ { \mathcal { S } } ^ { \mathbf { i } } \} _ { i = 1 } ^ { n }$ as source domain data and $\{ \dot { \mathbf { x } } _ { \mathcal { T } } ^ { \mathbf { i } } \} _ { i = n + 1 } ^ { N }$ as target domain data. We assume that each source domain data sample $\mathbf { x } _ { \mathcal { S } } ^ { \mathbf { i } }$ comes with $L$ labels $y _ { i } \in \{ 0 , 1 \} ^ { L }$ (for example, these labels may correspond to a clinical outcome such as mortality or ICD9 diagnosis codes), while target domain has no labeled data samples. We assign a domain label $d _ { i } \in \{ 0 , 1 \}$ to each data sample to indicate if it comes from the source or target domain. $d _ { i }$ will be used for adversarial training.
+
+# 3.2 VRADA
+
+The block diagram of our VRADA model is shown in Figure 2. To explicitly model the dependencies between the latent random variable across time steps, the VRADA model utilizes Variational Recurrent Neural Networks (VRNN) (Chung et al. (2016)). The VRNN effectively contains a Variational AutoEncoders (Kingma & Welling (2013)) at every time step, all of which are conditioned on previous auto-encoders via the hidden state $h _ { t - 1 }$ of an RNN, such as an LSTM (Hochreiter & Schmidhuber (1997)). Therefore, for each time-step of $\ v { x } _ { t } ^ { i }$ , we infer a latent random variable $z _ { t } ^ { i }$ via
+
+$$
+\begin{array} { r } { z _ { t } ^ { i } | x _ { t } ^ { i } \sim \mathcal { N } ( \mu _ { z , t } , \mathrm { d i a g } ( \sigma _ { z , t } ) ) , \quad \mathrm { w h e r e } \ [ \mu _ { z , t } , \sigma _ { z , t } ] = \varphi _ { \tau } ^ { e n c } ( \varphi _ { \tau } ^ { x } ( x _ { t } ^ { i } ) , h _ { t - 1 } ) } \end{array}
+$$
+
+with prior
+
+$$
+z _ { t } ^ { i } \sim \mathcal { N } ( \mu _ { 0 , t } , \mathrm { d i a g } ( \sigma _ { 0 , t } ) ) , \quad \mathrm { w h e r e } \ [ \mu _ { 0 , t } , \sigma _ { 0 , t } ] = \varphi _ { \tau } ^ { p r i o r } ( h _ { t - 1 } )
+$$
+
+where $\mu _ { * , t } , \sigma _ { * , t }$ denote parameters of a generating distribution, and $\boldsymbol { \varphi } _ { \tau } ^ { \ast }$ can be any highly flexible function such as deep neural networks. For each $z _ { t } ^ { i } , x _ { t } ^ { i }$ is generated via
+
+$$
+x _ { t } ^ { i } | z _ { t } ^ { i } \sim { \mathcal { N } } ( \mu _ { x , t } , \mathrm { d i a g } ( \sigma _ { x , t } ) ) , \quad { \mathrm { w h e r e ~ } } [ \mu _ { x , t } , \sigma _ { x , t } ] = \varphi _ { \tau } ^ { d e c } ( \varphi _ { \tau } ^ { z } ( z _ { t } ^ { i } ) , h _ { t - 1 } )
+$$
+
+and learned by optimizing the VRNN objective function:
+
+$$
+\overset { \cdot } { \underset { t = t } { \cdot } } ( x _ { t } ^ { i } ; \theta _ { \epsilon } , \theta _ { g } ) = E _ { q _ { \theta _ { \epsilon } } ( z _ { \leq T ^ { i } } ^ { i } | x _ { \leq T ^ { i } } ^ { i } ) } [ \underset { t = 1 } { \overset { T ^ { i } } { \sum } } ( - D ( q _ { \theta _ { \epsilon } } ( z _ { t } ^ { i } | x _ { \leq t } ^ { i } , z _ { < t } ^ { i } ) | | p ( z _ { t } ^ { i } | x _ { < t } ^ { i } , z _ { < t } ^ { i } ) ) + \log p _ { \theta _ { g } } ( x _ { t } ^ { i } | z _ { \leq t } ^ { i } , x _ { < t } ^ { i } ) ) ] .
+$$
+
+where $q _ { \theta _ { e } } \big ( z _ { t } ^ { i } | \boldsymbol { x } _ { \le t } ^ { i } , \boldsymbol { z } _ { < t } ^ { i } \big )$ is the inference model, $p ( z _ { t } ^ { i } | x _ { < t } ^ { i } , z _ { < t } ^ { i } )$ is the prior, $p _ { \theta _ { g } } ( x _ { t } ^ { i } | \boldsymbol { z } _ { \le t } ^ { i } , x _ { < t } ^ { i } )$ is the generative model, $\theta _ { e }$ is the parameters of the VRNN’s encoder, $\theta _ { g }$ the parameters of the VRNN’s decoder, and $D ( \cdot | | \cdot )$ refers to KL-Divergence. Note: $z _ { \le T }$ refers to the set of all $z _ { t }$ such that $t \leq T$ , likewise for $z _ { < T }$ . For each $\mathbf { x ^ { i } }$ , we use $\tilde { z } ^ { i } \sim q _ { \theta _ { e } } ( z _ { T ^ { i } } ^ { i } | x _ { \le T ^ { i } } ^ { i } , z _ { < T ^ { i } } ^ { i } )$ as our feature representation for source domain classification task since it captures temporal latent dependencies across the time-steps. Training the VRNN for the source domain classification involves solving the following optimization:
+
+$$
+\operatorname* { m i n } _ { \theta _ { e } , \theta _ { g } , \theta _ { y } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { 1 } { T ^ { i } } \mathcal { L } _ { r } ( \mathbf { x ^ { i } } ; \theta _ { e } , \theta _ { g } ) + \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathcal { L } _ { y } ( \mathbf { x ^ { i } } ; \theta _ { y } , \theta _ { e } ) + \lambda \mathcal { R } ( \theta _ { e } )
+$$
+
+where $\mathcal { R } ( \theta _ { e } )$ is a regularizer for the parameters of VRNN encoder (which is also the feature extractor of VRADA) with a tuning hyperparameter $\lambda$ .
+
+As we are interested in achieving domain adaptation via the latent representation $\tilde { z } ^ { i }$ (i.e. to make $\tilde { z } ^ { i }$ domain-invariant), we can adversarially train the above objective function (equation 1) by employing the domain adaptation idea proposed in Ganin et al. (2016). Let $G _ { y } ( \tilde { z } ^ { i } ; \theta _ { y } )$ and $G _ { d } ( \tilde { z } ^ { i } ; \theta _ { d } )$ represent the source label classifier (to predict source labels $y _ { i }$ ) and domain label classifier (to predict domain labels $d _ { i }$ ) respectively with parameters $\theta _ { y }$ and $\theta _ { d }$ for a given input $\tilde { z } ^ { i }$ . Here, $G _ { y } ( . )$ and $G _ { d } ( . )$ can be deep neural networks. Let us denote their loss functions respectively as
+
+$$
+\mathcal { L } _ { y } ( \mathbf { x ^ { i } } ; \theta _ { y } , \theta _ { e } ) = \mathcal { L } _ { B } ( G _ { y } ( V _ { e } ( \mathbf { x ^ { i } } ; \theta _ { e } ) ; \theta _ { y } ) , y _ { i } ) ; \quad \mathcal { L } _ { d } ( \mathbf { x ^ { i } } ; \theta _ { d } , \theta _ { e } ) = \mathcal { L } _ { B } ( G _ { d } ( V _ { e } ( \mathbf { x ^ { i } } ; \theta _ { e } ) ; \theta _ { d } ) , d _ { i } )
+$$
+
+where $\mathcal { L } _ { B }$ is the classification loss such as a binary or categorical cross-entropy loss function and $V _ { e } ( \mathbf { x } ^ { \mathbf { i } } ; \theta _ { e } )$ is the VRNN encoder that maps input $\mathbf { x ^ { i } }$ to $\tilde { z } ^ { i }$ .
+
+Now, for adversarial training, we consider the following domain adaptation term as the regularizer of equation 1.
+
+$$
+\mathcal { R } ( \theta _ { e } ) = \operatorname* { m a x } _ { \theta _ { d } } \Big [ - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathcal { L } _ { d } ( \mathbf { x } ^ { \mathbf { i } } ; \theta _ { d } , \theta _ { e } ) - \frac { 1 } { n ^ { \prime } } \sum _ { i = n + 1 } ^ { N } \mathcal { L } _ { d } ( \mathbf { x } ^ { \mathbf { i } } ; \theta _ { d } , \theta _ { e } ) \Big ]
+$$
+
+where $n ^ { \prime }$ is the number of target domain samples. As shown in Ganin et al. (2016), $\mathcal { R }$ is the domain regularizer and it is derived from the empirical $\varkappa -$ divergence between the source domain and target domain samples( Ben-David et al. (2010)).
+
+Combining the joint optimization problem of equations 1 and 2 leads to our VRADA model, where we minimize the source classification risk and at the same time achieve domain adaptation. Mathematically, we optimize the following complete objective function:
+
+$$
+\boldsymbol { \mathrm { 5 } } ( \theta _ { e } , \theta _ { g } , \theta _ { y } , \theta _ { d } ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \frac { 1 } { T ^ { i } } \mathcal { L } _ { r } ( \mathbf { x } ^ { \mathbf { i } } ; \theta _ { e } , \theta _ { g } ) + \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathcal { L } _ { y } ( \mathbf { x } ^ { \mathbf { i } } ; \theta _ { y } ) - \boldsymbol { \lambda } ( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \mathcal { L } _ { d } ( \mathbf { x } ^ { \mathbf { i } } ; \theta _ { d } ) + \frac { 1 } { n ^ { \prime } } \sum _ { i = n + 1 } ^ { N } \mathcal { L } _ { d } ( \mathbf { x } ^ { \mathbf { i } } ; \theta _ { d } ) ) = \frac { 1 } { n }
+$$
+
+where $\lambda$ is a trade-off between optimizing on making domain-invariant representations and optimizing source classification accuracy. Our optimization involves minimization with respect to some parameters, and maximization with respect to the others, i.e., we iteratively solve the following:
+
+$$
+\begin{array} { c } { { ( \hat { \theta } _ { g } , \hat { \theta } _ { y } , \hat { \theta } _ { e } ) = \arg \underset { \theta _ { g } , \theta _ { y } , \theta _ { e } } { \operatorname* { m i n } } E ( \theta _ { e } , \theta _ { g } , \theta _ { y } , \hat { \theta } _ { d } ) } } \\ { { \hat { \theta } _ { d } = \arg \underset { \theta _ { d } } { \operatorname* { m a x } } E ( \hat { \theta } _ { e } , \hat { \theta } _ { g } , \hat { \theta } _ { y } , \theta _ { d } ) } } \end{array}
+$$
+
+with the gradient updates calculated as:
+
+$$
+\begin{array} { r } { \theta _ { e }  \theta _ { e } - \eta ( \frac { \partial \mathcal { L } _ { r } } { \partial \theta _ { e } } + \frac { \partial \mathcal { L } _ { y } } { \partial \theta _ { y } } - \lambda \frac { \partial \mathcal { L } _ { d } } { \partial \theta _ { d } } ) } \\ { \theta _ { g }  \theta _ { g } - \eta \frac { \partial \mathcal { L } _ { r } } { \partial \theta _ { g } } } \\ { \theta _ { d }  \theta _ { d } - \eta \frac { \partial \mathcal { L } _ { d } } { \partial \theta _ { d } } } \\ { \theta _ { y }  \theta _ { y } - \eta \lambda \frac { \partial \mathcal { L } _ { y } } { \partial \theta _ { y } } } \end{array}
+$$
+
+where $\eta$ is the learning rate. We can use stochastic gradient descent (SGD) to solve the equations (5-7). To solve equation (4), we can use SGD and the gradient reversal layer (GRL)(Ganin et al. (2016)). The role of GRL is to reverse the gradient sign while performing backpropagation. This ensures that the domain classification loss is maximized which makes the feature representations domain-invariant.
+
+Thus, VRADA results in learning feature representations which are domain-invariant (due to domain regressor $\mathcal { R }$ ) and which capture the temporal latent dependencies (due to optimizing VRNN objective function $\mathcal { L } _ { r }$ ). These things combine to allow the VRADAs’ discriminative power on the source domain to transfer to the target domain.
+
+# 4 EXPERIMENTS
+
+We conduct experiments on two real-world health care datasets to answer the following questions: (a) How does our VRADA model perform when compared to the state-of-the-art domain adaptation and non-adaptation approaches? (b) How different are the domain-invariant representations learned by various domain adaptation methods? (c) How do we show that the temporal latent dependencies are transferred between domains? In the remainder of this section, we will describe the datasets, methods, empirical results, and show visualizations to answer the above questions.
+
+# 4.1 DATASET DESCRIPTION
+
+We conduct experiments on two health care datasets, including the MIMIC-III dataset and a Pediatric ICU (PICU) dataset from Children’s Hospital Los Angeles.
+
+MIMIC-III( Johnson et al. (2016)) is a public dataset with deidentified clinical care data collected at Beth Israel Deaconess Medical Center from 2001 to 2012. It contains over 58,000 hospital admission records of 38,645 adults and 7,875 neonates. For our experiments, we extracted the following two datasets:
+
+• Adult-AHRF dataset: To study domain adaptation for adult patients with acute hypoxemic respiratory failure (AHRF), we extracted 20 time series features (such as Base excess, blood pH value, Mean Air Pressure, PaO2, etc.) from 5527 admission records based on Khemani et al. (2009). We grouped the patients into 4 groups/cohorts based on their age[1] - Group 2: working-age adult (20 to 45 yrs, 508 patients); Group 3: old working-age adult (46 to 65 yrs, 1888 patients); Group 4: elderly (66 to 85 yrs, 2394 patients); Group 5: old elderly (85 yrs and up, 437 patients). We treated each group as a separate domain with which we could perform domain adaptation. For each patient, we used the first 4 day after admission (with each day serving as a single time-step) as time series data for training and testing our models.
+
+• ICD9 dataset: For this dataset we extracted 99 time series features from 19714 admission records from 4 modalities including input-events (fluids into patient, e.g., insulin), outputevents (fluids out of the patient, e.g., urine), lab-events (lab test results, e.g., blood pH values, platelet count, etc.) and prescription-events (drugs prescribed by doctors, e.g., aspirin, potassium chloride, etc.). These modalities are known to be extremely useful for monitoring ICU patients. All the time series are of more than 48 hours of duration, and only the first 24 hours (after admission) 2-hourly sampled time series data is used for training and testing our models. We use this dataset to predict the ICD9 Diagnosis code categories for each patient’s admission record.
+
+Child-AHRF dataset: This is a PICU dataset which contains health records of 398 children patient with acute hypoxemic respiratory failure in the intensive care unit at Children’s Hospital Los Angeles (CHLA)(Khemani et al. (2009)). Similar to Adult-AHRF, this dataset has 20 time series features collected for 4 days after ICU admission. This dataset is considered as one group (Group 1: children, age 0 to 19 yrs) and represents one domain.
+
+# 4.1.1 PREDICTION AND DOMAIN ADAPTATION TASKS
+
+Mortality Prediction: For Adult-AHRF and Child-AHRF datasets, we are interested in predicting mortality, i.e. whether a patient dies from AHRF during their hospital stay. $2 0 . 1 0 \%$ of all the patients in Child-AHRF and $1 3 . 8 4 \%$ of all patients in Adult-AHRF have a positive mortality label (i.e. the patients who die in hospital).
+
+ICD9 Code Prediction: Each admission record in MIMIC-III dataset has multiple ICD-9 diagnosis codes. We group all the occurrences of the ICD-9 codes into 20 diagnosis groups[2]. For the ICD9 dataset, we are interested in predicting these 20 ICD-9 Diagnosis Categories for each admission record. We treat this as a multi-task prediction problem.
+
+Domain Adaptation Tasks: We study unsupervised domain adaptation (i.e. target domain labels are unavailable during training and validation) task with-in age groups of Adult-AHRF dataset, ICD9 dataset and across Adult and Child-AHRF datasets. For Adult-AHRF and ICD9 datasets, we created 12 source-target domain pairs using the age groups, pairing up each domain $D _ { i }$ with another domain $D _ { j \neq i }$ , for example, the source-target pair 2-5 was used for adapting from group 2 (working-age adult) to group 5 (old elderly). We also created 4 source-target pairs for performing domain adaptation from 4 adult age-groups to 1 child age-group.
+
+# 4.2 METHODS AND IMPLEMENTATION DETAILS
+
+We categorize the methods used in our main experiments into the following groups:
+
+• Non-adaptive baseline methods: Logistic Regression (LR), Adaboost with decision regressors (Adaboost), and feed forward deep neural networks (DNN) Deep Domain adaptation methods: Domain Adversarial Neural Networks (DANN) (Ganin et al. (2016)); DANN with a RNN (LSTM) as feature extractor (R-DANN); Variational Fair Autocoder (VFAE)(Louizos et al. (2015))   
+• Our method: Variational Recurrent Adversarial Deep Domain Adaptation (VRADA)[3].
+
+In all our experiments, we conducted unsupervised domain adaptation where target domain labels are unavailable during training and validation. For R-DANN, we used LSTM(Hochreiter & Schmidhuber (1997)) as the feature extractor network instead of the feed-forward neural networks used in DANN. For VFAE, DANN and all the non-domain adaptive approaches we flattened the time series along time axis and treat it as the input to the model. For fairness, the classifier and feature extractors of the VRADA and R-DANN were equivalent in depth and both had the same model capacity. We also ensure that the size of latent feature representation $\tilde { z } ^ { i }$ are similar for VRADA and DANN models. The model capacity of VFAE was chosen to be similar to VRADA. All the deep domain adaptation models including ours had depth of size 8 (including output classifier layers). We used the Adam optimizer ( Kingma & Ba (2014)) and ran all models for 500 epochs with a learning rate of $3 e { - 4 }$ We set an early stopping criteria that the model does not experience a decrease in the validation loss for 20 epochs. Source domain data was split into train/validation subsets with a 70/30 ratio and target domain data into train/validation/test subsets with a 70/15/15 ratio. In order to compare all the methods, we report AUC scores on the entire target domain set, and the test subset for each target domain data of a source-target pair.
+
+# 4.3 QUANTITATIVE RESULTS
+
+In Table 1, we compare non domain adaptation and domain adaptation models’ performance on the target domain test subset for the AHRF mortality prediction task. It is immediately clear that domain adaptation methods consistently outperform non domain adaptation methods. We see that generally the VRADA outperforms both variants of the DANN with it consistently seeing scores $\sim 4 \%$ higher. While the standard deviation for the VRADA was about $1 \%$ , it was about $2 \%$ for the R-DANN, further showing our models efficacy as it converges to more stable local optima. Our model VRADA beats state-of-the-art DANN(Ganin et al. (2016)) and VFAE(Louizos et al. (2015)) on all the source-pair domain adaptation tasks for Adult-AHRF dataset. For the domain adaptation from Adult-AHRF to Child-AHRF dataset, we observe that VRADA mostly outperforms all the competing models. This shows that our model can perform well even for smaller target domain datasets.
+
+Table 1: AUC Comparison for AHRF Mortality Prediction task with and without Domain Adaptation   
+
+<table><tr><td>Source-Target</td><td>LR</td><td>Adaboost</td><td>DNN</td><td>DANN</td><td>VFAE</td><td>R-DANN</td><td>VRADA</td></tr><tr><td>3-2</td><td>0.555</td><td>0.562</td><td>0.569</td><td>0.572</td><td>0.615</td><td>0.603</td><td>0.654</td></tr><tr><td>4-2</td><td>0.624</td><td>0.645</td><td>0.569</td><td>0.589</td><td>0.635</td><td>0.584</td><td>0.656</td></tr><tr><td>5-2</td><td>0.527</td><td>0.554</td><td>0.551</td><td>0.540</td><td>0.588</td><td>0.611</td><td>0.616</td></tr><tr><td>2-3</td><td>0.627</td><td>0.621</td><td>0.550</td><td>0.563</td><td>0.585</td><td>0.708</td><td>0.724</td></tr><tr><td>4-3</td><td>0.681</td><td>0.636</td><td>0.542</td><td>0.527</td><td>0.722</td><td>0.821</td><td>0.770</td></tr><tr><td>5-3</td><td>0.655</td><td>0.706</td><td>0.503</td><td>0.518</td><td>0.608</td><td>0.769</td><td>0.782</td></tr><tr><td>2-4</td><td>0.585</td><td>0.591</td><td>0.530</td><td>0.560</td><td>0.582</td><td>0.716</td><td>0.777</td></tr><tr><td>3-4</td><td>0.652</td><td>0.629</td><td>0.531</td><td>0.527</td><td>0.697</td><td>0.769</td><td>0.764</td></tr><tr><td>5-4</td><td>0.689</td><td>0.699</td><td>0.538</td><td>0.532</td><td>0.614</td><td>0.728</td><td>0.738</td></tr><tr><td>2-5</td><td>0.565</td><td>0.543</td><td>0.549</td><td>0.526</td><td>0.555</td><td>0.659</td><td>0.719</td></tr><tr><td>3-5</td><td>0.576</td><td>0.587</td><td>0.510</td><td>0.526</td><td>0.533</td><td>0.630</td><td>0.721</td></tr><tr><td>4-5</td><td>0.682</td><td>0.587</td><td>0.575</td><td>0.548</td><td>0.712</td><td>0.747</td><td>0.775</td></tr><tr><td>5-1</td><td>0.502</td><td>0.573</td><td>0.557</td><td>0.563</td><td>0.618</td><td>0.563</td><td>0.639</td></tr><tr><td>4-1</td><td>0.565</td><td>0.533</td><td>0.572</td><td>0.542</td><td>0.668</td><td>0.577</td><td>0.636</td></tr><tr><td>3-1</td><td>0.500</td><td>0.500</td><td>0.542</td><td>0.535</td><td>0.570</td><td>0.591</td><td>0.631</td></tr><tr><td>2-1</td><td>0.520</td><td>0.500</td><td>0.534</td><td>0.559</td><td>0.578</td><td>0.630</td><td>0.637</td></tr></table>
+
+In the above table, we test classification without adaptation using Logistic Regression (LR), Adaboost with decision tree classifiers and Feed forward Deep Neural Networks (DNN); and with adaptation using Deep Domain Adversarial Neural Networks (DANN), a DANN with an LSTM in its feature extractor (R-DANN), Variational Fair Autoencoder (VFAE) and our Variational Adversarial Domain Adaptation Model (VRADA). All results are reported on the target domain test subset dataset.
+
+As the AHRF mortality prediction task made it clear that domain adaptation is necessary for intergroup adaptation, for the ICD9 multi-task prediction task that involved data with time-steps of length 12, we focused strictly on domain adaptive models (i.e. the DANN, R-DANN, and VRADA). Table 2 shows the aggregated AUC scores on the entire target domain dataset and test data of the target domain for the 20 tasks of the ICD9 Code Prediction task. Here, we clearly see that VRADA and
+
+Table 2: AUC Comparison for ICD9 Diagnosis Code Prediction task   
+
+<table><tr><td>Model</td><td></td><td>23</td><td>24</td><td>25</td><td>32</td><td>34</td><td>35</td><td>42 43</td><td>45</td><td>52</td><td>53</td><td></td><td>54</td></tr><tr><td rowspan="2">DANN</td><td>entire target</td><td>0.513</td><td>0.508</td><td>0.509</td><td>0.511</td><td>0.508</td><td>0.514</td><td>0.511</td><td>0.507</td><td>0.512</td><td>0.505</td><td>0.508</td><td>0.506</td></tr><tr><td>target test</td><td>0.509</td><td>0.513</td><td>0.531</td><td>0.527</td><td>0.515</td><td>0.531</td><td>0.515</td><td>0.521</td><td>0.521</td><td>0.518</td><td>0.514</td><td>0.519</td></tr><tr><td rowspan="2">R-DANN</td><td>entire target</td><td>0.608</td><td>0.581</td><td>0.562</td><td>0.618</td><td>0.610</td><td>0.586</td><td>0.604</td><td>0.607</td><td>0.575</td><td>0.573</td><td>0.558</td><td>0.566</td></tr><tr><td>target test</td><td>0.605</td><td>0.579</td><td>0.570</td><td>0.628</td><td>0.609</td><td>0.589</td><td>0.614</td><td>0.616</td><td>0.586</td><td>0.573</td><td>0.563</td><td>0.564</td></tr><tr><td rowspan="2">VRADA</td><td>entire target</td><td>0.620</td><td>0.564</td><td>0.557</td><td>0.611</td><td>0.617</td><td>0.580</td><td>0.598</td><td>0.615</td><td>0.588</td><td>0.571</td><td>0.582</td><td>0.576</td></tr><tr><td>target test</td><td>0.609</td><td>0.563</td><td>0.560</td><td>0.620</td><td>0.617</td><td>0.580</td><td>0.606</td><td>0.623</td><td>0.594</td><td>0.576</td><td>0.581</td><td>0.576</td></tr></table>
+
+Here, we compare results for the ICD9 Diagnosis Code Prediction task on the ICD9 dataset. For each model, the top row corresponds to the performance on the entire target domain dataset and the bottom row corresponds to performance on the test subset $( 1 5 \% )$ of the target domain dataset.
+
+R-DANN models outperform DANN Ganin et al. (2016) by significant margins. We also observe that VRADA outperforms R-DANN by $1 . 5 \sim 2 \%$ when averaged over all the source-target domain pairs.
+
+# 4.4 DISCUSSION
+
+Figure 3 shows the temporal latent dependencies captured by our VRADA as compared to the R-DANN for $_ { 3 - 4 }$ source-target pair. While both models learn temporal latent dependencies fairly well, the VRADA outperforms the R-DANN in two ways. First, the VRADA’s neurons learned stronger predictions of whether features are relevant towards modeling the data. If we look at the VRADA row, for both AHRF and ICD9 we see that the neural activation patterns are more consistent across time-steps than for R-DANN. Figure 4 shows the unrolled memory cell states (in the form Examples $\times$ (Time $^ *$ Neurons)) for all the source and target domain data points. We see a consistent activation firing patterns across all these data points for VRADA but not for R-DANN. Together with the stronger performance on 3-4 for AHRF and 2-5 for ICD9, this potentially indicates that VRADA is better learning the temporal dependencies.
+
+Second, nuanced values are consistent across time-steps for the VRADA, exhibiting a gradual transition towards stronger activation with time, whereas the temporal activation pattern of the RDANN seems somewhat sporadic. While activation gradients across time are consistent for both the R-DANN and VRADA, more consistent inhibitory and excitatory neuron firing patterns indicate that the VRADA better transfers knowledge. Another indication of domain adaptation was shown in Figure 1c. Looking at the t-SNE projections of feature representations of DNN, R-DANN, and VRADA we can see that the addition of temporal latent dependencies might help in better mixing of the domain distributions since we observe that the data is more evenly spread out. Figure 1c and Figure 3 together indicate that the VRADA’s temporal latent dependency capturing power and ability to create domain-invariant representations act synergistically. For plots of activation patterns without domain adaptation, please see appendix section 6.2.3.
+
+# 5 SUMMARY
+
+Because of its diverse range of patients and its episodic and longitudal nature, healthcare data provides a good platform to test domain adaptation techniques for temporal data. With it as our example, we showcase the Variational Recurrent Adversarial Domain Adaptation (VRADA) model’s ability to learn temporal latent representations that are domain-invariant. By comparing our model’s latent representations to others’, we show its ability to use variational methods to capture hidden factors of variation and produce more robust domain-invariant representations. We hope this work serves as a bedrock for future work capturing and adapting temporal latent representations across domains.
+
+# ACKNOWLEDGMENTS
+
+This material is based upon work supported by the NSF research grants IIS-1134990, IIS-1254206, Samsung GRO Grant and the NSF Graduate Research Fellowship Program under Grant No. DGE1418060. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding agencies. We also acknowledge Thailand’s Development and Promotion of Science and Technology Talents Project for financial support. We thank Dr. Robinder Khemani for sharing the Child-AHRF dataset.
+
+![](images/11fd6effccc792ff6631410da5d3ddcbc541c54cada9fc944919d2dd3bbc9116.jpg)  
+Figure 3: Cell states of memory cell for R-DANN and VRADA showing temporal latent dependencies captured by neurons of the R-DANN and VRADA for the source domain and transferred to the target domain. Each step along the y-axis refers to the activation of a single neuron with blue for strong inhibition and yellow for strong excitation. Step along the $\mathbf { X }$ -axis refers to activation per time-step. The left shows a single example in adapting 3-4 and the right for adapting 2-5.
+
+![](images/060dc22ac9abe81f633425af9fdaa8d121d2cdde4fbca74f78c03160c0517434.jpg)  
+Figure 4: Cell states of memory cell for R-DANN and VRADA showing activation for all ICD9 2-5 adaptation examples. Here, we show temporal dependencies learned across time, feature pairs for examples in a domain. The y-axis values refer to values per data point and the $\mathbf { X }$ -axis shows activation at time, feature pairs with the time and feature dimensions being flattened.
+
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+
+Table 3: AUC Comparison for AHRF Mortality Prediction task for different types of VRADA training   
+
+<table><tr><td>Training</td><td>23</td><td>24</td><td>25</td><td>32</td><td>34 35</td><td>42</td><td>43</td><td>45</td><td>52</td><td>53</td><td>54</td></tr><tr><td>I</td><td>0.704</td><td>0.777</td><td>0.682</td><td>0.540</td><td>0.764</td><td>0.721</td><td>0.603</td><td>0.727 0.710</td><td>0.616</td><td>0.782</td><td>0.738</td></tr><tr><td>II</td><td>0.724</td><td>0.656</td><td>0.719</td><td>0.627</td><td>0.748</td><td>0.683</td><td>0.656</td><td>0.770 0.755</td><td>0.595</td><td>0.736</td><td>0.732</td></tr><tr><td>ⅢI</td><td>0.721</td><td>0.688</td><td>0.656</td><td>0.654</td><td>0.757</td><td>0.691</td><td>0.609 0.766</td><td>0.775</td><td>0.602</td><td>0.709</td><td>0.714</td></tr></table>
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+
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+
+# 6 APPENDIX
+
+# 6.1 TRAINING VARIATIONS
+
+We tested 3 variations of training VRADA: (a) training VRADA regularly as discussed in Section 3 (denoted by I), (b) loading a pretrained VRNN encoder and optimizing strictly off the classification errors, i.e.
+
+$$
+E ( \theta _ { e } , \theta _ { y } , \theta _ { d } ) = { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } { \mathcal { L } } _ { y } ( \mathbf { x ^ { i } } ; \theta _ { y } ) - \lambda ( { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } { \mathcal { L } } _ { d } ( \mathbf { x ^ { i } } ; \theta _ { d } ) + { \frac { 1 } { n ^ { \prime } } } \sum _ { i = n + 1 } ^ { N } { \mathcal { L } } _ { d } ( \mathbf { x ^ { i } } ; \theta _ { d } ) ) )
+$$
+
+and (c) loading a pretrained VRNN encoder and using the objective as presented in equation 3 (denoted by $\mathbf { I I I }$ ). Key to note is that in method $\mathbf { I }$ , we do not apply variational methods towards learning the shared latent representation. This was done to test whether they were helpful or harmful towards the learned latent representation used for classification. In method III, we train VRADA as normal but load a pretrained encoder. We pretrain the encoder by training the VRNN on all source and target domain samples for a desired source-target adaptation pair. In order to choose how many samples would be used for training, we looked at which domain had more examples and chose the larger of the two. For example, if the source domain was group 2 with 508 patients and the target domain was group 5 with 437 patients, the VRNN would see 508 samples of each domain, with group 5 being sampled with replacement after seeing all its samples. As the encoder was used for learning latent representations, we thought it worth investigating whether if pretrained it better captured the latent representations that were being used by the domain classifier for adversarial training. We thought beginning domain classification at a better initialization point might help VRADA avoid local minima. For each method, we fed one source domain sample to $G _ { y }$ and either a source or target domain sample to $G _ { d }$ . (For this training and all training samples, order was randomized.) We only calculated the loss $\mathcal { L } _ { r }$ once for the $G _ { d }$ samples so as to not bias the optimization of the VRNN.
+
+Table 3 shows the results of AHRF Mortality Prediction task for different types of VRADA training. From these experiments, we found that jointly training VRADA (i.e method I) usually performed better than the other pretrained training approaches.
+
+# 6.2 MODEL VARIATIONS
+
+# 6.2.1 ADVERSARIAL TRAINING AT EVERY TIME-STEP
+
+A natural question is whether adversarial training at every time-step is more effective than adversarial training at the last time-step of a latent representation. If done at every time-step, the network learns to create domain-invariant representations of subsets of your input $x _ { \le T }$ . Do these domain-invariant representations help the network find more optimal domain-invariant representations of $x$ ? We empirically tested this scenario (Table 4) and found the results to be sub-optimal when compared to only performing adversarial training at the last time-step (Table 1). Below are results for the R-DANN and VRADA models for adversarial training at every time-step.
+
+Table 4: AUC Comparison for AHRF Mortality Prediction task with adversarial training done at every time-step   
+
+<table><tr><td>Model</td><td>23 24</td><td>25</td><td>34</td><td>35</td><td>42</td><td>43</td><td>45</td><td>52 53 54</td></tr><tr><td>R-DANN</td><td>.651 .599</td><td>.598</td><td>.557 .679</td><td>.534</td><td>.563 .768</td><td>.588</td><td>.528</td><td>.696 .669</td></tr><tr><td>VRADA</td><td>.681 .691</td><td>.643</td><td>.594 .733</td><td>.641</td><td>.733 .794</td><td>.675</td><td>.583</td><td>.755 .726</td></tr></table>
+
+# 6.2.2 EFFECT OF RECONSTRUCTION LOSS
+
+Table 5 shows the effect of reconstruction loss for our VRADA model. We observe that reconstructing the original data (i.e. using the decoder for reconstructing the data) helps in the overall performance improvement of our VRADA model.
+
+Table 5: AUC Comparison of VRADA model for AHRF Mortality Prediction task with and without reconstruction loss   
+
+<table><tr><td>Model</td><td>23</td><td>24</td><td>25</td><td>32</td><td>34</td><td>35</td><td>42</td><td>43</td><td>45</td><td>52</td><td>53</td><td>54</td></tr><tr><td>Without reconstruction</td><td>0.703</td><td>0.623</td><td>0.570</td><td>0.647</td><td>0.622</td><td>0.564</td><td>0.577</td><td>0.608</td><td>0.552</td><td>0.599</td><td>0.640</td><td>0.676</td></tr><tr><td>With reconstruction</td><td>0.724</td><td>0.777</td><td>0.719</td><td>0.654</td><td>0.764</td><td>0.721</td><td>0.656</td><td>0.770</td><td>0.775</td><td>0.616</td><td>0.782</td><td>0.738</td></tr></table>
+
+# 6.2.3 IMPACT OF ADVERSARIAL TRAINING
+
+In figures 5 and 6 we show the cell state activations for the VRADA and R-DANN without domain adaptation (i.e. no adversarial training). From these figures, we see that the dependencies between source and target domains are not transferred correctly since we do not perform adversarial training. On the otherhand, as discussed in section 4.4, figure 3 shows that adversarial training helps in transferring the dependencies between source and target domains efficiently.
+
+# 6.3 R-DANN MODEL INFORMATION
+
+Here we provide more details on the network architectures of the R-DANN and DANN. Please refer to Figure 7 for a diagram of the R-DANN model showing the dimensions of each layer and the connections between layers. The R-DANN and DANN were essentially identical except that, for the DANN, the first layer used a fully-connected layer instead of an RNN and took input flattened over the time-dimension. Thus the input dimensions corresponded to $f$ and $t \times f$ for the R-DANN and DANN, respectively, where $f$ is the number of features and $t$ is the length of the time-dimension.
+
+![](images/bee230ee86300b576363b47729e1f63bfc39e703a9aa8fcf9c19316fd4c71238.jpg)  
+Figure 5: Cell states of memory cell for R-DANN and VRADA showing temporal latent dependencies captured by neurons of the R-DANN and VRADA for the source domain and the target domain. Each step along the y-axis refers to the activation of a single neuron with blue for strong inhibition and yellow for strong excitation. Step along the $\mathbf { X }$ -axis refers to activation per time-step. The figure shows a single example in adapting 3-4 for AHRF dataset.
+
+![](images/d0627e3e2cc61001e649558d2740464a0862e463682008e147d7e287df284278.jpg)  
+Figure 6: Cell states of memory cell for R-DANN and VRADA showing temporal latent dependencies captured by neurons of the R-DANN and VRADA for the source domain and the target domain. Each step along the y-axis refers to the activation of a single neuron with blue for strong inhibition and yellow for strong excitation. Step along the $\mathbf { X }$ -axis refers to activation per time-step. The figure shows a single example in adapting 2-5 for ICD9 dataset.
+
+![](images/ab008a59e9a4e7df90a86064a5d4cefb15a372d0c8f2d11a890b871e7f24cd0f.jpg)  
+Figure 7: Block diagram of the R-DANN showing the number of neurons used in each layer and how the layers were connected. This model had a capacity of about 46, 000 parameters.

parse/train/rk9eAFcxg/rk9eAFcxg_content_list.json

@@ -0,0 +1,1609 @@
+[
+    {
+        "type": "text",
+        "text": "VARIATIONAL RECURRENT ADVERSARIALDEEP DOMAIN ADAPTATION",
+        "text_level": 1,
+        "bbox": [
+            173,
+            113,
+            683,
+            160
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sanjay Purushotham\\*, Wilka Carvalho\\*, Tanachat Nilanon, Yan Liu ",
+        "bbox": [
+            184,
+            184,
+            665,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Computer Science   \nUniversity of Southern California   \nLos Angeles, CA 90089, USA   \n{spurusho,wcarvalh,nilanon,yanliu.cs}@usc.edu ",
+        "bbox": [
+            184,
+            199,
+            624,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            290,
+            544,
+            305
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We study the problem of learning domain invariant representations for time series data while transferring the complex temporal latent dependencies between domains. Our model termed as Variational Recurrent Adversarial Deep Domain Adaptation (VRADA) is built atop a variational recurrent neural network (VRNN) and trains adversarially to capture complex temporal relationships that are domain-invariant. This is (as far as we know) the first to capture and transfer temporal latent dependencies of multivariate time-series data. Through experiments on real-world multivariate healthcare time-series datasets, we empirically demonstrate that learning temporal dependencies helps our model’s ability to create domain-invariant representations, allowing our model to outperform current state-of-the-art deep domain adaptation approaches. ",
+        "bbox": [
+            232,
+            323,
+            766,
+            476
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            505,
+            336,
+            520
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many real-world applications require effective machine learning algorithms that can learn invariant representations across related time-series datasets. For example, precision medicine for patients of various age groups, mobile application recommendation for users based on locations, and so on. In these examples, while the domains (i.e. age group and location) may vary, there exist common predictive patterns that can aid in inferring knowledge from one domain to another. More often than not, some domains have a significantly larger number of observations than others (e.g., respiratory failure in adults vs. children). Therefore effective domain adaption of time-series data is in great demand. ",
+        "bbox": [
+            174,
+            536,
+            825,
+            648
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The general approach to tackling domain adaptation has been explored under many facets which include reducing the domain discrepancy between the source and target domains(Ben-David et al. (2007)), instance re-weighting (Jiang & Zhai (2007)), subspace alignment (Fernando et al. (2013)), and deep learning (Tzeng et al. (2015); Ganin & Lempitsky (2014)). Many of these approaches work very well for non-sequential data but are not suitable for multivariate time-series data as they do not usually capture the temporal dependencies present in the data. For sequential data, earlier work has successfully used dynamic Bayesian Networks(Huang & Yates (2009)) and Recurrent Neural Networks (Socher et al. (2011)) to learn latent feature representations which were domaininvariant. Unfortunately, these works were not flexible enough to model non-linear dynamics or did not explicitly capture and transfer the complex latent dependencies needed to perform domain adaptation of time-series data. ",
+        "bbox": [
+            174,
+            655,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we address this problem with a model that learns temporal latent dependencies (i.e. dependencies between the latent variables across timesteps) that can be transferred across domains that experience different distributions in their features. We draw inspiration from the Variational Recurrent Neural Network (Chung et al. (2016)) and use variational methods to produce a latent representation that captures underlying temporal latent dependencies. Motivated by the theory of domain adaptation (Ben-David et al. (2010)), we perform adversarial training on this representation ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/5994455d34a7df0125ff448c1f0d78bf48aa6fe58d9be5e243ebb4424444504a.jpg",
+        "image_caption": [
+            "Figure 1: A Story of Temporal Dependency and Domain Invariance "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            123,
+            831,
+            248
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "t-SNE projections for the latent representations of DNN, R-DANN, and our VRADA model. We show adaption   \nfrom Adult-AHRF to Child-AHRF data. Source data is represented with red circles and target data with blue   \ncircles. From left to right, one can see that domain adaptation results in mixing the source and target domain data distributions. We can also see a story of how encoding more temporal dependency into the latent   \nrepresentation induces more domain-invariant representations. As models capture more underlying factors of   \nvariation, post domain adaptation representations gradually smoothen and become evenly dispersed, indicating that temporal dependency acts synergestically with domain adaptation. ",
+        "bbox": [
+            174,
+            252,
+            826,
+            340
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "similarly to the Domain Adversarial Neural Network (DANN) (Ganin et al. (2016)) to make the representations invariant across domains. We call our model the Variational Recurrent Adversarial Deep Domain Adaptation (VRADA) model. As far as we know, this is the first model capable of accomplishing unsupervised domain adaptation while transferring temporal latent dependencies for complex multivariate time-series data. Figure 1 shows an example of the domain invariant representations learned by different deep learning models including our VRADA model. From this figure, we can see that our model (VRADA) shows better mixing of the domain distributions than the competing models indicating that it learns better domain invariant representations. ",
+        "bbox": [
+            174,
+            367,
+            825,
+            479
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In order to prove the efficacy of our model, we perform domain adaptation using real-world healthcare time-series data. We choose healthcare data for two primary reasons. (1) Currently, a standard protocol in healthcare is to build, evaluate, and deploy machine learning models for particular datasets that may perform poorly on unseen datasets with different distributions. For example, models built around patient data from particular age groups perform poorly on other age groups because the features used to train the models have different distributions across the groups (Alemayehu & Warner (2004); Lao et al. (2004); Seshamani & Gray (2004)). Knowledge learned from one group is not transferrable to the other group. Domain adaptation seems like a natural solution to this problem as knowledge needs to be transferred across domains which share features that exhibit different distributions. (2) Healthcare data has multiple attributes recorded per patient visit, and it is longitudinal and episodic in nature. Thus, healthcare data is a suitable platform on which to study a model which seeks to capture complex temporal representations and transfer this knowledge across domains. ",
+        "bbox": [
+            174,
+            486,
+            825,
+            652
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The rest of the paper is structured as follows. In the following section, we briefly discuss the current state-of-the-art deep domain adaptation approaches. Afterwards, we present our model mathematically, detailing how it simultaneously learns to capture temporal latent dependencies and create domain-invariant representations. In Section 4, we compare and contrast the performance of proposed approach with other approaches on two real-world health care datasets, and provide analysis on our domain-invariant representations. ",
+        "bbox": [
+            174,
+            660,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            765,
+            343,
+            782
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Domain adaptation is a specific instance of transfer learning in which the feature spaces are shared but their marginal distributions are different. A good survey on the two has been done in several previous works (Pan & Yang (2009); Jiang (2008); Patel et al. (2015)). Domain adaptation has been thoroughly studied in computer vision(Saenko et al. (2010); Gong et al. (2012); Fernando et al. (2013)) and natural language processing (NLP) (Blitzer (2007); Foster et al. (2010)) applications. Recently, the deep learning paradigm has become popular in domain adaptation (Chen et al. (2012); Tzeng et al. (2015); Yang & Eisenstein; Long & Wang (2015)) due to its ability to learn rich, flexible, non-linear domain-invariant representations. Here, we briefly discuss two deep domain adaptation approaches which are closely related to our proposed model. Domain Adversarial Neural Networks (DANN) ",
+        "bbox": [
+            173,
+            797,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/cdec1408957e6500c20e8bea104c23b625844077c8fbf34a2b1ef94f9c4d6fa5.jpg",
+        "image_caption": [
+            "Figure 2: Block diagram of VRADA. Blue lines show the inference process, $q _ { \\theta _ { e } } \\left( z _ { t } | \\boldsymbol { x } _ { \\le t } , \\boldsymbol { z } _ { < t } \\right)$ . Brown lines show the generation process, $p _ { \\theta _ { g } } ( x _ { t } | \\boldsymbol { z } _ { \\le t } , \\boldsymbol { x } _ { < t } )$ . Red lines show the recurrence process where $h _ { t }$ is informed by $h _ { t - 1 }$ , which is informed by $z _ { t - 1 }$ and $x _ { t - 1 }$ . Black lines indicate classification. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            398,
+            98,
+            620,
+            219
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "(Ganin et al. (2016)) is a deep domain adaptation model which uses two core components to create domain-invariant representations, a feature extractor that produces the data’s latent representation, and an adversarial domain labeler that attempts to classify that data’s domain to help the feature extractor produce latent representations which are domain-invariant. In Louizos et al. (2015), the authors propose Variational Fair AutoEncoder, which uses Variational Autoencoding architecture (Kingma & Welling (2013)) to learn latent representations where most of the information about certain known factors of variation are purged from the representation while still retaining as much information about the data as possible. While, these deep learning approaches learn domain-invariant representations, they fail to capture and transfer the underlying complex temporal latent relationships from one domain to another as they use convolutional or feed forward neural networks which we claim are not suitable for multivariate time-series data. ",
+        "bbox": [
+            174,
+            308,
+            825,
+            462
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Other works such as Huang & Yates (2009); Xiao & Guo (2013) have used distributed representations for domain adaptation in NLP sequence labeling tasks. However, they either induce hidden states as latent features using dynamic Bayesian networks (DBNs) or learn generalizable distributed representations of words using Recurrent Neural Networks (RNN) (Socher et al. (2011)) to enable domain adaptation. These works either model the highly non-linear dynamics, as one can with RNN, or capture the complex latent dependencies present in sequential data, as one can with DBNs, but not both. To overcome the challenges of DBNs and RNNs, Variational Recurrent Neural Network (VRNN)( Chung et al. (2016)) was proposed recently to capture the complex relationship between the underlying hidden factors of variation and the output variables at different time-steps. The VRNN uses Variational Autoencoders (VAEs)( Kingma & Welling (2013); Goodfellow et al. (2016)) at each time-step to learn a complex relationship between the latent hidden factors across time-steps. Like the VAE, its latent variable is parametric. Combined, these things make it well-suited for multimodal sequential data such as multivariate time-series. In the following section, we discuss our approach, Variational Adversarial Deep Domain Adaptation (VRADA), which uses a VRNN to model and transfer complex domain-invariant temporal latent relationships for unsupervised domain adaptation of multivariate time-series. ",
+        "bbox": [
+            174,
+            468,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 VARIATIONAL RECURRENT ADVERSARIAL DEEP DOMAIN ADAPTATION",
+        "text_level": 1,
+        "bbox": [
+            173,
+            713,
+            802,
+            729
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present our Variational Recurrent Adversarial Deep Domain Adaptation (VRADA) model for the purpose of capturing and transferring temporal latent dependencies across domains via domain-invariant representations. First, we introduce the notations used in this paper and then discuss our VRADA model in detail. ",
+        "bbox": [
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+            747,
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+            803
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 NOTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            823,
+            295,
+            837
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let us denote a multivariate variable-length time series with $N$ data samples as $\\{ \\mathbf { x ^ { i } } = ( x _ { t } ^ { i } ) _ { t = 1 } ^ { T ^ { i } } \\} _ { i = 1 } ^ { N }$ where . (Note: in our experiments, for all data samples , but for generality we maintain $T ^ { i }$ ). We denote $\\{ \\mathbf { x } _ { \\mathcal { S } } ^ { \\mathbf { i } } \\} _ { i = 1 } ^ { n }$ as source domain data and $\\{ \\dot { \\mathbf { x } } _ { \\mathcal { T } } ^ { \\mathbf { i } } \\} _ { i = n + 1 } ^ { N }$ as target domain data. We assume that each source domain data sample $\\mathbf { x } _ { \\mathcal { S } } ^ { \\mathbf { i } }$ comes with $L$ labels $y _ { i } \\in \\{ 0 , 1 \\} ^ { L }$ (for example, these labels may correspond to a clinical outcome such as mortality or ICD9 diagnosis codes), while target domain has no labeled data samples. We assign a domain label $d _ { i } \\in \\{ 0 , 1 \\}$ to each data sample to indicate if it comes from the source or target domain. $d _ { i }$ will be used for adversarial training. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 VRADA ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            148,
+            276,
+            162
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The block diagram of our VRADA model is shown in Figure 2. To explicitly model the dependencies between the latent random variable across time steps, the VRADA model utilizes Variational Recurrent Neural Networks (VRNN) (Chung et al. (2016)). The VRNN effectively contains a Variational AutoEncoders (Kingma & Welling (2013)) at every time step, all of which are conditioned on previous auto-encoders via the hidden state $h _ { t - 1 }$ of an RNN, such as an LSTM (Hochreiter & Schmidhuber (1997)). Therefore, for each time-step of $\\ v { x } _ { t } ^ { i }$ , we infer a latent random variable $z _ { t } ^ { i }$ via ",
+        "bbox": [
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+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1aca6fc0b6463f559240ec7f7bb1571616d8e612e65e19fadd3b712660e9bbb5.jpg",
+        "text": "$$\n\\begin{array} { r } { z _ { t } ^ { i } | x _ { t } ^ { i } \\sim \\mathcal { N } ( \\mu _ { z , t } , \\mathrm { d i a g } ( \\sigma _ { z , t } ) ) , \\quad \\mathrm { w h e r e } \\ [ \\mu _ { z , t } , \\sigma _ { z , t } ] = \\varphi _ { \\tau } ^ { e n c } ( \\varphi _ { \\tau } ^ { x } ( x _ { t } ^ { i } ) , h _ { t - 1 } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            266,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with prior ",
+        "bbox": [
+            173,
+            291,
+            241,
+            305
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a2c92799c080948cf42a84415002817a311218fcf1401b68c41a26b805cd6e2c.jpg",
+        "text": "$$\nz _ { t } ^ { i } \\sim \\mathcal { N } ( \\mu _ { 0 , t } , \\mathrm { d i a g } ( \\sigma _ { 0 , t } ) ) , \\quad \\mathrm { w h e r e } \\ [ \\mu _ { 0 , t } , \\sigma _ { 0 , t } ] = \\varphi _ { \\tau } ^ { p r i o r } ( h _ { t - 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            303,
+            702,
+            320
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mu _ { * , t } , \\sigma _ { * , t }$ denote parameters of a generating distribution, and $\\boldsymbol { \\varphi } _ { \\tau } ^ { \\ast }$ can be any highly flexible function such as deep neural networks. For each $z _ { t } ^ { i } , x _ { t } ^ { i }$ is generated via ",
+        "bbox": [
+            176,
+            324,
+            825,
+            354
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/38a7bc11c9395469de7f9cca5ee424787755bc1c07b77b835186e5c1fd3b4c95.jpg",
+        "text": "$$\nx _ { t } ^ { i } | z _ { t } ^ { i } \\sim { \\mathcal { N } } ( \\mu _ { x , t } , \\mathrm { d i a g } ( \\sigma _ { x , t } ) ) , \\quad { \\mathrm { w h e r e ~ } } [ \\mu _ { x , t } , \\sigma _ { x , t } ] = \\varphi _ { \\tau } ^ { d e c } ( \\varphi _ { \\tau } ^ { z } ( z _ { t } ^ { i } ) , h _ { t - 1 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            359,
+            735,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and learned by optimizing the VRNN objective function: ",
+        "bbox": [
+            173,
+            385,
+            547,
+            400
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/69eed76fcccd834a86c05ab172732922432b8f6187fe7187aaacfe288aef4f44.jpg",
+        "text": "$$\n\\overset { \\cdot } { \\underset { t = t } { \\cdot } } ( x _ { t } ^ { i } ; \\theta _ { \\epsilon } , \\theta _ { g } ) = E _ { q _ { \\theta _ { \\epsilon } } ( z _ { \\leq T ^ { i } } ^ { i } | x _ { \\leq T ^ { i } } ^ { i } ) } [ \\underset { t = 1 } { \\overset { T ^ { i } } { \\sum } } ( - D ( q _ { \\theta _ { \\epsilon } } ( z _ { t } ^ { i } | x _ { \\leq t } ^ { i } , z _ { < t } ^ { i } ) | | p ( z _ { t } ^ { i } | x _ { < t } ^ { i } , z _ { < t } ^ { i } ) ) + \\log p _ { \\theta _ { g } } ( x _ { t } ^ { i } | z _ { \\leq t } ^ { i } , x _ { < t } ^ { i } ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            406,
+            839,
+            452
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $q _ { \\theta _ { e } } \\big ( z _ { t } ^ { i } | \\boldsymbol { x } _ { \\le t } ^ { i } , \\boldsymbol { z } _ { < t } ^ { i } \\big )$ is the inference model, $p ( z _ { t } ^ { i } | x _ { < t } ^ { i } , z _ { < t } ^ { i } )$ is the prior, $p _ { \\theta _ { g } } ( x _ { t } ^ { i } | \\boldsymbol { z } _ { \\le t } ^ { i } , x _ { < t } ^ { i } )$ is the generative model, $\\theta _ { e }$ is the parameters of the VRNN’s encoder, $\\theta _ { g }$ the parameters of the VRNN’s decoder, and $D ( \\cdot | | \\cdot )$ refers to KL-Divergence. Note: $z _ { \\le T }$ refers to the set of all $z _ { t }$ such that $t \\leq T$ , likewise for $z _ { < T }$ . For each $\\mathbf { x ^ { i } }$ , we use $\\tilde { z } ^ { i } \\sim q _ { \\theta _ { e } } ( z _ { T ^ { i } } ^ { i } | x _ { \\le T ^ { i } } ^ { i } , z _ { < T ^ { i } } ^ { i } )$ as our feature representation for source domain classification task since it captures temporal latent dependencies across the time-steps. Training the VRNN for the source domain classification involves solving the following optimization: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e86767b6ae9bf7044fd61d9fb8882b7a873c23ba4d2046bdf22a2722fd8b6c63.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta _ { e } , \\theta _ { g } , \\theta _ { y } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\frac { 1 } { T ^ { i } } \\mathcal { L } _ { r } ( \\mathbf { x ^ { i } } ; \\theta _ { e } , \\theta _ { g } ) + \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathcal { L } _ { y } ( \\mathbf { x ^ { i } } ; \\theta _ { y } , \\theta _ { e } ) + \\lambda \\mathcal { R } ( \\theta _ { e } )\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { R } ( \\theta _ { e } )$ is a regularizer for the parameters of VRNN encoder (which is also the feature extractor of VRADA) with a tuning hyperparameter $\\lambda$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As we are interested in achieving domain adaptation via the latent representation $\\tilde { z } ^ { i }$ (i.e. to make $\\tilde { z } ^ { i }$ domain-invariant), we can adversarially train the above objective function (equation 1) by employing the domain adaptation idea proposed in Ganin et al. (2016). Let $G _ { y } ( \\tilde { z } ^ { i } ; \\theta _ { y } )$ and $G _ { d } ( \\tilde { z } ^ { i } ; \\theta _ { d } )$ represent the source label classifier (to predict source labels $y _ { i }$ ) and domain label classifier (to predict domain labels $d _ { i }$ ) respectively with parameters $\\theta _ { y }$ and $\\theta _ { d }$ for a given input $\\tilde { z } ^ { i }$ . Here, $G _ { y } ( . )$ and $G _ { d } ( . )$ can be deep neural networks. Let us denote their loss functions respectively as ",
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+        "type": "equation",
+        "img_path": "images/bfb09283acec76b55c10b18f430667e290163af1bb9bcae4bfb6ce9fbeb98482.jpg",
+        "text": "$$\n\\mathcal { L } _ { y } ( \\mathbf { x ^ { i } } ; \\theta _ { y } , \\theta _ { e } ) = \\mathcal { L } _ { B } ( G _ { y } ( V _ { e } ( \\mathbf { x ^ { i } } ; \\theta _ { e } ) ; \\theta _ { y } ) , y _ { i } ) ; \\quad \\mathcal { L } _ { d } ( \\mathbf { x ^ { i } } ; \\theta _ { d } , \\theta _ { e } ) = \\mathcal { L } _ { B } ( G _ { d } ( V _ { e } ( \\mathbf { x ^ { i } } ; \\theta _ { e } ) ; \\theta _ { d } ) , d _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { B }$ is the classification loss such as a binary or categorical cross-entropy loss function and $V _ { e } ( \\mathbf { x } ^ { \\mathbf { i } } ; \\theta _ { e } )$ is the VRNN encoder that maps input $\\mathbf { x ^ { i } }$ to $\\tilde { z } ^ { i }$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "Now, for adversarial training, we consider the following domain adaptation term as the regularizer of equation 1. ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b7b667071730375097d8df8b1d3c78b88c362844af280e9e19d8a42f4bd6006.jpg",
+        "text": "$$\n\\mathcal { R } ( \\theta _ { e } ) = \\operatorname* { m a x } _ { \\theta _ { d } } \\Big [ - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathcal { L } _ { d } ( \\mathbf { x } ^ { \\mathbf { i } } ; \\theta _ { d } , \\theta _ { e } ) - \\frac { 1 } { n ^ { \\prime } } \\sum _ { i = n + 1 } ^ { N } \\mathcal { L } _ { d } ( \\mathbf { x } ^ { \\mathbf { i } } ; \\theta _ { d } , \\theta _ { e } ) \\Big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $n ^ { \\prime }$ is the number of target domain samples. As shown in Ganin et al. (2016), $\\mathcal { R }$ is the domain regularizer and it is derived from the empirical $\\varkappa -$ divergence between the source domain and target domain samples( Ben-David et al. (2010)). ",
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+    {
+        "type": "text",
+        "text": "Combining the joint optimization problem of equations 1 and 2 leads to our VRADA model, where we minimize the source classification risk and at the same time achieve domain adaptation. Mathematically, we optimize the following complete objective function: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/13d092bca1fef9c8954f2e87865068097bd904812ad9718712b90e9bcf876f28.jpg",
+        "text": "$$\n\\boldsymbol { \\mathrm { 5 } } ( \\theta _ { e } , \\theta _ { g } , \\theta _ { y } , \\theta _ { d } ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\frac { 1 } { T ^ { i } } \\mathcal { L } _ { r } ( \\mathbf { x } ^ { \\mathbf { i } } ; \\theta _ { e } , \\theta _ { g } ) + \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathcal { L } _ { y } ( \\mathbf { x } ^ { \\mathbf { i } } ; \\theta _ { y } ) - \\boldsymbol { \\lambda } ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathcal { L } _ { d } ( \\mathbf { x } ^ { \\mathbf { i } } ; \\theta _ { d } ) + \\frac { 1 } { n ^ { \\prime } } \\sum _ { i = n + 1 } ^ { N } \\mathcal { L } _ { d } ( \\mathbf { x } ^ { \\mathbf { i } } ; \\theta _ { d } ) ) = \\frac { 1 } { n }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda$ is a trade-off between optimizing on making domain-invariant representations and optimizing source classification accuracy. Our optimization involves minimization with respect to some parameters, and maximization with respect to the others, i.e., we iteratively solve the following: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/43014aa42913c5b9b3dd0dc18c89ef61f4c630b8900991755f1bf5033bca7f81.jpg",
+        "text": "$$\n\\begin{array} { c } { { ( \\hat { \\theta } _ { g } , \\hat { \\theta } _ { y } , \\hat { \\theta } _ { e } ) = \\arg \\underset { \\theta _ { g } , \\theta _ { y } , \\theta _ { e } } { \\operatorname* { m i n } } E ( \\theta _ { e } , \\theta _ { g } , \\theta _ { y } , \\hat { \\theta } _ { d } ) } } \\\\ { { \\hat { \\theta } _ { d } = \\arg \\underset { \\theta _ { d } } { \\operatorname* { m a x } } E ( \\hat { \\theta } _ { e } , \\hat { \\theta } _ { g } , \\hat { \\theta } _ { y } , \\theta _ { d } ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "with the gradient updates calculated as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d1c63df37793859e7f745b0aad0d34e389a98b6a784aa435cdcb0969e8239917.jpg",
+        "text": "$$\n\\begin{array} { r } { \\theta _ { e }  \\theta _ { e } - \\eta ( \\frac { \\partial \\mathcal { L } _ { r } } { \\partial \\theta _ { e } } + \\frac { \\partial \\mathcal { L } _ { y } } { \\partial \\theta _ { y } } - \\lambda \\frac { \\partial \\mathcal { L } _ { d } } { \\partial \\theta _ { d } } ) } \\\\ { \\theta _ { g }  \\theta _ { g } - \\eta \\frac { \\partial \\mathcal { L } _ { r } } { \\partial \\theta _ { g } } } \\\\ { \\theta _ { d }  \\theta _ { d } - \\eta \\frac { \\partial \\mathcal { L } _ { d } } { \\partial \\theta _ { d } } } \\\\ { \\theta _ { y }  \\theta _ { y } - \\eta \\lambda \\frac { \\partial \\mathcal { L } _ { y } } { \\partial \\theta _ { y } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            354,
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+            452
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta$ is the learning rate. We can use stochastic gradient descent (SGD) to solve the equations (5-7). To solve equation (4), we can use SGD and the gradient reversal layer (GRL)(Ganin et al. (2016)). The role of GRL is to reverse the gradient sign while performing backpropagation. This ensures that the domain classification loss is maximized which makes the feature representations domain-invariant. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Thus, VRADA results in learning feature representations which are domain-invariant (due to domain regressor $\\mathcal { R }$ ) and which capture the temporal latent dependencies (due to optimizing VRNN objective function $\\mathcal { L } _ { r }$ ). These things combine to allow the VRADAs’ discriminative power on the source domain to transfer to the target domain. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments on two real-world health care datasets to answer the following questions: (a) How does our VRADA model perform when compared to the state-of-the-art domain adaptation and non-adaptation approaches? (b) How different are the domain-invariant representations learned by various domain adaptation methods? (c) How do we show that the temporal latent dependencies are transferred between domains? In the remainder of this section, we will describe the datasets, methods, empirical results, and show visualizations to answer the above questions. ",
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+    {
+        "type": "text",
+        "text": "4.1 DATASET DESCRIPTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We conduct experiments on two health care datasets, including the MIMIC-III dataset and a Pediatric ICU (PICU) dataset from Children’s Hospital Los Angeles. ",
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+    {
+        "type": "text",
+        "text": "MIMIC-III( Johnson et al. (2016)) is a public dataset with deidentified clinical care data collected at Beth Israel Deaconess Medical Center from 2001 to 2012. It contains over 58,000 hospital admission records of 38,645 adults and 7,875 neonates. For our experiments, we extracted the following two datasets: ",
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+    {
+        "type": "text",
+        "text": "• Adult-AHRF dataset: To study domain adaptation for adult patients with acute hypoxemic respiratory failure (AHRF), we extracted 20 time series features (such as Base excess, blood pH value, Mean Air Pressure, PaO2, etc.) from 5527 admission records based on Khemani et al. (2009). We grouped the patients into 4 groups/cohorts based on their age[1] - Group 2: working-age adult (20 to 45 yrs, 508 patients); Group 3: old working-age adult (46 to 65 yrs, 1888 patients); Group 4: elderly (66 to 85 yrs, 2394 patients); Group 5: old elderly (85 yrs and up, 437 patients). We treated each group as a separate domain with which we could perform domain adaptation. For each patient, we used the first 4 day after admission (with each day serving as a single time-step) as time series data for training and testing our models. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• ICD9 dataset: For this dataset we extracted 99 time series features from 19714 admission records from 4 modalities including input-events (fluids into patient, e.g., insulin), outputevents (fluids out of the patient, e.g., urine), lab-events (lab test results, e.g., blood pH values, platelet count, etc.) and prescription-events (drugs prescribed by doctors, e.g., aspirin, potassium chloride, etc.). These modalities are known to be extremely useful for monitoring ICU patients. All the time series are of more than 48 hours of duration, and only the first 24 hours (after admission) 2-hourly sampled time series data is used for training and testing our models. We use this dataset to predict the ICD9 Diagnosis code categories for each patient’s admission record. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Child-AHRF dataset: This is a PICU dataset which contains health records of 398 children patient with acute hypoxemic respiratory failure in the intensive care unit at Children’s Hospital Los Angeles (CHLA)(Khemani et al. (2009)). Similar to Adult-AHRF, this dataset has 20 time series features collected for 4 days after ICU admission. This dataset is considered as one group (Group 1: children, age 0 to 19 yrs) and represents one domain. ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "4.1.1 PREDICTION AND DOMAIN ADAPTATION TASKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Mortality Prediction: For Adult-AHRF and Child-AHRF datasets, we are interested in predicting mortality, i.e. whether a patient dies from AHRF during their hospital stay. $2 0 . 1 0 \\%$ of all the patients in Child-AHRF and $1 3 . 8 4 \\%$ of all patients in Adult-AHRF have a positive mortality label (i.e. the patients who die in hospital). ",
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+    {
+        "type": "text",
+        "text": "ICD9 Code Prediction: Each admission record in MIMIC-III dataset has multiple ICD-9 diagnosis codes. We group all the occurrences of the ICD-9 codes into 20 diagnosis groups[2]. For the ICD9 dataset, we are interested in predicting these 20 ICD-9 Diagnosis Categories for each admission record. We treat this as a multi-task prediction problem. ",
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+    {
+        "type": "text",
+        "text": "Domain Adaptation Tasks: We study unsupervised domain adaptation (i.e. target domain labels are unavailable during training and validation) task with-in age groups of Adult-AHRF dataset, ICD9 dataset and across Adult and Child-AHRF datasets. For Adult-AHRF and ICD9 datasets, we created 12 source-target domain pairs using the age groups, pairing up each domain $D _ { i }$ with another domain $D _ { j \\neq i }$ , for example, the source-target pair 2-5 was used for adapting from group 2 (working-age adult) to group 5 (old elderly). We also created 4 source-target pairs for performing domain adaptation from 4 adult age-groups to 1 child age-group. ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "4.2 METHODS AND IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We categorize the methods used in our main experiments into the following groups: ",
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+    {
+        "type": "text",
+        "text": "• Non-adaptive baseline methods: Logistic Regression (LR), Adaboost with decision regressors (Adaboost), and feed forward deep neural networks (DNN) Deep Domain adaptation methods: Domain Adversarial Neural Networks (DANN) (Ganin et al. (2016)); DANN with a RNN (LSTM) as feature extractor (R-DANN); Variational Fair Autocoder (VFAE)(Louizos et al. (2015))   \n• Our method: Variational Recurrent Adversarial Deep Domain Adaptation (VRADA)[3]. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In all our experiments, we conducted unsupervised domain adaptation where target domain labels are unavailable during training and validation. For R-DANN, we used LSTM(Hochreiter & Schmidhuber (1997)) as the feature extractor network instead of the feed-forward neural networks used in DANN. For VFAE, DANN and all the non-domain adaptive approaches we flattened the time series along time axis and treat it as the input to the model. For fairness, the classifier and feature extractors of the VRADA and R-DANN were equivalent in depth and both had the same model capacity. We also ensure that the size of latent feature representation $\\tilde { z } ^ { i }$ are similar for VRADA and DANN models. The model capacity of VFAE was chosen to be similar to VRADA. All the deep domain adaptation models including ours had depth of size 8 (including output classifier layers). We used the Adam optimizer ( Kingma & Ba (2014)) and ran all models for 500 epochs with a learning rate of $3 e { - 4 }$ We set an early stopping criteria that the model does not experience a decrease in the validation loss for 20 epochs. Source domain data was split into train/validation subsets with a 70/30 ratio and target domain data into train/validation/test subsets with a 70/15/15 ratio. In order to compare all the methods, we report AUC scores on the entire target domain set, and the test subset for each target domain data of a source-target pair. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 QUANTITATIVE RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Table 1, we compare non domain adaptation and domain adaptation models’ performance on the target domain test subset for the AHRF mortality prediction task. It is immediately clear that domain adaptation methods consistently outperform non domain adaptation methods. We see that generally the VRADA outperforms both variants of the DANN with it consistently seeing scores $\\sim 4 \\%$ higher. While the standard deviation for the VRADA was about $1 \\%$ , it was about $2 \\%$ for the R-DANN, further showing our models efficacy as it converges to more stable local optima. Our model VRADA beats state-of-the-art DANN(Ganin et al. (2016)) and VFAE(Louizos et al. (2015)) on all the source-pair domain adaptation tasks for Adult-AHRF dataset. For the domain adaptation from Adult-AHRF to Child-AHRF dataset, we observe that VRADA mostly outperforms all the competing models. This shows that our model can perform well even for smaller target domain datasets. ",
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+    {
+        "type": "table",
+        "img_path": "images/b9cf5d68ef082b06e77d0c0410afaac8d7e84fd5b77091d93d0ddbeb7b1223a7.jpg",
+        "table_caption": [
+            "Table 1: AUC Comparison for AHRF Mortality Prediction task with and without Domain Adaptation "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source-Target</td><td>LR</td><td>Adaboost</td><td>DNN</td><td>DANN</td><td>VFAE</td><td>R-DANN</td><td>VRADA</td></tr><tr><td>3-2</td><td>0.555</td><td>0.562</td><td>0.569</td><td>0.572</td><td>0.615</td><td>0.603</td><td>0.654</td></tr><tr><td>4-2</td><td>0.624</td><td>0.645</td><td>0.569</td><td>0.589</td><td>0.635</td><td>0.584</td><td>0.656</td></tr><tr><td>5-2</td><td>0.527</td><td>0.554</td><td>0.551</td><td>0.540</td><td>0.588</td><td>0.611</td><td>0.616</td></tr><tr><td>2-3</td><td>0.627</td><td>0.621</td><td>0.550</td><td>0.563</td><td>0.585</td><td>0.708</td><td>0.724</td></tr><tr><td>4-3</td><td>0.681</td><td>0.636</td><td>0.542</td><td>0.527</td><td>0.722</td><td>0.821</td><td>0.770</td></tr><tr><td>5-3</td><td>0.655</td><td>0.706</td><td>0.503</td><td>0.518</td><td>0.608</td><td>0.769</td><td>0.782</td></tr><tr><td>2-4</td><td>0.585</td><td>0.591</td><td>0.530</td><td>0.560</td><td>0.582</td><td>0.716</td><td>0.777</td></tr><tr><td>3-4</td><td>0.652</td><td>0.629</td><td>0.531</td><td>0.527</td><td>0.697</td><td>0.769</td><td>0.764</td></tr><tr><td>5-4</td><td>0.689</td><td>0.699</td><td>0.538</td><td>0.532</td><td>0.614</td><td>0.728</td><td>0.738</td></tr><tr><td>2-5</td><td>0.565</td><td>0.543</td><td>0.549</td><td>0.526</td><td>0.555</td><td>0.659</td><td>0.719</td></tr><tr><td>3-5</td><td>0.576</td><td>0.587</td><td>0.510</td><td>0.526</td><td>0.533</td><td>0.630</td><td>0.721</td></tr><tr><td>4-5</td><td>0.682</td><td>0.587</td><td>0.575</td><td>0.548</td><td>0.712</td><td>0.747</td><td>0.775</td></tr><tr><td>5-1</td><td>0.502</td><td>0.573</td><td>0.557</td><td>0.563</td><td>0.618</td><td>0.563</td><td>0.639</td></tr><tr><td>4-1</td><td>0.565</td><td>0.533</td><td>0.572</td><td>0.542</td><td>0.668</td><td>0.577</td><td>0.636</td></tr><tr><td>3-1</td><td>0.500</td><td>0.500</td><td>0.542</td><td>0.535</td><td>0.570</td><td>0.591</td><td>0.631</td></tr><tr><td>2-1</td><td>0.520</td><td>0.500</td><td>0.534</td><td>0.559</td><td>0.578</td><td>0.630</td><td>0.637</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In the above table, we test classification without adaptation using Logistic Regression (LR), Adaboost with decision tree classifiers and Feed forward Deep Neural Networks (DNN); and with adaptation using Deep Domain Adversarial Neural Networks (DANN), a DANN with an LSTM in its feature extractor (R-DANN), Variational Fair Autoencoder (VFAE) and our Variational Adversarial Domain Adaptation Model (VRADA). All results are reported on the target domain test subset dataset. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As the AHRF mortality prediction task made it clear that domain adaptation is necessary for intergroup adaptation, for the ICD9 multi-task prediction task that involved data with time-steps of length 12, we focused strictly on domain adaptive models (i.e. the DANN, R-DANN, and VRADA). Table 2 shows the aggregated AUC scores on the entire target domain dataset and test data of the target domain for the 20 tasks of the ICD9 Code Prediction task. Here, we clearly see that VRADA and ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ca76fe348c43b1fa497e62dc68f6fc5c77b4e4af976ec24297d2af23699b97cf.jpg",
+        "table_caption": [
+            "Table 2: AUC Comparison for ICD9 Diagnosis Code Prediction task "
+        ],
+        "table_footnote": [
+            "Here, we compare results for the ICD9 Diagnosis Code Prediction task on the ICD9 dataset. For each model, the top row corresponds to the performance on the entire target domain dataset and the bottom row corresponds to performance on the test subset $( 1 5 \\% )$ of the target domain dataset. "
+        ],
+        "table_body": "<table><tr><td>Model</td><td></td><td>23</td><td>24</td><td>25</td><td>32</td><td>34</td><td>35</td><td>42 43</td><td>45</td><td>52</td><td>53</td><td></td><td>54</td></tr><tr><td rowspan=\"2\">DANN</td><td>entire target</td><td>0.513</td><td>0.508</td><td>0.509</td><td>0.511</td><td>0.508</td><td>0.514</td><td>0.511</td><td>0.507</td><td>0.512</td><td>0.505</td><td>0.508</td><td>0.506</td></tr><tr><td>target test</td><td>0.509</td><td>0.513</td><td>0.531</td><td>0.527</td><td>0.515</td><td>0.531</td><td>0.515</td><td>0.521</td><td>0.521</td><td>0.518</td><td>0.514</td><td>0.519</td></tr><tr><td rowspan=\"2\">R-DANN</td><td>entire target</td><td>0.608</td><td>0.581</td><td>0.562</td><td>0.618</td><td>0.610</td><td>0.586</td><td>0.604</td><td>0.607</td><td>0.575</td><td>0.573</td><td>0.558</td><td>0.566</td></tr><tr><td>target test</td><td>0.605</td><td>0.579</td><td>0.570</td><td>0.628</td><td>0.609</td><td>0.589</td><td>0.614</td><td>0.616</td><td>0.586</td><td>0.573</td><td>0.563</td><td>0.564</td></tr><tr><td rowspan=\"2\">VRADA</td><td>entire target</td><td>0.620</td><td>0.564</td><td>0.557</td><td>0.611</td><td>0.617</td><td>0.580</td><td>0.598</td><td>0.615</td><td>0.588</td><td>0.571</td><td>0.582</td><td>0.576</td></tr><tr><td>target test</td><td>0.609</td><td>0.563</td><td>0.560</td><td>0.620</td><td>0.617</td><td>0.580</td><td>0.606</td><td>0.623</td><td>0.594</td><td>0.576</td><td>0.581</td><td>0.576</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "R-DANN models outperform DANN Ganin et al. (2016) by significant margins. We also observe that VRADA outperforms R-DANN by $1 . 5 \\sim 2 \\%$ when averaged over all the source-target domain pairs. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.4 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows the temporal latent dependencies captured by our VRADA as compared to the R-DANN for $_ { 3 - 4 }$ source-target pair. While both models learn temporal latent dependencies fairly well, the VRADA outperforms the R-DANN in two ways. First, the VRADA’s neurons learned stronger predictions of whether features are relevant towards modeling the data. If we look at the VRADA row, for both AHRF and ICD9 we see that the neural activation patterns are more consistent across time-steps than for R-DANN. Figure 4 shows the unrolled memory cell states (in the form Examples $\\times$ (Time $^ *$ Neurons)) for all the source and target domain data points. We see a consistent activation firing patterns across all these data points for VRADA but not for R-DANN. Together with the stronger performance on 3-4 for AHRF and 2-5 for ICD9, this potentially indicates that VRADA is better learning the temporal dependencies. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Second, nuanced values are consistent across time-steps for the VRADA, exhibiting a gradual transition towards stronger activation with time, whereas the temporal activation pattern of the RDANN seems somewhat sporadic. While activation gradients across time are consistent for both the R-DANN and VRADA, more consistent inhibitory and excitatory neuron firing patterns indicate that the VRADA better transfers knowledge. Another indication of domain adaptation was shown in Figure 1c. Looking at the t-SNE projections of feature representations of DNN, R-DANN, and VRADA we can see that the addition of temporal latent dependencies might help in better mixing of the domain distributions since we observe that the data is more evenly spread out. Figure 1c and Figure 3 together indicate that the VRADA’s temporal latent dependency capturing power and ability to create domain-invariant representations act synergistically. For plots of activation patterns without domain adaptation, please see appendix section 6.2.3. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 SUMMARY ",
+        "text_level": 1,
+        "bbox": [
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+            667,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Because of its diverse range of patients and its episodic and longitudal nature, healthcare data provides a good platform to test domain adaptation techniques for temporal data. With it as our example, we showcase the Variational Recurrent Adversarial Domain Adaptation (VRADA) model’s ability to learn temporal latent representations that are domain-invariant. By comparing our model’s latent representations to others’, we show its ability to use variational methods to capture hidden factors of variation and produce more robust domain-invariant representations. We hope this work serves as a bedrock for future work capturing and adapting temporal latent representations across domains. ",
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+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This material is based upon work supported by the NSF research grants IIS-1134990, IIS-1254206, Samsung GRO Grant and the NSF Graduate Research Fellowship Program under Grant No. DGE1418060. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the funding agencies. We also acknowledge Thailand’s Development and Promotion of Science and Technology Talents Project for financial support. We thank Dr. Robinder Khemani for sharing the Child-AHRF dataset. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/11fd6effccc792ff6631410da5d3ddcbc541c54cada9fc944919d2dd3bbc9116.jpg",
+        "image_caption": [
+            "Figure 3: Cell states of memory cell for R-DANN and VRADA showing temporal latent dependencies captured by neurons of the R-DANN and VRADA for the source domain and transferred to the target domain. Each step along the y-axis refers to the activation of a single neuron with blue for strong inhibition and yellow for strong excitation. Step along the $\\mathbf { X }$ -axis refers to activation per time-step. The left shows a single example in adapting 3-4 and the right for adapting 2-5. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/060dc22ac9abe81f633425af9fdaa8d121d2cdde4fbca74f78c03160c0517434.jpg",
+        "image_caption": [
+            "Figure 4: Cell states of memory cell for R-DANN and VRADA showing activation for all ICD9 2-5 adaptation examples. Here, we show temporal dependencies learned across time, feature pairs for examples in a domain. The y-axis values refer to values per data point and the $\\mathbf { X }$ -axis shows activation at time, feature pairs with the time and feature dimensions being flattened. "
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+        "bbox": [
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+    {
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+            "Table 3: AUC Comparison for AHRF Mortality Prediction task for different types of VRADA training "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Training</td><td>23</td><td>24</td><td>25</td><td>32</td><td>34 35</td><td>42</td><td>43</td><td>45</td><td>52</td><td>53</td><td>54</td></tr><tr><td>I</td><td>0.704</td><td>0.777</td><td>0.682</td><td>0.540</td><td>0.764</td><td>0.721</td><td>0.603</td><td>0.727 0.710</td><td>0.616</td><td>0.782</td><td>0.738</td></tr><tr><td>II</td><td>0.724</td><td>0.656</td><td>0.719</td><td>0.627</td><td>0.748</td><td>0.683</td><td>0.656</td><td>0.770 0.755</td><td>0.595</td><td>0.736</td><td>0.732</td></tr><tr><td>ⅢI</td><td>0.721</td><td>0.688</td><td>0.656</td><td>0.654</td><td>0.757</td><td>0.691</td><td>0.609 0.766</td><td>0.775</td><td>0.602</td><td>0.709</td><td>0.714</td></tr></table>",
+        "bbox": [
+            173,
+            126,
+            879,
+            181
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Richard Socher, Cliff C Lin, Chris Manning, and Andrew Y Ng. Parsing natural scenes and natural language with recursive neural networks. In Proceedings of the $2 8 t h$ international conference on machine learning (ICML-11), pp. 129–136, 2011. ",
+        "bbox": [
+            174,
+            207,
+            825,
+            250
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Eric Tzeng, Judy Hoffman, Trevor Darrell, and Kate Saenko. Simultaneous deep transfer across domains and tasks. In Proceedings of the IEEE International Conference on Computer Vision, pp. 4068–4076, 2015. ",
+        "bbox": [
+            174,
+            258,
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+            301
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Min Xiao and Yuhong Guo. Domain adaptation for sequence labeling tasks with a probabilistic language adaptation model. In ICML (1), pp. 293–301, 2013. ",
+        "bbox": [
+            176,
+            311,
+            820,
+            340
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yi Yang and Jacob Eisenstein. Unsupervised multi-domain adaptation with feature embeddings. ",
+        "bbox": [
+            174,
+            349,
+            802,
+            364
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "6 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            392,
+            294,
+            409
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "6.1 TRAINING VARIATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            422,
+            377,
+            438
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We tested 3 variations of training VRADA: (a) training VRADA regularly as discussed in Section 3 (denoted by I), (b) loading a pretrained VRNN encoder and optimizing strictly off the classification errors, i.e. ",
+        "bbox": [
+            174,
+            449,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/209cdbdbe0715858272f421ad06689a2c2dad9262927cb258f5509d1c341e525.jpg",
+        "text": "$$\nE ( \\theta _ { e } , \\theta _ { y } , \\theta _ { d } ) = { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } { \\mathcal { L } } _ { y } ( \\mathbf { x ^ { i } } ; \\theta _ { y } ) - \\lambda ( { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ { n } { \\mathcal { L } } _ { d } ( \\mathbf { x ^ { i } } ; \\theta _ { d } ) + { \\frac { 1 } { n ^ { \\prime } } } \\sum _ { i = n + 1 } ^ { N } { \\mathcal { L } } _ { d } ( \\mathbf { x ^ { i } } ; \\theta _ { d } ) ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            497,
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+            542
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "and (c) loading a pretrained VRNN encoder and using the objective as presented in equation 3 (denoted by $\\mathbf { I I I }$ ). Key to note is that in method $\\mathbf { I }$ , we do not apply variational methods towards learning the shared latent representation. This was done to test whether they were helpful or harmful towards the learned latent representation used for classification. In method III, we train VRADA as normal but load a pretrained encoder. We pretrain the encoder by training the VRNN on all source and target domain samples for a desired source-target adaptation pair. In order to choose how many samples would be used for training, we looked at which domain had more examples and chose the larger of the two. For example, if the source domain was group 2 with 508 patients and the target domain was group 5 with 437 patients, the VRNN would see 508 samples of each domain, with group 5 being sampled with replacement after seeing all its samples. As the encoder was used for learning latent representations, we thought it worth investigating whether if pretrained it better captured the latent representations that were being used by the domain classifier for adversarial training. We thought beginning domain classification at a better initialization point might help VRADA avoid local minima. For each method, we fed one source domain sample to $G _ { y }$ and either a source or target domain sample to $G _ { d }$ . (For this training and all training samples, order was randomized.) We only calculated the loss $\\mathcal { L } _ { r }$ once for the $G _ { d }$ samples so as to not bias the optimization of the VRNN. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 3 shows the results of AHRF Mortality Prediction task for different types of VRADA training. From these experiments, we found that jointly training VRADA (i.e method I) usually performed better than the other pretrained training approaches. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "6.2 MODEL VARIATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            844,
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+            859
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "6.2.1 ADVERSARIAL TRAINING AT EVERY TIME-STEP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            871,
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+            886
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A natural question is whether adversarial training at every time-step is more effective than adversarial training at the last time-step of a latent representation. If done at every time-step, the network learns to create domain-invariant representations of subsets of your input $x _ { \\le T }$ . Do these domain-invariant representations help the network find more optimal domain-invariant representations of $x$ ? We empirically tested this scenario (Table 4) and found the results to be sub-optimal when compared to only performing adversarial training at the last time-step (Table 1). Below are results for the R-DANN and VRADA models for adversarial training at every time-step. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/f017ac91f9955da50fbec624d73f4f511231b6eee7b1071bac06b2e7944a9c7b.jpg",
+        "table_caption": [
+            "Table 4: AUC Comparison for AHRF Mortality Prediction task with adversarial training done at every time-step "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>23 24</td><td>25</td><td>34</td><td>35</td><td>42</td><td>43</td><td>45</td><td>52 53 54</td></tr><tr><td>R-DANN</td><td>.651 .599</td><td>.598</td><td>.557 .679</td><td>.534</td><td>.563 .768</td><td>.588</td><td>.528</td><td>.696 .669</td></tr><tr><td>VRADA</td><td>.681 .691</td><td>.643</td><td>.594 .733</td><td>.641</td><td>.733 .794</td><td>.675</td><td>.583</td><td>.755 .726</td></tr></table>",
+        "bbox": [
+            253,
+            228,
+            741,
+            272
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6.2.2 EFFECT OF RECONSTRUCTION LOSS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            294,
+            478,
+            309
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 5 shows the effect of reconstruction loss for our VRADA model. We observe that reconstructing the original data (i.e. using the decoder for reconstructing the data) helps in the overall performance improvement of our VRADA model. ",
+        "bbox": [
+            174,
+            318,
+            825,
+            361
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/4d8b589dedf23b101d58644263104ff07dd0649b45ce2be3add4c7616a109692.jpg",
+        "table_caption": [
+            "Table 5: AUC Comparison of VRADA model for AHRF Mortality Prediction task with and without reconstruction loss "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>23</td><td>24</td><td>25</td><td>32</td><td>34</td><td>35</td><td>42</td><td>43</td><td>45</td><td>52</td><td>53</td><td>54</td></tr><tr><td>Without reconstruction</td><td>0.703</td><td>0.623</td><td>0.570</td><td>0.647</td><td>0.622</td><td>0.564</td><td>0.577</td><td>0.608</td><td>0.552</td><td>0.599</td><td>0.640</td><td>0.676</td></tr><tr><td>With reconstruction</td><td>0.724</td><td>0.777</td><td>0.719</td><td>0.654</td><td>0.764</td><td>0.721</td><td>0.656</td><td>0.770</td><td>0.775</td><td>0.616</td><td>0.782</td><td>0.738</td></tr></table>",
+        "bbox": [
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+            416,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6.2.3 IMPACT OF ADVERSARIAL TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In figures 5 and 6 we show the cell state activations for the VRADA and R-DANN without domain adaptation (i.e. no adversarial training). From these figures, we see that the dependencies between source and target domains are not transferred correctly since we do not perform adversarial training. On the otherhand, as discussed in section 4.4, figure 3 shows that adversarial training helps in transferring the dependencies between source and target domains efficiently. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "6.3 R-DANN MODEL INFORMATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            593,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Here we provide more details on the network architectures of the R-DANN and DANN. Please refer to Figure 7 for a diagram of the R-DANN model showing the dimensions of each layer and the connections between layers. The R-DANN and DANN were essentially identical except that, for the DANN, the first layer used a fully-connected layer instead of an RNN and took input flattened over the time-dimension. Thus the input dimensions corresponded to $f$ and $t \\times f$ for the R-DANN and DANN, respectively, where $f$ is the number of features and $t$ is the length of the time-dimension. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/bee230ee86300b576363b47729e1f63bfc39e703a9aa8fcf9c19316fd4c71238.jpg",
+        "image_caption": [
+            "Figure 5: Cell states of memory cell for R-DANN and VRADA showing temporal latent dependencies captured by neurons of the R-DANN and VRADA for the source domain and the target domain. Each step along the y-axis refers to the activation of a single neuron with blue for strong inhibition and yellow for strong excitation. Step along the $\\mathbf { X }$ -axis refers to activation per time-step. The figure shows a single example in adapting 3-4 for AHRF dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            137,
+            282,
+            864,
+            661
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/d0627e3e2cc61001e649558d2740464a0862e463682008e147d7e287df284278.jpg",
+        "image_caption": [
+            "Figure 6: Cell states of memory cell for R-DANN and VRADA showing temporal latent dependencies captured by neurons of the R-DANN and VRADA for the source domain and the target domain. Each step along the y-axis refers to the activation of a single neuron with blue for strong inhibition and yellow for strong excitation. Step along the $\\mathbf { X }$ -axis refers to activation per time-step. The figure shows a single example in adapting 2-5 for ICD9 dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            138,
+            281,
+            862,
+            662
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab008a59e9a4e7df90a86064a5d4cefb15a372d0c8f2d11a890b871e7f24cd0f.jpg",
+        "image_caption": [
+            "Figure 7: Block diagram of the R-DANN showing the number of neurons used in each layer and how the layers were connected. This model had a capacity of about 46, 000 parameters. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    }
+]